![Mathematical
Structures
of the Universe
EDITED BY
Michal Eckstein
Michael Helier
Sebastian]. Szybka
0 Copemicus
Center
PRESS](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-5-320.jpg)
![14 Manuel Hohmann
manifold M equipped with a pseudo-Riemannian metric g of Lorentzian signa-
ture (-, +, +, +), an orientation and a time orientation. Observers are modeled
by world lines, which are smooth, future directed. timelike curves 1 : JR. --+ M.
Their tangent vectors satisfy
(1.1)
By a reparametrization we can always normalize the tangent vectors, so that
(1.2)
In this case we call the curve parameter the proper time along the world line 1
and denote it by the letter T instead oft. The proper time along a timelike curve
with arbitrary parametrization is given by the arc length integral
( 1.3)
The clock postulate of general relativity states that any clock moving along the
world line 1 measures the proper time, independent of the construction of the
clock. The prescription for the measurement of time is thus crucially linked to
the Lorentzian metric of spacetimc. Similarly, the metric provides a definition of
rulers and the length of spacelike curves by the same expression (1.3) of the arc
length integral. Finally, it also defines the angle q; between two tangent vectors
v, wE T.'"CM at the same point x E M as
(1.4)
In summary, the Lorentzian metric g defines the geometry ofspacetime.
Closely related to the geometry of spacetime is the notion of causality. It
answers the question which events on a spacetime manifold NI can have a causal
influence on which other events on iVJ. An event at x E l'vi can influence an
event x' E M if and only if there exists a continuous, future directed, causal (i.e.,
timelike or lightlike) curve from x to x'. All events which can be influenced by
x constitute the causal future of :r:. Conversely, all events which can influence x'
form the causal past of :r:'. This structure, called the causal structure of:,pacetime,
is defined by the metric geometry via the definition of causal curves.
The Lorentzian spacetime metric serves several further purposes besides pro-
viding a definition of spacetime geometry and causality. We have already seen
Observer dependent geometries
15
that it enters the definition of observer world lines as timelike curves, whose no-
tion is thus also relevant when we consider the measurements of observables by
these observers. Observables are modelcd by tensor fields, which are smooth
sections iJ? : .M ->- TT",s Af of a tensor bundle
yr,s1'vf = T 11°7
' 0 T* NI0
" ( 1.5)
over !VI. Their dynamics are consequently modeled by tensorial equations, which
are derived from a diffeomorphism-invariant action of the generic form
SM = ;· d4
xFYL(g, il?, ()cl?, ...),
.M
(1.6)
where the Lagrange function [, depends on the metric geometry, the (]elds and
their derivatives. Combining the notions of observers and observables we may
define an observation by an observer with world line 1 at proper time T as a
measurement of the field iJ?(:c) at the point :r = 1(-r). However, this definition
yields us an element of the tensor space 1-;:·s!VI, and not a set of numbers, as we
initially presumed. We further need to choose a frame, by which we denote a
basis .f of the tangent space T,'"CM. This frame allows us to express the tensor
i!?(:r:) in terms of its components with respect to f. The tensor components of
iJ?(x) are finally the numeric quantities which are measured in an experiment.
The frame f chosen by an observer to make measurements is usually not
completely arbitrary. Since the basis vectors fi are elements of the tangent space,
they are characterized as being timelike, lightlike or spacelike and possess units
of time or length. We can thus use the notions of time, length and angles de(]ned
by the spacetime metric to choose an orthonormal frame satisfying the condition
9 .fa fb - 'fj· .
. ab i. j - 'J
(1 .7)
with one unit timelike vector .fo and three unit spacelike vectors fee The clock
postulate, stating that proper time is measured by the arc length along the ob-
server world line 1 , further implies a canonical choice of the timelike vector fo
as the tangent vector "Y(T) to the observer world line. This observer adapted or-
thonormal frame is a convenient choice for most measurements.
It follows immediately from this model of observables and observations how
the measurements of the same observable made by two coincident observers,
whose world lines 1 and 1' meet at a common spacetime point x = 1(T) =
1'(T'), must be translated between their frames of reference. If both observer
frames f and .f' are orthonormalized, the condition (1.7) implies that they are](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-11-320.jpg)
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Manuel Hohmann
related by a Lorentz transform A. The same Lorentz transform must then be
applied to the tensor components measured by one observer in order to obtain the
tensor components measured by the other observer, using the standard formula
if>'": ...a.,.l I = A"I A"'· Ad! Ads c"F-CJ ...c,.
)J •·· Js Ct . · . C-r bt · · · b8 J! dJ ...d8
• (1.8)
This dose connection between observations made using different observer frames
constitutes the principle of local Lorentz invariance. It is a consequence of the
fact that we model the geometry of spacetime, which in turn defines the notion of
orthonormal frames, by a Lorentzian metric.
Even deeper implications arise from the fact that we model both observables
and geometry by tensor fields on the spacetime manifold !VI, and observations
by measurements of tensor components. If we introduce coordinates on M and
use their coordinate base in order to express the components of tensor fields, it
immediately follows how these components translate under a change of coordi-
nates. Moreover, since we model the dynamics of physical quantities by tensor
equations, they are independent of any choice of coordinates. This coordinate
freedom constitutes the principle of general covariance.
Besides its role in providing the background geometry which enters the def-
inition of observers, observations and causality, the Lorentzian metric of space-
time has a physical interpretation on its own, being the field which carries the
gravitational interaction. It does not only govern the dynamics of matter fields,
but is also influenced by their presence. This is reflected by the dynamics of
gravity, which is governed by the Einstein-Hilbert action
1 1 4
SEH = ;:;- d xFyR,
.c,K, M
(1.9)
which, together with the matter action (1.6), yields the Einstein equations
1
Rab - 2Rgab = K,Tab . ( l.lO)
Understanding the geometry of spacetime as a dynamical quantity, which mu-
tually interacts with matter fields, establishes a symmetric picture between both
matter and gravity.
However, it is exactly this symmetry between gravity and matter which may
lead us to new insights on the nature of spacetime geometry, and even question
its description i;J terms of a Lorentzian metric, from which we derived a number
of conclusions as stated above. This stems from the fact that all known matter
Observer dependent geometries 17
fields in the standard model are nowadays described by quantum theories. While
the process of quantization has been successfully applied to matter fields even
beyond the standard model, it is significantly harder in the case of gravity. This
difficulty has lead to a plethora of different approaches towards quantum grav-
ity, many of which suggest modifications to the geometry of spacctime, or even
resolve the unity of spacetime into a time evolution of spatial geometry. Main
contenders which fall into this class are given by geometrodynamic theories such
as loop quantum gravity [Ashtekar 1987, Thiemann 20071 and sum-over-histories
formulations such as spin foam models IRovelli & Smolin I995, Reisenberger &
Rovelli 1997, Baez 1998, Barrett & Crane 1998J or causal dynamical triangu-
lations [Ambj0rn & Loll 1998, Ambjorn, Jurkiewicz & Loll 2001,20051. The-
ories of this type introduce non-tensorial quantities, which may in turn suggest
a breaking of general covariance at least at the quantum level. Moreover, other
approaches to gravity may induce a breaking of local Lorentz invariance, for ex-
ample, by a preferred class of observers, or test particles, described by a future
unit timelike vector field [Brown & Kuchaf 1995, Jacobson & Mattingly 200 I].
The possible observer dependence of physical quantities beyond tensorial
transformations motivates the introduction of spacetime geometries obeying a
similar observer dependence, which generalize the well-known Lorentzian met-
ric geometry. In this work we review and discuss two different, albeit similar,
approaches to observer dependent geometries under the aspects of observers,
causality and gravity. In section 2 we review the concept of Finsler spacetimes
[Pfeifer & Wohlfarth 2011, 2012, Pfeifer 2013]. We show that it naturally gen-
eralizes the causal structure of Lorentzian spacetimes, provides clear definitions
of observers, observables and observations, serves as a background geometry for
field theories and constitutes a model for gravity. In section 3 we review the con-
cept of observer space in terms of Cartan geometry [Gielen & Wise 2013]. Our
discussion is based on the preceding discussion of Finsler spacetimes, from which
we translate the notions of observers and gravity to Cartan language [Hohmann
2013]. We finally ponder the question what implications do observer-dependent
geometries have on the nature of spacetime.
2. Geometry of the dock postulate:
Finsler spacetimes
As we have mentioned in the introduction, the metric geometry of spacetime
serves multiple roles: it provides a causal structure, crucially enters the defi-
nition of observers, defines measures for length, time and angles and mediates](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-12-320.jpg)
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Manuel Hohmann
the gravitational interaction. In this section we discuss a more general - non-
metric - spacetime geometry which is complete in the sense that it serves all of
these roles. This generalized geometry is based on the concept of Finsler geom-
etry lBao, Chern & Shen 2000, Bucataru & Miron 2007]. Models of this type
have been introduced as extensions to Einstein and string gravity [Horvath 1950,
Vacaru 2002,2007, 2012]. In this work we employ the Finsler spacetime frame-
work [Pfeifer & Wohlfarth 2011, 2012, Pfeifer 2013], which is an extension of
the well-known concept of Finsler geometry to Lorentzian signature, and review
some of its properties and physical applications. This framework is of particular
interest since, in addition to its aforementioned completeness, it can also be used
to model small deviations from metric geometry and provides a possible explana-
tion of the fly-by anomaly [Anderson, Campbell, Ekelund, Ellis & Jordan 2008].
2.1. Definition of Finsler spacetimes
The starting point of our discussion is the clock postulate, which states that the
time measured by an observer's clock moving along a timelike curve ry is the
proper timeT given by the arc length integral (1.3). The expression
(2.1)
under the integral depends on both the position 'Y(t) along the curve and the
tangent vector 'Y(t). Hence, it can be regarded as a function F : T M ---+ lR on the
tangent bundle. The clock postulate thus states that the proper time measured by
an observer's clock is given by the integral
1
t2
T2- Tl = F('Y(t), "y(t))di;'
tr
(2.2)
where F is the function on the tangent bundle given by equation (2.1 ).
For convenience we introduce a particular set (.Ta, ya) ofcoordinates on T j'vf.
Let (.ra) be coordinates on lvf. For y E Ta,lvi we then use the coordinates (ya)
defined by
a
Y = Ya fJxa .
We call these coordinates induced by the coordinates (:,;0·). As a further shorthand
notation we use
a
Oa, = EJxa ,
for the coordinate basis of T(x,y) T M.
Observer dependent geometries 19
We now introduce a different, non-metric geometry of spacetime which still
implements the clock postulate in the form of an arc length integral (2.2), but
with a more general function F on the tangent bundle. Geometries of this type
are known as Finsler geometries, and F is denoted the Finsler function. The
choice ofF we make here is not completely arbitrary. In order for the arc length
integral to be well-defined and to obtain a suitable notion of spacetime geometry
we need to preserve a few properties of the metric-induced Finsler function (2.1).
In particular we will consider only Finsler functions which satisfy the following:
Fl. F is non-negative, P(x, y) 2 0.
F2. Pis a continuous function on the tangent bundle TIVI and smooth where it
is non-vanishing, i.e., on TM {F = 0}.
F3. F is positively homogeneous of degree one in the fiber coordinates and
reversible, i.e.,
F(:,;,>.y) = j.AjF(x,;y) "f)., E lR. . (2.3)
Property Fl guarantees that the length of a curve is non-negative. We cannot
demand strict positivity here, since already in the metric case we have the no-
tion of lightlike curves "f, for which F('Y(t)/y(t)) = 0. For the same reason of
compatibility with the special case of a Lorentzian spacetime metric we cannot
demand that F is smooth on all ofTM, since the metric Finsler function (2.1)
does not satisfy this condition. It does, however, satisfy the weaker condition F2,
which guarantees that the arc length integral depends smoothly on deformations
of the curve "f, unless these pass the critical region where F = 0. Finally, we
demand that the arc length integral is invariant under changes of the parametriza-
tion and on the direction in which the curve is traversed, which is guaranteed by
condition F3.
One may ask whether the Lorentzian metric 9ab can be recovered in case the
Finsler function is given by (2.1 ). Indeed, the Finsler metric
F ( ) 1 .6 .6 p2 ( )
9ab X, Y = 2Ua,Ub :1:, Y ' (2.4)
which is defined everywhere on TM {F = 0}, agrees with 9ab whenever y
is spacelike and with -gab when y is timelike. However, for null vectors where
F = 0 we see that the Finsler metric g~ is not well-defined, since for a general
Finsler function P2
will not be differentiable. As a consequence any quanti-
ties derived from the metric, such as connections and curvatures, are not defined](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-13-320.jpg)
![20 Manue/ Hohmann
along the null structure, which renders this type of geometry useless for the de-
scription of lightlike geodesics. In the following we will therefore adopt the fol-
lowing definition of Finsler spacetimes which remedies this shortcoming [Pfeifer
& Wohlfarth 2011 ]:
Definition 2.1 (Finsler spacetime). A Finsler spacetime (M, L, F) is a four-
dimensional, connected, Hausdorff, paracompact, smooth manifold 111 equipped
with continuous real functions L, F on the tangent bundle T lvi which has the
following properties:
L I. L is smooth on the tangent bundle without the zero section T M {0}.
L2. L is positively homogeneous of real degree n 2: 2 with respect to the
fiber coordinates ofTlvf,
L(x,>..y) = >..nL(x,y) Y>.. > 0,
and defines the Finsler function F via F(x, y) = JL(x, y)J~.
L3. Lisreversible: JL(.T,-y)J = JL(x,y)J.
L4. The Hessian
L 1- -
Yab(x,y) = 20aObL(x,y)
of L with respect to the fiber coordinates is non-degenerate on TM X,
where X C T 111 has measure zero and does not contain the null set
{(.1:,y) E TMJL(x,y) = 0}.
L5. The unit timelike condition holds, i.e., for all x E M the set
Dx = {y E TxiVI /JL(.T, y)J = 1, g{;:b(x, y) has signature (E, -E, -E, -E)}
with E = L(x,y)/JL(x,y)J contains a non-empty closed connected com-
ponent Sx <:;; Dx C T."A1.
One can show that the Finsler function F induced from the fundamental ge-
ometry function L defined above indeed satisfies the conditions Fl to F3 we
required. Further, the Finsler metric (2.4) is defined on T M {L = 0} and is
non-degenerate on TM (X U {L = 0}), where X is the degeneracy set of the
Hessian g~b defined in condition L4 above. This definition in terms of the smooth
fundamental geometry function L will be the basis of our discussion of Finsler
spacetimes in the following sections, where we will see that it also extends the
Observer dependent 9eometries 21
sign [, = 1 sign /, = 1
sign/,= -1
Figure 1: Light cone and future unit timelike vectors Sa: in the tangent space of a
metric spacetime [Pfeifer & Wohlfarth 2011].
definitions of other geometrical structures such as connections and curvatures to
the null structure.
2.2. Causal structure and observers
The first aspect we discuss is the causal structure of Finsler spacetimes and the
definition of observer trajectories. For this purpose we first examine the causal
structure of metric spacetimes from the viewpoint of Finsler geometry, before
we come to the general case. We have already mentioned in the introduction
that the definition of causal curves is given by the split of the tangent spaces
into timelike, spacelike and lightlike vectors. Figure 1 shows this split induced
by the Lorentzian metric on the tangent space T",lvf. Solid lines mark the light
cone which is constituted by null vectors. In terms of the fundamental geometry
function
L(x, y) = Yab(:r;);~/yb
these are given by the condition L(x, y) = 0. Outside the light cone we have
spacelike vectors with L(:r.:, y) > 0, while inside the light cone we have timelike
vectors with L(x, y) < 0. The Hessian g{;:b = Yab therefore has the signature
indicated in condition L5 inside the light cone. In both the future and the past
light cones we find a closed subset with JL(x, y) J= 1. Using the time orientation
we pick one of these subsets and denote it the shell Sx of future unit timelike
vectors.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-14-320.jpg)
![22 Manuel Hohmann
The shell Sx has the important property that rescaling yields a convex cone
Cx = U >.Sa: C Txlvf.
.>0
(2.5)
The convexity of this cone is crucial for the interpretation of the elements of Sx
as tangent vectors to observer world lines, as it is closely linked to the hyperbol-
icity of the dispersion relations of massive particles and the positivity of particle
energies measured by an observer [Ratzel, Rivcra & Schuller :iOlll We require
this property also for the future light cone of a Finsler spacetime. In order to
find this structure in terms of the fundamental geometry function L consider the
simple bimetric example
with two Lorentzian metrics hab and kah where we assume that the light cone of
kab lies in the interior of the light cone of hab· The sign of L and the signature
of g{;b on the tangent space Txlvf are shown in figure 2. Solid lines mark the
null structure L = 0, while the dashed-dotted lines marks the degeneracy set
X n T.TM of L as defined in condition L4. The remaining dashed and dotted
lines mark the unit timelike vectors nx as defined in condition L5; for these only
the future directed tangent vectors are shown. The connected component marked
by the dashed line is closed, while the one marked with the dotted line is not.
Hence, the former marks the set Sx. As the figure indicates, the set (2.5) indeed
forms a convex cone for this simple bimetric example. It can be shown that
condition L5 always implies the existence of a convex cone of observers [Pfeifer
& Wohlfarth 2011 J, in consistency with the requirement stated above.
It is now straightforward to define:
Definition 2.2 (Observer world line). A physical observer world line on a Finsler
spacetime is a curve 1 : lR --+ Jvf such that at all times t the tangent vector "y(t)
lies inside the forward light cone cy(t)• or in the unit timelike shell s,(r) if the
curve parameter is given by the proper timeT.
In the following section we will discuss which of these observers are further
singled out by the Finsler spacetime geometry as being inertial observers.
2.3. Dynamics for point masses
In the preceding section we have seen which trajectories are allowed for physical
observers. We now turn our focus to a particular class of observers who follow
Observer dependent geometries
L = -1
S:r:
' ', L ~ 1
·· ...
...··
' '
(+,-·,-,-)
L>O /
/
/
/
/
/
/
/ ..· ..... ······
(-,+, 1,+)
L>O
··.,
··...
23
/'
Figure 2: Null structure and future unit timelike vectors Sx in the tangent space
of a bimetric Finsler spacetime [Pfeifer & Wohlfarth 2011 ].
the trajectories of freely falling test masses. These are denoted inertial observers,
since in their local frame of reference gravitational effects can be neglected. On
a metric spacetime they are given by those trajectories which extremizc the arc
Jenth integral (1.3). In Finsler geometry we can analogously obtain them from
extremizing the proper time integral (2.2). Variation with respect to the curve
yields the equation of motion
-·a Na ( ·) · b 0
I + b 1,11 = , (2.6)
where the coefficients Nab are given by the following definition:
Definition 2.3 (Cartan non-linear connection). The coefficients Nab of the Car-
tan non-linear connection are given by
(2.7)
and define a connection in the sense that they induce a split of the tangent bundle
over Tlvf,
TTM = HTM EB VTM, (2.8)
where HTM is spanned by Oa = Oa - NbaBb and VTJ"'vf is spanned by Ba.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-15-320.jpg)
![24
Manuel Hohmann
In the case of a metric-induced Fins!er function (2.1) the coefficients Nab are
given by
(2.9)
where f"bc denotes the Christoffel symbols. The split (2.8) of TTM into hori-
zontal and vertical subbundles plays an important role in Finsler geometry, as we
will see in the following sections. For convenience we use the following adapted
basisofTTM:
Definition 2.4 (Berwald basis). The Berwald basis is the basis
(2.10)
of TTl'vf which respects the split induced by the Cartan non-linear connection.
For the dual basis we use the notation
(2.11)
It induces a similar split of the cotangent bundle T*TM into the subbundles
T*TM = H*TM EB V*TM. (2.12)
We can now reformulate the geodesic equation (2.6) by making use of the geom-
etry on TTNI. For this purpose we canonically lift the curve 1 to a curve
(2.13)
in TTl'vf. The condition that 1 is a Finsler geodesic then translates into the con-
dition
r. ·a,;o, + ..a;:; ·a,;o, ·bNa ;:> ·a>
= I Ua I Ua = I ua - I bUa = I "a
Since "fa is simply the tangent bundle coordinate ya, it thus follows that the
canonical lift r of a Finsler geodesic must be an integral curve of the vector
field which is defined as follows:
Definition 2.5 (Geodesic spray). The geodesic ~prayS is the vector field on T A1
which is defined by
(2.14)
We now generalize this statement to null geodesics. Here we encounter two
problems. First, we see that the coefficients (2.7) of the non-linear connection
are not well-defined for null vectors where F = 0, since F is not differentiable
Observer dependent geometries 25
----------------
on the null structure. We therefore need to rewrite their definition in terms of the
fundamental geometry function L. It turns out that it takes the same form
N" = ~D [nLac('{lrlEJ [) L- D.L)]
b 4 b " ,} d c (. ) (2.15)
where .r/'' has been replaced by gL and F 2 by L. We can see that this is well-
defined whenever gL is non-degenerate, and thus in particular on the null struc-
ture. The second problem we encounter is that we derived the geodesic equation
from extremizing the action (2.2), which vanishes identically in the case of null
curves. We therefore need to use the constrained action
S[r, >-] = .{" (L(1 (t), "y(t)) + >-(t) [L('y(t), ''(t)) - h;]) dt. (2.16)
with a Lagrange multiplier A. and a constant "'· A thorough analysis shows that
the equations of motion derived from this action are equivalent to the geodesic
equation (2.6) also for null curves [Pfeifer & Wohlfarth 20 11].
The definitions of this and the preceding section provide us with the notions
of general and inertial observers. In the following section we will discuss how
these observers measure physical quantities and how the observations by different
observers can be related.
2.4. Observers and observations
As we have mentioned in the introduction, the notion of geometry in physics de-
fines not only causality and the allowed trajectories of observers, but also their
possible observations and the relation between observations made by different
observers. In the case of metric spacetime geometry we have argued that obser-
vations are constituted by measurements of the components of tensor fields at a
spacetime point x E l'vf with respect to a local frame f at :r. A particular class
of frames singled out by the geometry and most convenient for measurements is
given by the orthonormal frames. Different observations at the same spacetime
point, but made with different local orthonormal frames, are related by Lorentz
transforms. In this section we discuss a similar definition of observations on
Finsler spacetimes and relate the observations made by different observers.
As a first step we need to generalize the notion of observables from metric
spacetimes to Finsler spacetimes. In their definition in section 2.1 we have al-
ready seen that the geometry of Finsler spacetimes is defined by a homogeneous
function L : T M ---7 ~ on the tangent bundle, which in turn induces a Finsler](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-16-320.jpg)
![28 Manue/ Hohmann
Cartan linear connection on the tangent bundle TTli!J, which is defined as fol-
lows:
Definition 2.9 (Cartan linear connection). The Cartan linear connection 7 is the
connection on TT!vi defined by the covariant derivatives
7<>a Ob = F"a"Oc , 7",}J,, = F",,,J),, , '1'7 c; 'b - C" b:;: '1'7 fSb - C'c 1(;;;
u ,. u, •U • v,,au - a Uc, VOa'J - aJJc,
(2.19)
where the coefficients are given by
(2.20a)
(2.20b)
The Cartan linear connection is adapted to the Cartan non-linear connec-
tion (2.7) in the sense that it respects the split (2.8) into horizontal and verti-
cal components. By restriction, it thus provides a connection on the horizon-
tal tangent bundle. Given a curve v : [0, 1] -+ TM with v(O) = (:r, fo)
and v(l) = (:r, !6) we can then define a bijective map Pu from T(x,fo)'TM to
T(xJ£)TM by parallel transport: it maps the vector w to Pvw = w', which is
uniquely determined by the existence of a curve ·w : [0, 1] -+ TTM satisfying
{iJ(s)ETv(s)TM, <v(O)=w, u)(1)=v/, 7-uw=O.
However, this map Pv in general depends on the choice of the curve ·v. We
therefore restrict ourselves to a particular class of curves. Note that (x, Jo) and
(x, !6) have the same base point in M, and are thus elements of the same fiber
of the tangent bundle T !vi. Hence it suffices to consider only curves which are
entirely contained in the same fiber. Curves of this type are vertical, i.e., their
tangent vectors lie in the vertical tangent bundle VTlvi. We further impose the
condition that v is an autoparallel of the Cartan linear connection. This uniquely
fixes the curve v, provided that f~ is in a sufficiently small neighborhood of f0
.
Using the unique vertical autoparallel v defined above we can now generalize
the notion of Lorentz transformations to coincident observers on a Finsler space-
time. Consider two observers meeting at :1: E Jvi and using frames f and f', i.e.,
orthonormal bases of H(x,Jo) T Jvi and H(x,JfJ) T M. The map Pv maps the hori-
zontal basis vectors j; to horizontal vectors Pvfi, which constitute a basis Pvf
of H(xJ6)TlVI. Since f is orthonormal with respect to gf:t,(x, fo) and the Cartan
linear connection preserves the Finsler metric, it follows that Pvf is orthonormal
with respect to gf:t,(x, !0). Since also f' is orthonormal with respect to the same
Observer dependent geometries 29
metric, there exists a unique ordinary Lorentz transform mapping Pvf to .f'. The
combination of the parallel transport along v and this unique Lorentz transform
finally defines the desired generalized Lorentz transform.
The procedure to map bases of the horizontal tangent space between coin-
cident observers further allows us to compare horizontal tensor components be-
tween these observers, so that they can communicate and compare their mea-
surements of horizontal tensors. This corresponds to the transformation (1.8) of
tensor components of observables between different observer frames in metric
geometry. Since observables in metric geometry are modeled by spacetime ten-
sor fields, their observation in one frame determines the measured tensor compo-
nents in any other frame. This is not true on Finsler spacetimes, since we defined
observables as fields on the tangent bundle T1'd. They may therefore also pos-
sess a non-tensorial, explicit dependence on the four-velocity of the observer who
measures them.
As in metric geometry, also in Finsler geometry the dynamics of tensor fields
should be determined by a set of field equations which are derived from an action
principle. This will be discussed in the next section.
2..5. Field theory
In the preceding section we have argued that observables on a Finsler spacetime
are modeled by homogeneous horizontal tensor fields, which are homogeneous
sections of the horizontal tensor bundle (2.17). We will now discuss the dynamics
of these observable fields. For this purpose we will use a suitable generalization
of the action (1.6) to horizontal tensor fields on a Finsler spacetime. This will be
done in two steps. First we will lift the volume form from the spacetime manifold
Jvi to its tangent bundle T M, then we generalize the Lagrange function L to fields
on a Finsler spacetime.
In order to define a volume form on T Jvi we proceed in analogy to the volume
form of metric geometry, which means that we choose the volume form Vole of
a suitable metric G on TM. We have already partly obtained this metric in the
previous section when we discussed orthonormal observer frames. The definition
of orthonormality we introduced corresponds to lifting the Finsler metric gf:t, to
a horizontal metric on T M, which measures the length of horizontal vectors in
HTM. This metric needs to be complemented by a vertical metric, which anal-
ogously measures the length of vertical vectors in VTJvf. Both metrics together
constitute the desired metric on the tangent bundle. The canonical choice for this
metric is given by the Sasaki metric defined as follows:](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-18-320.jpg)
![30 Manuel Hohmann
Definition 2.10 (Sasaki metric). The Sasaki metric G is the metric on the tangent
bundle T M which is defined by
F'
G = -g"' dx" 0 d:r0 - !l_au oya 00 rl1,1'
ab p2 - . 1 ·
(2.21)
The factor p-z introduced here compensates for the intrinsic homogeneity of
degree 1ofthe one-forms oy", so that the Sasaki metric is homogeneous ofdegree
0. This intrinsic homogeneity becomes clear from the definition (2.11) of the dual
Berwald basis, taking into account that the coefficients N'"z, are homogeneous of
degree I, as can be seen from their definition (2.7). Using the volume form Vola
of the Sasaki metric one can now integrate functions .f on the tangent bundle,
{ Vola f(:r, y).
lrM
(2.22)
If one chooses the function .f to be a suitable Lagrange function L for a physical
field <P on a Finsler spacetime, one encounters another difficulty. Since all geo-
metric structures and matter fields <P are homogeneous, it is natural to demand
the same from the Lagrangc function. However, for a homogeneous function .f
the integral over the tangent bundle generically diverges, unless the function van-
ishes identically. This follows from the fact that along any ray (:r, Ay) with ).. > 0
in TM the value off is given by An f(x, y), where n is the degree of homogene-
ity. This difficulty can be overcome by integrating the function not over TM, but
over a smaller subset ofTM which intersects each ray only once, and which is
defined as follows:
Definition 2.11 (Unit tangent bundle). The unit tangent bundle ofa Finsler space-
time is the set I; C T NI on which the Finsler function takes the value F = 1.
Note that I; intersects each ray, which is not part of the null structure, exactly
once. This suffices since the null structure is of measure 0 and therefore does not
contribute to the integral (2.22) overTM. The canonical metric on I: is given by
the restriction
(2.23)
ofthe.Sasaki metric, which finally determines the volume form Vole;. This is the
volume form we will use in the generalized action integral.
In the second part of our discussion we generalize the Lagrange function L in
the metric matter action (1.6). For simplicity we restrict ourselves here top-form
fields <P whose Lagrange function depends only on the field itself and its first
derivatives d<D. These are of particular interest since, e.g., the Klein-Gordon and
Observer dependent geometries 31
Maxwell fields fall into this category. The most natural procedure to generalize
the dynamics of a given field theory from metric to Finsler geometry is then to
simply keep the formal structure of its Lagrange function £, but to replace the
Lorentzian metric g by the Sasaki metric G and to promote the p-form field <P to
a horizontal p-form field on T M. The generalized Lagrange function we obtain
from this procedure is now a function on TNI, which we can integrate over the
subset I; to form an action integral.
Using this procedure we encounter the problem that even though we have
chosen <P to be horizontal, d<l> will in general not be horizontal. In order to obtain
consistent field equations we therefore need to modify our procedure. Instead of
initially restricting ourselves to horizontal p-forms on the tangent bundle T lvf, we
let <P be an arbitrary p-form with both horizontal and vertical components. The
purely horizontal components can then be obtained by applying the horizontal
projector
(2.24)
In order to reduce the number of physical degrees of freedom to only these hor-
izontal components we dynamically impose that the non-horizontal components
vanish by introducing a suitable set of Lagrange multipliers /, so that the total
action reads
(2.25)
Variation with respect to the Lagrangc multipliers then yields the constraint that
the vertical components of <P vanish. Variation with respect to these vertical
components fixes the Lagrange multipliers. Finally, variation with respect lo the
horizontal components of <P yields the desired field equations. It can be shown
that in the metric limit they reduce to the usual tleld equations derived from the
action (1.6) for matter fields on a metric spacetime [Pfeifer & Wohlfarth 2012].
2.6. Gravity
In the previous sections we have considered the geometry of Finsler spacetimes
solely as a background geometry for observers, point masses and matter fields.
We now turn our focus to the dynamics of Finsler geometry itself. As it is also
the case for Lorentzian geometry, we will identify these dynamics with the dy-
namics of gravity. For this purpose we need to generalize the Einstein-Hilbcrt
action, from which the gravitational field equations are derived, and the energy-
momentum tensor, which acts as the source of gravity.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-19-320.jpg)
![32 Manue/ Hohmann
We start with a generalization of the Einstein-Hilbert action (1.9) to Finsler
spacetimes. As in the case of matter field theories detailed in the preceding sec-
tion this generalized action will be an integral not over spacetime !vi, but over the
unit tangent bundle .BC TM, since the geometry is defined in terms of the ho-
mogeneous fundamental geometry function L on TM. We have already seen that
a suitable volume form on .B is given by the volume form Vol0 of the restricted
Sasaki metric (2.23). This leaves us with the task of generalizing the Ricci scalar
R in terms on Finsler geometry.
The most natural and fundamental notion of curvature is defined by the Car-
tan non-linear connection (2.7), which we already encountered in the definition
ofFinsler geodesics in section 2.3 and which corresponds to the unique split (2.8)
of the tangent bundle TTlvf into horizontal and vertical components. This split
is also the basic ingredient for the following construction. The curvature of the
Cartan non-linear connection measures the non-integrability ofthe horizontal dis-
tribution HTlvf, i.e., the failure of the horizontal vector fields ba to be horizontal.
In fact, their Lie brackets are vertical vector fields, which are used in the follow-
ing definition:
Definition 2.12 (Non-linear curvature). The curvature ofthe non-linear connec-
tion is the quantity neab which measures the non-integrability of the horizontal
distribution induced by the Cartan non-linear connection,
(2.26)
The simplest scalar one can construct from the curvature coefficients defined
by (2.26) is the contraction RaabYb, so that the action for Finsler gravity takes the
form
11 a b
Sr = - Vol0
- R abY .
~;, B '
(2.27)
In the case of a metric-induced Finsler function, in which the non-linear connec-
tion coefficients Nab are given by (2.9), the expression under the integral indeed
reduces to the Ricci scalar, so that SF is a direct generalization ofthe Einstein-
Hilbert action (1.9). In order to obtain a full gravitational theory this action needs
to be complemented by a matter action, such as the field theory action (2.25) we
encountered in the previous section. This total action then needs to be varied
with respect to the mathematical object which fundamentally defines the space-
time geometry. On a Finsler spacetime this is the fundamental geometry function
L. Consequently, the gravitational field &lquations are not two-tensor equations as
Observer dependent geometries
in general relativity, but instead the scalar equation
[
Fab;- - r- . d .RaabYb
.1 !Jaih(R cdil ) - 6 p2-
33
-1- 2/"nb (vaSh -1- SaSb -1- 8a(y"l5cSb- Nc,s,J)] ~)~ = K:Til~ (2.28)
on the unit tangent bundle .B. Here T denotes the energy-momentum scalar ob-
tained by variation of the matter action SM with respect to the fundamental ge-
ometry function L. For the field theory action (2.25) it is given by
{
nL 15 [ /-:: ( (. _pH)">)]}
Tl2;= gnL v-c .c(c,iJ>,diJ>)+>. 1 'J. E
(2.29)
It can be shown that in the metric limit the resulting gravitational field equa-
tion (2.28) is equivalent to the Einstein equations (I. I0), whose free indices are
to be contracted with y0
fPfeifer & Wohlfarth 20121.
We finally remark that also the Cartan linear connection we used to define
generalized Lorentz transformations in section 2.4 defines a notion of curvature,
which may in principle be used to generalize the Einstein-Hilbert action. This
curvature is defined as follows:
Definition 2.13 (Linear curvature). The curvature of the Cartan linear connec-
tion is given by
(2.30)
for vector fields X, Y, Z on T M.
Using the action (2.19) of the Cartan linear connection on the vector fields
constituting the Berwald basis and the coefficients (2.20) one finds that its curva-
ture can be written in the form](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-20-320.jpg)
![34 Manuel Hohmann
where the coefficients are given by
Rdcab = 8,,1"dea - 8aFdcb + peca.Fdeb - pccbpdea + Cdce (obNea - 8aNeb) ,
(2.32a)
prlcab= [),prlea- 8a.Crlcb + peca.Cricb- C"cbprlm+ CdceDbNea, (2.32b)
c..,'fi - i) cd f)-:- cd ce cd ce cd
c ca.b - Ub ea - a cb + > ea cb - cb -' en · (2.32c)
ln the metric limit the coetlicient Rrlca.b reduces to the Riemann tensor, while
the remaining coefficients P'1
cab and 8dcab vanish. One may therefore consider
the term gF abReacb as another generalization of the Ricci scalar to generate the
gravitational dynamics on Finsler spacetimes. We do not pursue this idea further
here and only remark that also other choices are possible.
3. The local perspective:
Cartan geometry of observer space
In the previous section we have seen that on Finsler spacetimes the definitions
of observers and observables are promoted from geometrical structures on the
spacetime manifold lvf to homogeneous geometrical structures on its tangent
bundle T Jvf, and that this homogeneity fixes quantities on T M when they are
given on the unit tangent bundle ~- We have also seen that measurements by
an observer probe these structures along a lifted world liner = (r, "f) in T M.
However, it follows from the definition of physical observer trajectories that ev-
ery curve r is entirely confined to future unit timelike vectors, so that obser-
vations can be performed only on a smaller subset 0 c ~. which we denote
observer space. In this section we will therefore restrict our discussion to ob-
server space and equip it with a suitable geometrical structure in terms of Cartan
geometry [Cartan 1935, Sharpe 1997], which we derive from the previously de-
fined Finsler geometry [Hohmann 2013]. While Cmtan geometry turns out to be
useful already as a geometry for spacetime in the context of gravity [Wise 20 I0],
it becomes even more interesting as a geometry for observer space [Gielen &
Wise 2013] and provides a better insight into the role of Lorentz symmetry in
canonical quantum gravity [Gielen & Wise 2012a, 2012b].
3.1. Definition of observer space
We start our discussion with the definition of observer space as the space of
all tangent vectors to a Finsler spacetime which are allowed as tangent vectors
Observer dependent geometries
35
of normalized observer trajectories, i.e., observer trajectories which are para-
metrized by their proper time. This leads us to the definition:
Definition 3.1 (Observer space). The observer space 0 of a Finsler spacctime
(M, L, F) is the set of all future unit timelike vectors, i.e., the union
O=US:r (3.1)
xEII
of all unit shells inside the forward light cones.
Note that 0 is a seven-dimensional submanifold of TM and that its tangent
spaces T(x,y)O are spanned by the vectors v E T(x,y)T!vi which satisfy vF =: 0.
Further, there exists a canonical projection 1r
1
: 0 -+ NI onto the underlymg
spacetime manifold. The natural question arises which geometrical structure the
Finsler geometry on the spacetime manifold M induces on its observer space 0.
The structure which is most obvious already from our fmdings in the previous
section is the restricted Sasaki metric G, which we defined in (2.23) as the re-
striction of the full Sasaki metric G to ~ and which we now view as a metric on
the smaller set 0 c ~- It follows from the signature of G that Ghas Lorentzian
signature(-,+,+,+,+,+,+).
Another structure which we already encountered in the previous section is
the geodesic spray (2.14). Since it preserves the Finsler function, SF = 0, it is
tangent to the level sets ofF, and thus in particular tangent to observer space 0.
It therefore restricts to a vector field on 0, which we denote the Reeb vector field:
Definition 3.2 (Reeb vector field). The Reeb vectorfield r is the restriction of the
geodesic sprayS to 0,
(3.2)
We now have a metric and a vector field on 0. Combining these two struc-
tures we can form the dual one-form a of the Reeb vector field with respect to
the restricted Sasaki metric G, which we denote the contact form:
Definition 3.3 (Contact form). The contactfonn is the dual one-form of the Reeb
vector field r with respect to the restricted Sasaki metric G,
- F a bI 1[)7 p2 d a I
a= -G(r, .) = gabY d;-r; 0 = 2 a ,:r; 0 .
(3.3)
Conversely, the Reeb vector field is the unique vector field on 0 which is
normalized by a and whose flow preserves a, i.e., which satisfies
Lra=O and a(r)=l. (3.4)](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-21-320.jpg)
![36
Manue/ Hohmann
The naming of a and r originates from the notion of contact geometry. In this
context a contact form on a (2n +I)-dimensional manifold is defined as a one-
form a, which is maximally non-integrable in the sense that the (2n + 1)-fonn
u A du A ... A dais nowhere vanishing, hence defines a volume form, and the
Reeb vector field is the unique vector tield r satisfying (3.4). Indeed, it turns out
that the volume form defined by o: is simply the volume form of the Sasaki metric
GonO.
As we have seen in section 2.3the Finsler geometry induces a split (2.8) of the
eight-dimensional tangent bundle TTJv! into two four-dimensional subbundles
VTM and HTM, denoted the vertical and horizontal subbundles, respectively.
A similar split also applies to the tangent bundle TO of observer space. It splits
into the three subbundles
TO= VO EG HO= VO El) JJO CD H 00, (3.5)
which we denote the vertical, spatial and temporal subbundles, respectively. The
vertical bundle VO is defined in analogy to the vertical tangent bundle VTM
as the kernel of the differential 1r~ of the canonical projection 1r1
: () -r lvf. It
is constituted by the tangent spaces to the shells Sx of unit timelike vectors at
:1: E Jv! and hence three-dimensional. Its orthogonal complement with respect
to the Sasaki metric G is the four-dimensional horizontal bundle HO. One can
easily see that the contact form a vanishes on VO. Its kernel on HO defines
the three-dimensional spatial bundle HO. Finally, the orthogonal complement of
HO in HO is the one-dimensional temporal bundle H 00, which is spanned by
the Reeb vector field r.
The split of the tangent bundle TO has a clear physical interpretation. Verti-
cal vectors in VO correspond to infinitesimal generalized Lorentz boosts, which
change the velocity of an observer, but not his position. They are complemented
by horizontal vectors in HO, which change the observer's position, but not his
direction of motion. These further split into spatial translations in HO and tem-
poral translations in H 0
0 with respect to the observer's local frame. This inter-
pretation will become clear when we discuss the split of the tangent bundle from
a deeper geometric perspective using the language of Cartan geometry. We will
give a brief introduction to Cartan geometry in the following section.
3.2. Introduction to Cartan geometry
In order to describe the geometry of observer space, we make use of a frame-
work originally developed by Cartan under the name 'method of moving frames'
Observer dependent geometries 37
[Cartan 1935]. His description of the geometry of a manifold M is based on
a comparison to the geometry of a suitable model space. The latter is taken
to be a homogeneous space, i.e., the coset space GIH of a Lie group G and
a closed subgroup H c G. Homogeneous spaces were extensively studied in
Klein's Erlangen program and are hence also known as Klein geometries. Car-
tan's construction makes use of the fact that they carry the structure of a principal
If-bundle 1r : G -> GIIf and a connection given by the Maurer-Cartan one-f01m
A E D1
(G,g) on G taking values in the Lie algebra g of C. Using these struc-
tures in order to describe the local geometry of Jv!, a Cartan geometry is defined
as follows:
Definition 3.4 (Cartan geometry). Let G be a Lie group and H c G a closed
subgroup of G. A Cartan geometry modeled on the homogeneous space GIH
is a principal H-bundle 1r : P -r Jv! together with a g-valued one-form A E
D1
(P, g), called the Carum connection on P, such that
C I. For each p E P, A11 : T11P -r g is a linear isomorphism.
C2. A is H-equivariant: (Rh)* A= Ad(h-1
) o A :lh EH.
C3. A restricts to the Maurer-Cartan form on vertical vectors v E ker 1r*.
Instead of describing the Cartan geometry in terms of the Cartan connection
A, which is equivalent to specifying a linear isomorphism Ap : T11P -r g for all
p E P due to condition C I, we can use the inverse maps A11 = A;;- 1
: g -r T11P.
For each a E g they define a section A(a) of the tangent bundle, which we denote
a fundamental vector field:
Definition 3.5 (Fundamental vector fields). Let (1r : P -r M, A) be a Cartan
geometry modeled on GIH. For each a E g the fundamental vectorfield A(a) is
the unique vector field such that A(A(a)) =a.
We can therefore equivalently define a Cartan geometry in terms of its funda-
mental vector fields, due to the following proposition:
Proposition 3.1. Let (1r : P -r M, A) be a Cartan geometry modeled on GIfl
and A : g -r Vect Pits fundamental vector fields. Then the properties Cl to C3
of A are respectively equivalent to the following properties ofA:
Cl'. For each p E P, Ap : g --7 T11P is a linear isomorphism.
C2'. A is H-equivariant: Rh* o A= A o Ad(h-1
) :lh EH.
C3'. A restricts to the canonical vector.fzelds on [J.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-22-320.jpg)
![40
fv!anuel Hohmann
in the following diagram:
vpP (f} HpP TpP
wr +
ef Ar
b m 3 jJ
(3.13)
The vertical subbundle V P is constituted by the tangent spaces to the fibers of
the bundle if : P ---+ A1, which are given by the kernel of the differential if* of
the canonical projection. This is a direct consequence of condition C3 on the
Cartan connection. The elements of V P can be viewed as infinitesimal local
Lorentz transformations, which change only the local frame f and leave the base
point x unchanged. Conversely, the elements of the horizontal subbundle Hp
correspond to infinitesimal translations, which change the base point x without
changing the orientation of the local frame f. This follows from the fact that we
constructed the f)-valued part w of the Cartan connection from the Levi-Civita
connection.
3.3. Cartan geometry of observer space
We will now employ Cartan geometry in order to describe the geometry of ob-
server space. Hereby we will proceed in analogy to the metric spacetime example
discussed in the previous section, where we constructed a Cartan connection on
the orthonormal frame bundle. For this purpose we refer to the definition of or-
thonormal observer frames in section 2.4. If we translate this definition to the
context of observer space geometry, we find that an observer frame at (x, y) E 0
is a basis of the horizontal tangent space H(x,y)O such that w~fo = y and the
normalization (2.18) holds. Equivalently, we can make use of the differentialw~
of the canonical projection w' : 0 ---+ M, which isomorphically maps H(x,y)O
to Tx!VI, and regard frames as bases of Txllif, in analogy to the case of metric
geometry. Here we choose the latter and define:
Definition 3.6 (Observer frames). The space P of observer frames of a Finsler
spacetime (A1, L, F) with observer space 0 is the space of all oriented, time-
oriented tangent space bases f of llif, such that the basis vector fo lies in 0 and
the frame is orthonormal with respect to the Finsler metric,
One can now easily see that although there exists a canonical projection
if : P---+ NI, which assigns to an observer frame its base point on M, it does not
Observer dependent geometries 41
in general define a principal 1I-bundle, where H is the Lorentz group as in the
preceding section. This follows from the fact that the generalized Lorentz trans-
forms discussed in section 2.4 do not form a group, but only a grupoid. However,
this is not an obstruction, as it is our aim to construct a Cartan geometry on 0
and not on M. Indeed, the projection 1r : P ---+ 0, which simply discards the
spatial frame components, carries the structure of a principal J(-bundle, where
by J( we denote the rotation group S0(:3). It acts on P by rotating the spatial
frame components. The Cartan geometry on observer space will thus be modeled
on the homogeneous space GIf( instead of GIH.
We further need to equip 1r : P ---+ 0 with a Cartan connection which gener-
alizes the Cartan connection on the metric frame bundle displayed in the previous
section. Here we can proceed in full analogy and choose as the 3-valued parte of
the connection the solder form. The expression in component notation,
(3.14)
agrees with the analogous expression (3.1 0) in metric geometry. For the f)-valued
part w we generalize the Levi-Civita connection (3.12). Recall from section 2.4
that the tangent space T NI of a Finsler spacetime, and hence also its observer
space 0 c T M, is equipped with the Cartan linear connection (2.19). We can
therefore replace the projection if to !VI in (3.11) with the projection 1r to 0 and
define
(3.15)
where l now denotes the Cartan linear connection. In component notation this
yields the expression
i f-ljdr+"a f·-lj fb [Fa d c ea (Ne l .d + J'l'c.)]
w· 'i = a J i + . a. i be X + be d(,~L 1J 0
1 (>kd jk ) j·-ll df" 1 jkj·bfc(s: F s: F)d a
= 2 o; uz - 7) TJil , a . k + 27) i . k Ub.flac - Oc9ab ;r; ' (3.16)
where the coefficients cabc and Fabc are the coefficients for the Cartan linear
connection (2.20). From the Cartan connection (3.14) and (3.16) we then find the
fundamental vector fields
A(h) = (h;Jff- hioff fjCabc) &z,
A(z) = ziff (oa- f]Fc"J)1)
(3.17a)
(3.17b)](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-24-320.jpg)
![46 Manuel Hohmann
introducing the projectors
onto the subbundles of TO and obtain the form PJJf(T) = r(r(T)). Finally,
inserting the explicit formulas for PH and r we arrive at the reformulated defini-
tion:
Definition 3.8 (Observer trajectory). A physical observer trajectory is a curve
r on observer space whose horizontal components are given by the Reeb vector
field, i.e., which satisfies
A particular class of observers is given by inertial observers, whose trajecto-
ries follow those of freely falling test masses. In section 2.3 we have seen that
these are given by Finsler geodesics, or equivalently by curves whose complete
lift (1, '"') in TM is an integral curve of the geodesic spray (2.14). We have
further seen that the geodesic spray is tangent to observer space () c T A-1 and
defined the Reeb vector field r as its restriction to observer space. It thus im-
mediately follows that inertial observer trajectories on 0 are simply the integral
curves of the Reeb vector field. Comparing this finding with the aforementioned
definition we see that inertial observer trajectories arc exactly those observer tra-
jectories whose vertical tangent vector components vanish. We thus define, using
only Cartan geometry:
Definition 3.9 (Inertial observer trajectory). An inertial observer trajectory is an
integral curve of the Reeb vector field, i.e., a curve r on observer space which
satisfies
It appears now straightforward to translate the notions of observables and ob-
servations from Finsler geometry to Cartan geometry on observer space. A direct
translation yields observables as sections of a horizontal tensor bundle, which is
constructed from the horizontal subbundle HO in analogy to the horizontal ten-
sor bundle Hr,srNI. Observations by an observer at r(T) E 0 then translate
into measurements of the components of a horizontal tensor field with respect to
a basis of the corresponding horizontal tangent space Hl'(r)O, which can con-
veniently be expressed using the vector fields g_;. Finally, also a translation of
the matter action (2.25), where 1> is viewed as a one-form on 0 and the projec-
tors (3.26) are used, is straightforward. However, we do not pursue this topic
Observer dependent geometries
47
here. Instead we will directly move on to the gravitational dynamics in the next
section.
3.5. Gravity
As we have already done in the case of Finsler geometry in section 2.6, we now
focus on the dynamics of the Cartan geometry, which we identify with the grav-
itational dynamics. Since gravity is conventionally related to the curvature of
spacetime, we will first discuss the notion of curvature in Cartan geometry. We
will then derive dynamics for Cartan geometry from an action principle and see
how this notion of curvature is involved. For this purpose we will consider two
different actions, the first being the Finsler gravity action we encountered before
and which we now translate into Cartan language, and an action which is explic-
itly constructed in terms of Cartan geometric objects.
We start our discussion of curvature in Cartan geometry with its textbook
definition:
Definition 3.10 (Cartan curvature). The curvature of a Cm·tan geometry (if :
p -+ M, A) modeled on the homogeneous space GIHis the g-valued two-form
FE n2 (P, g) on P given by
(3.27)
The curvature has a simple interpretation in terms of the fundamental vector
fields £1: it measures the failure of A : g -+ Vect P to be a Lie algebra homo-
morphism. This can be seen from the relation
A([a, a'])- [ll(a),A(a')] = A(F(A(a),A(a'))), (3.28)
which can easily be derived from the definition (3.27) by making use of the stan-
dard formula
dO"(.X, Y) = X(O"(Y))- Y(O"(X))- O"([X, Y])
for any one-form O" and vector fields X, Y.
From this general definition we now turn our focus to the Cartan geometry
on observer space modeled on G1K, which we derived from Finsler geometry
in section 3.3. In this context the term [A(a), A(a')] for a, a' E 3 in the rela~
tion (3.28) reminds of the Lie bracket of horizontal vector fields [8a, 6b] in the](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-27-320.jpg)
![48 Manue/ Hohmann
definition of the non-linear curvature Reab on TTNI. Indeed, the similar expres-
sion on P given by
[ ] - j'bjcf·d(' pa s: F" +pc F" pe F" )<Sk
f;, f!.j - i j. k Oc bd- Ub cd bd ce- cd be Oa (3.29)
reproduces the components of the non-linear curvature (2.26), which can equiva-
lently be written in the form
Ra _ 'lJd(s: F" (" p,a , pe pa }i'e pa )
" be - . Uc bd - lb · cd T bd ce - cd ' be · (3.30)
We can directly apply this result to the Finsler gravity action (2.27) on the unit
tangent bundle I; C T NI. Since observer space is simply the connected com-
ponent of the unit tangent bundle constituted by the future timelike vectors, it is
straightforward to consider the restricted action
(3.31)
as a gravity action on 0. This action is still written in terms of Finsler geometric
objects, which we will now rewrite in terms of Cartan geometry. For the volume
form Vol0 of the Sasaki metric Gwe have already found the expression (3.25),
while for the non-linear curvature coefficients R"bc we can make use of the Lie
bracket (3.29) of horizontal vector fields on P together with the relation (3.30).
In order to reproduce the scalar quantity R"abYb in the Finsler gravity action from
this vector field we further apply the boost component b(J! of the Cartan connection
and contract appropriately, which yields
The last equality follows from the identification of the tangent vector y" with the
temporal frame component jg. Note that this expression is a scalar on P which
is constant along the fibers of 1r : P ---+ 0, and can thus be viewed as a scalar on
0. We thus finally obtain the gravitational action
(3.32)
which is now fully expressed in terms of Cartan geometry.
Another possible strategy to obtain gravitational dynamics on the observer
space Cartan geometry is to start from general relativity, rewrite the Einstein-
Hilbert action in terms of the Cartan connection derived from the metric ge-
Observer dependent geometries 49
ometry displayed in section 3.2, and finally transform the action to an integral
over observer space by introducing an appropriate volume form on the fibers of
1r' : 0---+ lvf. We will follow this procedure for the remainder of this section. The
starting point of this derivation is the action given by MacDowell and Mansouri
[1977!. In terms of spacetime Cartan geometry it takes the form !Wise 201 0]
(3.33)
Here r..:h is a non-degenerate inner product on ~. For simplicity we choose
where trn is the Killing form on ~ and* denotes a Hodge star operator. In com-
ponents we can write the Killing form as
and the Hodge star operator as
( )
i im ln 1k
*h j = 7J 7J Emjkl I. n ·
The two-form Ph is given hy the unique decomposition
F =Fry+ F;,
of the g-valued Cartan curvature (3.27) into parts with values in ~ and 3· Finally,
the tilde indicates that we need to lower this two-form Fn on the frame bundle P
to a two-form Ph on the base manifold M.
We now aim to lift the action (3.33) to observer space. For this purpose
we need to find a suitable volume form on the fibers of 7r
1
: 0 ---+ lvf. Recall
from definition (3.1) of observer space that these are given hy the future unit
timelike shells Sx for x E M, which are three-dimensional submanifolds of
T lvi. A natural metric on Sx is thus given by the restriction of the Sasaki metric
G on T ld, or equivalently Gon 0, to Sx. Using our results from section 3.3 on
the Cartan geometry of observer space we find that the tangent spaces to Sx are
spanned by the vertical vector fields ~"', so that the Sasaki metric (3.24) restricts
to the Euc!idean metric Ocv.ri.P· Q9 //3. Its volume form is given by
(3.34)](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-28-320.jpg)
![50
Manuel Hohmann
In combination with the action (3.33) lifted to observer space, which means that
P~ is now regarded as a two-form on 0, this yields the action
SMM = r r,;i)(Fr) A Fr)) A Vols.
.fo (3.35)
In order to analyze the terms in this action we make use of the algebra rela-
tions (3.9) to decompose F1
J in the form
1 1 1
FrJ = dw + 2[w, w] +
2[c, e] = Fw +
2
[e, e]
into the curvature Fw of wand a purely algebraic term ~[e, e]. Using the expres-
sions (3.14) fore and (3.16) for w these take the form
. 1 . (
J . _ _ _ .-!] c d .a. ,.b d .a ·b •d a b
Fw "- 2.f d.fi R caMh A dx + 2P cabdx A oj0 + S cabo.f0 A <'5f0
) ,
[e e]i - 2[-liJ-lk ., Ad .a Ad b
-, ' - . a b7Jik sgn . x ,, x ,
where we have introduced the shorthand notation of(f = dfff + Nabdxb. The
coefficients Rdcab• pdcab and Sdcab we find here are the coefficients (2.32) of
the curvature of the Cartan linear connection, which is not surprising, since we
used the Cartan linear connection in the definition (3.15) of w. The term [e, e]
depends on the choice of the group G, and thus on the sign of the cosmological
constant on the underlying homogeneous space. Applying this decomposition to
the expression r,;IJ (FIJ A FIJ) in the action (3.35) we obtain the following terms:
• A cosmological constant term:
• A curvature term:
• A Gauss-Bonnet term:
The ellipsis in the expressions above indicates that we have omitted terms which
are not horizontal, i.e., which contain the vertical one-form b. These terms do not
Observer dependent geometries 51
contribute to the total action since their wedge product with the vertical volume
form (3.34) vanishes. Note the appearance of the common term
. . ' l
Eijkte" A c1 A c" A c ,
which, when lowered to a four-form on 0, combines with the vertical volume
form (3.34) to the volume form (3.25) of the restricted Sasaki metric. The total
action thus takes the final form
l (1 A t•' ab c 1 R R a.bcf cdgh _ ( , A)2)
SMM = Vol0 i sgn g R a.cb-
96
a.bcd ·efghE E sgn . ·
'
0
( (3.37)
From this we see that we obtain an action based on the curvature of the Cartan
linear connection, as we have briefly discussed towards the end of section 2.6,
provided that we have chosen a model space G/ H for which A "I 0. We also
find that we always obtain a non-zero cosmological constant term. The magnitude
of the physical cosmological constant can be adjusted by introducing suitable nu-
merical factors into the algebra relations (3.9), which corresponds to a rcscaling
of the basis vectors Zi.
3.6. The role of spacetime
In the previous sections we have discussed the physics on Finsler spacetimes in
the language of Cartan geometry. For this purpose we considered a principal
K-bundle 7f : P -+ 0 over observer space 0 and equipped it with a Cmtan
connection A derived from Finsler geometry. This construction allowed us to
reformulate significant aspects of Finsler spacetime in purely Cartan geometric
terms: the definition of physical and inertial observers, the split of the tangent
bundle TO into horizontal and vertical components which crucially enters the
definition of observables and physical fields, the Sasaki metric and its volume
measure on 0 and finally the dynamics of gravity. It should be remarked that
these formulations can be applied to any Cartan geometry (7f : P -+ 0, A)
modeled on GjK, since they do not explicitly refer to the underlying Finsler
geometry, or even the spacetime manifold M. This observation stipulates the
question whether an underlying spacetime geometry is at all required, or may not
even exist, at least as a fundamental object. In this final section we will discuss
this question.
We first discuss whether and how we can reconstruct the Finsler spacetime
(M, L, F) if we are given only its observer space Cat-tan geometry (1r : P -+
0, A), together with the presumption that an underlying Finsler spacetime exists.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-29-320.jpg)
 may not be the Sasaki metric induced by Finsler
geometry.
The question arises whether we can still assign a meaningful physical in-
terpretation to an observer space Cartan geometry if its vertical distribution is
non-integrable, so that there is no underlying spacetime. Since any physical in-
terpretation should be given based on the measurement of dynamical, physical
quantities by observers, this amounts to the question whether these can meaning-
fully be defined on an arbitrary observer space Cartan geometry. We have pro-
vided these definitions throughout our discussion of observer space in section 3
of this work. Our findings suggest that the notion of spacetime is not needed
as a fundamental ingredient in the definition of physical observations, but rather
appears as a derived object for a restricted class of Cartan geometries.
Acknowledgments
The author is happy to thank Steffen Gielen, Christian Pfeifer and Derek Wise
for their helpful comments and discussions. He gratefully acknowledges the full
financial support of the Estonian Research Council through the Postdoctoral Re-
search GrantERMOS115.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-30-320.jpg)
![KrzysztofOrachal & Wieslaw Sasin
Warsaw University ofTechnology
Copernicus Center for Interdisciplinary Studies, Krak6w
Classification of classical singularities:
a differential spaces approach
T
HE aim of this work is to rethink the concept of Schmidt's b-boundary of a
spacetime in a more general framework. The original Schmidt's work is
done in the classical language of differential geometry. But, first of all, a mani-
fold (which is the mathematical formalisation of a spacetime) can be generalised.
One of the known generalisations is a differential space in the sense of Sikorski.
Secondly, differential geometry can be 'algebraised'. Then one works with tech-
niques of commutative algebra, for example in the spirit of the Nestruev group.
1. Motivation
It is known that the laws of General Relativity contain in their deepest core an
inevitable break down [Hawking & Ellis 19731. In this way a 'singularity' occurs
in the spacetime. Sometime it is said that a boundary is appended to the space-
time. However, even the notion of 'singular point' is not unique. As a result,
there exist various attempts towards this problem.
Starting from the theory of Sikorski's differential spaces, Helier and Sasin
(together with their collaborators) elaborated an interesting framework for study-
ing spacetime singularities !Helier, Sasin, Trafny & Zekanowski 1992, Helier
& Sasin 1994, !996, 1999, 2002, Helier, Odrzyg6zd:i., Pysiak & Sasin 2003].
Further, they developed a model based on the concept of groupoid and ideas of
non-commutative geometry [Helier et al. 2003]. In this paper we show that this
framework can be valuably enriched by some algebro-geometric methods ala
Nestruev [2002].](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-32-320.jpg)
![58 KrzysztofDrachal & Wieslaw Sasin
2. Fundamental concepts
This part is devoted to the basics of differential spaces in a sense of Sikorski
[1972]. Concise collection of basic notions can also be found in [Drachal 20131.
Let M be a set, M -:/= 0. Let A be a (possibly infinite) family of real functions
on M' i.e. A:= {fi, ·i E I IviE] J; : M--+ IR:}.
Definition 1. The weakest topology, for which all jimctions.from A are continu-
ous, is called the topology induced by A on l'vf. I11is topology is denoted by the
symbol TA.
Definition 2. Let f be a function defined on a subset A c M. If
where B is an open subset in topological space (A, T;t), then f is called the local
A-function. The set ofall local A~fimctions on a given set A c NI is denoted by
A;t. (TA:= {UnA 1 u ETA}.)
Definition 3. Superposition closure of a family offunctions A, denoted seA, is
defined as
seA:= {w 0 (h. ... ' fn) In EN' wE C00
(lRn)' .iJ, ... '.fn EA}
Definition 4. If A= scAo, then (M, A) is called the predifferential space.
Definition 5. A pair (lVI, A) such that M is an arbitrary set, and Ao is afamily of
functions, such that A= (scAo)M, is called the Sikorski differential space (gen-
erated by Ao). If Ao is finite, then (M, A) is called finitely generated. Functions
from Ao are called generators. A is called the differential structure.
Notice, that the above structures can, in certain cases, identify points of M. If
one would like to distinguish all points in M (i.e. have the Hausdorff property),
then one has to assume that Ao separates points of !VI. From now on, we adopt
this assumption.
Definition 6. For two predifferential spaces (M, A) and (N, D) the mapping
F : M --+ N is called smooth if VfED f oF E A. F is called a diffeomorphism
ifit is bijective and both F and p-l are smooth.
Definition 7. Let A be afunctionJR-algebra.
SpecA := {x: A-> lR IX E Horn(A, JR.) ' x(]_) = 1}
Classification of classical singularities: a differential spaces approach .______59
SpecA is called the spectrum of an algebra A.
Definition 8. Let M C JR.n. Then C00
(1VI) := (se{7ft[M, ... , 'lfniM})M, where
'If;, i = 1, ... ,n are projections.
Definition 9. Consider a differential space (M, A), with A= (Hc{f1, · · ·, .fn}) lvt·
A generator embedding is a smooth mapping
F: (M, A)--+ (F(M), C00
(1Rn)I,'(M)) ,
M3pH(fl(p), ... ,fn(P))ElRn ·
(F(M), C'Xl(JRn)F(IV/)) is then called a generator image.
The notation C00 (JRn) F(M) should be understood as j(Jllows:
where {7r;}·i=l,...,n are the projections, 'If; (xt, ... ,Xn) := x;for 'i = 1, · · . ,n.
Lemma 1 (lCukrowski, Pasternak-Winiarski & Sasin 2012]). Generator embed-
ding is a diffeomorphism.
Now, let A= scAo, where Ao = {ft, ... ,.fn I .fi: M--+ lR, i = 1, · · ·, n}.
Consider slightly modified construction of generator embedding, i.e.
F = (!1, ... ,.fn) ,
F: (M,A)--+ (F(M),C00
(1R
71
)1F(M))
Then F is a diffeomorphism from (1!I, A) to (F(M), C
00
(1Rn)IF(M))·
3. Spectral properties
In this part, algebraic techniques are presented in the context ofdifferential spaces
[Cukrowski, Pasternak-Winiarski & Sasin 2012]. A similar approach can be
found in [Nestruev 2002].
Definition 10. All evaluations ofalgebra A on set M are denoted by Ev(M, A),
i.e.
Ev(M,A) := {evp Ip EM} ,
where evp : A--+ R evpf := f(p) for f E A. Iffor a given predifferential
space (M, A) it is true that SpecA = Ev(M, A), then it is said that (M, A) has
spectral property.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-33-320.jpg)
![6_0___________________.Krzysztof0rachal & Wieslaw Sasin
Example 1. The d{fferential space (JR:n, c=(JR:n)) has spectral property.
Example 2. If M C JR:n, then the differential space (M, C00 (M)) has spectral
property.
Example 3. The d{fferential space (.NI, AM) has spectral property.
Lemma 2 ([Drachal & Sasin 20 13]). Let A= sc{f1, ... , j~,}, where J; : M-+
lRfor every i = 1, ... , n, be such that A f= AM. Then there exists a monomor-
phism
SpccAM "-+ SpecA
Lemma 3 ([Drachal & Sasin 2013]). Spectral property is invariant under diffeo-
morphisms.
Definition 11. Consider mappings a : SpecA -+ JR:, such that a(x) = x(a).
The weakest topology on SpccA,for which aare continuous for every a E A, is
calle:!. the Gelfand topology. The family of all such mappings is denoted by A,
i.e. A:= {a: SpecA-+ lR Ia EA}.
Lemma 4~([Drachal & Sasin 2013]). If (M, A) is a predif.ferential space, then
(SpecA, A) is also a predif.ferential space.
Lemma 5 ([Drachal & Sasin 2013]). Let A = sc{f1, •.• , fn}, fi : M -+
lR ,fi=l,...,n· The predif.ferential space (SpecA, A) has the spectral property.
Proposition 1 ([Drachal & Sasin 2013]). Let M c JR:n and M = M. Then
sc{1Tl!M,···,1Tn!M} = (sc{1TJ!M, ... ,1Tn!M})M.
The above Proposition is a direct consequence of the previously presented
Lemmas. Moreover, the following Theorem can be proved.
Theorem 1 ([Drachal & Sasin 2013]). If M c JR:n, then Spec(C00 (JR:n)IM) =
Ev(M, coo(JR:n) !M) ifand only (f M = ld.
Definition 12. Let F = U1, ... ,fn) be a generator embedding. Consider an
Euclidean metric don F(Jv!). Using F, an inducedmetric on M may be obtained
by the formula
p(p, q) := d(F(p), F(q))
forp, q EM and F(p) = (h(p), ... , fn(P)), F(q) = (JI(q), ... Jn(q)). Then
(M, p) is a metric space homeomorphic to (F(M), d) and F is an isometry. pis
called a generator metric.
Classification of classical singularities: a differential spaces approach._____ 61
Definition 13. Similarly, a generator metric is introduced on (Spec:A, A) by the
formula P(XI,X2) := ~1 (h(XJ)- h(X2))2·, where Xl,X2 E SpeeA.
Basing on the definitions and results presented above, one can derive the
following result:
Theorem 2. For the pred{fferential space (!vi, A) the _f()/lowing statements are
equivalent:
• (M, A) has the spectral property.
• The generator image F(Jvi) is closed with respect to the Euclidean metric.
• (lvf, A) is complete in a sense of Cauchy with respect to the generator
metric p.
• (M, A) and (SpecA, A) are d{ffeomorphic.
The detailed proof, which is straightforward but technical, will be presented
in a forthcoming paper.
Definition 14. Let M c JR:n and let (M, AM) he a finitely generated differential
space (i.e. A = sc{.f1 , ... , j~} ). A generator boundary (or gen-boundary) ofthis
differential space (M, AM), denoted by OgenM, is defined as
DgenM := SpecA SpecAM .
Such an approach towards the spacetime boundary has been proposed for the
first time in [Drachal & Sasin 2013]. It can be easily observed that
Proposition 2. If M c JR:n and A = se {?T1jM, ... ,1TnIM}, I}:en the generator
boundary of M, DgenM, is IVfM. I.e. (SpecASpecANr, AspecASpecAM) is
d{ffeomorphic and isometric to (M M, C00
( MM) ). The first space is equipped
with the Euclidean metric, the second one- with the generator metric.
Definition 15. Generators ft, ... ,fn and g1, ... , Dm are called equivalent, (f
sc{fi, ... , fn} = sc{gt, ... ,.IJm}.
It can be noticed that for equivalent generators the gen-boundary is well de-
fined (up to a diffeomorphism). However, by including the requirement of isom-
etry, the unique family of generators is obtained.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-34-320.jpg)
![62
KrzysztofDrachal &Wieslaw Sasin
4. 8-boundary
In this part, Schmidt's b-boundary [Schmidt 1971 Jis constructed in the language
of spectrums.
Let (M, g) be a spacetime. I.e. M is a four dimensional, smooth manifold
and g is Lorentzian metric. Let 0(1!I) be the connected component of the fibre
~undle of orthonormal frames over 1II. lt is known that O(l!I) is a smooth man-
Ifold, i.e. it is the differential space (O(M), C00 ( 0(1!J))). It is also known that
~he metric connection on l!I gives parallelization of O(.NI). This parallelization
mduces a Riemannian metric gu on O(M) [Schmidt 1971].
For sufficiently large n E N, on the strength of Nash embedding theorems,
O(M) may be isometrically embedded in JR.n [Nash 1956], i.e. in the differential
space (JR.n, coo(JR")).
Therefore, let
F: (O(M), C
00
(0(M)))---+ (F(O(M)), C00
(!R")F(O(M)))
be the generator embedding, such that .9R is the generator metric and let
F=(fl, ... ,fn).
The C~uchy completion (O(M), C00
(0(M))) of O(M) is diffeomorphic to
(S~ec.A, ..4), where .A~c{ft, ... , fn}. (O(lVf), C00 (0(lvl))) is ditfeomor-
phic to (Spec.Ao(M), .Ao(M))·
It is known that the action of the structural group 0(3, 1) may be extended
from O(M) to O(M). Then consider the orbit spaces
1r(O(M)) := O(M)/0(3, 1)
and
1r(O(M)) := O(M)/0(3, 1) 9! M .
Let us note that (O(M),1rM,M) is a principal 0(3,1)-bundle and that
O(M)/0(3, 1) can be identified with M.
These spaces are differential spaces, i.e.
and
Classification of classical singularities: a differential spaces approac~=-------6~3
where Ainv denotes 0(3, I)-invariant functions from c=(o(M)) and .A~)(M)
denotes 0(3, I)-invariant functions from Ao(M)· f is called H-invariant, if
fhEHf(xh) = .f(:r).
Moreover, being finitely generated, they have a spectral property [Cukrowski,
Pasternak-Winiarski & Sasin 2012], so they are diffcomorphic to
(s , .-.Ainv -..4-inv)
pcc ,
and
--
(Spcc.A~(!vl)' .A2)(M))
respectively.
Of course, one may consider various differential structures on orbit spaces,
but the proposed ones are maximal, which keep 1r smooth lSasin 19881. More-
over, one would like to have 1rlo(M) = 1rMio(M)·
The original Schmidt b-boundary [Schmidt 1971] was defined as Dbl!I
1r(O(M)) 1r(O(M)). It is diffeomorphic to
. -:-- --...
(s -.Amv -.Amv) (S ..4inv .Ainv )
pcc , pec O(M)' o(M)
Therefore in the language of spectrums the b-boundary is given by
S -.Ainv n ..4inv
pec >:lpec O(M) , (I)
i.e a slight modification of gen-boundary including 0(3, I)-invariance problem.
5. Singularities
In this part, some classification of spacetime singularities is proposed. Some
considerations on this topic can also be found e.g. in [Helier & Sasin 1999] and
[Helier, Sasin, Trafny & Zekanowski 19921. Moreover, [Gruszczak 1990, 20 14]
can provide interesting applications of the below considerations.
In the below considerations one can substitute E = O(M) and G = 0(~~, 1).
Lemma 6 ([Sniatycki 2013]). Let E 7rM Af be a fibre bundle and G be a topo-
logical group, i.e. (E, 1rM, M) is a principal G-bundle. Then (EjG, C00
(E/G))
is dijfeomorphic to (M, C00
(1vf) ).
Space Eisa sum of orbits of the right action E x G ---+ E of the group G on
E, i.e. E = UxEM 7r"N/(x).](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-35-320.jpg)
![64
KrzysztofDracha/ & Wieslaw Sasin
~onsider the Cauchy completion E of the space E (with respect to a fixed
metnc). The mentioned right action may be prolonged to E x G ---+ E by the
formula
PoD := lirn PnU ,
n-><XJ .
where p0 E E, p0
g E G.
lim,HooPn for the sequence (Pn), Pn E E for ·n E N,
Consider now two algebras of G-invariant functions on E and on E, i.e.
Fe(E):= {.f E coo(E) I'igEG,pEFJ f(pg) = .f(p)}
and
Fc(E) := {71 f E F2,(E)} ,
where F8(E) is subalgebra of Fc(E) consisting of functions which can be con-
tinuously prolonged on E, i.e. } : E---+ !l{ is defined by the formula
f(Po) := lirn f(Pn) ,
n----1-oo
where Po E E, Po = lirnn->oo Pn for the sequence (Pn), Pn E E for n E N.
Of course, one can always choose the generators fi of coo (E) such that they
are ~rolongable. This is because by the Nash theorem (E, C00 (E)) can be (iso-
rnetncally) embedded in some (!l{n C00 (!l{n)). Let F = (f f ) b h.·
' 1, • · ·, n e t IS
embedding, then F is a diffeomorphism and fi = ni oF.
It should ~e noticed that in order to make sense, the notion of prolongability
has to be constdered with respect to some topology.
Th~efore, the algebra of G-invariant functions from C00 (E) is isomorphic
to Fc(E).
Lemma 7. Fc(E) = n!v1(C
00
(M)) = (sc{n!v1h1, ... , n!v1
hn})E, where hi are
generators of(M, C 00
(M)).
, The detailed proof, which is straightforward but technical, can be found in
[Sniatycki 2013].
The linear operator of prolongation is denoted by P, i.e.
P : FS(E) ---+ Fc(E) , P(f) := f .
Note that P is an isomorphism of algebras .Fg(E) and Fc(E), so it may be
used in classifying singularities of b-boundary, basing on the relation between
algebras Fe;(E) and Fc(E) (equivalently: FS(E) and Fe(E)).
Classification of classical singularities: a differe_n_~I_2E~_c:es a_eproach ----~-
Because !l{ <;;:; :F2:(E) <;;:; Fc:(E), there are three cases:
I. .Fg(E) = Fe( E) and then SpeeFn(E) = Spec.lJ;(E),
2. !l{ ~ Fg( E) s-_;; Fe;(E),
3. J1:(E) ~ !l{ and then SpecF~;(E) = {*}.
(!l{ above represents constant functions.)
In the first case, the boundary given by Eq. (I) is 0.
Notice that by detinition, the algebra Fc;(E) = (t->e{nA!/hl,··. ,n/.,1h,})E
might consist also of functions not prolongable to E. One can restrict considera-
tions only to prolongable functions, i.e. t->c{n;;1h1, . .. , n;;1hn}.
Therefore in Case 2 there are two subcases:
2a. (Fg(E)) r:; = Fc(E) and then the boundary given by Eq. (I) corresponds
to AJM,
Case 2b is possible only if some nonprolongability emerges at the level of the gen-
erators h1, ... , hn of NI. This can happen if the topology on O(M) is stronger
than the topology induced by n.
The third case is a sign of very strong problems with prolongability. For
example, the topology onE has to be coarser than the topology on M. Of course,
one should keep in mind that nM is assumed to be continuous, so these two
topologies are linked with each other in some sense. It seems that this case should
be tackled with sheaf methods and some further research in this direction will be
presented elsewhere.
But even now, one can easily see that generators can be understood as coor-
dinates in a classical sense. Therefore, the presented classification gives a nice
pictorial tool: in Case 2b there are both kind of 'directions' -ones in which the
prolongation is possible and ones in which the prolongation is definitely impossi-
ble. In Case 3, in every direction the prolongation is impossible. From the other
(spectral) point of view the diffeomorphic spectral space is the space over one
point. It is obvious that the geometry over a point is trivial.
Case 3 is also one where the only prolongable functions are constant ones.
Therefore, it might be interesting to compare the above classification with the
results of [Heller, Odrzyg6idi, Pysiak & Sasin 2003, 20071, where similar results
Were obtained though in a slightly different framework.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-36-320.jpg)
![70 Jacek Gruszczak
us consider points at which this axiom is not satisfied. The set of these points
forms the so-called differential boundary (d-boundary). This issue is discussed
in Section 2. The construction of this type of boundary is also described.
In Section 3, the construction from Section 2 is used for building the differ-
ential space (d-space) with a d-boundary for the flat FRW model. The issue of
the prolongation of cosmic time and time-orientability to the d-boundary for this
model is discussed in Section 4. In Section 5, we describe the concept of the
smooth evolution of this model starting from the initial singularity. We find the
simplest smoothly evolved flat FRW models in Section 6. Some of them have the
kinematical properties which agree qualitatively with the recent cosmological ob-
servations [Riess et al. 1998, Perlmutter et al. 1999]. Namely, these models begin
their evolution with a decelerated expansion which changes into an accelerated
one.
The discussion of properties of the cosmological fluid which causes that type
of smooth evolution for the simplest cosmological models is carried out in Sec-
tion 7. Behaviour of any smoothly evolving flat model in a neighbourhood of the
initial singularity is investigated in Section 8. Finally, in Section 9, we summarize
the main results of this paper
2. Sikorski's differential spaces and GR
Space-time is a 4-dimensional, Lorentzian and time-orientable d-manifold of the
class coo [Beem & Ehrlich 1981]. This definition is a mathematical synthesis of
what is known about gravitational fields. It implicitly includes the equivalence
principle [Weinberg 1972, Raine & Helier 1981, Torretti 1983].
The equivalence principle, in its non-metric version, is implicit in the axiom
stating that space-time is a d-manifold which means that it is locally homeomor-
phic to an open set in JR.n, where n = 4. DitTerential properties of these home-
omorphisms (maps) are determined by the atlas axioms [Kobayashi & Nomizu
1962Jthrough the assumption, that the composition of two maps IPl o IP2 as
a mapping between open subsets of JR.n, is a diffeomorphism of the class Ck,
k E N. In the present paper it is assumed that these diffeomorphisms are smooth.
It is worth noting, that the non-metric version of the equivalence principle is
encoded into two levels of the classical d-manifold definition, namely in the as-
sumption of the existence of local homeomorphisms to JR.n and in the axioms of
atlases.
One can look at space-time, or more generally at ad-manifold, from the view-
point of theories that are more general than classical differential geometry. In
The smooth beginning of the Universe
71
----
these theories·d-manifolds are special examples of more general objects. Exam-
ples of theories of this type are: the theory of sub-cartesian spaces by Aronszajn
and Marshall [Aronszajn 1967, Mm·shall 1975] or the theory of Mostow's spaces
LMostov 1979]. In the present paper we will study the problem of the begin-
ning of cosmological time with help of Sikorski's differential spaces (d-spaces
for brevity) 1Sikorski 1967, 1971, 1972]. In this theory, the generalization level
does not lead to an excessive abstraction and therefore the d-space's theory may
have applications in physics.
Generally speaking, ad-space is a pair (M, /vi), where M is any set, and
M is a family of functions lP: M ---t JR. (see Appendix A). This family satis-
fies the following conditions: a) M is closed with respect to superposition with
smooth functions from C00 (!R",JR), and b) M is closed with respect to local-
ization. The precise sense of these notions is not important at the moment. The
d-space (lvf, M) is also a topological space (IV!, rM), where 'M denotes the in-
duced topology on M given by the family JVI. The topology 'M is the weakest
topology in which all functions from JVI are continuous. With in the theory of
d-spaces the d-manifold definition assumes the form
Definition 2.1. Non empty d-space (M, M) is n-dimensional d-manifold, ifthe
following condition is satisfied
(*)for every p E M, there are neighbourhoods Up E TM and 0 E TJRn,
such thatd-subspaces (Up,MuP) and (O,Eg
1
)) ojd-.1paces (M, M) and
(JR7!' c:(n)) respectively are diffeomorphic, where [(n) := C
00
(lRn' JR.).
Briefly speaking, d-space (M, M) is an-dimensional d-manifold if it is lo-
cally diffeomorphic to open subspaces of JR.n which are treated as d-spaces (see
Definitions A.8 and A.9). Diffeomorphisms appearing in condition (*) are gen-
eralized maps. Unlike in the classical definition, these maps are automatically
diffeomorphisms. Definition 2.1 is equivalent to the classical d-manifold defini-
tion [Sikorski 1972].
In the d-spaces theory, the non-metric version of the equivalence principle is
'localized' in the single axiom(*). This enables us to give the following definition
of space-time:
Definition 2.2. A space-time is a non empty d-space (lv!, A1) such that
a) (M, M) satisfies the non-metric equivalence principle as expressed by the
condition (*)for n=4,
b) a Lorentzian metric form g is defined on 1II,
c) Lorentzian d-manifold (M, M, g) is time orientable.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-39-320.jpg)
![72 Jacek Gruszczak
If the non-metric equivalence principle in the classical definition of space-
time is thrown out, the time-orientability, the Lorentzian metric structure, the
classical differential structure and the topological manifold structure are auto-
matically destroyed. Only the topological space structure survives.
In the case of Definition 2.2 the situation is different. Throwing out the non-
metric equivalence principle in the form of axiom (*) leaves a rich and workable
structure on which one can define generalized counterparts of classical notions
such as orientability, Riemannian and pseudo-Riemannian metrics, tensors, etc
[Sikorski 1967, 197 I, 1972, Gruszczak, Helier & Multarzynski 1988, Sasin 1991,
Buchner 1997, Helier & Sasin 1999, Abdel~Megied & Gad 2005, Rosinger 2007].
Looking at space-time manifolds from the d-spaces perspective, one can
imagine a situation when a background of our considerations is a sufficiently
:broad' d-space (B, E). Let us additionally suppose that in B there exists a sub-
set lvf C B having a structure of a 4-dimensional d-manifold (M, M) satisfying
the condition(*) for n =4, regarded as a d-subspace of (B, E). Naturally, at each
point p E M the non-metric equivalence principle is satisfied. Among points not
belonging to A!J the most interesting are accumulation points of the set M in the
.topological space (B, Ts). One can call a set of all accumulation points of M a
boundary of the space-timed-manifold (M, M). At the points of the boundary
the non-metric equivalence principle can be violated.
Definition 2.3. Let (M, M) and (Mo, Mo) be Sikorski's d-spaces, such that
a) (Mo, Mo) is an-dimensional d-manifold,
b) Nio C NI and Mo is a dense set in the topological space (M, TM),
c) (Mo, Mo) is a d-subspace ofthe d-space (M, M).
The set DdMo := NI - Nio is said to be differential boundary (d-boundary for
brevity) ofd-manifold (Mo, Mo), ifthere is a point p E odMo such that the (*)
property is not satisfied. Then, the d-space (M, M) is said to bed-space with
differential boundary (or cl-space with cl-boundary for brevity) associated with
the d-manifold (Mo, Mo).
In the following we shall restrict our considerations to the case of Defini-
tion 2.3 for n = 4. The d-boundary definition for a space-time's d-manifold
(lllfo, Mo) depends on the choice of the d-space (M, M). Therefore, a reason-
able method of the d-space (NI, M) construction is necessary. Fortunately, the
idea described above suggests such a construction. Namely, one chooses a suf-
ficiently 'broad' d-space 'well' surrounding the whole investigated space-time
The smooth beginning of the Universe
73
Mo and then one carries out the process of determination of all ac~umulation
points for Mo. These points form the d-boundary iJr1Mo of space-time. Next,
one defines the set M as Nf := Mo u 8r1Mo. By treating M as a d-subspa~e
of the sufficiently 'broad' d-space, the d-space with the d-boundary (M, .M) IS
uniquely determined. On can easily check that the pair of d-s~aces (NI, A1) a~d
(Mo, Mo), where (M, M) is determined by the above descnbed method, satis-
fies Definition 2.3.
The choice of a sufficiently 'broad' d-space which 'well' surrounds the whole
investigated space-time is almost obvious. As it is well known [Clarke 197~1,
every space-time (Mo,Mo,g) can be globally and isometrically embedded m
a sufficiently dimensional pseudo-Euclidean space EM, where J>, q depends on
space-time model. The space EP,lJ is also the d-space (~PN,c=(~P+'I,~))
which 'well' surrounds the studied space-timed-manifold (Mo, Mo).
Such a choice of the surrounding d-space is well motivated since all causal
properties of the studied space-time are taken into account, even if we do _not
refer to them directly. This is because of the isometricity of the embeddmg.
Furthermore, the concrete form of the isometrical embedding provides us with a
practical method of constructing the d-structure M for the the d-space (M, M).
Namely, M= c=(JR;P+<l,JR;)M·
To test this method of constructing (M, M) we shall apply it to the flat FRW
cosmological model.
3. A differential space for the flat FRW d-manifold
Let us consider the flat FRW model with the metric
g = -dt2 + a6(t)(dx2
+ dy
2
+ dz
2
),
(I)
h (I
. ) E wO ·= no X :m;3 and Do is a domain of the scale factor. The
W ere ,, X,'!), Z VI · a a .
I f t
Do 3 t ---t a (t) E :m;+ ao E C00
(D0
), is an even real functiOn
sea e ac or ao : a , o ' a 0 + .
of the cosmological timet. A domain of the scale factor Da C :m; , IS an o~en
and connected set. For convenience, let us assume that t = 0 is an accumulatl~n
point of the set D~ and 0 t/:. D~. In addition, the model has an initial singulanty
at t = 0 i.e. limt-;o+ ao(t) = 0.. In the next parts of the ~aper, the symb~~
is reserved for the following functiOn: a: Da---+ R a(t) - ao(t) fortE a•
a(t) = 0 fort= 0 where Da := D~ U {0}.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-40-320.jpg)
![74 Jacek Gruszczak
The construction of a d-space with a cl-boundary for the flat FRW model is
s_imilar_ to the construction of a differential space for the string-generated space
time wtth a conical singularity described in [Gruszczak 2008].
Every 4-dimensional Lorentzian manifold can be isometrically embedded in
apseudo-Euclidean space [Ciarke 1970]. In particular, the manifold p:vo,g) of
mode~ (I) can be isometrically embedded in (~5
, r/")) by means of the following
mapptng
FP(t, x, y, z) = ~ao(t)(x2 + y2 + z2 + 1) + ~ rt .,..QI_
0
2 .Jo a.o(r)'
F2 (t, x, y, z) = ~a0 (t)(:r2
+ y2
+ z2
- 1) + ~ j0
'
1
· :!#:_:'
0 2 ILOTJ'
F3 (t, x, y, z) = ao(t):r, F2(t, :r, y, z) = a0 (t)y, F~(t;, x, y, z) = a0 (t)z,
(2)
where .,,(s) = d'iag(-1 1 1 1 1)
' ' ' ' .
As ever~ manif~ld, space-time (W0 , g) is also a differential space (W0
, wo),
where the differential structure W0 := d4
~ is a family of local £(4)_functions
on W° C ~4, where £(4) = C00 (~4). wOn the other hand, the set F 0
(W0 )
can be equipped, in a natural way, with a differential structure treated as a dif-
:eren~~~l subspace of th; (~5
, ~(5
l), where £(5) = C00
(~5). Then the fam-
tly £ po(wo) of local £(")-functiOns on F 0 (W0 ) is a differential structure and
the pair (F°CW0
),£~"Jcwo)) is a differential space. In addition, if the inte-
gral J~ drIao (r) is convergent for every t E Do, then the mapping pO is a
diff~omorphism of the differential spaces (FO (WO),£}.~ wo ) and (WO, WO)
[Sasm 1988, Gruszczak 2008]. ( )
The process of attaching the initial singularity depends on the completion
of the set F 0 (W0 ) by means of points from the surrounding space JR.5
in the
way controlled by the isometry F 0. It enables us to define the d-space with a
d-boundary for the flat FRW model.
Let the mapping F: W ---t JR.5
denotes a prolongation of F 0 to the set W :=
Dn X 1R3
• The values ofF are given by formulae (2) changing the symbol pO
onto F. Then for every x, y, z E lR the value of Pat the initial moment t = 0
is F(O, x, y, z) = (0, 0, 0, 0, 0), since a(O) = 0 by assumption. Therefore, the cl-
boundary distinguished by the embedding procedure is represented by the single
point: odF
0
(W0) = {(0,0,0,0,0)} c JR.5.
Th_e pseu~o-Euclidean space (JR.5, r/5l) is a differential space (JR. £(5)).
!he_ dtfferenttal structure £(5) of this space is finitely generated. The pro-
Jections on the axes of the Cartesian system 1r·: JR.5 ---t m ~- (z z ) -
. ~ I&,"z 1, z, ... ,Z5-
Z;, 2 = 1, 2, ... , 5, are generators of this structure and therefore: £(5) =
The smooth beginning of the Universe
75
-----
Gcn(1r
1
, 1r
2
, ... , 1r
5
). As is well known [Gruszczak, Heller & Multarzynski 1988,
Gruszczak 2008], every subset A of a support M of a differential space (M, C) is
a differential subspace with the following differential structure: Gen(CJl). For
finitely generated differential spaces (NI, C), every differential subspace with a
support A c M is also finitely generated. Generators of the differential structure
on A are generators of the differential structure Crestricted to A.
The differential space (F (W), Ei~{w)) is a differential subspace of the differ-
ential space (JR. £(5)) and represents the d-space with a cl-boundary for model
(I). Since £(5) is finitely generated, the cl-structure £1,~(W)' induced on F(W), is
also finitely generated and
The d-space (F(W), E?,{w)) is not convenient for further discussion. Let us
define ad-space (1V, W) diffeomorphic to (F(W), Ej,?(w)) which will enable us
to apply the Sikorski's geometry in a form similar to the standard differential
geometry.
Let us consider an auxiliary d-space (W, W), W = Gen(!:h, /32,. · ·, /35),
where
f3i: W ----t IR, f3i(t, x, y, z) =11'i o F(t, x, y, z), ·i = 1, 2, ... , 5. (3)
This space is not diffeomorphic to the d-space with cl-boundary (F(W), E~?(w))'
since the function F: W ----t ~5 is not one to one. Additionally, the generators
/3; do not distinguish the following points p E oW := W - W
0
. Therefore, the
d-space (W, W) is not a Hausdorff space. With the help of this space one can
build a cl-space with cl-boundary (l;V, W) diffeomorphic to (F(W), t:J,?(w)).
Let eH be the following equivalence relation
Pf2FN {o} V/3 E W: f3(p) = f3(q).
The quotient space W = W/p11 can be equipped with a cl-structure W := W IPH
coinduced from W
where](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-41-320.jpg)
![76
Jacek Gruszczak
The symbol [p] denotes the equivalence class ofa point p E W with respect to (}JI.
The pair (vV, W) is ad-space [Waliszewski I975, Sasin 1988, Gruszczak 2008].
Let us define the following mapping
P: Vll--+ F(W), F([p]) := F(p).
Theorem 3.1. The d-space (W, W) is diffeomorphic to the d-.1pace with ad-
boundary (F(W), E~?(w)). 11
Proof. The mapping F: vV --7 F(W) is a bijection. In addition, W is
by construction a Hausdorff topological space with the topology given by the
generators /J1, /J2, ... ,/35[Sikorski 1967, I97 I, 1972, Gruszczak, Helier & Mul-
tarzynski I988]. The mapping
is a diffeomorphism of the d-space (W, W) onto its image (F(W),41
(vr)
(F(W), 41
(w)) (see Theorem A. I). 0
According to Definition 2.3, (W, W) is the d-space with d-boundary for the
flat FRW cl-manifold (W0
, W 0
), where the d-boundary is represented by the set
8dvV
0
:= W- W
0
. The function F 0
plays an important role in the construc-
tion of (vV, W). A deeper analysis of this role can be found in my next paper
[Gruszczak 20 I4].
The cl-space with cl~boundary (W, W) has been constructed with help of the
cl-space (W, W) and the relation QH. Generally speaking, cl-spaces of the type
of (lvl,C) and (MIQIT,CIQII) have a lot of common features because of the
isomorphism of the algebras C i CIeII. In particular, modules of smooth vector
fields X(M) and X(M IQII) are isomorphic [Sasin 1988, Gruszczak 2008]. This
property will, in the next parts of the paper, enable us to work with the help of
the more convenient cl-space (Hi'; W) instead of (W, W).
The cl-structure W of the cl-space (W, W) is finitely generated by means of
functions (3;, i = 1, 2, ... , 5. However, in the next parts of the paper, we use a
different, but equivalent, system of generators
(4)
Then the cl-structure W has the form
The smooth beginning of the Universe 77
----
4. Time orientability
The flat FRW model is a time orientable Lorentzian manifold M. By definition,
there is a timelike directional field generated by a nowhere vanishing timelike
vector field X. If X generates the directional field then the field .X generates it
also, where A is a nowhere vanishing scalar field on J'vf. The field X caries a part
of information included in the casual structure of !VI, which enables us to define
the direction of the stream of time and the succession of events IGeroch 1971 ].
The manifold structure and the casual structures of space-time are broken
down at the initial singularity. In the hierarchy of space-time structures, the
Sikorski's cl-structure is placed below the casual structure lGruszczak, Helier &
Multarzynski 1988]. Therefore, the cl-space with d-boundary (W, W) of the flat
FRW model is timeless independently of the fact that one of coordinates is called
time and the moment t = 0 is named the beginning of time. In this situation
one cannot say that the d-bounclary advV is an initial or final state of the cosmic
evolution. It is necessary to introduce a notion which would be a substitute of
time orientability.
Let (W0 , Wo) and (W, W) be the pair of cl-spaces described in the section
3. For convenience we can consider the d-space (W, W) instead of the cl-space
with cl-boundary (W, W) according to the remark after Theorem 3. I. In this
representation the set of not Hausdorff separated points 8W := W - W 0
is a
counterpart of the cl-boundary 8dW 0 for the flat FRW cl-manifold.
Let in aclclition, X 0 : W 0 --+ TW0 be a timelike and smooth vector field
without critical points, tangent to the manifold (W0 , W 0
), fixing the time-
orientability on the manifold (W0 , g).
Definition 4.1. The d-space (W, W) is said to be time oriented by means of a
vectorfield X if
a) there is a nonzero vector field X: W --+ TW tangent to (W, W) given
by the formula
{
X 0
(p)(aiwo) for Ji E W 0
fa E w: X(p)(a) := 1" xo( )( I ) for- p E 8W, q E wo
lmq-+p q a wo
b) and there is a.function A E W, A(q) > 0 (or A(q) < O)for q E W° C W
such that the vector field V := AX is smooth on (W, W).
A coordinate defined by means ofX is called time and the moment t = 0 the
beginning oftimet. We also say that the d-space (W, W) is oriented with respect
to timet.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-42-320.jpg)
![80
Jacek Gruszczak
6. The simplest smoothly evolving models
According to Definition 5.1 the scale factor a(t) for smoothly evolving models
satisfies the condition
a E W, a(t) > 0 for t E Da. (7)
For simplicity, let us confine our considerations to the following subalgebra of
the algebra W
where
l
L dr
f.(t,x,y,z) = a(t) -.-.
· . o a(r)
The explicit form of the generators a 1, a 2 , ... , a.5 can be obtained with the help
of formulae (4), (3) and (2). :Por functions belonging to W1
, the smoothness
condition (7) has the following form
i
t dr
a(t) = f(a(t), a(t) ~(
) ),
. o a r
(8)
where
f(O, 0) > 0, f E [(2). (9)
Formula (8) is an equation for a(t) with an initial condition a(O) = 0. The
function f E E(
2
), is in a principle, arbitrary. The only restriction on j is con-
dition (9) which is a consequence of Definition 5.1, and the physical assumption
that the real universe expands from the initial singularity. The simplest choice is
the following function
Now, the smoothness equation (8) for a(t) has the form
a(t) = f3 +!'la(t) +!'2a(t) -.-.
. lot dr
o a(r)
(10)
Solutions of (10) have to satisfy the following conditions
a(O) = 0, a(O) = {3 > 0, a(t) > 0, a(l;) > 0 fort> 0. (11)
The smooth beginning of the Universe 81
Proposition 6.1. When ry2 = 0 then solutions(){ the smoothness equation (10)
satisfying conditions (11) have the form
and
a(t) = {Jt, t E [O,oo), for /'1 = 0,
a(t) = j~-(e1' 1 -1), t E [O,oo), foT /'1 =I 0.
"(!
Proof. Obvious calculus.
(12)
(13)
11
0
Solution ( 12) represents the well known model of the universe which expands
with the constant velocity it = /3 and which is a solution of the Friedman's equa-
tions with the following equation of state p = -e/:3. For /'1 > 0 solution (13) is
the universe model which is asymptotically ( t --+ oo) the de-Sitter model. The
model expands from the very beginning with a positive acceleration. The pa-
rameter /'t can be asymptotically interpreted as a cosmological constant. When
ry1 < 0 cosmological model (13) describes an expanding universe, and the ex-
pansion slows down from the very beginning. For great t, the size of universe
fixes on the level a(t) ~ limt--+ooa(t) = f3/b1 1 and a and ii tend to zero when
t --+ oo. Such a universe asymptotically becomes the Minkowski space-time.
In the case 1'2 f 0, let us introduce the following auxiliary symbols
/'1 := i1i2/Vf>, /'2 := sgn(!'2)f3iV3, K := Vf>Hh2, i2 > 0,
a(K) := i2
{3~), t(K) := i2 t(K)jJ3,
where H(t) := a(t)ja(t).
(14)
Proposition 6.2. If ry2 > 0 then solutions ofsmoothness equation (I 0) have the
form
·t- K - ~-oo a(y)ydy
a(K) = (K- .:Y1 - arccothK)-1
, ( )
I - K y2 -1 '
where K E (Kf, oo), and]{f is a solution ofthe following equation
Kt- arccothKJ = 1'1, Kt E (1, oo).
Proof. Solution of an elementary differential equation.
(15)
(16)
11
0](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-44-320.jpg)
(19)
11
0
Proposition 6.5. /f/2 < Oand i1 < rr/2, then cosmological model (I) has a
final curvature singularity at t8 < oo. The set [0, ts) is a domain of the scale
factor a(t), where t8 := t(O) and thefunction t(K) is given byformulae (18) and
(14). Ill
Proof. Proposition is the result of a fact that some of the components of the
curvature tensor are undefinite at t5 , because a(t) --+ -oo when t--+ t-;. 0
83
The smooth beginning ofthe Universe -----------
Fiuure ]· Scale factor ii(t) for smooth solutions with /2 > ~l. When idl ~. 01
o · . - 0 . 1 t. · initially expan w1l1
d 1 d ith acceleratton. For /'I < so u Jons
mo e s expan w .. 1, th ,·aph an
a ne ative acceleration but at a moment, denoted by a cnc e _o~ .. e _gr . ' ..
g . .. .t.ated The black point denotes the mttlal smgulanty.
accelerated expanstOll ts 1111 1 .
. p .t. s 6 2 and 6 4 solutions of smoothness equation (1 0)
Accordmg to ropos1 ton. . · , . . . .
. h d . (f'/ )()) But in the case of soluttons discussed m the
aredefinedmt e omam '-!•' · . . . . . fa'K)
Proposition 6.5, there appears an additional restnctwn for the domams o . (
. . f fa (K) as the scale tactor
and t(K) coming from the geometncal mterpreta IO~ o , ·. .
for a flat FRW model. The final curvature singulanty ends the evolutwn of the
model.
Proposition 6.6. If ~/2 < 0 then
I. ifi1
~ rr/2, acceleration ii(t,) > Ofort E (O,oo),
2. ifO < il < 7f/2, the scale factor a(t) has the inflexion point at t. (ii(t,.) =
0), a(t) > Ofort E [O,t.) andii(t) < Ofort E (t.,ts),
3. ifil ( 0, accelerationii(t) < Ofort E [O,ts),
where t* := t(K.). The quantity I<. is a solution of the following equation
___!!_::_ - arctanK* + 1r/2 = i1· (20)
K; + 1 111
Proof. By properties of the function given by formulae (18).
0](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-45-320.jpg)
![84
Jacek Gruszczak
. I~ /'2 < 0, the smooth evolution of the universe with respect to the cosmolog-
tcal ttme t can proceed on three different ways:
a) If i't ?:: 1r/2, a smooth accelerated evolution starts from the initial singu-
larity. The acceleration goes on continuously during an infinite petiod of
time.
b) For i'1 E (0, 1r/2), these smoothly evolving universes initially accelerate
but the acceleration is slowing down so as to change, at t = t*, into a de-
celeration. Smooth evolution ends at the final curvature singularity within
the finite period of time [0, ts). These models have two singnlarities: the
initial and final one.
c) Models with i'1 :( 0 start their evolution in the Big-Bang and decelerate.
The rate of expansion slows down strongly and these models end their
evolution at curvature singularities in a finite time t8
• Models of this class
have also two singularities.
ii(t) for 12 < 0
0.5 1.0 1.5
t
Figure 2: Scale factors a(i) for smoothly evolving models with 1'2
< 0. Curves
on the graph with i'1 ?:: 1r/2 represent accelerated solutions. For models with
i'1 E (0, 7r/2) initially accelerating expansion is slowing down. At the moment t*
(circles on the graph) the acceleration changes into a strong deceleration. When
i'1 :( 0 solutions expand with negative acceleration. The black points on the
graph denote initial and final curvature singularities.
The smooth beginning of the Universe 85
7. Interpretation
Let us assume that solutions of the smoothness equation represent cosmological
models. Then the Friedman equations
can serve as a definition of a pressure p and energy density {j of a kind of cos-
mological fluid which causes the smooth evolution of models, where p = "-P and
{! = n,(j. In the present paper this fluid is called the cosmological primordial fluid,
or primordial fluid for brevity.
Proposition 7.1. lff'J E lR and /'2 = 0 then the equation o.lstatej(Jr the primor-
dial fluid has the form
(21)
In addition
a) the energy density Q is a decreasing function of cosmoh;gical time and
limt-->o+ g(t) = oo,
b) at the initial moment p(O) = lim~-ro+ p(t) = -oo, and the remaining
details ofthe dependence p(t) are shown in Figure 3,
c) limt-+oo g(t) = 3rf
limt-+00 Q(t) = 0
Proof. An elementary calculus.
]J
limt-rooP(t) = -3')'f
limt-tooP(t) = 0
11111
0
The abbreviations SEC and WEC on the above and next tables denote the
strong and weak energy conditions and the statements below are answers to the
question of whether the strong or the weak energy conditions are satisfied.
In the case 1'2 =/= 0 it is convenient to introduce the following abbreviations
- -·-2
P := P/'2 ,](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-46-320.jpg)
![86
p
z•n-r-~--~---~-----~
p(t) for '1:2 = 0
-zs::o_
.........-<-L~~--,Ui.o:=--~---.,.s-;----::':z.o:--_jz.s
t
Jacek Gruszczak
~igure 3: Dependence p(t) for 1'2 = 0. In the case 1'1 < 0 the pressure has a
smgle positive maximum.
Proposition 7.2. Ifrv > 0 d - Tfl) •
f
.
1
. ,2 an '11 E !ft then the equatwn ofstate ofthe primor-
{ za flwd has the form ·
In addition
a) energy density i2 is an increasing function of time and lim -(t)
+oo, lim -(t) _ ](2 . . HO " f2
t-+oo f2 - f where Kf zs a solutum ofequation (16),
b) pressure at the beginning and end ofthe evolution is lim p-(t) -
lim -(t) _ 2 t.-+0+ · - -oo,
H~P . - -J(f' and the remaining details of dependence p(t) are
shown m Fzgure 4,
c) the week energy condition is satisfied during the whole evolution,
d) for "Yr ~ 0 the strong energy condition is broken down,
e) if "Yl .~ 0, the strong energy condition is satisfied for t E (0 t ). Fl. r
remammg t > t th ·. d" . . b ' * 0
. . . * zs con ltwn zs roken down. The moment t. is defined
m Proposmon 6.3
Proof. An elementary calculus.
0
The smooth beginning of the Universe 87
p(t) for 1'2 > 0
Figure 4: For )'1 2:, -1.35, pressure is a decreasing function of cosmological
time. For remaining )'1, function p(t) has both the maximum and minimum. The
minimum is not well visible on the graph.
Proposition 7.3. If 1'2 < 0 and )'1 E JR. then the equation ofstate ofthe primor-
dial fluid has the following form
j5 = +~ - ~g- ~(1'1 - arccot IQ )(IQ + 1/VU ).
3 3 3
Additionally
a) energy density g is a decreasing function of the cosmological time, and
limHo+ g(t) = +oo,
b) initial pressure is p(O) := limt-+D+ p(t) = -oo, and the remaining details
ofdependence p(t) are shown in the Figure 5,
~<0 I· SEC I wEt]
c) ;h):7r/2 limHoo g(t) = I<J limHooP(t) = -I<j no yes
0 < ;h < 7r/2 lilllHt.. Q(t) = 0 lilllHt, p(t) = --CO no/yes yes
il ::;;; 0 limHt, §(t) = 0 limHt.. p(t) =+eo yes yes
d) if 0 < 1'1 < 1r/2, the cosnwlogical fluid violates the strong energy con-
dition fort E (0, t,). In the remaining range t E (t*, t8 ), the SEC is
satisfied.](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-47-320.jpg)
![90 Jacek Gruszczak
A mixture of1'1-matter and 1'2-matter (/'1 ::( 0, 1'2 < 0). This kind of the pri-
mordial fluid satisfies the strong energy condition during the whole finite period
of evolution. The mixture is a fluid with interacting components. At the final sin-
gularity /'z-matter absorbs, in a sense, 1'1-matter and finally the mixture vanishes,
{!J = 0 at an infinite pressure. An admixture of 1'1-matter into /'2-matter shortens
the lifetime of the cosmological model.
A mixture of/'t -matter and 'Tz-energy (1'1 ::( 0, 1'2 > 0). Components of the
mixture are interacting fluids. The beginning of the model evolution is dominated
by 1'1-matter. The universe expands with a negative acceleration and the mixture
satisfies the strong energy condition. But the influence of1'2-energy is still rising.
Fort > t. the evolution is dominated by /':renergy. Since t = t., the mixture
has properties of a dark energy and changes the further evolution into accelerated
expansion. During the final stages of the evolution the equation of state has the
following form {!J = -pf = ;y~Kj, where K1 is a solution of equation (16).
The evolutionary behaviour of the model is extremely interesting because
such an evolution is qualitatively consistent with the observational data of la
type supernovae [Riess et al. 1998, Perlmutter et al. 1999]. Preliminary qucm-
titative investigations of the consistency of the discussed model have been car-
ried out with the help of the Habs(z) dependence published in [Simon, Verde
& Jimenez 2005, Ma & Zhang 2011, Yu, Lan, Wan, Zhang & Wang 2011,
Zhang, Ma & Lan 201 1]. Results of the best-fit procedure depend on H0 . For
70.6 ::( Ho ::( 77.8 km ~:>-1
Mpc-1 lRiess et al. 2009] the best-fit parameters
are in the range '11 E [-1.829, -1.075], ;y2 E [2.82, 4.011] x1o-1Mpc-1 and
x;,.in E [8.66, 9.76] (see also Figure 6). Values of parameters ;y1 and ;y2 en-
able us to find: an age of universe [0 E [13.561, 14.241] x 109
y, the moment of
the acceleration beginning l. E [7.427, 8.471] x 109
y, the Hubble constant in
the acceleration moment H. E [106.31, 108.86] x km s-1Mpc1
and redshift
z. E [0.648, 0.743], where
(}/'") ao'fz - -1 - -1 v'3 -
z ". := v'3{Ja(K) - 1, to= c t(K0 ), t. = c t(K.), K0 := c'f
2
H0.
Quantities K, a(K), t(K) are given by formulas (14) and (15), and K. is a
solution of equation (17). In the present Section bar over quantities denotes that
we use the systems of units in which c =J. 1.
A mixture of /'1-energy and 'fz-matter (/'I > 0, /'z < 0). Fort E [0, t.) the
mixture has properties of a dark energy and therefore this model accelerates from
The smooth beginning ofthe Universe
250
~ '
~150
s -
~ H. //
'-' 100 ?~ :1
lg so 1 :
H0=72.0
H .=106.87
z.=0.67
Ob---~~:~'~··~----._--~----~
0.0 0.5 1.0 1.5 2.0 2.5
z
91
Figure 6: Comparison of the theoretical__ ff(z) dep~ndence for th~-~moot~l(
evolved model ('Yt < 0 and/'2 > 0) with Ilobs(z) for H~ = 72.0 km_c; .~pc ,·
The solid line represents the smoothly evolved model wtth the best-tit par~meter
I
- 1 241 - - 0 000312Mpc-l and x·2
. = 8.79. In thts case
va ues /'1 = - · , /'2 - · _ mm _ 1 _ 1 ,
fo = 14.103 x 109y, T. = 8.241 x 109y, H. = 106.87 km s Mpc and
z. = 0.668. The dashed line represents the best-fit of the ACDM model.
the very beginning. But later, 1'2-matter component begins to play a bigger and
bigger role. The further evolution depends on the value of the parameter /'1 ·
When ;y1
E (0, 1r/2), the repulsive properties of ~~1-energy are not able to
dominate the evolution and the acceleration is gradually stopped because of. a
greater and greater attractive influence of 1'2-matter. The momen_t t., defined m
the Proposition 6.6, is the end of acceleration. Starting from_ thts mo~1
cnt, t~~
expansion slows strongly till the final singularity. The behaviOur of ptes_sure IS
interesting, Figure 5, fort > t•. After t. the mixture behaves as ~ dus~ (j5 :::::: 0).
But later, pressure rapidly increases to infinity at the final singulanty. Smtultane-
ously, because of expansion, the energy density decreases to zero.
When ;y1
:;::: 1r/2, repulsive influence of 1'1-energy is domi_nating during the
whole pe~iod the infinitely long evolution. The mixture acts llke a dark ~nergy
causing acceleration, independently of the 'Yz-matter presence. As for previOusly
considered accelerating models, the equation of state, in the last stages of the
-2 J{~
evolution is as the one for the cosmological constant, i.e., f2J = -pf = 1'2 ]•
'
where K
1
is a solution of equation (19).](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-49-320.jpg)
![92
Jacek Gruszczak
8. Smoothly evolved models in a
neighbourhood of singularity
The function f appearing in the smoothness equation (8) can be expanded into a
seri.es in a neighbourhood of the point (0, 0): f(z1 , z2) = f3 + DJ}'(O, O) . z
1
+
fh/(0, 0) · Zz +· ... Then the smoothness equation assumes the following form
h(t) = {-J + lh.f(O, 0) · a(t) + Ehf(O, O). a(t) l ar~~) + ....
For sufficiently small t, or equivalently in a small neighbourhood of the initial
singularity, one can omit the higher powers of the expansion and consider the
sm~othness equation in the linear approximation. [f one assumes that l'l :=
D1j(O, 0) and /''2 := Ehf(O, 0), the above equation. in the linear approximation
is identical with the smoothness equation (I 0) for the simplest smoothly evolve~
models (Section 6).
The above observation leads to the conclusion that properties of the solutions
studied in Sections 6 and 7, for small t, are typical for every smoothly evolvino-
llat FRW model in a small neighbourhood of the initial singularity. In particula;,
every smoothly evolving model during the initial stages of its evolution is filled
with a cosmological fluid which is /';-matter or /';-energy, i = 1, 2, or one of the
mixtures described in Section 7. These fields satisfy the following approximate
equation of state
9. Summary
a) The discussion of the equivalence principle and its breaking down has
led to the formulation of the d-boundary notion which is, roughly speakina, an
'optimal' set of points at which the equivalence principle is broken. A space~ime
cl-manifold with the attached d-boundary is not only a topological space but also
an object with a rich geometrical structure called differential space (see Appendix
A). The effective construction of both d-boundary and cl-space with ad-boundary
for any space-timed-manifold has also been described (see Section 2).
The smooth beginning ofthe Universe 93
b) For every flat FRW cosmological model with the initial singularity the d-
space with d-boundary (W, W) has been constructed. In this case the d-bounclary
is a single point (see Section 3).
c) In the d-space formalism it is possible to extend the concept of time ori-
entability to the d-boundary of the FRW models. In this way, the intuitive un-
derstanding of the beginning of the cosmological time obtains a precise mathe-
matical form (see Section 4). However, not every flat FRW model with the initial
singularity has a well defined beginning of cosmological time.
d) In the whole class of flat FRW models with a well defined beginning of
cosmological time we distinguish the large subclass of the so-called smoothly
evolved models (see Definition 5.1 ). The condition defining this subclass is called
the smoothness equation. The most practical form of this equation is given by
formula (8).
e) The simplest flat FRW models with a well defined beginning of cosmo-
logical time have been found in the explicit form. This set of models can be
divided into six qualitatively different classes (Lemmas 6.1, 6.2 and 6.4). The
most important classes are
• solutions with parameters ,:Y1 < 1r/2 and /'2 < 0 which have two curvature
singularities: the initial singularity and the final singularity (see Figure 2),
•• the subfamily with ,:Y1 < 0 and 1'2 > 0, being qualitatively consistent with
the observational flobs(z) data from [Simon, Verde & Jimenez 2005]. The
quantitative consistency with the data is discussed in Section 7 and is
shown in Figure 6. The level of the consistency is similar to that for the
ACDM model.
f) The Friedman equation without the cosmological constant, in application to
the simplest solutions, may serve as a definition of energy density 1j and pressure
fi of a cosmological fluid (primordial fluid) which causes the smooth evolution.
This strategy enables us to find main phenomenological properties (in particular,
the equation of state) of this primordial fluid (see Lemmas 7.1, 7.2 i 7.3).
g) In Section 7 we present an interpretation of the primordial fluid as a mix-
ture of more elementary interacting primordial fluids l'i-matter and l'i-energy,
i =1, 2.
h) From the analysis of smoothness equation (8) in a neighbourhood of the
initial singularity, i.e for small t, one can conclude (see Section 8) that in the
earlier stages of the evolution every smoothly evolving flat FRW model is filled
With a primordial cosmological fluid with properties characteristic for particular
solutions found in this paper (Sections 6 and 7).](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-50-320.jpg)
![96 Jacek Gruszczak
Sometimes one uses the following abbreviation: C = Gen(Co) := lc(sc(C0))
= (sc(Co))M.
Definition A.S. The set Coin Lemma A. I is said to be a set ofgenerators. Func-
tions r.p E Co are called generators of the cl-structure C := lc(sc(Co)). JfCo is
finite then the cl-structure C is calledfinitely generated.
The method of constructing ad-structure with the help of a set ofgenerators is
the greatest advantage of Sikorski's theory, especially in the case of finitely gen-
erated d-spaces such as, for example, the d-space with ad-boundary associated
with the flat FRW world model.
Definition A.6. If (M, C) is ad-space and A C M then the d-space (A, CA) is
said to be a differential subspace (d-subspace) ofthe d-space (M, C).
The above definition enables us to determine a d-structure for any subset A
of NI. It is enough to 'localize' every function from the d-structure C to A. In
the case of a finitely generated d-spaces (M, C), a simpler situation occurs. Then
C = Gcn(Co), Co := {/31, /32, ... ,f3n} , n E N, where f3r, /32, ... ,f3n are given
functions. The d-structure CA is given in terms of generators Co =ColA which is
a set of restrictions of the set Co to A. Then CA = Gen(C0).
Definition A.7. Let (M, C) i (N, D) bead-spaces.
I. A mapping f: M-:+ N is said to be smooth if:1(3 E V : f3 of E C.
2. A mapping f: NI -:+ N is said to be a dijfeomorphism from a d-space
(M, C) to a d-space (N, D), if it is a bijection from M to N and both
mappings f: M-:+ Nand j-1 : N ->-NI are smooth.
In this case, we say that (NI, C) and (N, D) are dijfeomorphic.
A smooth mapping f transforms smooth functions on N onto smooth func-
tions on Af. The notion of d-spaces diffeomorphism is the key notion from the
point of view of the present paper. If there is a diffeomorphism f between d-
spaces (NI, C) and (N, V) then these d-spaces, from the viewpoint the d-spaces
theory, are equivalent.
Definition A.8. Let (M, C) i (N, D) bed-spaces. A d-space (M, C) is said to
be locally dijfeomorphic to the d-space (N, D) iffor every p E M there is Up E
Tc and a mapping JP: Up -:+ fv(Up) E TD such that fp is a dijfeomorphism
between the d-subspaces (Up, CuP) and (jp(Up), V fp(Up)) ofthe d-spaces (M, C)
i (N, D), respectively.
The smooth beginning of the Universe 97
Definition A.9. Ad-space (M, C) is said to be ann-dimensional cl-manifold, ifit
is locally diffeomorphic to the d-space (lRn, £(n)).
Applying Definition A.8 to Definition A.9, leads to condition(*) in Definition
2. 1. Local diffeomorphisms fp are obviously maps. A set of maps forms an
atlas. It turns out that Definition A.9 is equivalent to the classical definition of
d-mariifold [Sikorski 1972].
Theorem A.l. Let a Hausdorfl d-space (M, C) be a finite fly generated d-space
with the d-structure C generated by a.finite set of.fimctions: Co := {J:h,. ··, f1n},
C = Gen(Co). Then the mapping F: Af -:+ lR", F(p) := (/-31 (p),. · ·, f3n(P)) is
a diffeomorphism of the d-space (M, C) onto the d-subspace (F(M), E;;?M)) of
the d-space (lR71
, £(n) ).
Proof, see !Sasin & Zekanowski 19871.
The d-subspace (F(M), £~~)M)) of (lRn, £(")) is an image of th~ d-space
(M, C) in the mapping F. Theorem A.l is called the theorem on a dJffeomor-
phism onto the image.
Definition A.lO. A mapping v: C -:+ lR is said to be a tangent vector to a
d-space (M,C) atapointp EM if"
I. :la, f3 E C:!a, b E lR : v(ac~ + b/3) = av(a) +bv(/3),
2. :!a.,/3 E C: v(af3) = v(a.)/)(p) + a(p)v(f3), p EM.
The set ofall tangent vectors to (!11, C) at p E NI is said to be a tangent vector
space to (M, C) at p E M and is denoted by TpM. The symbol T M denotes the
following disjoint sum:
Tld := U TpM·
pEl>I
Definition A.ll. Let (M, C) bead-space. The mapping X: M ----7 T M such
that :!p E M, X (p) E TpM is said to be a vector field tangent to (M, C).
With help ofa vector field X: M -:+ T J1 one can define the following map-
ping
X: c-:+ JRM, X(a.)(p) := X(p)(a.),
where a E C i p E M. The mapping is linear and satisfies the Leibnitz rule
Va,/) E C : X(a./)) = X(a)f3 + aX(f3). Therefore, it is a derivation and a
global alternative for the definition of a vector field (A. l I).](https://image.slidesharecdn.com/5590127-250520140410-cc3d8b3b/85/Mathematical-Structures-Of-The-Universe-1st-Edition-Micha-Heller-52-320.jpg)
Mathematical Structures Of The Universe 1st Edition Micha Heller
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- 5. Mathematical Structures of the Universe EDITED BY Michal Eckstein Michael Helier Sebastian]. Szybka 0 Copemicus Center PRESS
- 6. ©Copyright by Copernicus Center Press, 2014 Editing: AeddanShaw Cover design: Mariusz Banochowicz BibTeX: Dominika Hunik, Pawel Kostyra Publication Supported by the John Templeton Foundation Grant "The Limits of Scientific Explanation" ISBN.978-83-7886-1 07-2 Krak6w 2014 8 Copernicus Center PRESS I Publisher: Copernicus Center Press Sp. z o.o., pi. Szczepanski 8, 31-011 Krak6w, tel/fax (+48) 12 430 63 00 e-mail: marketing@ccpress.pl www.ccpress.pl Table of Contents Michal Eckstein, Michae/ Helier, Sebastian J. Szybka Introduction 9 Part I General Relativity and Cosmology Manue/ Hohmann Observer dependent geometries 13 KrzysztofDrachal & Wieslaw Sasin Classification of classical singularities: a differential spaces approach 57 Jacek Gruszczak The smooth beginning of the Universe 69 Mariusz P. Dqbrowski Are singularities the limits of cosmology? . 101 Boudewljn F. Roukema Simplicity in cosmology: add virialisation, remove A, keep classical GR . 119 Andrzej Woszczyna & Zdzislaw A. Golda Computer algebra tests physical theories: the case of relativistic astrophysics . . . . . . . . . . . . . . . . 127 Sebastian J. Szybka On gravitational interactions between two bodies 137
- 7. 6 MarekKus Part 11 Quantum Geometries Table of Contents Geometry of quantum correlations . . . . . . . . . . . . . 155 Jordan Fran~ois, Serge Lazzarini & Thierry Masson Gauge field theories: various mathematical approaches . . . . . . . . . 177 Haraid Grosse & Raimar Wulkenhaar Towards a construction of a quantum field theory in four dimensions . 227 Mairi Sakellariadou Unweaving the fabric ofthe Universe: the interplay between mathematics and physics . . . . . . . . . . . . . 259 Jerzy Lukierski Quantum gravity models- a brief conceptual summary . . . . . . . . . 277 Andrzej Sitarz Pointless geometry . Nicolas Franco & Michal Eckstein . . . . . . . . . . . . . . . . . . . . . . 301 Noncommutative geometry, Lorentzian structures and causality . . . . 315 Michael Helier & Oominique Lambert Ontology and noncommutative geometry . . . . . . . . . . . . . . . . 341 ShahnMajid Part Ill Overviews The self-representing Universe . . . . . . . . . . . . . . . . . . . . . . . 357 Ma/colm A.H. MacCallum Reflections on the geometrization of physics . . . . . . . . . . . . . . . 389 Bernard Carr Metacosmology and the limits of science . . . . . . . . . . . . . . . . . 407 Table of Contents 7 Jerzy Kowalski-Giikman The price for mathematics . . . . . . . . . . . . . . . . . . . · . · · · · 433 Michael Helier The field of rationality and category theory . . . . . . . . . . . . . . . . 441
- 8. It seems to be one of the fundamental features of nature that fundamen- tal physical laws arc described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the uni- verse. Our feeble auernpts at mathematics enable us to understand a hit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better. Paul A.M. Dirae* Introduction A s the mathematical-empirical method deeply underlies the foundations of the modern natural sciences, we have simply grown accustomed to the idea that mathematical structures are indeed inherent in the Universe. This idea has guided successive generations of scientists, starting with some of its most famous precursors such as Copernicus, Galileo or Newton. However, the use of the in- trinsic interplay between mathematics and physics in scientific discourse can be traced back even further- to the time of Ancient Greece and the pioneering works ofArchimedes. One of the main goals of the natural sciences appears to be the search for the mathematical language of physical phenomena. This aim has to be defined precisely. Contemporary mathematics encompasses an abundance of different structures which, moreover, are linked with one another in larger structures and meta-structures. However, only a small number of these turn out to be suitable for physical models. To identify this tiny fraction, scientists have to explore vast areas of mathematics. Some of them cover known mathematical structures, others explore new territories. The somewhat mythical quantum gravity is a typical example of a domain for which the correct mathematical architecture still needs to be fathomed out. In the first two parts of the volume, the Reader will meet various mathemat- ical structures. Some of them do indeed model certain aspects of the Universe, *''The evolution of the physicist's picture of nature." Scientific American 208(5):45-53, 1963.
- 9. Michal Eckstein, Michael Helier & Sebastian J. Szybka --------------------------------~--- 10 as they correctly predict the outcomes of experiments and observations. Among these, one may find differential geometry and the theory of Hilbert spaces, which lie at the heart of General Relativity and Quantum Mechanics, respectively. The second group of mathematical structures described in the book, such as noncom- mutative geometry, are designed to model the aspects of the Universe not covered by known theories. These still await an experimental confirmation to merit the name of 'mathematical structures of the Universe'. As one explores the mathematical structures of the Universe, one cannot escape deeper philosophical reflection. Why is Nature constructed along these lines? What is the actual relation between mathematics and the real World? Do the structures describe the Universe, model it or perhaps they are just the outcome of our minds whereas 'the Universe' itself remains inconceivable (if it can be said to exist in an absolute sense at all)? lf one accepts the idea of a mathematical Uni- verse, then what kind of methodological assumptions does one make on the way and what are the limits of the method? Can the whole Universe be encompassed in a single, consistent mathematical structure? These are the questions addressed in the third part of the book- a philosophically flavoured overview. Michal Eckstein Michael Helier Sebastian J. Szybka Part I General Relativity and Cosmology
- 10. Manuel Hohmann Fuusika lnstituut, Tartu Observer dependent geometries FROM general relativity we have learned the principles of general covariance and local Lorentz invariance, which follow from the fact that we consider observables as tensors on a spacetime manifold whose geometry is modeled by a Lorentzian metric. Approaches to quantum gravity, however, hint towards a breaking of these symmetries and the possible existence of more general, non- tensorial geometric structures. Possible implications of these approaches are non-tensorial transformation laws between different observers and an observer- dependent notion of geometry. ln this work we review two different frameworks for observer dependent geometries, which may provide hints towards a quanti- zation of gravity and possible explanations for so far unexplained phenomena: Finsler spacetimes and Cartan geometry on observer space. We discuss their def- initions, properties and applications to observers, field theories and gravity. 1. Geometry for observers and observables In order to establish a link with experiments, every physical theory needs to de- fine the notions of observers and observables. From an experimentalist's point of view, an observation is the process of an observer performing an experiment in which he measures a number of physical quantities, called observables. Each measured observable is expressed by a single number or a set of numbers. In order to understand the meaning of these numbers from a theorist's point of view, and thus in a mathematical language, observers and observables must be mod- elect by mathematical objects, which can in turn be related to the outcomes of measurements. This model determines how the result of an observation depends on the observer who is performing it, and how the results obtained by different observers can be related to each other. In this work we will focus on geometric models for these relations. We start our discussion from the viewpoint of general relativity. The most ba- sic notion of general relativity is that of spacetime, which is modeled by a smooth
- 11. 14 Manuel Hohmann manifold M equipped with a pseudo-Riemannian metric g of Lorentzian signa- ture (-, +, +, +), an orientation and a time orientation. Observers are modeled by world lines, which are smooth, future directed. timelike curves 1 : JR. --+ M. Their tangent vectors satisfy (1.1) By a reparametrization we can always normalize the tangent vectors, so that (1.2) In this case we call the curve parameter the proper time along the world line 1 and denote it by the letter T instead oft. The proper time along a timelike curve with arbitrary parametrization is given by the arc length integral ( 1.3) The clock postulate of general relativity states that any clock moving along the world line 1 measures the proper time, independent of the construction of the clock. The prescription for the measurement of time is thus crucially linked to the Lorentzian metric of spacetimc. Similarly, the metric provides a definition of rulers and the length of spacelike curves by the same expression (1.3) of the arc length integral. Finally, it also defines the angle q; between two tangent vectors v, wE T.'"CM at the same point x E M as (1.4) In summary, the Lorentzian metric g defines the geometry ofspacetime. Closely related to the geometry of spacetime is the notion of causality. It answers the question which events on a spacetime manifold NI can have a causal influence on which other events on iVJ. An event at x E l'vi can influence an event x' E M if and only if there exists a continuous, future directed, causal (i.e., timelike or lightlike) curve from x to x'. All events which can be influenced by x constitute the causal future of :r:. Conversely, all events which can influence x' form the causal past of :r:'. This structure, called the causal structure of:,pacetime, is defined by the metric geometry via the definition of causal curves. The Lorentzian spacetime metric serves several further purposes besides pro- viding a definition of spacetime geometry and causality. We have already seen Observer dependent geometries 15 that it enters the definition of observer world lines as timelike curves, whose no- tion is thus also relevant when we consider the measurements of observables by these observers. Observables are modelcd by tensor fields, which are smooth sections iJ? : .M ->- TT",s Af of a tensor bundle yr,s1'vf = T 11°7 ' 0 T* NI0 " ( 1.5) over !VI. Their dynamics are consequently modeled by tensorial equations, which are derived from a diffeomorphism-invariant action of the generic form SM = ;· d4 xFYL(g, il?, ()cl?, ...), .M (1.6) where the Lagrange function [, depends on the metric geometry, the (]elds and their derivatives. Combining the notions of observers and observables we may define an observation by an observer with world line 1 at proper time T as a measurement of the field iJ?(:c) at the point :r = 1(-r). However, this definition yields us an element of the tensor space 1-;:·s!VI, and not a set of numbers, as we initially presumed. We further need to choose a frame, by which we denote a basis .f of the tangent space T,'"CM. This frame allows us to express the tensor i!?(:r:) in terms of its components with respect to f. The tensor components of iJ?(x) are finally the numeric quantities which are measured in an experiment. The frame f chosen by an observer to make measurements is usually not completely arbitrary. Since the basis vectors fi are elements of the tangent space, they are characterized as being timelike, lightlike or spacelike and possess units of time or length. We can thus use the notions of time, length and angles de(]ned by the spacetime metric to choose an orthonormal frame satisfying the condition 9 .fa fb - 'fj· . . ab i. j - 'J (1 .7) with one unit timelike vector .fo and three unit spacelike vectors fee The clock postulate, stating that proper time is measured by the arc length along the ob- server world line 1 , further implies a canonical choice of the timelike vector fo as the tangent vector "Y(T) to the observer world line. This observer adapted or- thonormal frame is a convenient choice for most measurements. It follows immediately from this model of observables and observations how the measurements of the same observable made by two coincident observers, whose world lines 1 and 1' meet at a common spacetime point x = 1(T) = 1'(T'), must be translated between their frames of reference. If both observer frames f and .f' are orthonormalized, the condition (1.7) implies that they are
- 12. 16 Manuel Hohmann related by a Lorentz transform A. The same Lorentz transform must then be applied to the tensor components measured by one observer in order to obtain the tensor components measured by the other observer, using the standard formula if>'": ...a.,.l I = A"I A"'· Ad! Ads c"F-CJ ...c,. )J •·· Js Ct . · . C-r bt · · · b8 J! dJ ...d8 • (1.8) This dose connection between observations made using different observer frames constitutes the principle of local Lorentz invariance. It is a consequence of the fact that we model the geometry of spacetime, which in turn defines the notion of orthonormal frames, by a Lorentzian metric. Even deeper implications arise from the fact that we model both observables and geometry by tensor fields on the spacetime manifold !VI, and observations by measurements of tensor components. If we introduce coordinates on M and use their coordinate base in order to express the components of tensor fields, it immediately follows how these components translate under a change of coordi- nates. Moreover, since we model the dynamics of physical quantities by tensor equations, they are independent of any choice of coordinates. This coordinate freedom constitutes the principle of general covariance. Besides its role in providing the background geometry which enters the def- inition of observers, observations and causality, the Lorentzian metric of space- time has a physical interpretation on its own, being the field which carries the gravitational interaction. It does not only govern the dynamics of matter fields, but is also influenced by their presence. This is reflected by the dynamics of gravity, which is governed by the Einstein-Hilbert action 1 1 4 SEH = ;:;- d xFyR, .c,K, M (1.9) which, together with the matter action (1.6), yields the Einstein equations 1 Rab - 2Rgab = K,Tab . ( l.lO) Understanding the geometry of spacetime as a dynamical quantity, which mu- tually interacts with matter fields, establishes a symmetric picture between both matter and gravity. However, it is exactly this symmetry between gravity and matter which may lead us to new insights on the nature of spacetime geometry, and even question its description i;J terms of a Lorentzian metric, from which we derived a number of conclusions as stated above. This stems from the fact that all known matter Observer dependent geometries 17 fields in the standard model are nowadays described by quantum theories. While the process of quantization has been successfully applied to matter fields even beyond the standard model, it is significantly harder in the case of gravity. This difficulty has lead to a plethora of different approaches towards quantum grav- ity, many of which suggest modifications to the geometry of spacctime, or even resolve the unity of spacetime into a time evolution of spatial geometry. Main contenders which fall into this class are given by geometrodynamic theories such as loop quantum gravity [Ashtekar 1987, Thiemann 20071 and sum-over-histories formulations such as spin foam models IRovelli & Smolin I995, Reisenberger & Rovelli 1997, Baez 1998, Barrett & Crane 1998J or causal dynamical triangu- lations [Ambj0rn & Loll 1998, Ambjorn, Jurkiewicz & Loll 2001,20051. The- ories of this type introduce non-tensorial quantities, which may in turn suggest a breaking of general covariance at least at the quantum level. Moreover, other approaches to gravity may induce a breaking of local Lorentz invariance, for ex- ample, by a preferred class of observers, or test particles, described by a future unit timelike vector field [Brown & Kuchaf 1995, Jacobson & Mattingly 200 I]. The possible observer dependence of physical quantities beyond tensorial transformations motivates the introduction of spacetime geometries obeying a similar observer dependence, which generalize the well-known Lorentzian met- ric geometry. In this work we review and discuss two different, albeit similar, approaches to observer dependent geometries under the aspects of observers, causality and gravity. In section 2 we review the concept of Finsler spacetimes [Pfeifer & Wohlfarth 2011, 2012, Pfeifer 2013]. We show that it naturally gen- eralizes the causal structure of Lorentzian spacetimes, provides clear definitions of observers, observables and observations, serves as a background geometry for field theories and constitutes a model for gravity. In section 3 we review the con- cept of observer space in terms of Cartan geometry [Gielen & Wise 2013]. Our discussion is based on the preceding discussion of Finsler spacetimes, from which we translate the notions of observers and gravity to Cartan language [Hohmann 2013]. We finally ponder the question what implications do observer-dependent geometries have on the nature of spacetime. 2. Geometry of the dock postulate: Finsler spacetimes As we have mentioned in the introduction, the metric geometry of spacetime serves multiple roles: it provides a causal structure, crucially enters the defi- nition of observers, defines measures for length, time and angles and mediates
- 13. 18 Manuel Hohmann the gravitational interaction. In this section we discuss a more general - non- metric - spacetime geometry which is complete in the sense that it serves all of these roles. This generalized geometry is based on the concept of Finsler geom- etry lBao, Chern & Shen 2000, Bucataru & Miron 2007]. Models of this type have been introduced as extensions to Einstein and string gravity [Horvath 1950, Vacaru 2002,2007, 2012]. In this work we employ the Finsler spacetime frame- work [Pfeifer & Wohlfarth 2011, 2012, Pfeifer 2013], which is an extension of the well-known concept of Finsler geometry to Lorentzian signature, and review some of its properties and physical applications. This framework is of particular interest since, in addition to its aforementioned completeness, it can also be used to model small deviations from metric geometry and provides a possible explana- tion of the fly-by anomaly [Anderson, Campbell, Ekelund, Ellis & Jordan 2008]. 2.1. Definition of Finsler spacetimes The starting point of our discussion is the clock postulate, which states that the time measured by an observer's clock moving along a timelike curve ry is the proper timeT given by the arc length integral (1.3). The expression (2.1) under the integral depends on both the position 'Y(t) along the curve and the tangent vector 'Y(t). Hence, it can be regarded as a function F : T M ---+ lR on the tangent bundle. The clock postulate thus states that the proper time measured by an observer's clock is given by the integral 1 t2 T2- Tl = F('Y(t), "y(t))di;' tr (2.2) where F is the function on the tangent bundle given by equation (2.1 ). For convenience we introduce a particular set (.Ta, ya) ofcoordinates on T j'vf. Let (.ra) be coordinates on lvf. For y E Ta,lvi we then use the coordinates (ya) defined by a Y = Ya fJxa . We call these coordinates induced by the coordinates (:,;0·). As a further shorthand notation we use a Oa, = EJxa , for the coordinate basis of T(x,y) T M. Observer dependent geometries 19 We now introduce a different, non-metric geometry of spacetime which still implements the clock postulate in the form of an arc length integral (2.2), but with a more general function F on the tangent bundle. Geometries of this type are known as Finsler geometries, and F is denoted the Finsler function. The choice ofF we make here is not completely arbitrary. In order for the arc length integral to be well-defined and to obtain a suitable notion of spacetime geometry we need to preserve a few properties of the metric-induced Finsler function (2.1). In particular we will consider only Finsler functions which satisfy the following: Fl. F is non-negative, P(x, y) 2 0. F2. Pis a continuous function on the tangent bundle TIVI and smooth where it is non-vanishing, i.e., on TM {F = 0}. F3. F is positively homogeneous of degree one in the fiber coordinates and reversible, i.e., F(:,;,>.y) = j.AjF(x,;y) "f)., E lR. . (2.3) Property Fl guarantees that the length of a curve is non-negative. We cannot demand strict positivity here, since already in the metric case we have the no- tion of lightlike curves "f, for which F('Y(t)/y(t)) = 0. For the same reason of compatibility with the special case of a Lorentzian spacetime metric we cannot demand that F is smooth on all ofTM, since the metric Finsler function (2.1) does not satisfy this condition. It does, however, satisfy the weaker condition F2, which guarantees that the arc length integral depends smoothly on deformations of the curve "f, unless these pass the critical region where F = 0. Finally, we demand that the arc length integral is invariant under changes of the parametriza- tion and on the direction in which the curve is traversed, which is guaranteed by condition F3. One may ask whether the Lorentzian metric 9ab can be recovered in case the Finsler function is given by (2.1 ). Indeed, the Finsler metric F ( ) 1 .6 .6 p2 ( ) 9ab X, Y = 2Ua,Ub :1:, Y ' (2.4) which is defined everywhere on TM {F = 0}, agrees with 9ab whenever y is spacelike and with -gab when y is timelike. However, for null vectors where F = 0 we see that the Finsler metric g~ is not well-defined, since for a general Finsler function P2 will not be differentiable. As a consequence any quanti- ties derived from the metric, such as connections and curvatures, are not defined
- 14. 20 Manue/ Hohmann along the null structure, which renders this type of geometry useless for the de- scription of lightlike geodesics. In the following we will therefore adopt the fol- lowing definition of Finsler spacetimes which remedies this shortcoming [Pfeifer & Wohlfarth 2011 ]: Definition 2.1 (Finsler spacetime). A Finsler spacetime (M, L, F) is a four- dimensional, connected, Hausdorff, paracompact, smooth manifold 111 equipped with continuous real functions L, F on the tangent bundle T lvi which has the following properties: L I. L is smooth on the tangent bundle without the zero section T M {0}. L2. L is positively homogeneous of real degree n 2: 2 with respect to the fiber coordinates ofTlvf, L(x,>..y) = >..nL(x,y) Y>.. > 0, and defines the Finsler function F via F(x, y) = JL(x, y)J~. L3. Lisreversible: JL(.T,-y)J = JL(x,y)J. L4. The Hessian L 1- - Yab(x,y) = 20aObL(x,y) of L with respect to the fiber coordinates is non-degenerate on TM X, where X C T 111 has measure zero and does not contain the null set {(.1:,y) E TMJL(x,y) = 0}. L5. The unit timelike condition holds, i.e., for all x E M the set Dx = {y E TxiVI /JL(.T, y)J = 1, g{;:b(x, y) has signature (E, -E, -E, -E)} with E = L(x,y)/JL(x,y)J contains a non-empty closed connected com- ponent Sx <:;; Dx C T."A1. One can show that the Finsler function F induced from the fundamental ge- ometry function L defined above indeed satisfies the conditions Fl to F3 we required. Further, the Finsler metric (2.4) is defined on T M {L = 0} and is non-degenerate on TM (X U {L = 0}), where X is the degeneracy set of the Hessian g~b defined in condition L4 above. This definition in terms of the smooth fundamental geometry function L will be the basis of our discussion of Finsler spacetimes in the following sections, where we will see that it also extends the Observer dependent 9eometries 21 sign [, = 1 sign /, = 1 sign/,= -1 Figure 1: Light cone and future unit timelike vectors Sa: in the tangent space of a metric spacetime [Pfeifer & Wohlfarth 2011]. definitions of other geometrical structures such as connections and curvatures to the null structure. 2.2. Causal structure and observers The first aspect we discuss is the causal structure of Finsler spacetimes and the definition of observer trajectories. For this purpose we first examine the causal structure of metric spacetimes from the viewpoint of Finsler geometry, before we come to the general case. We have already mentioned in the introduction that the definition of causal curves is given by the split of the tangent spaces into timelike, spacelike and lightlike vectors. Figure 1 shows this split induced by the Lorentzian metric on the tangent space T",lvf. Solid lines mark the light cone which is constituted by null vectors. In terms of the fundamental geometry function L(x, y) = Yab(:r;);~/yb these are given by the condition L(x, y) = 0. Outside the light cone we have spacelike vectors with L(:r.:, y) > 0, while inside the light cone we have timelike vectors with L(x, y) < 0. The Hessian g{;:b = Yab therefore has the signature indicated in condition L5 inside the light cone. In both the future and the past light cones we find a closed subset with JL(x, y) J= 1. Using the time orientation we pick one of these subsets and denote it the shell Sx of future unit timelike vectors.
- 15. 22 Manuel Hohmann The shell Sx has the important property that rescaling yields a convex cone Cx = U >.Sa: C Txlvf. .>0 (2.5) The convexity of this cone is crucial for the interpretation of the elements of Sx as tangent vectors to observer world lines, as it is closely linked to the hyperbol- icity of the dispersion relations of massive particles and the positivity of particle energies measured by an observer [Ratzel, Rivcra & Schuller :iOlll We require this property also for the future light cone of a Finsler spacetime. In order to find this structure in terms of the fundamental geometry function L consider the simple bimetric example with two Lorentzian metrics hab and kah where we assume that the light cone of kab lies in the interior of the light cone of hab· The sign of L and the signature of g{;b on the tangent space Txlvf are shown in figure 2. Solid lines mark the null structure L = 0, while the dashed-dotted lines marks the degeneracy set X n T.TM of L as defined in condition L4. The remaining dashed and dotted lines mark the unit timelike vectors nx as defined in condition L5; for these only the future directed tangent vectors are shown. The connected component marked by the dashed line is closed, while the one marked with the dotted line is not. Hence, the former marks the set Sx. As the figure indicates, the set (2.5) indeed forms a convex cone for this simple bimetric example. It can be shown that condition L5 always implies the existence of a convex cone of observers [Pfeifer & Wohlfarth 2011 J, in consistency with the requirement stated above. It is now straightforward to define: Definition 2.2 (Observer world line). A physical observer world line on a Finsler spacetime is a curve 1 : lR --+ Jvf such that at all times t the tangent vector "y(t) lies inside the forward light cone cy(t)• or in the unit timelike shell s,(r) if the curve parameter is given by the proper timeT. In the following section we will discuss which of these observers are further singled out by the Finsler spacetime geometry as being inertial observers. 2.3. Dynamics for point masses In the preceding section we have seen which trajectories are allowed for physical observers. We now turn our focus to a particular class of observers who follow Observer dependent geometries L = -1 S:r: ' ', L ~ 1 ·· ... ...·· ' ' (+,-·,-,-) L>O / / / / / / / / ..· ..... ······ (-,+, 1,+) L>O ··., ··... 23 /' Figure 2: Null structure and future unit timelike vectors Sx in the tangent space of a bimetric Finsler spacetime [Pfeifer & Wohlfarth 2011 ]. the trajectories of freely falling test masses. These are denoted inertial observers, since in their local frame of reference gravitational effects can be neglected. On a metric spacetime they are given by those trajectories which extremizc the arc Jenth integral (1.3). In Finsler geometry we can analogously obtain them from extremizing the proper time integral (2.2). Variation with respect to the curve yields the equation of motion -·a Na ( ·) · b 0 I + b 1,11 = , (2.6) where the coefficients Nab are given by the following definition: Definition 2.3 (Cartan non-linear connection). The coefficients Nab of the Car- tan non-linear connection are given by (2.7) and define a connection in the sense that they induce a split of the tangent bundle over Tlvf, TTM = HTM EB VTM, (2.8) where HTM is spanned by Oa = Oa - NbaBb and VTJ"'vf is spanned by Ba.
- 16. 24 Manuel Hohmann In the case of a metric-induced Fins!er function (2.1) the coefficients Nab are given by (2.9) where f"bc denotes the Christoffel symbols. The split (2.8) of TTM into hori- zontal and vertical subbundles plays an important role in Finsler geometry, as we will see in the following sections. For convenience we use the following adapted basisofTTM: Definition 2.4 (Berwald basis). The Berwald basis is the basis (2.10) of TTl'vf which respects the split induced by the Cartan non-linear connection. For the dual basis we use the notation (2.11) It induces a similar split of the cotangent bundle T*TM into the subbundles T*TM = H*TM EB V*TM. (2.12) We can now reformulate the geodesic equation (2.6) by making use of the geom- etry on TTNI. For this purpose we canonically lift the curve 1 to a curve (2.13) in TTl'vf. The condition that 1 is a Finsler geodesic then translates into the con- dition r. ·a,;o, + ..a;:; ·a,;o, ·bNa ;:> ·a> = I Ua I Ua = I ua - I bUa = I "a Since "fa is simply the tangent bundle coordinate ya, it thus follows that the canonical lift r of a Finsler geodesic must be an integral curve of the vector field which is defined as follows: Definition 2.5 (Geodesic spray). The geodesic ~prayS is the vector field on T A1 which is defined by (2.14) We now generalize this statement to null geodesics. Here we encounter two problems. First, we see that the coefficients (2.7) of the non-linear connection are not well-defined for null vectors where F = 0, since F is not differentiable Observer dependent geometries 25 ---------------- on the null structure. We therefore need to rewrite their definition in terms of the fundamental geometry function L. It turns out that it takes the same form N" = ~D [nLac('{lrlEJ [) L- D.L)] b 4 b " ,} d c (. ) (2.15) where .r/'' has been replaced by gL and F 2 by L. We can see that this is well- defined whenever gL is non-degenerate, and thus in particular on the null struc- ture. The second problem we encounter is that we derived the geodesic equation from extremizing the action (2.2), which vanishes identically in the case of null curves. We therefore need to use the constrained action S[r, >-] = .{" (L(1 (t), "y(t)) + >-(t) [L('y(t), ''(t)) - h;]) dt. (2.16) with a Lagrange multiplier A. and a constant "'· A thorough analysis shows that the equations of motion derived from this action are equivalent to the geodesic equation (2.6) also for null curves [Pfeifer & Wohlfarth 20 11]. The definitions of this and the preceding section provide us with the notions of general and inertial observers. In the following section we will discuss how these observers measure physical quantities and how the observations by different observers can be related. 2.4. Observers and observations As we have mentioned in the introduction, the notion of geometry in physics de- fines not only causality and the allowed trajectories of observers, but also their possible observations and the relation between observations made by different observers. In the case of metric spacetime geometry we have argued that obser- vations are constituted by measurements of the components of tensor fields at a spacetime point x E l'vf with respect to a local frame f at :r. A particular class of frames singled out by the geometry and most convenient for measurements is given by the orthonormal frames. Different observations at the same spacetime point, but made with different local orthonormal frames, are related by Lorentz transforms. In this section we discuss a similar definition of observations on Finsler spacetimes and relate the observations made by different observers. As a first step we need to generalize the notion of observables from metric spacetimes to Finsler spacetimes. In their definition in section 2.1 we have al- ready seen that the geometry of Finsler spacetimes is defined by a homogeneous function L : T M ---7 ~ on the tangent bundle, which in turn induces a Finsler
- 17. 26 Manuel Hohmann function F and a Finsler metric g~. These geometric objects explicitly depend not only on the manifold coordinates :ra, but also on the coordinates ya along the fibers of the tangent bundle T M. It therefore appears natural that also ob- servables should not be functions on the spacetime manifold, but homogeneous functions Oll its tangent bundle. A straightforward idea might thus be to model observables as homogeneous tensor fields overTM, i.e., as sections of a tensor bundle However, since T lvf is an eight-dimensional manifold, each tensor index would then take eight values, so that the number of components of a tensor of rank (r, s) would increase by a factor of 2r+s. Since we do not observe these additional tensor components in nature, we will not follow this idea. Instead we define observables as tensor fields with respect to a different vector bundle over T lvf, whose fibers are four-dimensional vector spaces generalizing the tangent spaces of M. In the preceding section we have seen that the Cartan non-linear connec- tion (2.7) of a Finsler spacetime equips the tangent bundle TTlvf of TM with a split (2.8) into a horizontal subbundle HTM and a vertical subbundle VTM. The fibers of both subbundles are four-dimensional vector spaces. A particular section of HTM, which we have already encountered and which is closely con- nected to Finsler geodesics, is the geodesic spray (2.14). We therefore choose HTM as the bundle from which we define observables as follows: Definition 2.6 (Observable). The observables on a Finsler spacetime are modeled by homogeneous horizontal tensor fields, i.e., sections <P of the tensor bundle (2.17) over the tangent bundle T M of AI. Consequently we define observations in full analogy to the case of metric spacetime geometry: Definition 2.7 (Observation). An observation of an observable <I> by an observer with world line "( at proper time T is a measurement of the components of the horizontal tensor <P(x, y) with respect to a basis f of the horizontal tangent space H(x,y)TM at :r = "f(T), y = 1(T). As we have argued in the introduction, the most natural frame f an observer on a metric spacetime can choose is an orthonormal frame whose temporal com- ponent fo agrees with his four-velocity 1(T ). If we wish to generalize this concept Observer dependent geometries 27 to Finsler spacetimes, we first need to map the basis vectors .fi, which are now elements of HTlvf, to T M. For this purpose we use the differential 7f* of the tangent bundle map r. : T lvf -t M, which isomorphically maps every horizontal tangent space H(x,y)TNI to TxA1. We can then orthonormalize the frame using the Finsler metric g~·;,, which now explicitly depends on the observer's four-velocity y = 1f*fo. Taking into account the signature (+, -, -·,-) of the Finsler metric on timelike vectors inside the forward light cone we arrive at the following definition: Definition 2.8 (Orthonormal observer frame). An orthonormal observer .frame on an observer world line 'Y at proper timeT is a basis f of the horizontal tangent space H(x,y)TM at x = "f(T), y = 1(T) which has y = 1r*.fo and is orthonormal with respect to the Finsler metric, F ( ) fa.fb _ .. Yab X, Y . i j - -ThJ · (2.18) An important property of metric spacetimes is the fact that any two orthonor- mal observer frames f, .f' at the same spacetime point :x; E M are related by a unique Lorentz transform. Together with the dellnition that observations yield tensor components this property implies local Lorentz invariance, which means that the outcomes of measurements are related by the standard formula (1.8). We now generalize this concept to Finsler spacetimes. For this purpose we consider two coincident observers whose world lines 'Y, "( 1meet at x = "!(T) = "! 1 (T') together with orthonormal frames j, .f' at x. One immediately encounters the difficulty that f and .f' are now bases of different vector spaces H(xJo)T l'vf and H(xJh)TM. We therefore need to tlnd a map between these vector spaces which in particular preserves the notion of orthonormality. The canonical map given by the isomorphisms 7f* : H(x,Jo)TM -t TxM and 7f* : H(xJh)TM -t T.'~)1, however, does not have this property. ln the following we will therefore discuss a different map which will yield the desired generalization of Lorentz transforma- tions. In order to construct a map between the horizontal tangent spaces H(x,Jo)Tlvf and H(x,Jh)T iW we employ the concept of parallel transport. We thus need a con- nection on the horizontal tangent bundle HTM with respect to which the Fins!er metric is covariantly constant, so that the notion of orthonormality is preserved. In Finsler geometry an appropriate choice which satisfies these conditions is the
- 18. 28 Manue/ Hohmann Cartan linear connection on the tangent bundle TTli!J, which is defined as fol- lows: Definition 2.9 (Cartan linear connection). The Cartan linear connection 7 is the connection on TT!vi defined by the covariant derivatives 7<>a Ob = F"a"Oc , 7",}J,, = F",,,J),, , '1'7 c; 'b - C" b:;: '1'7 fSb - C'c 1(;;; u ,. u, •U • v,,au - a Uc, VOa'J - aJJc, (2.19) where the coefficients are given by (2.20a) (2.20b) The Cartan linear connection is adapted to the Cartan non-linear connec- tion (2.7) in the sense that it respects the split (2.8) into horizontal and verti- cal components. By restriction, it thus provides a connection on the horizon- tal tangent bundle. Given a curve v : [0, 1] -+ TM with v(O) = (:r, fo) and v(l) = (:r, !6) we can then define a bijective map Pu from T(x,fo)'TM to T(xJ£)TM by parallel transport: it maps the vector w to Pvw = w', which is uniquely determined by the existence of a curve ·w : [0, 1] -+ TTM satisfying {iJ(s)ETv(s)TM, <v(O)=w, u)(1)=v/, 7-uw=O. However, this map Pv in general depends on the choice of the curve ·v. We therefore restrict ourselves to a particular class of curves. Note that (x, Jo) and (x, !6) have the same base point in M, and are thus elements of the same fiber of the tangent bundle T !vi. Hence it suffices to consider only curves which are entirely contained in the same fiber. Curves of this type are vertical, i.e., their tangent vectors lie in the vertical tangent bundle VTlvi. We further impose the condition that v is an autoparallel of the Cartan linear connection. This uniquely fixes the curve v, provided that f~ is in a sufficiently small neighborhood of f0 . Using the unique vertical autoparallel v defined above we can now generalize the notion of Lorentz transformations to coincident observers on a Finsler space- time. Consider two observers meeting at :1: E Jvi and using frames f and f', i.e., orthonormal bases of H(x,Jo) T Jvi and H(x,JfJ) T M. The map Pv maps the hori- zontal basis vectors j; to horizontal vectors Pvfi, which constitute a basis Pvf of H(xJ6)TlVI. Since f is orthonormal with respect to gf:t,(x, fo) and the Cartan linear connection preserves the Finsler metric, it follows that Pvf is orthonormal with respect to gf:t,(x, !0). Since also f' is orthonormal with respect to the same Observer dependent geometries 29 metric, there exists a unique ordinary Lorentz transform mapping Pvf to .f'. The combination of the parallel transport along v and this unique Lorentz transform finally defines the desired generalized Lorentz transform. The procedure to map bases of the horizontal tangent space between coin- cident observers further allows us to compare horizontal tensor components be- tween these observers, so that they can communicate and compare their mea- surements of horizontal tensors. This corresponds to the transformation (1.8) of tensor components of observables between different observer frames in metric geometry. Since observables in metric geometry are modeled by spacetime ten- sor fields, their observation in one frame determines the measured tensor compo- nents in any other frame. This is not true on Finsler spacetimes, since we defined observables as fields on the tangent bundle T1'd. They may therefore also pos- sess a non-tensorial, explicit dependence on the four-velocity of the observer who measures them. As in metric geometry, also in Finsler geometry the dynamics of tensor fields should be determined by a set of field equations which are derived from an action principle. This will be discussed in the next section. 2..5. Field theory In the preceding section we have argued that observables on a Finsler spacetime are modeled by homogeneous horizontal tensor fields, which are homogeneous sections of the horizontal tensor bundle (2.17). We will now discuss the dynamics of these observable fields. For this purpose we will use a suitable generalization of the action (1.6) to horizontal tensor fields on a Finsler spacetime. This will be done in two steps. First we will lift the volume form from the spacetime manifold Jvi to its tangent bundle T M, then we generalize the Lagrange function L to fields on a Finsler spacetime. In order to define a volume form on T Jvi we proceed in analogy to the volume form of metric geometry, which means that we choose the volume form Vole of a suitable metric G on TM. We have already partly obtained this metric in the previous section when we discussed orthonormal observer frames. The definition of orthonormality we introduced corresponds to lifting the Finsler metric gf:t, to a horizontal metric on T M, which measures the length of horizontal vectors in HTM. This metric needs to be complemented by a vertical metric, which anal- ogously measures the length of vertical vectors in VTJvf. Both metrics together constitute the desired metric on the tangent bundle. The canonical choice for this metric is given by the Sasaki metric defined as follows:
- 19. 30 Manuel Hohmann Definition 2.10 (Sasaki metric). The Sasaki metric G is the metric on the tangent bundle T M which is defined by F' G = -g"' dx" 0 d:r0 - !l_au oya 00 rl1,1' ab p2 - . 1 · (2.21) The factor p-z introduced here compensates for the intrinsic homogeneity of degree 1ofthe one-forms oy", so that the Sasaki metric is homogeneous ofdegree 0. This intrinsic homogeneity becomes clear from the definition (2.11) of the dual Berwald basis, taking into account that the coefficients N'"z, are homogeneous of degree I, as can be seen from their definition (2.7). Using the volume form Vola of the Sasaki metric one can now integrate functions .f on the tangent bundle, { Vola f(:r, y). lrM (2.22) If one chooses the function .f to be a suitable Lagrange function L for a physical field <P on a Finsler spacetime, one encounters another difficulty. Since all geo- metric structures and matter fields <P are homogeneous, it is natural to demand the same from the Lagrangc function. However, for a homogeneous function .f the integral over the tangent bundle generically diverges, unless the function van- ishes identically. This follows from the fact that along any ray (:r, Ay) with ).. > 0 in TM the value off is given by An f(x, y), where n is the degree of homogene- ity. This difficulty can be overcome by integrating the function not over TM, but over a smaller subset ofTM which intersects each ray only once, and which is defined as follows: Definition 2.11 (Unit tangent bundle). The unit tangent bundle ofa Finsler space- time is the set I; C T NI on which the Finsler function takes the value F = 1. Note that I; intersects each ray, which is not part of the null structure, exactly once. This suffices since the null structure is of measure 0 and therefore does not contribute to the integral (2.22) overTM. The canonical metric on I: is given by the restriction (2.23) ofthe.Sasaki metric, which finally determines the volume form Vole;. This is the volume form we will use in the generalized action integral. In the second part of our discussion we generalize the Lagrange function L in the metric matter action (1.6). For simplicity we restrict ourselves here top-form fields <P whose Lagrange function depends only on the field itself and its first derivatives d<D. These are of particular interest since, e.g., the Klein-Gordon and Observer dependent geometries 31 Maxwell fields fall into this category. The most natural procedure to generalize the dynamics of a given field theory from metric to Finsler geometry is then to simply keep the formal structure of its Lagrange function £, but to replace the Lorentzian metric g by the Sasaki metric G and to promote the p-form field <P to a horizontal p-form field on T M. The generalized Lagrange function we obtain from this procedure is now a function on TNI, which we can integrate over the subset I; to form an action integral. Using this procedure we encounter the problem that even though we have chosen <P to be horizontal, d<l> will in general not be horizontal. In order to obtain consistent field equations we therefore need to modify our procedure. Instead of initially restricting ourselves to horizontal p-forms on the tangent bundle T lvf, we let <P be an arbitrary p-form with both horizontal and vertical components. The purely horizontal components can then be obtained by applying the horizontal projector (2.24) In order to reduce the number of physical degrees of freedom to only these hor- izontal components we dynamically impose that the non-horizontal components vanish by introducing a suitable set of Lagrange multipliers /, so that the total action reads (2.25) Variation with respect to the Lagrangc multipliers then yields the constraint that the vertical components of <P vanish. Variation with respect to these vertical components fixes the Lagrange multipliers. Finally, variation with respect lo the horizontal components of <P yields the desired field equations. It can be shown that in the metric limit they reduce to the usual tleld equations derived from the action (1.6) for matter fields on a metric spacetime [Pfeifer & Wohlfarth 2012]. 2.6. Gravity In the previous sections we have considered the geometry of Finsler spacetimes solely as a background geometry for observers, point masses and matter fields. We now turn our focus to the dynamics of Finsler geometry itself. As it is also the case for Lorentzian geometry, we will identify these dynamics with the dy- namics of gravity. For this purpose we need to generalize the Einstein-Hilbcrt action, from which the gravitational field equations are derived, and the energy- momentum tensor, which acts as the source of gravity.
- 20. 32 Manue/ Hohmann We start with a generalization of the Einstein-Hilbert action (1.9) to Finsler spacetimes. As in the case of matter field theories detailed in the preceding sec- tion this generalized action will be an integral not over spacetime !vi, but over the unit tangent bundle .BC TM, since the geometry is defined in terms of the ho- mogeneous fundamental geometry function L on TM. We have already seen that a suitable volume form on .B is given by the volume form Vol0 of the restricted Sasaki metric (2.23). This leaves us with the task of generalizing the Ricci scalar R in terms on Finsler geometry. The most natural and fundamental notion of curvature is defined by the Car- tan non-linear connection (2.7), which we already encountered in the definition ofFinsler geodesics in section 2.3 and which corresponds to the unique split (2.8) of the tangent bundle TTlvf into horizontal and vertical components. This split is also the basic ingredient for the following construction. The curvature of the Cartan non-linear connection measures the non-integrability ofthe horizontal dis- tribution HTlvf, i.e., the failure of the horizontal vector fields ba to be horizontal. In fact, their Lie brackets are vertical vector fields, which are used in the follow- ing definition: Definition 2.12 (Non-linear curvature). The curvature ofthe non-linear connec- tion is the quantity neab which measures the non-integrability of the horizontal distribution induced by the Cartan non-linear connection, (2.26) The simplest scalar one can construct from the curvature coefficients defined by (2.26) is the contraction RaabYb, so that the action for Finsler gravity takes the form 11 a b Sr = - Vol0 - R abY . ~;, B ' (2.27) In the case of a metric-induced Finsler function, in which the non-linear connec- tion coefficients Nab are given by (2.9), the expression under the integral indeed reduces to the Ricci scalar, so that SF is a direct generalization ofthe Einstein- Hilbert action (1.9). In order to obtain a full gravitational theory this action needs to be complemented by a matter action, such as the field theory action (2.25) we encountered in the previous section. This total action then needs to be varied with respect to the mathematical object which fundamentally defines the space- time geometry. On a Finsler spacetime this is the fundamental geometry function L. Consequently, the gravitational field &lquations are not two-tensor equations as Observer dependent geometries in general relativity, but instead the scalar equation [ Fab;- - r- . d .RaabYb .1 !Jaih(R cdil ) - 6 p2- 33 -1- 2/"nb (vaSh -1- SaSb -1- 8a(y"l5cSb- Nc,s,J)] ~)~ = K:Til~ (2.28) on the unit tangent bundle .B. Here T denotes the energy-momentum scalar ob- tained by variation of the matter action SM with respect to the fundamental ge- ometry function L. For the field theory action (2.25) it is given by { nL 15 [ /-:: ( (. _pH)">)]} Tl2;= gnL v-c .c(c,iJ>,diJ>)+>. 1 'J. E (2.29) It can be shown that in the metric limit the resulting gravitational field equa- tion (2.28) is equivalent to the Einstein equations (I. I0), whose free indices are to be contracted with y0 fPfeifer & Wohlfarth 20121. We finally remark that also the Cartan linear connection we used to define generalized Lorentz transformations in section 2.4 defines a notion of curvature, which may in principle be used to generalize the Einstein-Hilbert action. This curvature is defined as follows: Definition 2.13 (Linear curvature). The curvature of the Cartan linear connec- tion is given by (2.30) for vector fields X, Y, Z on T M. Using the action (2.19) of the Cartan linear connection on the vector fields constituting the Berwald basis and the coefficients (2.20) one finds that its curva- ture can be written in the form
- 21. 34 Manuel Hohmann where the coefficients are given by Rdcab = 8,,1"dea - 8aFdcb + peca.Fdeb - pccbpdea + Cdce (obNea - 8aNeb) , (2.32a) prlcab= [),prlea- 8a.Crlcb + peca.Cricb- C"cbprlm+ CdceDbNea, (2.32b) c..,'fi - i) cd f)-:- cd ce cd ce cd c ca.b - Ub ea - a cb + > ea cb - cb -' en · (2.32c) ln the metric limit the coetlicient Rrlca.b reduces to the Riemann tensor, while the remaining coefficients P'1 cab and 8dcab vanish. One may therefore consider the term gF abReacb as another generalization of the Ricci scalar to generate the gravitational dynamics on Finsler spacetimes. We do not pursue this idea further here and only remark that also other choices are possible. 3. The local perspective: Cartan geometry of observer space In the previous section we have seen that on Finsler spacetimes the definitions of observers and observables are promoted from geometrical structures on the spacetime manifold lvf to homogeneous geometrical structures on its tangent bundle T Jvf, and that this homogeneity fixes quantities on T M when they are given on the unit tangent bundle ~- We have also seen that measurements by an observer probe these structures along a lifted world liner = (r, "f) in T M. However, it follows from the definition of physical observer trajectories that ev- ery curve r is entirely confined to future unit timelike vectors, so that obser- vations can be performed only on a smaller subset 0 c ~. which we denote observer space. In this section we will therefore restrict our discussion to ob- server space and equip it with a suitable geometrical structure in terms of Cartan geometry [Cartan 1935, Sharpe 1997], which we derive from the previously de- fined Finsler geometry [Hohmann 2013]. While Cmtan geometry turns out to be useful already as a geometry for spacetime in the context of gravity [Wise 20 I0], it becomes even more interesting as a geometry for observer space [Gielen & Wise 2013] and provides a better insight into the role of Lorentz symmetry in canonical quantum gravity [Gielen & Wise 2012a, 2012b]. 3.1. Definition of observer space We start our discussion with the definition of observer space as the space of all tangent vectors to a Finsler spacetime which are allowed as tangent vectors Observer dependent geometries 35 of normalized observer trajectories, i.e., observer trajectories which are para- metrized by their proper time. This leads us to the definition: Definition 3.1 (Observer space). The observer space 0 of a Finsler spacctime (M, L, F) is the set of all future unit timelike vectors, i.e., the union O=US:r (3.1) xEII of all unit shells inside the forward light cones. Note that 0 is a seven-dimensional submanifold of TM and that its tangent spaces T(x,y)O are spanned by the vectors v E T(x,y)T!vi which satisfy vF =: 0. Further, there exists a canonical projection 1r 1 : 0 -+ NI onto the underlymg spacetime manifold. The natural question arises which geometrical structure the Finsler geometry on the spacetime manifold M induces on its observer space 0. The structure which is most obvious already from our fmdings in the previous section is the restricted Sasaki metric G, which we defined in (2.23) as the re- striction of the full Sasaki metric G to ~ and which we now view as a metric on the smaller set 0 c ~- It follows from the signature of G that Ghas Lorentzian signature(-,+,+,+,+,+,+). Another structure which we already encountered in the previous section is the geodesic spray (2.14). Since it preserves the Finsler function, SF = 0, it is tangent to the level sets ofF, and thus in particular tangent to observer space 0. It therefore restricts to a vector field on 0, which we denote the Reeb vector field: Definition 3.2 (Reeb vector field). The Reeb vectorfield r is the restriction of the geodesic sprayS to 0, (3.2) We now have a metric and a vector field on 0. Combining these two struc- tures we can form the dual one-form a of the Reeb vector field with respect to the restricted Sasaki metric G, which we denote the contact form: Definition 3.3 (Contact form). The contactfonn is the dual one-form of the Reeb vector field r with respect to the restricted Sasaki metric G, - F a bI 1[)7 p2 d a I a= -G(r, .) = gabY d;-r; 0 = 2 a ,:r; 0 . (3.3) Conversely, the Reeb vector field is the unique vector field on 0 which is normalized by a and whose flow preserves a, i.e., which satisfies Lra=O and a(r)=l. (3.4)
- 22. 36 Manue/ Hohmann The naming of a and r originates from the notion of contact geometry. In this context a contact form on a (2n +I)-dimensional manifold is defined as a one- form a, which is maximally non-integrable in the sense that the (2n + 1)-fonn u A du A ... A dais nowhere vanishing, hence defines a volume form, and the Reeb vector field is the unique vector tield r satisfying (3.4). Indeed, it turns out that the volume form defined by o: is simply the volume form of the Sasaki metric GonO. As we have seen in section 2.3the Finsler geometry induces a split (2.8) of the eight-dimensional tangent bundle TTJv! into two four-dimensional subbundles VTM and HTM, denoted the vertical and horizontal subbundles, respectively. A similar split also applies to the tangent bundle TO of observer space. It splits into the three subbundles TO= VO EG HO= VO El) JJO CD H 00, (3.5) which we denote the vertical, spatial and temporal subbundles, respectively. The vertical bundle VO is defined in analogy to the vertical tangent bundle VTM as the kernel of the differential 1r~ of the canonical projection 1r1 : () -r lvf. It is constituted by the tangent spaces to the shells Sx of unit timelike vectors at :1: E Jv! and hence three-dimensional. Its orthogonal complement with respect to the Sasaki metric G is the four-dimensional horizontal bundle HO. One can easily see that the contact form a vanishes on VO. Its kernel on HO defines the three-dimensional spatial bundle HO. Finally, the orthogonal complement of HO in HO is the one-dimensional temporal bundle H 00, which is spanned by the Reeb vector field r. The split of the tangent bundle TO has a clear physical interpretation. Verti- cal vectors in VO correspond to infinitesimal generalized Lorentz boosts, which change the velocity of an observer, but not his position. They are complemented by horizontal vectors in HO, which change the observer's position, but not his direction of motion. These further split into spatial translations in HO and tem- poral translations in H 0 0 with respect to the observer's local frame. This inter- pretation will become clear when we discuss the split of the tangent bundle from a deeper geometric perspective using the language of Cartan geometry. We will give a brief introduction to Cartan geometry in the following section. 3.2. Introduction to Cartan geometry In order to describe the geometry of observer space, we make use of a frame- work originally developed by Cartan under the name 'method of moving frames' Observer dependent geometries 37 [Cartan 1935]. His description of the geometry of a manifold M is based on a comparison to the geometry of a suitable model space. The latter is taken to be a homogeneous space, i.e., the coset space GIH of a Lie group G and a closed subgroup H c G. Homogeneous spaces were extensively studied in Klein's Erlangen program and are hence also known as Klein geometries. Car- tan's construction makes use of the fact that they carry the structure of a principal If-bundle 1r : G -> GIIf and a connection given by the Maurer-Cartan one-f01m A E D1 (G,g) on G taking values in the Lie algebra g of C. Using these struc- tures in order to describe the local geometry of Jv!, a Cartan geometry is defined as follows: Definition 3.4 (Cartan geometry). Let G be a Lie group and H c G a closed subgroup of G. A Cartan geometry modeled on the homogeneous space GIH is a principal H-bundle 1r : P -r Jv! together with a g-valued one-form A E D1 (P, g), called the Carum connection on P, such that C I. For each p E P, A11 : T11P -r g is a linear isomorphism. C2. A is H-equivariant: (Rh)* A= Ad(h-1 ) o A :lh EH. C3. A restricts to the Maurer-Cartan form on vertical vectors v E ker 1r*. Instead of describing the Cartan geometry in terms of the Cartan connection A, which is equivalent to specifying a linear isomorphism Ap : T11P -r g for all p E P due to condition C I, we can use the inverse maps A11 = A;;- 1 : g -r T11P. For each a E g they define a section A(a) of the tangent bundle, which we denote a fundamental vector field: Definition 3.5 (Fundamental vector fields). Let (1r : P -r M, A) be a Cartan geometry modeled on GIH. For each a E g the fundamental vectorfield A(a) is the unique vector field such that A(A(a)) =a. We can therefore equivalently define a Cartan geometry in terms of its funda- mental vector fields, due to the following proposition: Proposition 3.1. Let (1r : P -r M, A) be a Cartan geometry modeled on GIfl and A : g -r Vect Pits fundamental vector fields. Then the properties Cl to C3 of A are respectively equivalent to the following properties ofA: Cl'. For each p E P, Ap : g --7 T11P is a linear isomorphism. C2'. A is H-equivariant: Rh* o A= A o Ad(h-1 ) :lh EH. C3'. A restricts to the canonical vector.fzelds on [J.
- 23. 38 Manuel Hohmann We illustrate these definitions using a physically motivated example. Let 7r : P -t lvf be the oriented, time-oriented, orthonormal frame bundle of a Lorentzian manifold (M, g). It carries the structure of a principal H -bundle, where H = SOo(3, 1) is the proper orthochronous Lorentz group. The homoge- neous space GIH can be any of the maximally symmetric de Sitter, Minkowski or anti-de Sitter spacetimes, which is achieved by choosing the group G to be { SOo(4, 1) for A> 0 9 de Sitter spacetime, G = ISOo(:l, 1) for A= 0 9 Minkowski spacetime, SOo(3, 2) for A < 0 9 anti-de Sitter spacetime, (3.6) where ISOo(3, 1) = SOo(:~, 1) 1>< JR3•1 is the proper orthochronous Poincare group and the subscript 0 indicates the connected component of the correspond- ing group. Here A denotes the cosmological constant on the respective maximally symmetric spacetime and does not necessarily agree with the physical cosmolog- ical constant. We further need to equip the frame bundle 7r : P -t lvf with a Cartan connec- tion. For this purpose we introduce a component notation for elements of the Lie algebra g = Lie G and its subalgebras. First observe that g splits into irreducible subrepresentations of the adjoint representation of H c G, (3.7) These subspaces correspond to infinitesimal Lorentz transforms b = Lie H and infinitesimal translations 3 ~ fllfl of the homogeneous spacetimes GIH. We can use this split to uniquely decompose any algebra element a E g in the form 1 . . . a= h + z = '2h'/H/ + z''Zi, (3.8) where 1-l;J are the generators of b = .so(3, 1) and Z; are the generators of trans- lations on GIH. They satisfy the algebra relations (3.9) The last expression explicitly depends on the choice of the group G, which can conveniently be expressed using the sign of the cosmological constant A. Observer dependent geometries 39 We can now apply this component notation to the Cartan connection A. We first split A = w + c into a [J-valued part w and a 3-valued part c. The latter we set equal to the solder form, which in component notation can be written as (3.1 0) where the coordinates (.fj") on the fibers of P are defined as the components of the frames .f; in the coordinate basis of the manifold coordinates (:r;a), and f-1 ;, denote the corresponding inverse frame components. For the [J-valued part w we choose the Levi-Civita connection. Given a curve T H (x(T), f (T)) on P it measures the covariant derivative of the frame vectors fi along the projected curve T H :r(T) on M. For a tangent vector v E T P this yields (3.11) Using the same component notation as above it reads .i. = f-l.id'fa + f-l.i {ITa dxc W ,_ a 1 . . a. 1 be ' (3.12) where rabc denotes the Christoffel symbols. It is not difficult to check that the g- valued one-form .4 defined above indeed satisfies conditions C I to C3 of a Cartan connection, and thus defines a Cartan geometry modeled on GIH. Equivalently, we can describe the Cartan geometry in terms of the fundamental vector fields. Using the notation (3.8) they take the form A( ) 1i faiSj i fa("' fbrc iSj) _ a = ! j i Ua +Z . i ua - j abUc , where we have introduced the notation for tangent vectors to the frame bundle P. A well-known result of Cartan geom- etry states that the metric g can be reconstructed from the Cartan connection, up to a global scale factor. We finally remark that the Cartan geometry provides a split of the tangent bundle T P which has a similar physical interpretation as the split (3.5) of TO. This split is induced by the decomposition (3.7) of the Lie algebra g, which is carried over to the tangent spaces TpP by the isomorphic mappings Ap as shown
- 24. 40 fv!anuel Hohmann in the following diagram: vpP (f} HpP TpP wr + ef Ar b m 3 jJ (3.13) The vertical subbundle V P is constituted by the tangent spaces to the fibers of the bundle if : P ---+ A1, which are given by the kernel of the differential if* of the canonical projection. This is a direct consequence of condition C3 on the Cartan connection. The elements of V P can be viewed as infinitesimal local Lorentz transformations, which change only the local frame f and leave the base point x unchanged. Conversely, the elements of the horizontal subbundle Hp correspond to infinitesimal translations, which change the base point x without changing the orientation of the local frame f. This follows from the fact that we constructed the f)-valued part w of the Cartan connection from the Levi-Civita connection. 3.3. Cartan geometry of observer space We will now employ Cartan geometry in order to describe the geometry of ob- server space. Hereby we will proceed in analogy to the metric spacetime example discussed in the previous section, where we constructed a Cartan connection on the orthonormal frame bundle. For this purpose we refer to the definition of or- thonormal observer frames in section 2.4. If we translate this definition to the context of observer space geometry, we find that an observer frame at (x, y) E 0 is a basis of the horizontal tangent space H(x,y)O such that w~fo = y and the normalization (2.18) holds. Equivalently, we can make use of the differentialw~ of the canonical projection w' : 0 ---+ M, which isomorphically maps H(x,y)O to Tx!VI, and regard frames as bases of Txllif, in analogy to the case of metric geometry. Here we choose the latter and define: Definition 3.6 (Observer frames). The space P of observer frames of a Finsler spacetime (A1, L, F) with observer space 0 is the space of all oriented, time- oriented tangent space bases f of llif, such that the basis vector fo lies in 0 and the frame is orthonormal with respect to the Finsler metric, One can now easily see that although there exists a canonical projection if : P---+ NI, which assigns to an observer frame its base point on M, it does not Observer dependent geometries 41 in general define a principal 1I-bundle, where H is the Lorentz group as in the preceding section. This follows from the fact that the generalized Lorentz trans- forms discussed in section 2.4 do not form a group, but only a grupoid. However, this is not an obstruction, as it is our aim to construct a Cartan geometry on 0 and not on M. Indeed, the projection 1r : P ---+ 0, which simply discards the spatial frame components, carries the structure of a principal J(-bundle, where by J( we denote the rotation group S0(:3). It acts on P by rotating the spatial frame components. The Cartan geometry on observer space will thus be modeled on the homogeneous space GIf( instead of GIH. We further need to equip 1r : P ---+ 0 with a Cartan connection which gener- alizes the Cartan connection on the metric frame bundle displayed in the previous section. Here we can proceed in full analogy and choose as the 3-valued parte of the connection the solder form. The expression in component notation, (3.14) agrees with the analogous expression (3.1 0) in metric geometry. For the f)-valued part w we generalize the Levi-Civita connection (3.12). Recall from section 2.4 that the tangent space T NI of a Finsler spacetime, and hence also its observer space 0 c T M, is equipped with the Cartan linear connection (2.19). We can therefore replace the projection if to !VI in (3.11) with the projection 1r to 0 and define (3.15) where l now denotes the Cartan linear connection. In component notation this yields the expression i f-ljdr+"a f·-lj fb [Fa d c ea (Ne l .d + J'l'c.)] w· 'i = a J i + . a. i be X + be d(,~L 1J 0 1 (>kd jk ) j·-ll df" 1 jkj·bfc(s: F s: F)d a = 2 o; uz - 7) TJil , a . k + 27) i . k Ub.flac - Oc9ab ;r; ' (3.16) where the coefficients cabc and Fabc are the coefficients for the Cartan linear connection (2.20). From the Cartan connection (3.14) and (3.16) we then find the fundamental vector fields A(h) = (h;Jff- hioff fjCabc) &z, A(z) = ziff (oa- f]Fc"J)1) (3.17a) (3.17b)
- 25. 42 Manuel Hohmann for hE ~and z E J. One easily checks that indeed A11 = A;;-1 for all p E P, so that condition C I is satisfied. Another simple calculation shows that also condi- tions C2 and C3 are satisfied, so that A defines a Cartan geometry. The Cartan geometry on the observer frame bundle 1r : P -+ 0 induces a split of the tangent bundle T Pin analogy the split (3.13) we observed for the Cartan geometry of a metric spacetime. Since the observer space Cartan geometry is modeled on GIK instead of GIH we first decompose the Lie algebra g into irreducible subrepresentations of the adjoint representation of K c G, (3.18) The subspaces we encounter here are the rotation algebra£ = Lie K, the rotation- free Lorentz boosts IJ ~ ~I£, as well as the spatial and temporal translations J = 3ED 3° of the homogeneous spacetimes. We can decompose the Cartan connection accordingly and obtain the following split of the tangent spaces TpP: (3. 19) g The elements of these subbundles correspond to infinitesimal rotations of ob- server frames in RP, infinitesimal rotation-free Lorentz boosts in BP as well as translations along the spatial and temporal frame directions in fj p and HoP, respectively. For convenience we introduce a component notation for the algebra- valued one-forms DE D1 (P, £),bE D1 (P, IJ), eE D1 (P,3) and eo E Dl(P,Jo) in the form (3.20) where Ra, Ln, Za, Zo are the generators of rotations, Lorentz boosts as well as spatial and temporal translations. The ten components na, ba, e'-', eo are ordinary one-forms on P. Note that for each p E P they are linearly independent and thus constitute a basis ofT;P. In a similar fashion we will write the fundamental vector fields A in the decomposed form where the ten components D.n,12w~w~o are ordinary vector fields on P. They constitute bases of the tangent spaces TpP which respect the split into the re- Observer dependent geometries 43 ----- spective subspaces R11P, BpP, ilpP, I-I2P and are dual to the aforementioned cotangent space bases. Recall from section 3.1 that the tangent bundle TO of observer space features a split (3.5) into Lorentz boosts and spatial and temporal translations which is similar to the split (3.19). In fact these two splits arc closely related. For each frame p E P the differential n, of the bundle projection isomorphically maps th~ subspaces of TpP, except the kernel RpP, to the corresponding subspaces of T7C(Pp' as shown in the following diagram: RpP Efl :r Efl :r (j) H 0 P 7C, 1 7C,t 0 v7C(vP Efl ilTI(pp fl) H~(pJo We see that we obtain the split of TO, which we previously derived directly from Finsler geometry, also by using Cartan geometry. This observation brings us to the question of whether the observer space Cartan geometry also yields us the geometric structures on observer space we defined in section 3.1 - the Sasaki. metric, the contact form and the Reeb vector field.. In order to relate geometric objects on () to the Cartan connection A and the fundamental vector fields A on P, one naturally makes use of the bundle pro- jection 1r : P -+ 0. Its pushforward 1r* maps tangent vectors on P to tangent vectors on 0, as displayed also in diagram (3.22). However, since 1r is not injec- tive, and thus fails to be a diffeomorphism, it does not allow us to carry vector fields or differential forms from P to 0. We therefore need to enhance the re- lation between these spaces with a section s : 0 -+ P. It allows us evaluate the fundamental vector fields A(a) for a E g on the image of s and apply the differentialn., which yields us vector fields .fi(a) = rr. o A(a) os on 0. Note that these depend on the choice of the section .s. Using the component notation (3.21) we can define component vector fields on 0 by It follows from (3.22) that ne> vanishes, since the vector fields D_C> lie inside the ro- tation subbundle RP and thus in the kernel of n•. Further we find that the remain- ing vector fields f2a, ~a' ~0 constitute bases of the subspaces VaO, HaG, sgo of
- 26. 44 Manuel Hohmann ToO for each o E 0. This shows that the fundamental vector fields 1L evaluated at o isomorphically map the vector space 1J EB 3EB 3° to T0 0 while respecting the split into subspaces. The inverse maps A0 = J~ 1 therefore constitute a one-form 1 -/-a[. -az -Oz n 1(0 - 0) • =1 a+e a+e OEH ,!Jffi3ED3, whose components are the pullbacks of the components b", e'", e0 on the image of the section s. Since the one-form Aand fundamental vector fields Adefined above depend on the choice of the section, we now pose the question how they are related if we choose different sections s and s'. Recall that 1r : P -+ 0 is a principal J(- bundle, so that any two sections are related by a local gauge transform, i.e., by a function k : 0 -+ K. Under this gauge transform the fundamental vector fields transform as using the irreducible subrepresentations of the adjoint representation of J( on g. Similarly, the one-forms transform as Since the adjoint representation of J( acts trivially on the subspace 3° it immedi- ately follows that the component fields e0 and fo are independent of the choice of the sections. From the expressions (3.16) and (3.14) of the Cartan connection and the fundamental vector fields (3.17) in terms of FinsIer geometry we see that these are simply the contact form (3.3) and the Reeb vector field (3.2), We have thus expressed these structures on 0 in terms of the Cartan connection on P. It further turns out that the Sasaki metric takes the form (3.24) and is thus also expressed in terms of the Cartan connection. Note that also this is invariant under changes of the section, which act as a local rotation of the component fields. The same applies to its volume form (3.25) Observer dependent geometrie,s_____________ 45 In the following sections we will make use of these structures are their expres- sions in terms of Cartan geometry in order to provide definitions for observers and observations in analogy to those given in section 2 using Finsler geometry. 3.4. Observers and observations We now come to the description of observers and their measurements in the lan- guage of Cartan geometry on observer space. In the following we will discuss which curves on observer space correspond to the trajectories of physical ob- servers. In particular we will define the notion of inertial observers using ele- ments of Cartan geometry. In section 2.2 we have discussed the notion of physical observers on a Finsler spacetime. We have defined the trajectories of physical observers as those curves 7 H I(T) on a Finsler spacetime, whose tangent vectors 'Y(T) in arc length parametrization lie in the future unit timelike shell S"f(r) c T"f(r)M. If we lift these curves canonically to curves T H ('-y(T),'Y(T)) on TM, we thus see that they are entirely contained in observer space 0 C T NI. This leads to a very simple definition of physical trajectories on observer space: Definition 3.7 (Observer trajectory). A physical observer trajectory is a curve r on observer space which is the canonical lift r = (1, 'Y) of an observer world line 1 on the underlying Finsler spacetime. We will now rewrite this condition in terms of Cartan geometry. First observe that canonical lifts in 0 are exactly those curves r such that the tangent vector of the projected curve 1r 1 or in Af reproduces r, One can easily see that this condition does not restrict the vertical components of i'(T), which lie inside the kernel VO of 1r~ according to the split (3.5), and fully determines itS horizontal COmponents as a function of the position r(T) in observer space. It therefore defines a horizontal vector field h on 0, i.e. a sec- tion h : 0 -+ HO of the horizontal tangent bundle which has the property that 1r~ o h : 0 -+ T M is the identity on 0. The unique vector field which satisfies this condition is the Reeb vector field r = fo defined in (3.2). Hence, observer trajectories are those curves r on 0 whose horizontal tangent vector compo- nents are given by the Reeb vector field. We can further rewrite this condition by
- 27. 46 Manuel Hohmann introducing the projectors onto the subbundles of TO and obtain the form PJJf(T) = r(r(T)). Finally, inserting the explicit formulas for PH and r we arrive at the reformulated defini- tion: Definition 3.8 (Observer trajectory). A physical observer trajectory is a curve r on observer space whose horizontal components are given by the Reeb vector field, i.e., which satisfies A particular class of observers is given by inertial observers, whose trajecto- ries follow those of freely falling test masses. In section 2.3 we have seen that these are given by Finsler geodesics, or equivalently by curves whose complete lift (1, '"') in TM is an integral curve of the geodesic spray (2.14). We have further seen that the geodesic spray is tangent to observer space () c T A-1 and defined the Reeb vector field r as its restriction to observer space. It thus im- mediately follows that inertial observer trajectories on 0 are simply the integral curves of the Reeb vector field. Comparing this finding with the aforementioned definition we see that inertial observer trajectories arc exactly those observer tra- jectories whose vertical tangent vector components vanish. We thus define, using only Cartan geometry: Definition 3.9 (Inertial observer trajectory). An inertial observer trajectory is an integral curve of the Reeb vector field, i.e., a curve r on observer space which satisfies It appears now straightforward to translate the notions of observables and ob- servations from Finsler geometry to Cartan geometry on observer space. A direct translation yields observables as sections of a horizontal tensor bundle, which is constructed from the horizontal subbundle HO in analogy to the horizontal ten- sor bundle Hr,srNI. Observations by an observer at r(T) E 0 then translate into measurements of the components of a horizontal tensor field with respect to a basis of the corresponding horizontal tangent space Hl'(r)O, which can con- veniently be expressed using the vector fields g_;. Finally, also a translation of the matter action (2.25), where 1> is viewed as a one-form on 0 and the projec- tors (3.26) are used, is straightforward. However, we do not pursue this topic Observer dependent geometries 47 here. Instead we will directly move on to the gravitational dynamics in the next section. 3.5. Gravity As we have already done in the case of Finsler geometry in section 2.6, we now focus on the dynamics of the Cartan geometry, which we identify with the grav- itational dynamics. Since gravity is conventionally related to the curvature of spacetime, we will first discuss the notion of curvature in Cartan geometry. We will then derive dynamics for Cartan geometry from an action principle and see how this notion of curvature is involved. For this purpose we will consider two different actions, the first being the Finsler gravity action we encountered before and which we now translate into Cartan language, and an action which is explic- itly constructed in terms of Cartan geometric objects. We start our discussion of curvature in Cartan geometry with its textbook definition: Definition 3.10 (Cartan curvature). The curvature of a Cm·tan geometry (if : p -+ M, A) modeled on the homogeneous space GIHis the g-valued two-form FE n2 (P, g) on P given by (3.27) The curvature has a simple interpretation in terms of the fundamental vector fields £1: it measures the failure of A : g -+ Vect P to be a Lie algebra homo- morphism. This can be seen from the relation A([a, a'])- [ll(a),A(a')] = A(F(A(a),A(a'))), (3.28) which can easily be derived from the definition (3.27) by making use of the stan- dard formula dO"(.X, Y) = X(O"(Y))- Y(O"(X))- O"([X, Y]) for any one-form O" and vector fields X, Y. From this general definition we now turn our focus to the Cartan geometry on observer space modeled on G1K, which we derived from Finsler geometry in section 3.3. In this context the term [A(a), A(a')] for a, a' E 3 in the rela~ tion (3.28) reminds of the Lie bracket of horizontal vector fields [8a, 6b] in the
- 28. 48 Manue/ Hohmann definition of the non-linear curvature Reab on TTNI. Indeed, the similar expres- sion on P given by [ ] - j'bjcf·d(' pa s: F" +pc F" pe F" )<Sk f;, f!.j - i j. k Oc bd- Ub cd bd ce- cd be Oa (3.29) reproduces the components of the non-linear curvature (2.26), which can equiva- lently be written in the form Ra _ 'lJd(s: F" (" p,a , pe pa }i'e pa ) " be - . Uc bd - lb · cd T bd ce - cd ' be · (3.30) We can directly apply this result to the Finsler gravity action (2.27) on the unit tangent bundle I; C T NI. Since observer space is simply the connected com- ponent of the unit tangent bundle constituted by the future timelike vectors, it is straightforward to consider the restricted action (3.31) as a gravity action on 0. This action is still written in terms of Finsler geometric objects, which we will now rewrite in terms of Cartan geometry. For the volume form Vol0 of the Sasaki metric Gwe have already found the expression (3.25), while for the non-linear curvature coefficients R"bc we can make use of the Lie bracket (3.29) of horizontal vector fields on P together with the relation (3.30). In order to reproduce the scalar quantity R"abYb in the Finsler gravity action from this vector field we further apply the boost component b(J! of the Cartan connection and contract appropriately, which yields The last equality follows from the identification of the tangent vector y" with the temporal frame component jg. Note that this expression is a scalar on P which is constant along the fibers of 1r : P ---+ 0, and can thus be viewed as a scalar on 0. We thus finally obtain the gravitational action (3.32) which is now fully expressed in terms of Cartan geometry. Another possible strategy to obtain gravitational dynamics on the observer space Cartan geometry is to start from general relativity, rewrite the Einstein- Hilbert action in terms of the Cartan connection derived from the metric ge- Observer dependent geometries 49 ometry displayed in section 3.2, and finally transform the action to an integral over observer space by introducing an appropriate volume form on the fibers of 1r' : 0---+ lvf. We will follow this procedure for the remainder of this section. The starting point of this derivation is the action given by MacDowell and Mansouri [1977!. In terms of spacetime Cartan geometry it takes the form !Wise 201 0] (3.33) Here r..:h is a non-degenerate inner product on ~. For simplicity we choose where trn is the Killing form on ~ and* denotes a Hodge star operator. In com- ponents we can write the Killing form as and the Hodge star operator as ( ) i im ln 1k *h j = 7J 7J Emjkl I. n · The two-form Ph is given hy the unique decomposition F =Fry+ F;, of the g-valued Cartan curvature (3.27) into parts with values in ~ and 3· Finally, the tilde indicates that we need to lower this two-form Fn on the frame bundle P to a two-form Ph on the base manifold M. We now aim to lift the action (3.33) to observer space. For this purpose we need to find a suitable volume form on the fibers of 7r 1 : 0 ---+ lvf. Recall from definition (3.1) of observer space that these are given hy the future unit timelike shells Sx for x E M, which are three-dimensional submanifolds of T lvi. A natural metric on Sx is thus given by the restriction of the Sasaki metric G on T ld, or equivalently Gon 0, to Sx. Using our results from section 3.3 on the Cartan geometry of observer space we find that the tangent spaces to Sx are spanned by the vertical vector fields ~"', so that the Sasaki metric (3.24) restricts to the Euc!idean metric Ocv.ri.P· Q9 //3. Its volume form is given by (3.34)
- 29. 50 Manuel Hohmann In combination with the action (3.33) lifted to observer space, which means that P~ is now regarded as a two-form on 0, this yields the action SMM = r r,;i)(Fr) A Fr)) A Vols. .fo (3.35) In order to analyze the terms in this action we make use of the algebra rela- tions (3.9) to decompose F1 J in the form 1 1 1 FrJ = dw + 2[w, w] + 2[c, e] = Fw + 2 [e, e] into the curvature Fw of wand a purely algebraic term ~[e, e]. Using the expres- sions (3.14) fore and (3.16) for w these take the form . 1 . ( J . _ _ _ .-!] c d .a. ,.b d .a ·b •d a b Fw "- 2.f d.fi R caMh A dx + 2P cabdx A oj0 + S cabo.f0 A <'5f0 ) , [e e]i - 2[-liJ-lk ., Ad .a Ad b -, ' - . a b7Jik sgn . x ,, x , where we have introduced the shorthand notation of(f = dfff + Nabdxb. The coefficients Rdcab• pdcab and Sdcab we find here are the coefficients (2.32) of the curvature of the Cartan linear connection, which is not surprising, since we used the Cartan linear connection in the definition (3.15) of w. The term [e, e] depends on the choice of the group G, and thus on the sign of the cosmological constant on the underlying homogeneous space. Applying this decomposition to the expression r,;IJ (FIJ A FIJ) in the action (3.35) we obtain the following terms: • A cosmological constant term: • A curvature term: • A Gauss-Bonnet term: The ellipsis in the expressions above indicates that we have omitted terms which are not horizontal, i.e., which contain the vertical one-form b. These terms do not Observer dependent geometries 51 contribute to the total action since their wedge product with the vertical volume form (3.34) vanishes. Note the appearance of the common term . . ' l Eijkte" A c1 A c" A c , which, when lowered to a four-form on 0, combines with the vertical volume form (3.34) to the volume form (3.25) of the restricted Sasaki metric. The total action thus takes the final form l (1 A t•' ab c 1 R R a.bcf cdgh _ ( , A)2) SMM = Vol0 i sgn g R a.cb- 96 a.bcd ·efghE E sgn . · ' 0 ( (3.37) From this we see that we obtain an action based on the curvature of the Cartan linear connection, as we have briefly discussed towards the end of section 2.6, provided that we have chosen a model space G/ H for which A "I 0. We also find that we always obtain a non-zero cosmological constant term. The magnitude of the physical cosmological constant can be adjusted by introducing suitable nu- merical factors into the algebra relations (3.9), which corresponds to a rcscaling of the basis vectors Zi. 3.6. The role of spacetime In the previous sections we have discussed the physics on Finsler spacetimes in the language of Cartan geometry. For this purpose we considered a principal K-bundle 7f : P -+ 0 over observer space 0 and equipped it with a Cmtan connection A derived from Finsler geometry. This construction allowed us to reformulate significant aspects of Finsler spacetime in purely Cartan geometric terms: the definition of physical and inertial observers, the split of the tangent bundle TO into horizontal and vertical components which crucially enters the definition of observables and physical fields, the Sasaki metric and its volume measure on 0 and finally the dynamics of gravity. It should be remarked that these formulations can be applied to any Cartan geometry (7f : P -+ 0, A) modeled on GjK, since they do not explicitly refer to the underlying Finsler geometry, or even the spacetime manifold M. This observation stipulates the question whether an underlying spacetime geometry is at all required, or may not even exist, at least as a fundamental object. In this final section we will discuss this question. We first discuss whether and how we can reconstruct the Finsler spacetime (M, L, F) if we are given only its observer space Cat-tan geometry (1r : P -+ 0, A), together with the presumption that an underlying Finsler spacetime exists.
- 30. 52 Manuel Hohmann Recall from its definition (3.1) that the observer space 0 of a Finsler spacetime is the (disjoint) union of the future unit timelike shells Sx for all spacetime points x E 11/1. Every spacetime point :1: thus corresponds to a non-empty subset S.'" of 0. Reconstructing the spacetime manifold from its observer space therefore amounts to specifying an equivalence relation which decomposes 0 into subsets, and to equipping the resulting set of equivalence classes with the structure of a differentiable manifold. This can be done by making use of the vertical distri- bution VO, which is tangent to the shells S"' and can be expressed completely in terms of Cartan geometry as the span of the vector fields La defined in (3.23). From our presumption that an underlying spacetime manifold exists it follows that VO is integrable. The Frobenius theorem then guarantees that VO can be integrated to a foliation of 0, with projection 1r' : 0 -t M onto its leaf space, and further that lv1 carries the structure of a differentiable manifold so that 1r' becomes a smooth submersion. The aforementioned procedure allows us to reconstruct the spacetime mani- fold 11/1 from observer space Cartan geometry. If w~ now aim to reconstruct also its Finsler geometry on T M, we immediately see that this will be possible at most for vectors which lie inside the forward light cones Cx. This comes from the fact that in the construction of the Cartan geometry on 0 we used only the Finsler geometry on the shells Sx, which yields the Finsler geometry on Cx by rescaling and using its homogeneity properties. This means that we cannot reconstruct the Finsler geometry on spacelike or lightlike vectors, and in particular we cannot reconstruct the null structure of a Finsler spacetime. In order to reconstruct the Finsler function F' on the future light cones we need to reconstruct the embedding a- : () -t T 11/1 of observer space into the tangent bundle of the spacetime manifold M. For this purpose we make use of tbe properties of observer trajectories. Recall that in section 2.3 we applied the canonical lift (2.13) to a curve "! on 11/1 in order to obtain a curve r on T lv1, and concluded that the canonical lifts of observer trajectories on 0 are exactly those curves r whose horizontal tangent vector components are given by the Reeb vector field (3.2) in section 3.4. We can therefore proceed as follows. For 0 E 0 we choose an observer trajectory r in 0 so that r(T) = 0. We then project r to a curve "f on J11 using the projection 1r 1 . The tangent vector 1'(T ), which we identify with o via the embedding a-, is then related to f(T) via the differential 1r~. This relation yields the formula Observer dependent geometries 53 where we have used the fact that 1r: isomorphically maps the horizontal tangent space Hqr)O to Ty(r)M. The embedding a- is thus simply given by Finally, we obtain the Finsler function on timelike vectors by imposing F' = 1 on the image a-(0) c T M and the homogeneity (2.3). Note that L can be any homogeneous function L = pn here, since F is smooth when restricted to the timelike vectors. We now turn our focus to a general Cartan geometry (7f : P -t 0, A) mocl- eled on G/](for which we do not presume the existence of an underlying Finsler geometry or even a spacetime manifold. Indeed, the latter will not in general exist, as we can already deduce from the reconstruction of a Finsler spacetime detailed above. There we have seen that spacetime naturally appears as the leaf space of a foliation, which we obtained by integrating the vertical distribution VO on observer space. This procedure fails if VO is non-integrable. Further, even if VO integrates to a foliation of 0, this foliation may not be strictly simple, i.e., its leaf space may not carry the structure of a differentiable manifold. This means that only a limited class of observer space Cartan geometries, including those derived from Finsler spacetimes, admit for an underlying spacetime mani- fold. Further, even if a spacetime exists, it may not be a Finsler spacetime, since the reconstructed metric (3.24) may not be the Sasaki metric induced by Finsler geometry. The question arises whether we can still assign a meaningful physical in- terpretation to an observer space Cartan geometry if its vertical distribution is non-integrable, so that there is no underlying spacetime. Since any physical in- terpretation should be given based on the measurement of dynamical, physical quantities by observers, this amounts to the question whether these can meaning- fully be defined on an arbitrary observer space Cartan geometry. We have pro- vided these definitions throughout our discussion of observer space in section 3 of this work. Our findings suggest that the notion of spacetime is not needed as a fundamental ingredient in the definition of physical observations, but rather appears as a derived object for a restricted class of Cartan geometries. Acknowledgments The author is happy to thank Steffen Gielen, Christian Pfeifer and Derek Wise for their helpful comments and discussions. He gratefully acknowledges the full financial support of the Estonian Research Council through the Postdoctoral Re- search GrantERMOS115.
- 31. 54 Manuel Hohmann References Ambj0rn, .1., Jurkiewicz, J. & Loll, R. 200 I. "Dynamically triangulating Lorentzian quantum gravity." Nuclear Physics B 6IO(l):347-382. Ambj!')rn, J., .Jurkiewicz, J. & Loll, R. 2005. "Reconstructing the Universe." Physical Review D 72(6):064014. Ambj0rn, J. & Loll, R. 1998. "Non-pcrturbative Lorcntzian quantum gravity, causality and topol- ogy change." Nuclear Physics B 536( I):407-434. Anderson, J.D., Campbell, J.K., Ekelund, J.E., Ellis, J. & Jordan, J.F. 2008. "Anomalous orhital-energy changes observed during spacecraft llybys of Earth." Physical Review Letters I00(9):091102. Ashtekar, A. 1987. "New Hamiltonian formulation of general relativity." Physical Review D 36(6): 1587. Baez, J.C. I998. "Spin foam models." Classical and Quantum Gravity 15(7):1827. Bao, D., Chcrn, S.-S. & Shcn, Z. 2000. An lntmduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics. Springer. Barrctt, J.W. & Crane, L. 1998. "Relativistic spin networks and quantum gravity." Journal of Mathematical Physics 39(6):3296-3302. Brown, J.D. & Kuchaf, K.Y. 1995. "Dust as a standard of space and time in canonical quantum gravity." Physical Review [) 51 (I 0):5600. Bucataru, I. & Miron, R. 2007. Finsler-L.agrange Geometry: Applications to Dynamical Systems. Editura Academiei Romane Bucurcsti. Cartan, E. 1935. La m.ethode de repere mobile, la theorie des groupes continus, et les espaces generalises. Actualites scientifiques et industrielles. Hermann. Gielen, S. & Wise, D.K. 2012a. "Linking covariant and canonical general relativity via local observers." General Relativity and Gravitation 44( 12):31 03-3109. Gielen, S. & Wise, D.K. 20 12b. "Spontaneously broken Lorentz symmetry for Hamiltonian grav- ity." Physical Review [) 85( I0): 104013. Gielen, S. & Wise, D.K. 2013. "Lifting general relativity to observer space." Journal ofMathe- matical Physics 54(5):05250J. Hohmann, M. 2013. "Extensions of Lorentzian spacctime geometry: from Finsler to Cartan and vice versa." Physical Review[) 87(12): J24034. Horvath, J.l. 1950. "A geometrical model for the unified theory of physical fields." Physical Review 80(5):901. Jacobson, T. & Mattingly, D. 200 l. "Gravity with a dynamical preferred frame." Physical Review D 64(2):024028. MacDowell, S.W. & Mansouri, F. 1977. "Unified geometric theory of gravity and supergravily." Physical Review Letters 38(14):739. Pfeifcr, C. 2013. The Finsler Spacetime Framework: Backgrounds for Physics Beyond Metric Geometry. PhD thesis. Hamburg University. Pfeifer, C. & Wohlfarth, M.N.R. 20 I I. "Causal structure and electrodynamics on Finsler space- times." Physical Review D 84(4):044039. Observer dependent geometries 55 · · & w hit' tl M N I' 2012 "Finsler geometric ·extension of Einstein gravity." Physical Pletfer, C. o ar 1• · · '· · Review[) 85(6):064009. .. · , s & s 1 11, 1o p 20 I 1 "Geometrv of physical dispersion relations." Physical Ratzel, D., Rtvcra, . cm t:r, . · · - Review D 83(4):044047. · · I' 1 • llttm gravitv" Physi- . M p & R ·11' C 1997 "Sum over >urlaces tonn o oop quat -· Retsenberger, . . ove 1, . • cat Review D 56(6):3490. Rovelli, C. & Smolin, L. 1995. "Spin networks and quantum gravity." Physical Review D 52( I0):5743. Sh , , R w !997 Di"erential Geonzetl)'. Vol. 166 of Graduate Texts in Mathematics. Springcr- ttrpc, . . - · JJ Ycrlag. G • lJ> 1 ( 'ty Cambridge Monographs on Thicmann, T. 2007. Modem Canonical Quantum enera ,ea tl'l · Mathematical Physics. Cambridge University Press. ' .. . s 1 7002 "(Non) commutative Finslcr geometry from String/M-theory." Preprint VdCdtll, ~. . ..... . arXiv:hep-th/0211068. . . d d· .l th. ·ies in physics· new methods 111 Vacaru, S.l. 2007. "Finsler-Lagrange geometnes an stan att cot · '· Einstein and string gravity." ?reprint arXiv:gr-qc/0707.1524 · , · · d ··ctives in modern cosmolo<>y. Yaearu, S.l. 2012. "Principles of Einstein-Fmsier gravtty an perspc , " [nternational.Tournal ofModern Physics /J 21(09). · d c try" Classical and Quantum Wise, D.K. 2010. "MacDowell-Mansouri gravtty an artan gcome · · Gravity 27(15): 1550 I0.
- 32. KrzysztofOrachal & Wieslaw Sasin Warsaw University ofTechnology Copernicus Center for Interdisciplinary Studies, Krak6w Classification of classical singularities: a differential spaces approach T HE aim of this work is to rethink the concept of Schmidt's b-boundary of a spacetime in a more general framework. The original Schmidt's work is done in the classical language of differential geometry. But, first of all, a mani- fold (which is the mathematical formalisation of a spacetime) can be generalised. One of the known generalisations is a differential space in the sense of Sikorski. Secondly, differential geometry can be 'algebraised'. Then one works with tech- niques of commutative algebra, for example in the spirit of the Nestruev group. 1. Motivation It is known that the laws of General Relativity contain in their deepest core an inevitable break down [Hawking & Ellis 19731. In this way a 'singularity' occurs in the spacetime. Sometime it is said that a boundary is appended to the space- time. However, even the notion of 'singular point' is not unique. As a result, there exist various attempts towards this problem. Starting from the theory of Sikorski's differential spaces, Helier and Sasin (together with their collaborators) elaborated an interesting framework for study- ing spacetime singularities !Helier, Sasin, Trafny & Zekanowski 1992, Helier & Sasin 1994, !996, 1999, 2002, Helier, Odrzyg6zd:i., Pysiak & Sasin 2003]. Further, they developed a model based on the concept of groupoid and ideas of non-commutative geometry [Helier et al. 2003]. In this paper we show that this framework can be valuably enriched by some algebro-geometric methods ala Nestruev [2002].
- 33. 58 KrzysztofDrachal & Wieslaw Sasin 2. Fundamental concepts This part is devoted to the basics of differential spaces in a sense of Sikorski [1972]. Concise collection of basic notions can also be found in [Drachal 20131. Let M be a set, M -:/= 0. Let A be a (possibly infinite) family of real functions on M' i.e. A:= {fi, ·i E I IviE] J; : M--+ IR:}. Definition 1. The weakest topology, for which all jimctions.from A are continu- ous, is called the topology induced by A on l'vf. I11is topology is denoted by the symbol TA. Definition 2. Let f be a function defined on a subset A c M. If where B is an open subset in topological space (A, T;t), then f is called the local A-function. The set ofall local A~fimctions on a given set A c NI is denoted by A;t. (TA:= {UnA 1 u ETA}.) Definition 3. Superposition closure of a family offunctions A, denoted seA, is defined as seA:= {w 0 (h. ... ' fn) In EN' wE C00 (lRn)' .iJ, ... '.fn EA} Definition 4. If A= scAo, then (M, A) is called the predifferential space. Definition 5. A pair (lVI, A) such that M is an arbitrary set, and Ao is afamily of functions, such that A= (scAo)M, is called the Sikorski differential space (gen- erated by Ao). If Ao is finite, then (M, A) is called finitely generated. Functions from Ao are called generators. A is called the differential structure. Notice, that the above structures can, in certain cases, identify points of M. If one would like to distinguish all points in M (i.e. have the Hausdorff property), then one has to assume that Ao separates points of !VI. From now on, we adopt this assumption. Definition 6. For two predifferential spaces (M, A) and (N, D) the mapping F : M --+ N is called smooth if VfED f oF E A. F is called a diffeomorphism ifit is bijective and both F and p-l are smooth. Definition 7. Let A be afunctionJR-algebra. SpecA := {x: A-> lR IX E Horn(A, JR.) ' x(]_) = 1} Classification of classical singularities: a differential spaces approach .______59 SpecA is called the spectrum of an algebra A. Definition 8. Let M C JR.n. Then C00 (1VI) := (se{7ft[M, ... , 'lfniM})M, where 'If;, i = 1, ... ,n are projections. Definition 9. Consider a differential space (M, A), with A= (Hc{f1, · · ·, .fn}) lvt· A generator embedding is a smooth mapping F: (M, A)--+ (F(M), C00 (1Rn)I,'(M)) , M3pH(fl(p), ... ,fn(P))ElRn · (F(M), C'Xl(JRn)F(IV/)) is then called a generator image. The notation C00 (JRn) F(M) should be understood as j(Jllows: where {7r;}·i=l,...,n are the projections, 'If; (xt, ... ,Xn) := x;for 'i = 1, · · . ,n. Lemma 1 (lCukrowski, Pasternak-Winiarski & Sasin 2012]). Generator embed- ding is a diffeomorphism. Now, let A= scAo, where Ao = {ft, ... ,.fn I .fi: M--+ lR, i = 1, · · ·, n}. Consider slightly modified construction of generator embedding, i.e. F = (!1, ... ,.fn) , F: (M,A)--+ (F(M),C00 (1R 71 )1F(M)) Then F is a diffeomorphism from (1!I, A) to (F(M), C 00 (1Rn)IF(M))· 3. Spectral properties In this part, algebraic techniques are presented in the context ofdifferential spaces [Cukrowski, Pasternak-Winiarski & Sasin 2012]. A similar approach can be found in [Nestruev 2002]. Definition 10. All evaluations ofalgebra A on set M are denoted by Ev(M, A), i.e. Ev(M,A) := {evp Ip EM} , where evp : A--+ R evpf := f(p) for f E A. Iffor a given predifferential space (M, A) it is true that SpecA = Ev(M, A), then it is said that (M, A) has spectral property.
- 34. 6_0___________________.Krzysztof0rachal & Wieslaw Sasin Example 1. The d{fferential space (JR:n, c=(JR:n)) has spectral property. Example 2. If M C JR:n, then the differential space (M, C00 (M)) has spectral property. Example 3. The d{fferential space (.NI, AM) has spectral property. Lemma 2 ([Drachal & Sasin 20 13]). Let A= sc{f1, ... , j~,}, where J; : M-+ lRfor every i = 1, ... , n, be such that A f= AM. Then there exists a monomor- phism SpccAM "-+ SpecA Lemma 3 ([Drachal & Sasin 2013]). Spectral property is invariant under diffeo- morphisms. Definition 11. Consider mappings a : SpecA -+ JR:, such that a(x) = x(a). The weakest topology on SpccA,for which aare continuous for every a E A, is calle:!. the Gelfand topology. The family of all such mappings is denoted by A, i.e. A:= {a: SpecA-+ lR Ia EA}. Lemma 4~([Drachal & Sasin 2013]). If (M, A) is a predif.ferential space, then (SpecA, A) is also a predif.ferential space. Lemma 5 ([Drachal & Sasin 2013]). Let A = sc{f1, •.• , fn}, fi : M -+ lR ,fi=l,...,n· The predif.ferential space (SpecA, A) has the spectral property. Proposition 1 ([Drachal & Sasin 2013]). Let M c JR:n and M = M. Then sc{1Tl!M,···,1Tn!M} = (sc{1TJ!M, ... ,1Tn!M})M. The above Proposition is a direct consequence of the previously presented Lemmas. Moreover, the following Theorem can be proved. Theorem 1 ([Drachal & Sasin 2013]). If M c JR:n, then Spec(C00 (JR:n)IM) = Ev(M, coo(JR:n) !M) ifand only (f M = ld. Definition 12. Let F = U1, ... ,fn) be a generator embedding. Consider an Euclidean metric don F(Jv!). Using F, an inducedmetric on M may be obtained by the formula p(p, q) := d(F(p), F(q)) forp, q EM and F(p) = (h(p), ... , fn(P)), F(q) = (JI(q), ... Jn(q)). Then (M, p) is a metric space homeomorphic to (F(M), d) and F is an isometry. pis called a generator metric. Classification of classical singularities: a differential spaces approach._____ 61 Definition 13. Similarly, a generator metric is introduced on (Spec:A, A) by the formula P(XI,X2) := ~1 (h(XJ)- h(X2))2·, where Xl,X2 E SpeeA. Basing on the definitions and results presented above, one can derive the following result: Theorem 2. For the pred{fferential space (!vi, A) the _f()/lowing statements are equivalent: • (M, A) has the spectral property. • The generator image F(Jvi) is closed with respect to the Euclidean metric. • (lvf, A) is complete in a sense of Cauchy with respect to the generator metric p. • (M, A) and (SpecA, A) are d{ffeomorphic. The detailed proof, which is straightforward but technical, will be presented in a forthcoming paper. Definition 14. Let M c JR:n and let (M, AM) he a finitely generated differential space (i.e. A = sc{.f1 , ... , j~} ). A generator boundary (or gen-boundary) ofthis differential space (M, AM), denoted by OgenM, is defined as DgenM := SpecA SpecAM . Such an approach towards the spacetime boundary has been proposed for the first time in [Drachal & Sasin 2013]. It can be easily observed that Proposition 2. If M c JR:n and A = se {?T1jM, ... ,1TnIM}, I}:en the generator boundary of M, DgenM, is IVfM. I.e. (SpecASpecANr, AspecASpecAM) is d{ffeomorphic and isometric to (M M, C00 ( MM) ). The first space is equipped with the Euclidean metric, the second one- with the generator metric. Definition 15. Generators ft, ... ,fn and g1, ... , Dm are called equivalent, (f sc{fi, ... , fn} = sc{gt, ... ,.IJm}. It can be noticed that for equivalent generators the gen-boundary is well de- fined (up to a diffeomorphism). However, by including the requirement of isom- etry, the unique family of generators is obtained.
- 35. 62 KrzysztofDrachal &Wieslaw Sasin 4. 8-boundary In this part, Schmidt's b-boundary [Schmidt 1971 Jis constructed in the language of spectrums. Let (M, g) be a spacetime. I.e. M is a four dimensional, smooth manifold and g is Lorentzian metric. Let 0(1!I) be the connected component of the fibre ~undle of orthonormal frames over 1II. lt is known that O(l!I) is a smooth man- Ifold, i.e. it is the differential space (O(M), C00 ( 0(1!J))). It is also known that ~he metric connection on l!I gives parallelization of O(.NI). This parallelization mduces a Riemannian metric gu on O(M) [Schmidt 1971]. For sufficiently large n E N, on the strength of Nash embedding theorems, O(M) may be isometrically embedded in JR.n [Nash 1956], i.e. in the differential space (JR.n, coo(JR")). Therefore, let F: (O(M), C 00 (0(M)))---+ (F(O(M)), C00 (!R")F(O(M))) be the generator embedding, such that .9R is the generator metric and let F=(fl, ... ,fn). The C~uchy completion (O(M), C00 (0(M))) of O(M) is diffeomorphic to (S~ec.A, ..4), where .A~c{ft, ... , fn}. (O(lVf), C00 (0(lvl))) is ditfeomor- phic to (Spec.Ao(M), .Ao(M))· It is known that the action of the structural group 0(3, 1) may be extended from O(M) to O(M). Then consider the orbit spaces 1r(O(M)) := O(M)/0(3, 1) and 1r(O(M)) := O(M)/0(3, 1) 9! M . Let us note that (O(M),1rM,M) is a principal 0(3,1)-bundle and that O(M)/0(3, 1) can be identified with M. These spaces are differential spaces, i.e. and Classification of classical singularities: a differential spaces approac~=-------6~3 where Ainv denotes 0(3, I)-invariant functions from c=(o(M)) and .A~)(M) denotes 0(3, I)-invariant functions from Ao(M)· f is called H-invariant, if fhEHf(xh) = .f(:r). Moreover, being finitely generated, they have a spectral property [Cukrowski, Pasternak-Winiarski & Sasin 2012], so they are diffcomorphic to (s , .-.Ainv -..4-inv) pcc , and -- (Spcc.A~(!vl)' .A2)(M)) respectively. Of course, one may consider various differential structures on orbit spaces, but the proposed ones are maximal, which keep 1r smooth lSasin 19881. More- over, one would like to have 1rlo(M) = 1rMio(M)· The original Schmidt b-boundary [Schmidt 1971] was defined as Dbl!I 1r(O(M)) 1r(O(M)). It is diffeomorphic to . -:-- --... (s -.Amv -.Amv) (S ..4inv .Ainv ) pcc , pec O(M)' o(M) Therefore in the language of spectrums the b-boundary is given by S -.Ainv n ..4inv pec >:lpec O(M) , (I) i.e a slight modification of gen-boundary including 0(3, I)-invariance problem. 5. Singularities In this part, some classification of spacetime singularities is proposed. Some considerations on this topic can also be found e.g. in [Helier & Sasin 1999] and [Helier, Sasin, Trafny & Zekanowski 19921. Moreover, [Gruszczak 1990, 20 14] can provide interesting applications of the below considerations. In the below considerations one can substitute E = O(M) and G = 0(~~, 1). Lemma 6 ([Sniatycki 2013]). Let E 7rM Af be a fibre bundle and G be a topo- logical group, i.e. (E, 1rM, M) is a principal G-bundle. Then (EjG, C00 (E/G)) is dijfeomorphic to (M, C00 (1vf) ). Space Eisa sum of orbits of the right action E x G ---+ E of the group G on E, i.e. E = UxEM 7r"N/(x).
- 36. 64 KrzysztofDracha/ & Wieslaw Sasin ~onsider the Cauchy completion E of the space E (with respect to a fixed metnc). The mentioned right action may be prolonged to E x G ---+ E by the formula PoD := lirn PnU , n-><XJ . where p0 E E, p0 g E G. lim,HooPn for the sequence (Pn), Pn E E for ·n E N, Consider now two algebras of G-invariant functions on E and on E, i.e. Fe(E):= {.f E coo(E) I'igEG,pEFJ f(pg) = .f(p)} and Fc(E) := {71 f E F2,(E)} , where F8(E) is subalgebra of Fc(E) consisting of functions which can be con- tinuously prolonged on E, i.e. } : E---+ !l{ is defined by the formula f(Po) := lirn f(Pn) , n----1-oo where Po E E, Po = lirnn->oo Pn for the sequence (Pn), Pn E E for n E N. Of course, one can always choose the generators fi of coo (E) such that they are ~rolongable. This is because by the Nash theorem (E, C00 (E)) can be (iso- rnetncally) embedded in some (!l{n C00 (!l{n)). Let F = (f f ) b h.· ' 1, • · ·, n e t IS embedding, then F is a diffeomorphism and fi = ni oF. It should ~e noticed that in order to make sense, the notion of prolongability has to be constdered with respect to some topology. Th~efore, the algebra of G-invariant functions from C00 (E) is isomorphic to Fc(E). Lemma 7. Fc(E) = n!v1(C 00 (M)) = (sc{n!v1h1, ... , n!v1 hn})E, where hi are generators of(M, C 00 (M)). , The detailed proof, which is straightforward but technical, can be found in [Sniatycki 2013]. The linear operator of prolongation is denoted by P, i.e. P : FS(E) ---+ Fc(E) , P(f) := f . Note that P is an isomorphism of algebras .Fg(E) and Fc(E), so it may be used in classifying singularities of b-boundary, basing on the relation between algebras Fe;(E) and Fc(E) (equivalently: FS(E) and Fe(E)). Classification of classical singularities: a differe_n_~I_2E~_c:es a_eproach ----~- Because !l{ <;;:; :F2:(E) <;;:; Fc:(E), there are three cases: I. .Fg(E) = Fe( E) and then SpeeFn(E) = Spec.lJ;(E), 2. !l{ ~ Fg( E) s-_;; Fe;(E), 3. J1:(E) ~ !l{ and then SpecF~;(E) = {*}. (!l{ above represents constant functions.) In the first case, the boundary given by Eq. (I) is 0. Notice that by detinition, the algebra Fc;(E) = (t->e{nA!/hl,··. ,n/.,1h,})E might consist also of functions not prolongable to E. One can restrict considera- tions only to prolongable functions, i.e. t->c{n;;1h1, . .. , n;;1hn}. Therefore in Case 2 there are two subcases: 2a. (Fg(E)) r:; = Fc(E) and then the boundary given by Eq. (I) corresponds to AJM, Case 2b is possible only if some nonprolongability emerges at the level of the gen- erators h1, ... , hn of NI. This can happen if the topology on O(M) is stronger than the topology induced by n. The third case is a sign of very strong problems with prolongability. For example, the topology onE has to be coarser than the topology on M. Of course, one should keep in mind that nM is assumed to be continuous, so these two topologies are linked with each other in some sense. It seems that this case should be tackled with sheaf methods and some further research in this direction will be presented elsewhere. But even now, one can easily see that generators can be understood as coor- dinates in a classical sense. Therefore, the presented classification gives a nice pictorial tool: in Case 2b there are both kind of 'directions' -ones in which the prolongation is possible and ones in which the prolongation is definitely impossi- ble. In Case 3, in every direction the prolongation is impossible. From the other (spectral) point of view the diffeomorphic spectral space is the space over one point. It is obvious that the geometry over a point is trivial. Case 3 is also one where the only prolongable functions are constant ones. Therefore, it might be interesting to compare the above classification with the results of [Heller, Odrzyg6idi, Pysiak & Sasin 2003, 20071, where similar results Were obtained though in a slightly different framework.
- 37. 66 KrzysztofDrachal &Wieslaw Sasin Acknowledgment The research of K.D. was funded by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST 1/00256. Classification of classical singularities: a diff~rential spaces approach 67 ----- References Cukrowski, M.J., Pasternak-Winiarski. Z. & Sasin, W. 2012. "On real-valued homomorphisms in countably generated differential structures." Demons/ratio Mathematica 45(3):665--676. Drachal, K. 2013. "Introduction to d-spaces theory." Mathematica Aetema 3:753-770. Drachal, K. & Sasin, W. 2013. Construction of a generator boundary. In WDS'/3 Proceedings of Contributed Papers: Part I - Mathematics and Computer Sciences, eel. J. Sal"ntnkov{t, J. Pavlu. Matfyzprcss, pp. 69-73. Gruszczak, J. 1990. "Cauchy boundaries of space-limes." International Journal of Theoretical Physics 29( I ):37-43. Gruszcz.ak, J. 2014. "Smooth beginning of the Universe." In this volume. Hawking, S.W. & Ellis, G.FR. 1973. The Large Scale Structure (!f"Space-Jlmc. Cambridge Uni- versity Press. Helier, M., Odr<:yg6zdz, Z., Pysiak, L. & Sasin. W. 2003. "Structure of malicious singularities." International Journal of71zeoretical Physics 42(3):427-441. Helier, M., Odrzyg6zd:i., Z., Pysiak, L. & Sasin, W. 2007. "Anatomy of malicious singularitics." Journal ofMathematical Physics 4~(9):092504. Helier, M. & Sasin, W. 1994. "The structure of the b-complction of space-time." General Relativity and Gravitation 26(8):797-811. Helier, M. & Sasin, W. 1996. "Noncommutative structure ot: singularities in general relativity." Journal of"Mathematical Physics 37( I I):5665-5671. Helier, M.·& Sasin, W. 1999. "Origin of cla<;sical singularities." General Relativity and Gravitation (4):555-570. Helier, M. & Sasin, W. 2002. "Differential groupoids and their application to the theory of space- time singularities." International Journal of7/zeoretical Physics 41 (5):919-937. Helier, M., Sasin, W., Trafny, A. & Zekanowski, Z. !992. "Differential spaces and new aspects of Schmidt's b-boundary of spacetime." Acta Cosmologica XVIII:57-75. Nash, J. 1956. "The imbedding problem for Riemannian manifolds." Annals of Mathematics 63( I):20-63. Nestruev, J. 2002. Smooth manifolds and observables. Springer. Sasin. W. 1988. "On equivalence relations on a differential space." Commemationes Mathematicae Universitatis Carolinae 29(3):529-539. Schmidt, B.G. 1971. "A new definition of singular points in general relativity." General Relativity and Gravitation 1(3):269-280. Sikorski, R. 1972. Wstfp do geometrii r6iniczkowej. Panstwowe Wydawnictwo Naukowe. Sniatycki, J. 2013. Differential Geometry ofSingular Spaces and Reduction ofSymmetry. Vol. 23 of New Mathematical Monographs. Cambridge University Press.
- 38. Jacek Gruszczak Pedagogical University, Krak6w Copernicus Center for Interdisciplinary Studies, Krak6w The smooth beginning of the Universe T HE breaking down of the equivalence principle, when discussed in the context of Sikorski's differential space theory, leads to the definition of the so-called differential boundary (d-boundary) and to the concept of differential space with ad-boundary associated with a given space-time differential manifold. This en- ables us to define the time orientability, the beginning of the cosmological time and the smooth evolution for the flat FRW world model. The simplest smoothly evolved models are studied. Among all of the investigated smoothly evolved so- lutions, models quantitatively consistent with the observational data of type [a supernovae have been found. 1. Introduction It is generally believed that the Universe had a beginning and everything indicates that it did so. According to contemporary ideas, the Planck era is the beginning of the Universe. During the flow of cosmic time the known Universe emerges from the Planck era. Certainly it is not an immediate process and one can imagine that individual space-time structures emerge from the Planck era gradually. One can expect that the simplest structures, such as a set structure or a topology on this set, can emerge first. The appearance of the manifold structure is the key event in the process. Then the equivalence principle, in its non metric form, appears and makes possible a creation of 'higher' structures such as, for example, the Lorentzian structure. From this perspective it is interesting to ask how space- time geometry is 'nested' in theories more general than the differential geometry since the emerging process of gravitation has to be associated with a theory which is more general than the geometry of space-time manifold. In this paper the Sikorski's differential spaces theory (see Appendix A) is ap- plied to the discussion of the breaking down process of the equivalence principle. Within this theory the non-metric version of the equivalence principle is one of the axioms of the space-time differential manifold (d-manifold) definition. Let
- 39. 70 Jacek Gruszczak us consider points at which this axiom is not satisfied. The set of these points forms the so-called differential boundary (d-boundary). This issue is discussed in Section 2. The construction of this type of boundary is also described. In Section 3, the construction from Section 2 is used for building the differ- ential space (d-space) with a d-boundary for the flat FRW model. The issue of the prolongation of cosmic time and time-orientability to the d-boundary for this model is discussed in Section 4. In Section 5, we describe the concept of the smooth evolution of this model starting from the initial singularity. We find the simplest smoothly evolved flat FRW models in Section 6. Some of them have the kinematical properties which agree qualitatively with the recent cosmological ob- servations [Riess et al. 1998, Perlmutter et al. 1999]. Namely, these models begin their evolution with a decelerated expansion which changes into an accelerated one. The discussion of properties of the cosmological fluid which causes that type of smooth evolution for the simplest cosmological models is carried out in Sec- tion 7. Behaviour of any smoothly evolving flat model in a neighbourhood of the initial singularity is investigated in Section 8. Finally, in Section 9, we summarize the main results of this paper 2. Sikorski's differential spaces and GR Space-time is a 4-dimensional, Lorentzian and time-orientable d-manifold of the class coo [Beem & Ehrlich 1981]. This definition is a mathematical synthesis of what is known about gravitational fields. It implicitly includes the equivalence principle [Weinberg 1972, Raine & Helier 1981, Torretti 1983]. The equivalence principle, in its non-metric version, is implicit in the axiom stating that space-time is a d-manifold which means that it is locally homeomor- phic to an open set in JR.n, where n = 4. DitTerential properties of these home- omorphisms (maps) are determined by the atlas axioms [Kobayashi & Nomizu 1962Jthrough the assumption, that the composition of two maps IPl o IP2 as a mapping between open subsets of JR.n, is a diffeomorphism of the class Ck, k E N. In the present paper it is assumed that these diffeomorphisms are smooth. It is worth noting, that the non-metric version of the equivalence principle is encoded into two levels of the classical d-manifold definition, namely in the as- sumption of the existence of local homeomorphisms to JR.n and in the axioms of atlases. One can look at space-time, or more generally at ad-manifold, from the view- point of theories that are more general than classical differential geometry. In The smooth beginning of the Universe 71 ---- these theories·d-manifolds are special examples of more general objects. Exam- ples of theories of this type are: the theory of sub-cartesian spaces by Aronszajn and Marshall [Aronszajn 1967, Mm·shall 1975] or the theory of Mostow's spaces LMostov 1979]. In the present paper we will study the problem of the begin- ning of cosmological time with help of Sikorski's differential spaces (d-spaces for brevity) 1Sikorski 1967, 1971, 1972]. In this theory, the generalization level does not lead to an excessive abstraction and therefore the d-space's theory may have applications in physics. Generally speaking, ad-space is a pair (M, /vi), where M is any set, and M is a family of functions lP: M ---t JR. (see Appendix A). This family satis- fies the following conditions: a) M is closed with respect to superposition with smooth functions from C00 (!R",JR), and b) M is closed with respect to local- ization. The precise sense of these notions is not important at the moment. The d-space (lvf, M) is also a topological space (IV!, rM), where 'M denotes the in- duced topology on M given by the family JVI. The topology 'M is the weakest topology in which all functions from JVI are continuous. With in the theory of d-spaces the d-manifold definition assumes the form Definition 2.1. Non empty d-space (M, M) is n-dimensional d-manifold, ifthe following condition is satisfied (*)for every p E M, there are neighbourhoods Up E TM and 0 E TJRn, such thatd-subspaces (Up,MuP) and (O,Eg 1 )) ojd-.1paces (M, M) and (JR7!' c:(n)) respectively are diffeomorphic, where [(n) := C 00 (lRn' JR.). Briefly speaking, d-space (M, M) is an-dimensional d-manifold if it is lo- cally diffeomorphic to open subspaces of JR.n which are treated as d-spaces (see Definitions A.8 and A.9). Diffeomorphisms appearing in condition (*) are gen- eralized maps. Unlike in the classical definition, these maps are automatically diffeomorphisms. Definition 2.1 is equivalent to the classical d-manifold defini- tion [Sikorski 1972]. In the d-spaces theory, the non-metric version of the equivalence principle is 'localized' in the single axiom(*). This enables us to give the following definition of space-time: Definition 2.2. A space-time is a non empty d-space (lv!, A1) such that a) (M, M) satisfies the non-metric equivalence principle as expressed by the condition (*)for n=4, b) a Lorentzian metric form g is defined on 1II, c) Lorentzian d-manifold (M, M, g) is time orientable.
- 40. 72 Jacek Gruszczak If the non-metric equivalence principle in the classical definition of space- time is thrown out, the time-orientability, the Lorentzian metric structure, the classical differential structure and the topological manifold structure are auto- matically destroyed. Only the topological space structure survives. In the case of Definition 2.2 the situation is different. Throwing out the non- metric equivalence principle in the form of axiom (*) leaves a rich and workable structure on which one can define generalized counterparts of classical notions such as orientability, Riemannian and pseudo-Riemannian metrics, tensors, etc [Sikorski 1967, 197 I, 1972, Gruszczak, Helier & Multarzynski 1988, Sasin 1991, Buchner 1997, Helier & Sasin 1999, Abdel~Megied & Gad 2005, Rosinger 2007]. Looking at space-time manifolds from the d-spaces perspective, one can imagine a situation when a background of our considerations is a sufficiently :broad' d-space (B, E). Let us additionally suppose that in B there exists a sub- set lvf C B having a structure of a 4-dimensional d-manifold (M, M) satisfying the condition(*) for n =4, regarded as a d-subspace of (B, E). Naturally, at each point p E M the non-metric equivalence principle is satisfied. Among points not belonging to A!J the most interesting are accumulation points of the set M in the .topological space (B, Ts). One can call a set of all accumulation points of M a boundary of the space-timed-manifold (M, M). At the points of the boundary the non-metric equivalence principle can be violated. Definition 2.3. Let (M, M) and (Mo, Mo) be Sikorski's d-spaces, such that a) (Mo, Mo) is an-dimensional d-manifold, b) Nio C NI and Mo is a dense set in the topological space (M, TM), c) (Mo, Mo) is a d-subspace ofthe d-space (M, M). The set DdMo := NI - Nio is said to be differential boundary (d-boundary for brevity) ofd-manifold (Mo, Mo), ifthere is a point p E odMo such that the (*) property is not satisfied. Then, the d-space (M, M) is said to bed-space with differential boundary (or cl-space with cl-boundary for brevity) associated with the d-manifold (Mo, Mo). In the following we shall restrict our considerations to the case of Defini- tion 2.3 for n = 4. The d-boundary definition for a space-time's d-manifold (lllfo, Mo) depends on the choice of the d-space (M, M). Therefore, a reason- able method of the d-space (NI, M) construction is necessary. Fortunately, the idea described above suggests such a construction. Namely, one chooses a suf- ficiently 'broad' d-space 'well' surrounding the whole investigated space-time The smooth beginning of the Universe 73 Mo and then one carries out the process of determination of all ac~umulation points for Mo. These points form the d-boundary iJr1Mo of space-time. Next, one defines the set M as Nf := Mo u 8r1Mo. By treating M as a d-subspa~e of the sufficiently 'broad' d-space, the d-space with the d-boundary (M, .M) IS uniquely determined. On can easily check that the pair of d-s~aces (NI, A1) a~d (Mo, Mo), where (M, M) is determined by the above descnbed method, satis- fies Definition 2.3. The choice of a sufficiently 'broad' d-space which 'well' surrounds the whole investigated space-time is almost obvious. As it is well known [Clarke 197~1, every space-time (Mo,Mo,g) can be globally and isometrically embedded m a sufficiently dimensional pseudo-Euclidean space EM, where J>, q depends on space-time model. The space EP,lJ is also the d-space (~PN,c=(~P+'I,~)) which 'well' surrounds the studied space-timed-manifold (Mo, Mo). Such a choice of the surrounding d-space is well motivated since all causal properties of the studied space-time are taken into account, even if we do _not refer to them directly. This is because of the isometricity of the embeddmg. Furthermore, the concrete form of the isometrical embedding provides us with a practical method of constructing the d-structure M for the the d-space (M, M). Namely, M= c=(JR;P+<l,JR;)M· To test this method of constructing (M, M) we shall apply it to the flat FRW cosmological model. 3. A differential space for the flat FRW d-manifold Let us consider the flat FRW model with the metric g = -dt2 + a6(t)(dx2 + dy 2 + dz 2 ), (I) h (I . ) E wO ·= no X :m;3 and Do is a domain of the scale factor. The W ere ,, X,'!), Z VI · a a . I f t Do 3 t ---t a (t) E :m;+ ao E C00 (D0 ), is an even real functiOn sea e ac or ao : a , o ' a 0 + . of the cosmological timet. A domain of the scale factor Da C :m; , IS an o~en and connected set. For convenience, let us assume that t = 0 is an accumulatl~n point of the set D~ and 0 t/:. D~. In addition, the model has an initial singulanty at t = 0 i.e. limt-;o+ ao(t) = 0.. In the next parts of the ~aper, the symb~~ is reserved for the following functiOn: a: Da---+ R a(t) - ao(t) fortE a• a(t) = 0 fort= 0 where Da := D~ U {0}.
- 41. 74 Jacek Gruszczak The construction of a d-space with a cl-boundary for the flat FRW model is s_imilar_ to the construction of a differential space for the string-generated space time wtth a conical singularity described in [Gruszczak 2008]. Every 4-dimensional Lorentzian manifold can be isometrically embedded in apseudo-Euclidean space [Ciarke 1970]. In particular, the manifold p:vo,g) of mode~ (I) can be isometrically embedded in (~5 , r/")) by means of the following mapptng FP(t, x, y, z) = ~ao(t)(x2 + y2 + z2 + 1) + ~ rt .,..QI_ 0 2 .Jo a.o(r)' F2 (t, x, y, z) = ~a0 (t)(:r2 + y2 + z2 - 1) + ~ j0 ' 1 · :!#:_:' 0 2 ILOTJ' F3 (t, x, y, z) = ao(t):r, F2(t, :r, y, z) = a0 (t)y, F~(t;, x, y, z) = a0 (t)z, (2) where .,,(s) = d'iag(-1 1 1 1 1) ' ' ' ' . As ever~ manif~ld, space-time (W0 , g) is also a differential space (W0 , wo), where the differential structure W0 := d4 ~ is a family of local £(4)_functions on W° C ~4, where £(4) = C00 (~4). wOn the other hand, the set F 0 (W0 ) can be equipped, in a natural way, with a differential structure treated as a dif- :eren~~~l subspace of th; (~5 , ~(5 l), where £(5) = C00 (~5). Then the fam- tly £ po(wo) of local £(")-functiOns on F 0 (W0 ) is a differential structure and the pair (F°CW0 ),£~"Jcwo)) is a differential space. In addition, if the inte- gral J~ drIao (r) is convergent for every t E Do, then the mapping pO is a diff~omorphism of the differential spaces (FO (WO),£}.~ wo ) and (WO, WO) [Sasm 1988, Gruszczak 2008]. ( ) The process of attaching the initial singularity depends on the completion of the set F 0 (W0 ) by means of points from the surrounding space JR.5 in the way controlled by the isometry F 0. It enables us to define the d-space with a d-boundary for the flat FRW model. Let the mapping F: W ---t JR.5 denotes a prolongation of F 0 to the set W := Dn X 1R3 • The values ofF are given by formulae (2) changing the symbol pO onto F. Then for every x, y, z E lR the value of Pat the initial moment t = 0 is F(O, x, y, z) = (0, 0, 0, 0, 0), since a(O) = 0 by assumption. Therefore, the cl- boundary distinguished by the embedding procedure is represented by the single point: odF 0 (W0) = {(0,0,0,0,0)} c JR.5. Th_e pseu~o-Euclidean space (JR.5, r/5l) is a differential space (JR. £(5)). !he_ dtfferenttal structure £(5) of this space is finitely generated. The pro- Jections on the axes of the Cartesian system 1r·: JR.5 ---t m ~- (z z ) - . ~ I&,"z 1, z, ... ,Z5- Z;, 2 = 1, 2, ... , 5, are generators of this structure and therefore: £(5) = The smooth beginning of the Universe 75 ----- Gcn(1r 1 , 1r 2 , ... , 1r 5 ). As is well known [Gruszczak, Heller & Multarzynski 1988, Gruszczak 2008], every subset A of a support M of a differential space (M, C) is a differential subspace with the following differential structure: Gen(CJl). For finitely generated differential spaces (NI, C), every differential subspace with a support A c M is also finitely generated. Generators of the differential structure on A are generators of the differential structure Crestricted to A. The differential space (F (W), Ei~{w)) is a differential subspace of the differ- ential space (JR. £(5)) and represents the d-space with a cl-boundary for model (I). Since £(5) is finitely generated, the cl-structure £1,~(W)' induced on F(W), is also finitely generated and The d-space (F(W), E?,{w)) is not convenient for further discussion. Let us define ad-space (1V, W) diffeomorphic to (F(W), Ej,?(w)) which will enable us to apply the Sikorski's geometry in a form similar to the standard differential geometry. Let us consider an auxiliary d-space (W, W), W = Gen(!:h, /32,. · ·, /35), where f3i: W ----t IR, f3i(t, x, y, z) =11'i o F(t, x, y, z), ·i = 1, 2, ... , 5. (3) This space is not diffeomorphic to the d-space with cl-boundary (F(W), E~?(w))' since the function F: W ----t ~5 is not one to one. Additionally, the generators /3; do not distinguish the following points p E oW := W - W 0 . Therefore, the d-space (W, W) is not a Hausdorff space. With the help of this space one can build a cl-space with cl-boundary (l;V, W) diffeomorphic to (F(W), t:J,?(w)). Let eH be the following equivalence relation Pf2FN {o} V/3 E W: f3(p) = f3(q). The quotient space W = W/p11 can be equipped with a cl-structure W := W IPH coinduced from W where
- 42. 76 Jacek Gruszczak The symbol [p] denotes the equivalence class ofa point p E W with respect to (}JI. The pair (vV, W) is ad-space [Waliszewski I975, Sasin 1988, Gruszczak 2008]. Let us define the following mapping P: Vll--+ F(W), F([p]) := F(p). Theorem 3.1. The d-space (W, W) is diffeomorphic to the d-.1pace with ad- boundary (F(W), E~?(w)). 11 Proof. The mapping F: vV --7 F(W) is a bijection. In addition, W is by construction a Hausdorff topological space with the topology given by the generators /J1, /J2, ... ,/35[Sikorski 1967, I97 I, 1972, Gruszczak, Helier & Mul- tarzynski I988]. The mapping is a diffeomorphism of the d-space (W, W) onto its image (F(W),41 (vr) (F(W), 41 (w)) (see Theorem A. I). 0 According to Definition 2.3, (W, W) is the d-space with d-boundary for the flat FRW cl-manifold (W0 , W 0 ), where the d-boundary is represented by the set 8dvV 0 := W- W 0 . The function F 0 plays an important role in the construc- tion of (vV, W). A deeper analysis of this role can be found in my next paper [Gruszczak 20 I4]. The cl-space with cl~boundary (W, W) has been constructed with help of the cl-space (W, W) and the relation QH. Generally speaking, cl-spaces of the type of (lvl,C) and (MIQIT,CIQII) have a lot of common features because of the isomorphism of the algebras C i CIeII. In particular, modules of smooth vector fields X(M) and X(M IQII) are isomorphic [Sasin 1988, Gruszczak 2008]. This property will, in the next parts of the paper, enable us to work with the help of the more convenient cl-space (Hi'; W) instead of (W, W). The cl-structure W of the cl-space (W, W) is finitely generated by means of functions (3;, i = 1, 2, ... , 5. However, in the next parts of the paper, we use a different, but equivalent, system of generators (4) Then the cl-structure W has the form The smooth beginning of the Universe 77 ---- 4. Time orientability The flat FRW model is a time orientable Lorentzian manifold M. By definition, there is a timelike directional field generated by a nowhere vanishing timelike vector field X. If X generates the directional field then the field .X generates it also, where A is a nowhere vanishing scalar field on J'vf. The field X caries a part of information included in the casual structure of !VI, which enables us to define the direction of the stream of time and the succession of events IGeroch 1971 ]. The manifold structure and the casual structures of space-time are broken down at the initial singularity. In the hierarchy of space-time structures, the Sikorski's cl-structure is placed below the casual structure lGruszczak, Helier & Multarzynski 1988]. Therefore, the cl-space with d-boundary (W, W) of the flat FRW model is timeless independently of the fact that one of coordinates is called time and the moment t = 0 is named the beginning of time. In this situation one cannot say that the d-bounclary advV is an initial or final state of the cosmic evolution. It is necessary to introduce a notion which would be a substitute of time orientability. Let (W0 , Wo) and (W, W) be the pair of cl-spaces described in the section 3. For convenience we can consider the d-space (W, W) instead of the cl-space with cl-boundary (W, W) according to the remark after Theorem 3. I. In this representation the set of not Hausdorff separated points 8W := W - W 0 is a counterpart of the cl-boundary 8dW 0 for the flat FRW cl-manifold. Let in aclclition, X 0 : W 0 --+ TW0 be a timelike and smooth vector field without critical points, tangent to the manifold (W0 , W 0 ), fixing the time- orientability on the manifold (W0 , g). Definition 4.1. The d-space (W, W) is said to be time oriented by means of a vectorfield X if a) there is a nonzero vector field X: W --+ TW tangent to (W, W) given by the formula { X 0 (p)(aiwo) for Ji E W 0 fa E w: X(p)(a) := 1" xo( )( I ) for- p E 8W, q E wo lmq-+p q a wo b) and there is a.function A E W, A(q) > 0 (or A(q) < O)for q E W° C W such that the vector field V := AX is smooth on (W, W). A coordinate defined by means ofX is called time and the moment t = 0 the beginning oftimet. We also say that the d-space (W, W) is oriented with respect to timet.
- 43. 78 Jacek Gruszczak In the flat FRW model (I) the cosmological time t is a time variable. The vector field of the form (5) where a 0 E W 0 ,p E W 0 , establishes the time orientation on (WO, g). The vector f~ld is smooth on the manifold (W0 , wo) since derivation go: wo -----+ JR:.W0 X 0 (a 0 )(p) := X 0 (p)(a0 ) satisfies the condition _io(WD) C W 0 (Definitio~ A.l2). In the next parts of the paper, cosmological models for which the vector field X 0 can be extended on (W, W) are discussed. Then if remaining conditions of Definition 4.1 are satisfied, (W, W) is ad-space with boundary DW of the flat FRW d-manifold which is time oriented with help of the following vector field X: W-----+ TW, X(p)(i:t) := 0 ~~), (6) ~h~re a~ W, p E W. In other words, the d-space (H'; W) with boundary oW IS time onented with respect to the cosmological time t. ~emma 4.1. The mapping X: W-----+ TW, X(p)(a) := oa(p)lfJt is a vector field tangentto (W, W) ifand only iffor every t E Da, a(t) isfinite and a(t) i= 0. 11 Proof. A vector field is tangent to (W, W) if its value X (p) (a) is finite for every a E W and P E W. lt is enough to check this property on the generators ar, a2, ···,as since the d-space (W, W) is finitely generated. Straightforward calculations show that the value of the field X (p) (a 2 ) is finite for p E w iff a(t) is finite and a(t) i= 0 fort E Da. Then, the value of X on the remaininu generators is always finite. ~ Lemma 4.2. If V := .X, A E W, is a smooth vector field tangent to the d- space (W, W) and for every p E W 0 the value of the jimction .(p) i= 0 then .(p) = 0 for P E fJliV. In other words, the smooth vectorfield V has a critical point at the boundary fJW. 11 _P~·~of. Every smooth function 1 E W is a local W-function on W (see Defimt10n A.}). This means that for every p E W there is Up E 7 w and JP E coo(JR:.'>, JR:.) such that 1(q) = JP (a 1 (q), a2 (q), ... ,a5 (q)) for q E Up· Therefore, the value of 1 in p = (0, x, y, z) E DW, where x, y, z are any, is The smooth beginning of the Universe 79 a constant function of x, y, z: 1(0, x, y, z) = fp(al (p), (t2(p), ... , aG(p)) = .fp(O, 0, 0, 0) = const. Now, let us suppose that .(p) i= 0 for p E oW also. Smoothness conditions for the field V have the form: V(o:;) E W, i = 1, 2, ... , 5. In particular, the conditions V(a1) E Wand V(a2 ) E W lead to a E Wand a:/=- 0 for p E W. Then, for example, the function rJ := V(a.3) IAiL, rJ(p) = :1:, is a smooth function (rJ E W) since, E Wand .(q) i= 0 for q E W. This is a contradiction since this function is not a constant function on aw and therefore it is not generated by means of a;, 'i = 1, 2, ... , 5. 0 5. Asmooth evolution with respect to cosmological time The vector field X defined by formula (6) is, in general, not smooth on (W, W). This means, that there are functions a E W such that a value of the derivation X: W-----+ JR:..w, X(oo)(p) := X(p)(o:), is not a smooth function (X(o:) ~ W). According to the definition of X, its restriction X 0 =Xlwo is a smooth vector field on (wo, W 0 ), and therefore its value a~ := X0 (a.~) = ao on the smooth function a2 := a 1 Jwo is smooth: a£ E W 0 . The generator 001 = a is by definition a smooth function on (W, W). According to the earlier argumentation, the value of the derivation X on the function oo1, a1 := X(n1) = a, is not necessarily smooth in the Sikorski sense. But, from the physical point of view, it is natural that the velocity aof the expansion of the universe is a positive function and is a smooth function even at the beginning of timet = 0 since a moment later, it is positive and smooth ( a~ E wo )both in the classical sense and Sikorski's sense. Let us distinguish a class of cosmological models with such a property. Definition 5.1. An evolution ofcosmological model (I) is said to be smoothfrom thebeginningofcosmologicaltimeifa1 := X(oo1) = {;, E Wanda(t) > Ofor t E Da.. We shall also say that the cosmological model is smoothly evolving or smoothly evolved. Theorem 5.1. If the fiat FRW model is smoothly evolving, then the d-space (lV, W) with boundary oH' ofthis model is time oriented with respect to the cos- mological time t. A smooth vector field V defining time orientability on CW, W) has the form v:w -----+ w, v:= aifJ1at. 11 Proof. Proof consists of verification whether the following inclusion is sat- isfied V(~·)_ ",2{2Qi E w 7..- 1 2 5 0 ._,, - '-"1 i)t ' ' - ' , ... , .
- 44. 80 Jacek Gruszczak 6. The simplest smoothly evolving models According to Definition 5.1 the scale factor a(t) for smoothly evolving models satisfies the condition a E W, a(t) > 0 for t E Da. (7) For simplicity, let us confine our considerations to the following subalgebra of the algebra W where l L dr f.(t,x,y,z) = a(t) -.-. · . o a(r) The explicit form of the generators a 1, a 2 , ... , a.5 can be obtained with the help of formulae (4), (3) and (2). :Por functions belonging to W1 , the smoothness condition (7) has the following form i t dr a(t) = f(a(t), a(t) ~( ) ), . o a r (8) where f(O, 0) > 0, f E [(2). (9) Formula (8) is an equation for a(t) with an initial condition a(O) = 0. The function f E E( 2 ), is in a principle, arbitrary. The only restriction on j is con- dition (9) which is a consequence of Definition 5.1, and the physical assumption that the real universe expands from the initial singularity. The simplest choice is the following function Now, the smoothness equation (8) for a(t) has the form a(t) = f3 +!'la(t) +!'2a(t) -.-. . lot dr o a(r) (10) Solutions of (10) have to satisfy the following conditions a(O) = 0, a(O) = {3 > 0, a(t) > 0, a(l;) > 0 fort> 0. (11) The smooth beginning of the Universe 81 Proposition 6.1. When ry2 = 0 then solutions(){ the smoothness equation (10) satisfying conditions (11) have the form and a(t) = {Jt, t E [O,oo), for /'1 = 0, a(t) = j~-(e1' 1 -1), t E [O,oo), foT /'1 =I 0. "(! Proof. Obvious calculus. (12) (13) 11 0 Solution ( 12) represents the well known model of the universe which expands with the constant velocity it = /3 and which is a solution of the Friedman's equa- tions with the following equation of state p = -e/:3. For /'1 > 0 solution (13) is the universe model which is asymptotically ( t --+ oo) the de-Sitter model. The model expands from the very beginning with a positive acceleration. The pa- rameter /'t can be asymptotically interpreted as a cosmological constant. When ry1 < 0 cosmological model (13) describes an expanding universe, and the ex- pansion slows down from the very beginning. For great t, the size of universe fixes on the level a(t) ~ limt--+ooa(t) = f3/b1 1 and a and ii tend to zero when t --+ oo. Such a universe asymptotically becomes the Minkowski space-time. In the case 1'2 f 0, let us introduce the following auxiliary symbols /'1 := i1i2/Vf>, /'2 := sgn(!'2)f3iV3, K := Vf>Hh2, i2 > 0, a(K) := i2 {3~), t(K) := i2 t(K)jJ3, where H(t) := a(t)ja(t). (14) Proposition 6.2. If ry2 > 0 then solutions ofsmoothness equation (I 0) have the form ·t- K - ~-oo a(y)ydy a(K) = (K- .:Y1 - arccothK)-1 , ( ) I - K y2 -1 ' where K E (Kf, oo), and]{f is a solution ofthe following equation Kt- arccothKJ = 1'1, Kt E (1, oo). Proof. Solution of an elementary differential equation. (15) (16) 11 0
- 45. 82 Jacek Gruszczak Proposition 6.3. If12 > 0 then f. iFh ~ 0, acceleration ii(t) > Ofort E (0, oo), 2. ifi1 < 0, acceleration a(t) < 0fort E (0, t.), a(t.) = 0 and a(t) > 0 fort C: (t., oo), where t, := t(K.) and K. is a solution of the following equation - K. 'Yl + !(2 _ 1 + arccothK. = 0, K. E (K1,oo). (17) * Proof. By obvious calculation. 11 0 In the case /2 > 0, there are two essentially different scenmios of a smooth evolution with respect to the cosmological timet: a) If i1 ~ 0 the model accelerates from the very beginning and expands indefinitely. b) In the case iJ < 0, initially the expansion slows down, but at the m0ment t. the unlimited and infinitely long accelerated expansion is initiated (see Figure 1). Proposition 6.4. If /2 < 0, solutions ofsmoothness equation (10) have the fol- lowing form ii(K) = (K- i1 - arctanK + n-j2) -l, where K E (K 1, oo) and](f is a solution ofthe following equation Proof. Solution of an elementary differential equation. (18) (19) 11 0 Proposition 6.5. /f/2 < Oand i1 < rr/2, then cosmological model (I) has a final curvature singularity at t8 < oo. The set [0, ts) is a domain of the scale factor a(t), where t8 := t(O) and thefunction t(K) is given byformulae (18) and (14). Ill Proof. Proposition is the result of a fact that some of the components of the curvature tensor are undefinite at t5 , because a(t) --+ -oo when t--+ t-;. 0 83 The smooth beginning ofthe Universe ----------- Fiuure ]· Scale factor ii(t) for smooth solutions with /2 > ~l. When idl ~. 01 o · . - 0 . 1 t. · initially expan w1l1 d 1 d ith acceleratton. For /'I < so u Jons mo e s expan w .. 1, th ,·aph an a ne ative acceleration but at a moment, denoted by a cnc e _o~ .. e _gr . ' .. g . .. .t.ated The black point denotes the mttlal smgulanty. accelerated expanstOll ts 1111 1 . . p .t. s 6 2 and 6 4 solutions of smoothness equation (1 0) Accordmg to ropos1 ton. . · , . . . . . h d . (f'/ )()) But in the case of soluttons discussed m the aredefinedmt e omam '-!•' · . . . . . fa'K) Proposition 6.5, there appears an additional restnctwn for the domams o . ( . . f fa (K) as the scale tactor and t(K) coming from the geometncal mterpreta IO~ o , ·. . for a flat FRW model. The final curvature singulanty ends the evolutwn of the model. Proposition 6.6. If ~/2 < 0 then I. ifi1 ~ rr/2, acceleration ii(t,) > Ofort E (O,oo), 2. ifO < il < 7f/2, the scale factor a(t) has the inflexion point at t. (ii(t,.) = 0), a(t) > Ofort E [O,t.) andii(t) < Ofort E (t.,ts), 3. ifil ( 0, accelerationii(t) < Ofort E [O,ts), where t* := t(K.). The quantity I<. is a solution of the following equation ___!!_::_ - arctanK* + 1r/2 = i1· (20) K; + 1 111 Proof. By properties of the function given by formulae (18). 0
- 46. 84 Jacek Gruszczak . I~ /'2 < 0, the smooth evolution of the universe with respect to the cosmolog- tcal ttme t can proceed on three different ways: a) If i't ?:: 1r/2, a smooth accelerated evolution starts from the initial singu- larity. The acceleration goes on continuously during an infinite petiod of time. b) For i'1 E (0, 1r/2), these smoothly evolving universes initially accelerate but the acceleration is slowing down so as to change, at t = t*, into a de- celeration. Smooth evolution ends at the final curvature singularity within the finite period of time [0, ts). These models have two singnlarities: the initial and final one. c) Models with i'1 :( 0 start their evolution in the Big-Bang and decelerate. The rate of expansion slows down strongly and these models end their evolution at curvature singularities in a finite time t8 • Models of this class have also two singularities. ii(t) for 12 < 0 0.5 1.0 1.5 t Figure 2: Scale factors a(i) for smoothly evolving models with 1'2 < 0. Curves on the graph with i'1 ?:: 1r/2 represent accelerated solutions. For models with i'1 E (0, 7r/2) initially accelerating expansion is slowing down. At the moment t* (circles on the graph) the acceleration changes into a strong deceleration. When i'1 :( 0 solutions expand with negative acceleration. The black points on the graph denote initial and final curvature singularities. The smooth beginning of the Universe 85 7. Interpretation Let us assume that solutions of the smoothness equation represent cosmological models. Then the Friedman equations can serve as a definition of a pressure p and energy density {j of a kind of cos- mological fluid which causes the smooth evolution of models, where p = "-P and {! = n,(j. In the present paper this fluid is called the cosmological primordial fluid, or primordial fluid for brevity. Proposition 7.1. lff'J E lR and /'2 = 0 then the equation o.lstatej(Jr the primor- dial fluid has the form (21) In addition a) the energy density Q is a decreasing function of cosmoh;gical time and limt-->o+ g(t) = oo, b) at the initial moment p(O) = lim~-ro+ p(t) = -oo, and the remaining details ofthe dependence p(t) are shown in Figure 3, c) limt-+oo g(t) = 3rf limt-+00 Q(t) = 0 Proof. An elementary calculus. ]J limt-rooP(t) = -3')'f limt-tooP(t) = 0 11111 0 The abbreviations SEC and WEC on the above and next tables denote the strong and weak energy conditions and the statements below are answers to the question of whether the strong or the weak energy conditions are satisfied. In the case 1'2 =/= 0 it is convenient to introduce the following abbreviations - -·-2 P := P/'2 ,
- 47. 86 p z•n-r-~--~---~-----~ p(t) for '1:2 = 0 -zs::o_ .........-<-L~~--,Ui.o:=--~---.,.s-;----::':z.o:--_jz.s t Jacek Gruszczak ~igure 3: Dependence p(t) for 1'2 = 0. In the case 1'1 < 0 the pressure has a smgle positive maximum. Proposition 7.2. Ifrv > 0 d - Tfl) • f . 1 . ,2 an '11 E !ft then the equatwn ofstate ofthe primor- { za flwd has the form · In addition a) energy density i2 is an increasing function of time and lim -(t) +oo, lim -(t) _ ](2 . . HO " f2 t-+oo f2 - f where Kf zs a solutum ofequation (16), b) pressure at the beginning and end ofthe evolution is lim p-(t) - lim -(t) _ 2 t.-+0+ · - -oo, H~P . - -J(f' and the remaining details of dependence p(t) are shown m Fzgure 4, c) the week energy condition is satisfied during the whole evolution, d) for "Yr ~ 0 the strong energy condition is broken down, e) if "Yl .~ 0, the strong energy condition is satisfied for t E (0 t ). Fl. r remammg t > t th ·. d" . . b ' * 0 . . . * zs con ltwn zs roken down. The moment t. is defined m Proposmon 6.3 Proof. An elementary calculus. 0 The smooth beginning of the Universe 87 p(t) for 1'2 > 0 Figure 4: For )'1 2:, -1.35, pressure is a decreasing function of cosmological time. For remaining )'1, function p(t) has both the maximum and minimum. The minimum is not well visible on the graph. Proposition 7.3. If 1'2 < 0 and )'1 E JR. then the equation ofstate ofthe primor- dial fluid has the following form j5 = +~ - ~g- ~(1'1 - arccot IQ )(IQ + 1/VU ). 3 3 3 Additionally a) energy density g is a decreasing function of the cosmological time, and limHo+ g(t) = +oo, b) initial pressure is p(O) := limt-+D+ p(t) = -oo, and the remaining details ofdependence p(t) are shown in the Figure 5, ~<0 I· SEC I wEt] c) ;h):7r/2 limHoo g(t) = I<J limHooP(t) = -I<j no yes 0 < ;h < 7r/2 lilllHt.. Q(t) = 0 lilllHt, p(t) = --CO no/yes yes il ::;;; 0 limHt, §(t) = 0 limHt.. p(t) =+eo yes yes d) if 0 < 1'1 < 1r/2, the cosnwlogical fluid violates the strong energy con- dition fort E (0, t,). In the remaining range t E (t*, t8 ), the SEC is satisfied.
- 48. 88 Jacek Gruszczak Quantities Kf, ts and t* are defined in Propositions 6.4, 6.5 and 6.6. Proof. Elementary calculations. p(i) fop 12 < 0 0 <it < 7r/2 4 i 0 Figure 5: Th_e graph shows the great qualitative differences in the jj(i) depen- dence for various ranges of 1'1 . The simplest solution, 11 = /2 = 0, of smoothness equation (I 0) represents a model filled with the primordial fluid with the equation of state p = -l/3g. In the present paper, this fluid is called a /a-matter. In the case of the following parameters system h1 i- 0,/2 = 0}, the primordial fluid consists of the {o- matt~r enriched by a material ingredient connected with the generator a 1 (t) = a(t) m formula (10). This enriched primordial fluid we call a 11 -matter when 'Yl < 0, or a 11-energy when /1 > 0. Similarly, in the case of the following parameter system {/J = 0, /2 i- 0}, the primordial fluid composed of the ~to matter and a matter connected with the generator a 2, through the function ~ in formula (10) , we call a 12-matter when 12 < 0, or a 12-energy when 12 > 0. Taking into account Proposition 7.1 11-energy has the properties of a dark energy. This energy causes the expansion to accelerate. During the evolution acceleration grows to infinity. After an infinitely long evolution pressure and energy density reach the finite values P! = -31f and PI= 31f respectively. Let us notice that then the following equation of state for the cosmological constant The smooth beginning of the Un_i_ve_rs_e___ 89 is satisfied, p1 = - o.r. The 11-energy reaches this property at last stage of the evolution. fn contrast to /J -energy, ~(1-matter satisfies the strong energy condition dur- ing the whole evolution. It has an attraction property. Therefore the expansion of the cosmological model is slowing down in such a manner that at the last stages of the evolution the model becomes static. The 1 1-matter changes its properties during the evolution. initially, it has a negative pressure. But later it transforms itself into a kind of matter with a positive pressure. At the last stages of the evo- lution, the pressure and the energy density of /I -matter become zero: p1 = 0 and Of = 0. After an infinite period of time since the Big-Bang this cosmological model becomes, in an asymptotic sense, the Minkowski space-time. A model of the universe tilled with 12-energy monotonically accelerates. Ini- tially, the jostling property of 12-energy has a small influence on the expansion but its inflationary power is disclosed at the last stages of evolution of the model. The initially infinite energy density strongly decreases and at the end of the evo- lution is on the level of o.r = i3K6.r, where K01 ~ 1.19 is a solution of equation (16) for ;y1 = 0. A negative pressure rapidly grows from -oo to a finite level of pf = -i~Kg1. At the end of the evolution the equation of state is as for the cosmological constant: P! = -o1. During the whole evolution the strong energy condition is violated. 12-energy can be interpreted as a dark energy of different kind then /J-energy. Details can be found in Figures I, 4 and in Proposition 7.2. 12-matter has a strong attraction property and therefore the expansion of a model with such a fluid is rapidly slowing down. Acceleration quickly decreases from 0 to -oo in a finite period. At the end of the evolution the model is stopped a(tf) = 0, and its scale factor reaches the maximal, finite value. The final curva- ture singularity ends the evolution. Properties of 12-matter are changing during the evolution. Pressure rapidly increases from -oo to +oo in the finite period [0, t f). Simultaneously, the energy density decreases from +oo to zero indepen- dently of the fact that the scale factor is finite (a(tf) < oo) at tf. This kind of matter has very interesting properties at the end of the evolution: it has a slight energy density but simultaneously a huge positive pressure. Details can be found in Figures 2, 5 and in Proposition 7.3. A mixture of11-energy and 12-energy (/1 > 0, /2 > 0). During the whole period of the evolution of this model the mixture has the properties of a dark energy. Both components interact with each other causing increased acceleration. In the last stages of the infinitely long evolution, the equation of state for the mixture has the form of equation of state for the cosmological constant {!J -pf = i~KJ, where Kf is a solution of equation (16).
- 49. 90 Jacek Gruszczak A mixture of1'1-matter and 1'2-matter (/'1 ::( 0, 1'2 < 0). This kind of the pri- mordial fluid satisfies the strong energy condition during the whole finite period of evolution. The mixture is a fluid with interacting components. At the final sin- gularity /'z-matter absorbs, in a sense, 1'1-matter and finally the mixture vanishes, {!J = 0 at an infinite pressure. An admixture of 1'1-matter into /'2-matter shortens the lifetime of the cosmological model. A mixture of/'t -matter and 'Tz-energy (1'1 ::( 0, 1'2 > 0). Components of the mixture are interacting fluids. The beginning of the model evolution is dominated by 1'1-matter. The universe expands with a negative acceleration and the mixture satisfies the strong energy condition. But the influence of1'2-energy is still rising. Fort > t. the evolution is dominated by /':renergy. Since t = t., the mixture has properties of a dark energy and changes the further evolution into accelerated expansion. During the final stages of the evolution the equation of state has the following form {!J = -pf = ;y~Kj, where K1 is a solution of equation (16). The evolutionary behaviour of the model is extremely interesting because such an evolution is qualitatively consistent with the observational data of la type supernovae [Riess et al. 1998, Perlmutter et al. 1999]. Preliminary qucm- titative investigations of the consistency of the discussed model have been car- ried out with the help of the Habs(z) dependence published in [Simon, Verde & Jimenez 2005, Ma & Zhang 2011, Yu, Lan, Wan, Zhang & Wang 2011, Zhang, Ma & Lan 201 1]. Results of the best-fit procedure depend on H0 . For 70.6 ::( Ho ::( 77.8 km ~:>-1 Mpc-1 lRiess et al. 2009] the best-fit parameters are in the range '11 E [-1.829, -1.075], ;y2 E [2.82, 4.011] x1o-1Mpc-1 and x;,.in E [8.66, 9.76] (see also Figure 6). Values of parameters ;y1 and ;y2 en- able us to find: an age of universe [0 E [13.561, 14.241] x 109 y, the moment of the acceleration beginning l. E [7.427, 8.471] x 109 y, the Hubble constant in the acceleration moment H. E [106.31, 108.86] x km s-1Mpc1 and redshift z. E [0.648, 0.743], where (}/'") ao'fz - -1 - -1 v'3 - z ". := v'3{Ja(K) - 1, to= c t(K0 ), t. = c t(K.), K0 := c'f 2 H0. Quantities K, a(K), t(K) are given by formulas (14) and (15), and K. is a solution of equation (17). In the present Section bar over quantities denotes that we use the systems of units in which c =J. 1. A mixture of /'1-energy and 'fz-matter (/'I > 0, /'z < 0). Fort E [0, t.) the mixture has properties of a dark energy and therefore this model accelerates from The smooth beginning ofthe Universe 250 ~ ' ~150 s - ~ H. // '-' 100 ?~ :1 lg so 1 : H0=72.0 H .=106.87 z.=0.67 Ob---~~:~'~··~----._--~----~ 0.0 0.5 1.0 1.5 2.0 2.5 z 91 Figure 6: Comparison of the theoretical__ ff(z) dep~ndence for th~-~moot~l( evolved model ('Yt < 0 and/'2 > 0) with Ilobs(z) for H~ = 72.0 km_c; .~pc ,· The solid line represents the smoothly evolved model wtth the best-tit par~meter I - 1 241 - - 0 000312Mpc-l and x·2 . = 8.79. In thts case va ues /'1 = - · , /'2 - · _ mm _ 1 _ 1 , fo = 14.103 x 109y, T. = 8.241 x 109y, H. = 106.87 km s Mpc and z. = 0.668. The dashed line represents the best-fit of the ACDM model. the very beginning. But later, 1'2-matter component begins to play a bigger and bigger role. The further evolution depends on the value of the parameter /'1 · When ;y1 E (0, 1r/2), the repulsive properties of ~~1-energy are not able to dominate the evolution and the acceleration is gradually stopped because of. a greater and greater attractive influence of 1'2-matter. The momen_t t., defined m the Proposition 6.6, is the end of acceleration. Starting from_ thts mo~1 cnt, t~~ expansion slows strongly till the final singularity. The behaviOur of ptes_sure IS interesting, Figure 5, fort > t•. After t. the mixture behaves as ~ dus~ (j5 :::::: 0). But later, pressure rapidly increases to infinity at the final singulanty. Smtultane- ously, because of expansion, the energy density decreases to zero. When ;y1 :;::: 1r/2, repulsive influence of 1'1-energy is domi_nating during the whole pe~iod the infinitely long evolution. The mixture acts llke a dark ~nergy causing acceleration, independently of the 'Yz-matter presence. As for previOusly considered accelerating models, the equation of state, in the last stages of the -2 J{~ evolution is as the one for the cosmological constant, i.e., f2J = -pf = 1'2 ]• ' where K 1 is a solution of equation (19).
- 50. 92 Jacek Gruszczak 8. Smoothly evolved models in a neighbourhood of singularity The function f appearing in the smoothness equation (8) can be expanded into a seri.es in a neighbourhood of the point (0, 0): f(z1 , z2) = f3 + DJ}'(O, O) . z 1 + fh/(0, 0) · Zz +· ... Then the smoothness equation assumes the following form h(t) = {-J + lh.f(O, 0) · a(t) + Ehf(O, O). a(t) l ar~~) + .... For sufficiently small t, or equivalently in a small neighbourhood of the initial singularity, one can omit the higher powers of the expansion and consider the sm~othness equation in the linear approximation. [f one assumes that l'l := D1j(O, 0) and /''2 := Ehf(O, 0), the above equation. in the linear approximation is identical with the smoothness equation (I 0) for the simplest smoothly evolve~ models (Section 6). The above observation leads to the conclusion that properties of the solutions studied in Sections 6 and 7, for small t, are typical for every smoothly evolvino- llat FRW model in a small neighbourhood of the initial singularity. In particula;, every smoothly evolving model during the initial stages of its evolution is filled with a cosmological fluid which is /';-matter or /';-energy, i = 1, 2, or one of the mixtures described in Section 7. These fields satisfy the following approximate equation of state 9. Summary a) The discussion of the equivalence principle and its breaking down has led to the formulation of the d-boundary notion which is, roughly speakina, an 'optimal' set of points at which the equivalence principle is broken. A space~ime cl-manifold with the attached d-boundary is not only a topological space but also an object with a rich geometrical structure called differential space (see Appendix A). The effective construction of both d-boundary and cl-space with ad-boundary for any space-timed-manifold has also been described (see Section 2). The smooth beginning ofthe Universe 93 b) For every flat FRW cosmological model with the initial singularity the d- space with d-boundary (W, W) has been constructed. In this case the d-bounclary is a single point (see Section 3). c) In the d-space formalism it is possible to extend the concept of time ori- entability to the d-boundary of the FRW models. In this way, the intuitive un- derstanding of the beginning of the cosmological time obtains a precise mathe- matical form (see Section 4). However, not every flat FRW model with the initial singularity has a well defined beginning of cosmological time. d) In the whole class of flat FRW models with a well defined beginning of cosmological time we distinguish the large subclass of the so-called smoothly evolved models (see Definition 5.1 ). The condition defining this subclass is called the smoothness equation. The most practical form of this equation is given by formula (8). e) The simplest flat FRW models with a well defined beginning of cosmo- logical time have been found in the explicit form. This set of models can be divided into six qualitatively different classes (Lemmas 6.1, 6.2 and 6.4). The most important classes are • solutions with parameters ,:Y1 < 1r/2 and /'2 < 0 which have two curvature singularities: the initial singularity and the final singularity (see Figure 2), •• the subfamily with ,:Y1 < 0 and 1'2 > 0, being qualitatively consistent with the observational flobs(z) data from [Simon, Verde & Jimenez 2005]. The quantitative consistency with the data is discussed in Section 7 and is shown in Figure 6. The level of the consistency is similar to that for the ACDM model. f) The Friedman equation without the cosmological constant, in application to the simplest solutions, may serve as a definition of energy density 1j and pressure fi of a cosmological fluid (primordial fluid) which causes the smooth evolution. This strategy enables us to find main phenomenological properties (in particular, the equation of state) of this primordial fluid (see Lemmas 7.1, 7.2 i 7.3). g) In Section 7 we present an interpretation of the primordial fluid as a mix- ture of more elementary interacting primordial fluids l'i-matter and l'i-energy, i =1, 2. h) From the analysis of smoothness equation (8) in a neighbourhood of the initial singularity, i.e for small t, one can conclude (see Section 8) that in the earlier stages of the evolution every smoothly evolving flat FRW model is filled With a primordial cosmological fluid with properties characteristic for particular solutions found in this paper (Sections 6 and 7).
- 51. 94 Jacek Gruszczak i) It is very surprising that without any assumptions concerning the physical nature of the cosmological fluid, among the simplest solutions of the smoothness equation (8), it is possible to find models consistent with the observed evolution of the Universe (see Figure 6 ). The existence of a well defined beginning of cosmological time was the only requirement which has been assumed. A primor- dial mixture of fluids ')'1-matter and ')'2-energy is a consequence of this simple assumption. However, we cannot expect that every detail of cosmic evolution can be determined in this way. The material ingredients such as radiation, dust and dark matter should also be taken into account. Acknowledgments We would like to thank Professor Michael Helier for the valuable discussions we had when preparing this paper. This publication was made possible through the support of a grant from the John Templeton Foundation. A. Sikorski's differential spaces Let C be a non empty family of real functions defined on a set M. The family C generates on M a topology denoted by the symbol Tc. It is the weakest topology on !vi in which every function from C is continuous. Let A C M be a subset of M and let the symbol CIA denotes the set of all functions belonging to C restricted to A. On A one can define the induced topol- ogy Tc n A= TCIA· The topological space (A, Tq11 ) is a topological subspace of (M, Tc). Next, we introduce two key notions in d-spaces theory: a) the closure of C with respect to localization and b) the closure of C with respect to superposition with smooth functions from £(m) := C00 (JR.m, JR.), rn = 0, 1, 2, ... Definition A.l. A function ')': A ---7 JR. is said to be a local C-junction on a ~ubs~t A C M if, for every p E A, there is a neighbourhood Up E TC/A and a functzon c/J E C such that ')'IUP = c/JIUP. The set ofall local C-functions Oil A is denoted by CA. It easy to check that in general CIA c CA. In particular Cc CM. Definition A.2. A family C ofrea/junctions on a set M is said to be closed with respect to localization if C =CM. The smooth beginning of the Universe 95 Definition A.3. A family of functions C is closed with respect to superposi- tion with smooth functions jimn £(rn), m = 0, 1, 2, ..., ilfor every function wE £(m) andfor every system ofmfunctions 'PL,'P'2,···,'Prn E C, the su- perposition w('Pl, cp2, ... , 'Pm.) is a function from C; w('Pl, 'P2, · · · , 'Pm) E C. The above described system of concepts makes it possible to define an object (a d-spacc) which is a commutative generalisation of the d-manifold concept. Definition A.4. A pair (M, C), where M is a set (~{points and C a family ofreal functions on M, is said to be a differential space (d-space for brevity) if J. C is closed with re:.pect to localization, C =CM, 2. C is closed with respect to supe1position with smooth functions from £(m)_ The family C is called a differential structure on NI (d-structure for brevity) and the set NI a support (~l the d-structure C. Functions cp E C are called smooth functions. Every d-space is simultaneously a topological space with the topology Tc given by the d-structure C in the standard way. The d-structure itself, with the usual multiplication, is a commutative algebra. The notion of smoothness, given by the condition cp E C, is an abstract generalization of the smoothness notion for functions defined on JR.n. Differential structure, by definition, is a set of all smooth functions on NI. There are no other smooth functions on M. This class of smooth functions may consists of functions which are not smooth in the tra- ditional sense. This is a great advantage of the d-spaces theory. The simplest example of a d-space is the n-dimensional Euclidean d-space (JR.n, £(n) ), where £(n) = COO(JR.n; JR.). . · There exists a procedure to construct a d-structure with the help of a chosen set of real functions on M. Let us denote it as C0 . The method consists in adding to a given Co missing functions so as to satisfy the axioms of the closure with respect superposition with smooth functions and the closure with respect to lo- calization. The closure with respect to superposition with smooth real functions on JR.n, is denoted by mathematicians by se(C0) and the closure with respect to localization is denoted by (Co)Ivi (see Definition A.J) or lc(Co). It is easy to check that Lemma A.l. Let Co be a set of real functions defined on a set M. The family of functions C := lc(sc(Co)) = (sc(Co))M is the smallest, in the sense ofinclusion, d-structure on M containing Co.
- 52. 96 Jacek Gruszczak Sometimes one uses the following abbreviation: C = Gen(Co) := lc(sc(C0)) = (sc(Co))M. Definition A.S. The set Coin Lemma A. I is said to be a set ofgenerators. Func- tions r.p E Co are called generators of the cl-structure C := lc(sc(Co)). JfCo is finite then the cl-structure C is calledfinitely generated. The method of constructing ad-structure with the help of a set ofgenerators is the greatest advantage of Sikorski's theory, especially in the case of finitely gen- erated d-spaces such as, for example, the d-space with ad-boundary associated with the flat FRW world model. Definition A.6. If (M, C) is ad-space and A C M then the d-space (A, CA) is said to be a differential subspace (d-subspace) ofthe d-space (M, C). The above definition enables us to determine a d-structure for any subset A of NI. It is enough to 'localize' every function from the d-structure C to A. In the case of a finitely generated d-spaces (M, C), a simpler situation occurs. Then C = Gcn(Co), Co := {/31, /32, ... ,f3n} , n E N, where f3r, /32, ... ,f3n are given functions. The d-structure CA is given in terms of generators Co =ColA which is a set of restrictions of the set Co to A. Then CA = Gen(C0). Definition A.7. Let (M, C) i (N, D) bead-spaces. I. A mapping f: M-:+ N is said to be smooth if:1(3 E V : f3 of E C. 2. A mapping f: NI -:+ N is said to be a dijfeomorphism from a d-space (M, C) to a d-space (N, D), if it is a bijection from M to N and both mappings f: M-:+ Nand j-1 : N ->-NI are smooth. In this case, we say that (NI, C) and (N, D) are dijfeomorphic. A smooth mapping f transforms smooth functions on N onto smooth func- tions on Af. The notion of d-spaces diffeomorphism is the key notion from the point of view of the present paper. If there is a diffeomorphism f between d- spaces (NI, C) and (N, V) then these d-spaces, from the viewpoint the d-spaces theory, are equivalent. Definition A.8. Let (M, C) i (N, D) bed-spaces. A d-space (M, C) is said to be locally dijfeomorphic to the d-space (N, D) iffor every p E M there is Up E Tc and a mapping JP: Up -:+ fv(Up) E TD such that fp is a dijfeomorphism between the d-subspaces (Up, CuP) and (jp(Up), V fp(Up)) ofthe d-spaces (M, C) i (N, D), respectively. The smooth beginning of the Universe 97 Definition A.9. Ad-space (M, C) is said to be ann-dimensional cl-manifold, ifit is locally diffeomorphic to the d-space (lRn, £(n)). Applying Definition A.8 to Definition A.9, leads to condition(*) in Definition 2. 1. Local diffeomorphisms fp are obviously maps. A set of maps forms an atlas. It turns out that Definition A.9 is equivalent to the classical definition of d-mariifold [Sikorski 1972]. Theorem A.l. Let a Hausdorfl d-space (M, C) be a finite fly generated d-space with the d-structure C generated by a.finite set of.fimctions: Co := {J:h,. ··, f1n}, C = Gen(Co). Then the mapping F: Af -:+ lR", F(p) := (/-31 (p),. · ·, f3n(P)) is a diffeomorphism of the d-space (M, C) onto the d-subspace (F(M), E;;?M)) of the d-space (lR71 , £(n) ). Proof, see !Sasin & Zekanowski 19871. The d-subspace (F(M), £~~)M)) of (lRn, £(")) is an image of th~ d-space (M, C) in the mapping F. Theorem A.l is called the theorem on a dJffeomor- phism onto the image. Definition A.lO. A mapping v: C -:+ lR is said to be a tangent vector to a d-space (M,C) atapointp EM if" I. :la, f3 E C:!a, b E lR : v(ac~ + b/3) = av(a) +bv(/3), 2. :!a.,/3 E C: v(af3) = v(a.)/)(p) + a(p)v(f3), p EM. The set ofall tangent vectors to (!11, C) at p E NI is said to be a tangent vector space to (M, C) at p E M and is denoted by TpM. The symbol T M denotes the following disjoint sum: Tld := U TpM· pEl>I Definition A.ll. Let (M, C) bead-space. The mapping X: M ----7 T M such that :!p E M, X (p) E TpM is said to be a vector field tangent to (M, C). With help ofa vector field X: M -:+ T J1 one can define the following map- ping X: c-:+ JRM, X(a.)(p) := X(p)(a.), where a E C i p E M. The mapping is linear and satisfies the Leibnitz rule Va,/) E C : X(a./)) = X(a)f3 + aX(f3). Therefore, it is a derivation and a global alternative for the definition of a vector field (A. l I).
- 53. 98 Jacek Gruszczak Definition~A.l2. A vector field tangent to (M, C) is said to be smooth if the mapping X satisfies the condition: X(C) cC. Definition A.13. Let f: M ---t N be a smooth mapping. The mapping f,p: TpM ---t Tf(p)N, given by the formula is said to be differential ofthe mapping fat the point p E M. Let us define the following mapping idA: A ---t M, idA(P) A cM. p, where Definition A.l4. A vector field Y: M ---t T M on (M, C) is said to be tangent to a d-subspace (A, CA), ifthere is a vector.field X: A-? TA on (A, CA) such that The smooth beginning of the Universe 99 References Abdel-Megied, M. & Gad, R.M. 2005. "On the singularities of Reissner-Nordstriim space-time." Chaos, Solitons & Fracra!s 23(l)}I3-320. Aronszajn. N. 1967. "Subcrutesian and subriemannian spaces." Notices of the American Mathe- matical Society 14: Ill. Beem, J.K. & Ehrlich, P.E. 1981. Global Lorentzian Geometry. Dekkcr. Buchner, K. 1997. "DiiTerential spaces and singularities of space-time." General Mathemalics 5:53-66. Clarke, C.J.S. 1970. "On the global isometric embedding ofpseudo-Riemannian manifolds." Pro- ceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 314( 1518):417--428. Gcroch, R. 1971. Space-time structure from a global viewpoint. In General Relativity and Cosrnol- ogy. Proceedings of the International School ofPhysics 'Enrico Fermi' Cow:1·e 47, ed. R.K. Sachs. Academic Press pp. 71--103. Gruszczak, J. 2008. "Discrete spectrum of the deficit angle and the differential structure or a cosmic string." International Journal of1/1eoretical Physics 47( 11 ):2911-2923. Gruszczak, J. 2014. "Cosmological models .in terms of di!Terential spaces (I) -llat FLRW models." In preparalion. Gruszczak, J., Heller, M. & Multarzyriski, P. 1988. "A generalization of manil"olds as space-time models." Journal ofMathematical Physics 29(12):2576-2580. Helier, M. & Sasin, W. 1999. "Origin of classical singularitics." General RelaLivity and Gravilmion 31(4):555-570. Kobayashi, S. & Nomi;:u, K. 1962. Foundations of Differential Geometry. lnterscience Publishers. Ma, C. & Zhang, T.-J. 2011. "Power of observational Bubble parameter data: a figure of merit exploration." The Astrophysical Journal730(2):74. Marshall, C.D. 1975. "Calculus on subcartesian spaces." Journal of Dijferential Geometry 10(4):551-573. Mostov, M.A. 1979. "The differentiable space structures of Milnor classifying spaces- simplicial complex and geometric relations." Journal of Differential Geometry 14:255-293. Perlmuttcr, S., Aldering, G., Goldhaber, G., Knop, R.A., Nugent, P., Castro, P.G.. Deustua, S., Fabbro, S., Goobar, A.• Groom, D.E., Hook, J.M., Kim, A.G., Kim, M.Y., Lee, J.C.. Nunes, N.J., Pain, R., Pennypaeker, C.R., Quimby, R., Lidman, C., Ellis, R.S., lrwin, M., McMahon, R.G., Ruiz-Lapuente. P.. Walton, N.. Schacfer, 13., Boyle, B.J., Filippenko, A. V., Matbeson, T., Fruchter, A.S., Panagia, N., Newberg, H..l.M., Couch, W.J. & The Supernova Cosmology Project. 1999. "Measurements of [2 and A from 42 high-redshift supernovae." The Astro- physical.Tournal 517(2):565. Raine, D.J. & Heller, M. 1981. The Science ofSpace-1ime. Pachart Publishing House. Ricss, A.G., Filippenko. A.V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P.M., Gilliland, R.L., Hogan, C.J., Jha, S., Kirshncr, R.P., Leibundgut, B., Phillips. M.M., Reiss, D., Schmidt, B.P., Schommer, R.A., Smith, R.C., Spyromilio, J., Stubbs. C., Suntzcff, N.B. & Tonry, J. 1998. "Observational evidence from supernovae for an accelerating Universe and a cosmo- logical constant." The Astronomical.loumal116(3): I009.
- 54. 100 Jacek Gruszczak Ricss, A.G., Macri, L., Cascrtano, S., Sosey, M., Lampeitl, H., Fcrguson, H.C., Filippcnko, A.V., Jha, S.W., Li, W., Chornock, R. & Sarkar, D. 2009. "A redetermination of the Hubble con- stant with the Hubble space telescope from a differential distance ladder." The Astrophysical Jouma/699( I):539. Rosinger, E.E. 2007. "Differential algebras with dense singularities on manifolds." Acta Applican- dae Mathematicae 95(3):233-256. Sasin, W. I98S. "On equivalence relations on a differential space." Commentationes Mathematicae Universitatis Caro/inae 29(3):529-539. Sasin, W. 1991. "DifTcrcntial spaces and singularities in differential spacetimc." Demons/ratio Mathematica 24(3-4):60 I-634. Sasin, W. & Zckanowski, Z. 1987. "On locally tlnitely generated differential spaces." Demons/ratio Mathematica 20:477-486. Sikorski, R. 1967. Abstract covariant derivative. In Colloquium Mathematicae. Vol. I8. Institute of Mathematics Polish Academy of Sciences pp. 251-272. Sikorski, R. 1971. Differential modules. In Colloquium Mathematicae. Vol. 24. Institute of Math- ematics Polish Academy of Sciences pp. 45-79. Sikorski, R. 1972. Wstcp do geometrii r6iniczkowej. Panstwowe Wydawnictwo Naukowc. Simon, J., Vcrdc, L. & Jimenez, R. 2005. "Constraints on the redshift dependence of· the dark energy potential." Physical Review D 71:12300 I. Torretti, R. l983. Relativity and Geometry. Pergamon Press. Waliszcwski, W. I975. "Regular and corcgular mappings of dill'erential spaces." Annates Polonici Mathematic! 30:263-281. Weinbcrg, S. 1972. Gravitation and Cosmology: Principle and Applications ofGeneral Theory of Relativity. John Wilcy and Sons, Inc. • Yu, H.-R., Lan, T., Wan, H.-Y., Zhang, T.-J. & Wang, B.-Q. 20 I I. "Constraints on smoothness parameter and dark energy using observational H(z) data." Research in Astronomy and As- trophysics 1I(2): 125. Zhang, T.-J., Ma, C. & Lan, T. 201 I. "Constraints on the dark side of the Universe and observa- tional Hubble parameter data." Advances in Astronomy 20 I0: I84284. Mariusz P. Dqbrowski University of Szczecin Copernicus Center for Interdisciplinary Studies, Krak6w Are singularities the limits of cosmology? WE.refer to the classic d.efi.nition of a singularity in Einstein's general.rel~tiv tty (based on geodesic mcompleteness) as well as to some other cntena to evaluate the nature of singularities in cosmology. We review what different (non- Big-Bang) types of singularities are possible even in the simplest cosmological framework of Friedmann cosmology. We also show that various cosmological singularities may be removed or changed due to the variability of physical con- stants. ----·------------ .----- 1. What are singularities? Asking about the limits in cosmology is almost the same as asking about singu- larities. They are one of the most intriguing objects since they open the way to the new physics- the physics which cannot be described by actual theories of the uni- verse. The singularities are just some infinities of the physical and mathematical quantities which lead to experimental or observational problems since one cannot practically measure any infinite quantity with any type of a realistic device. This is why we say that our (whatever) theory fails, once it possesses a singularity. However, the singularities appear in physics and are formally described by math- ematics so that we have to somehow deal with them. In fact, they appear in all physical theories. For example, in Newton's theory of gravity there is a singu- larity when a spherical shell collapses to a point at its center. It is obvious for cosmologists that we also experience singularities in Einstein's gravity and the best-known example is a Big-Bang singularity (corresponding to the beginning of the evolution of the universe) and a black hole singularity (corresponding to the collapsed matter as a result of gravitational attraction). In some intuitive approach we can talk about singularity as a 'place' in which some kind of a 'pathology' is observed. We know that physical fields, such as for example the electric field, can be singular at the place where a point charge is situated. This is a physical field singularity which resides in space. However,
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- 56. PLATE VI.—KING COPHETUA AND THE BEGGAR MAID (The Tate Gallery) The old story of the king who succumbed to the charms of a simple beggar maid has inspired many artists, but none have rivalled Burne-Jones in appreciation of the artistic possibilities of the subject.
- 57. His picture on its appearance at the Grosvenor Gallery in 1884 set the seal on his reputation, and put an end to whatever doubts remained then in the public mind as to his right to serious consideration. It is in many ways the finest of all his works, the most ambitious and the most exacting in the technical problems presented, and it is certainly the most notable in accomplishment. This point needs to be elaborated for the sake of clearing up any misapprehensions which might arise from his more or less erratic way of exhibiting his work. As an example, when he exhibited for the first time in 1864 in the gallery of the Royal Society of Painters in Water Colours, he showed the "Fair Rosamond," painted in 1862, with the "Annunciation" and "The Merciful Knight," both of which belong to 1863; but in 1865 he sent "A Knight and a Lady," finished just before the exhibition opened, "Green Summer," painted in 1863, and "The Enchantments of Nimue," which was one of the things he produced in 1861 while he was still frankly and unreservedly an imitator of Rossetti. Such an inversion in the order in which his works were set before the public might cause some perplexity to students of his art if they did not realise what was his custom in this matter. He exhibited in the gallery of the Royal Water Colour Society in 1869 a painting, "The Wine of Circe," which was not only the most important work he had produced up to that time but is also to be counted as one of the most admirable of all his performances; and he showed there in 1870 two other notable works, "Love Disguised as Reason" and "Phyllis and Demophoon." It was over this last painting that the dispute arose which led to his resignation of his membership of the Society; and one of the results of this dispute was that for a space of seven years hardly any of his pictures were seen in public. Indeed, the only things he exhibited during this period were a couple of water-colours, "The Garden of the Hesperides" and "Love among the Ruins," which appeared at the
- 58. Dudley Gallery in 1873. Both were important additions to the list of his achievements, and the "Love among the Ruins" especially was a painting of exquisite beauty and significance. He repeated this subject in oil some twenty years later, because the original water- colour had been damaged somewhat seriously, and was not, as he considered, capable of repair. The opening of the Grosvenor Gallery in 1877 gave him his first great opportunity of setting before the mass of art lovers his claims to special attention. Hitherto he had counted in the minds of a few men of taste and sound judgment as an artist of remarkable gifts who promised before long to take high rank in his profession, but by the larger public interested in art matters he was practically undiscovered. That he would have won his way step by step to the position he deserved cannot be doubted; if there had been no break in his activity as an exhibiting painter his successive contributions to the Royal Water Colour Gallery could not have failed to make him widely known. But his reappearance at the Grosvenor Gallery was so dramatic, and so convincing in its proof of the amazing development of his powers, that he leaped at one bound into the place among the greatest of his artistic contemporaries, which he was able to hold for the rest of his life without the possibility of dispute. For he had not been idle during this seven years of abstention from exhibitions; the period had been rather one of strenuous activity and unceasing production. It saw the completion of several important canvases on which he had laboured long and earnestly, and it saw the commencement of many others which were in later years to be added to the list of his more memorable achievements. In some ways, indeed, it was a fortunate break; it saved him from the need to strive year by year to get pictures finished for specific exhibitions, and it allowed him time for calm reflection about the schemes he desired to work out. It freed him, too, from the temptation—one to which all artists are exposed—to modify the character of his art so that his pictures might be sufficiently effective in the incongruous atmosphere of the ordinary public gallery. He was able to form his
- 59. style and develop his individuality in the manner he thought best; and then at last to come before the public fully matured and with his æsthetic purpose absolutely defined. When the first fruits of this long spell of assiduous effort were seen at the Grosvenor Gallery, Burne-Jones became instantly a power in the art world. The judgment of the few connoisseurs who had hailed "The Wine of Circe" and "Love among the Ruins" as works of the utmost significance, and as revelations of real genius, received wide endorsement; and though some people who were out of sympathy with the spirit of his art were quite ready to attack what they did not understand, their voices were scarcely heard amid the general chorus of approval. Indeed, for such pictures as "The Days of Creation," "The Mirror of Venus," and "The Beguiling of Merlin," exhibited in 1877; "Laus Veneris," "Chant d'Amour," and "Pan and Psyche," which with some others were shown in 1878; the series of four subjects from the story of "Pygmalion and the Image," and the magnificent "Annunciation," in 1879; and that exquisite composition, "The Golden Stairs," which was his sole contribution to the Grosvenor Gallery in 1880, nothing but enthusiastic approval was to be expected from all sincere art lovers; to carp at work so noble in conception and so personal in manner implied an entire want of artistic discretion. There were two exhibitions at the Grosvenor Gallery in 1881. In the summer one Burne-Jones was not represented, but the winter show included a number of his studies and decorative drawings, among them the large circular panel, "Dies Domini," a water-colour of rare beauty which can be reckoned as one of the most admirable of his designs. In 1882, however, he showed "The Mill," "The Tree of Forgiveness," "The Feast of Peleus," and several smaller paintings; and in 1883 that splendid piece of symbolism, "The Wheel of Fortune," and "The Hours." The following year is memorable for the appearance of the important canvas, "King Cophetua and the Beggar Maid," and the less ambitious but even more fascinating "Wood Nymph," in both of which the artist touched quite his highest level of
- 60. achievement, and gave the most ample proof of the maturity of his powers. PLATE VII.—DANAE (The Tower of Brass) (Glasgow Corporation Art Gallery)
- 61. Like the "Sibylla Delphica" this canvas shows how Burne-Jones was accustomed to treat subjects from the classic myths in the mediæval spirit to which he inclined by habit and association. In his illustration of a subject from the story of Danae, where she stands watching in wonder the building of the tower of brass which was to be her prison, he has looked at Greek tradition in a way that was partly his own and partly a reflection of William Morris; but the result is none the less persuasive because it does not conform to the Greek convention. His election as an Associate of the Royal Academy came in 1885. That he coveted this particular distinction can scarcely be said; indeed, he was at first unwilling to accept it, and it was only in response to a personal request from Leighton that he finally decided to take his place in the ranks of the Associates. But he exhibited a picture at Burlington House in 1886, "The Depths of the Sea," and then, feeling that his work was unsuited for the Academy galleries, he sent nothing else there, and in 1893 resigned his Associateship. His contributions to the Grosvenor Gallery in 1886 were "The Morning of the Resurrection," "Sibylla Delphica," and "Flamma Vestalis"; and in 1887 "The Baleful Head," "The Garden of Pan," and some other canvases. After this year he ceased to exhibit at the Grosvenor Gallery, as he was one of the chief members of the group of artists who supported Mr. Comyns Carr and Mr. C. E. Hallé in the founding of the New Gallery, and he sent there nearly all the works he produced during the rest of his life. The most important exceptions were the magnificent "Briar Rose" series of pictures, which were shown in 1890 by Messrs. Agnew at their gallery in Bond Street, and "The Bath of Venus," which went straight from the artist's studio to the Glasgow Institute in 1888. The first exhibition at the New Gallery was opened in 1888, and it included several of his oil-paintings, among them "The Tower of
- 62. Brass," an enlarged repetition of an earlier picture, and two canvases, "The Rock of Doom" and "The Doom Fulfilled," from the "Story of Perseus" series, to which also belonged "The Baleful Head," shown in the previous year. To the succeeding shows there he sent much besides that can be taken as representing his soundest convictions. There were the large water-colour, "The Star of Bethlehem," and the "Sponsa di Libano," in 1891; "The Pilgrim at the Gate of Idleness" and "The Heart of the Rose" in 1893; "Vespertina Quies" and the oil version of "Love among the Ruins" in 1894; "The Wedding of Psyche" in 1895; "Aurora" and "The Dream of Launcelot at the Chapel of the San Graal" in 1896; "The Pilgrim of Love" in 1897; and "The Prioress' Tale" and "St. George" in 1898. In all of these his consistent pursuit of definite ideals, his love of poetic fantasy, and his admirable perception of the decorative possibilities of the subjects he selected are as evident as in any of his earlier works; as years went on he relaxed neither his steadfastness of purpose nor his sincerity of method. To the last he remained unspoiled by success and unaffected by the popularity which came to him in such ample measure—it may be safely said that with his temperament and his artistic creed he would have continued on the course he had marked out for himself even if the effect of his persistence had been to rouse the bitterest opposition of the public, and he was as little inclined to trade on his success as he would have been to tout for attention if his efforts had been ignored. There was no waning of his powers as his career drew towards its close. It was not his fate to be compelled by failing vitality to be content with achievements that lacked the force and freshness by which the work of his vigorous maturity was distinguished, for he died before advancing years had begun in any way to dull his faculties. Only a few weeks after the opening of the 1898 exhibition at the New Gallery he was seized with a sudden illness, which had a fatal termination on the morning of June 17. Really robust health he had never enjoyed, and on several occasions serious breakdowns had hampered his activity; but his devotion to his art was so sincere, and his determination so strong, that these interruptions did not
- 63. perceptibly affect the continuity of his work. Towards the end of his life, however, he suffered from an affection of the heart, and the demands which he made upon his strength helped, no doubt, to exhaust his vitality. At the time of his death he was striving to complete one of the most important and ambitious pictures he ever planned—"Arthur in Avalon," a vast canvas which, even in its unfinished condition, must be reckoned as an amazing performance, and worthy of a distinguished place in the record of modern art. One of the most interesting things in the life-story of Edward Burne- Jones is the manner of his advance, within some twenty years only, from a position of obscurity to one of exceptional authority in the British school. The young student, who in 1855 had just discovered his vocation and was beginning to feel his way under the guidance of Rossetti, had become in 1877 one of the most discussed of British artists, and had with dramatic suddenness entered into the company of the greatest of the nineteenth-century painters. With no effort on his part to attract attention, without having recourse to any of those devices by which in the ordinary way popularity is won, he secured, practically at the first time of asking, all that other men have had to strive for laboriously through a long period of probation. Although the few things he exhibited while he was a member of the Royal Water Colour Society were sufficient to rouse in the few real judges a deep interest in his future achievement, it was the singular merit of his contributions to the first exhibition at Grosvenor Gallery that made him instantly famous. The wider public realised then, and realised most forcibly, that he was an artist to be reckoned with, and that his work, whether people liked it or not, could by no means be ignored.
- 64. PLATE VIII.—THE ENCHANTMENTS OF NIMUE (South Kensington Museum) Painted, like the "Sidonia von Bork," while Burne-Jones was still under the influence of Rossetti, "The Enchantments of Nimue" is interesting as an example of his earliest methods. It was finished in 1861, but it
- 65. was not exhibited until 1865, when it was hung in the Gallery of the Royal Society of Painters in Water Colours; it was bought for the South Kensington Museum in 1896. The painting shows how Nimue "caused Merlin to pass under a heaving-stone into a grave" by the power of her enchantments. From that time onwards there was for him no looking back. The twenty years of preparation, which were spent mainly in ceaseless seeking after completer knowledge and in careful study of the practical details of his profession, were followed by another twenty years of strenuous production, in which he worked out more and more effectively the ideas formed in his extraordinarily active mind. In the series of his paintings there is a very perceptible advance year by year in technical facility, but to suggest that they show also a growth of imaginative power would scarcely be correct, because there seems to have been no moment in his career when he did not possess in fullest measure the faculty of poetic invention and the capacity to put his mental images into an exquisite and persuasive shape. What he acquired as a result of his exhaustive study was a closer agreement between mind and hand, the skill to convey to others what he himself felt. But he had no need to labour to make his intelligence more keen or his fancies more varied; nature had endowed him with a temperament perfectly adapted for every demand which he could make upon it in the pursuit of his art. That he did not at first secure the unanimous approval of art lovers is scarcely surprising. The markedly individual artist who cares nothing for popular favour and is more anxious to satisfy his own conscience than to gather round him possible clients is never likely to become a favourite offhand. Burne-Jones by the brilliancy of his ability silenced all opposition long before his death, and gained over the bulk of the doubters who questioned his right to the admiration he received when he first began to exhibit at the Grosvenor Gallery. But for some while the unusual character of his art caused it to be much misunderstood by people who had not taken the trouble to
- 66. analyse his intentions. He was accused of affectation, of deliberate imitation of the early Italians; he was attacked for his indifference to realism and for his decorative preferences. Even the genuineness of his poetic feeling was suspected, and his love of symbolism was ridiculed as the aberration of a warped mind. Much of this misconception was cleared away by the collected exhibition of his works which was held at the New Gallery in the winter of 1892- 1893, for this show, by bringing together the best of his productions and by summing up all phases of his practice, proved emphatically that he had been as sincere and logical in his aims as he had been consistent in his expression. It was no longer possible to attack him out of mere prejudice; the verdict given fifteen years before on his art by those who understood him best was seen to be just. When a second collection was shown at the New Gallery—a memorial exhibition arranged in 1898, a few months after his death—few people remained who were prepared to dispute his mastery. It is fortunate that justice should have been done to him by his contemporaries and that there should have been really so little delay in the wider acknowledgment of his claims. If appreciation had been withheld from him while he lived, if it had been his fate to secure only a posthumous reputation, there would have been some diminution of his influence, and his art would have lost some of its authority. But as a right estimate of his position was arrived at during his lifetime, when he was at the height of his activity as an exponent of an exceptionally intelligent æsthetic creed, he was able to make his beliefs effective in bringing about the conversion of a large section of the public to a truer understanding of the value of decorative qualities in pictorial art. He proved emphatically that decoration does not imply, as is popularly supposed, the abandonment of the characteristics which make a picture interesting; he showed that a subject can be legitimately treated so that it engages fully the sympathies of the average man, and yet can be kept from any descent into obviousness or commonplace conventionality. The painted story in his hands was no trivial anecdote; it was a motive by means of which he conveyed not only
- 67. moral lessons but artistic truths as well, something didactically valuable but at the same time capable of appealing to the senses with exquisite daintiness and charm. Indeed, he can best be summed up as a teacher who clothed the lessons of life with noble beauty and with dignity that was commanding without being forbidding. There was human sympathy in everything he painted—a tender, gentle sentiment which escaped entirely the taint of sentimentality and which, tinged as it always was with a kind of quiet sadness, never became morbid or unwholesome. He was too truly a poet to dwell upon the ugly side of existence, just as he was too sincerely a decorator to insist unnecessarily upon common realities. That he searched deeply into facts is made clear by the mass of preparatory work he produced to guide him in his paintings, by the enormous array of drawings and studies which he executed to satisfy the demand he made upon himself for exactness and accuracy in the building up of his designs. But in his studies, as in his pictures, the intention to express a personal feeling is never absent. He selected, modified, re-arranged as his temperament suggested; he omitted unimportant things and amplified those which were of dominant interest; he sought for what was helpful to his artistic purpose and passed by what would have seemed in wrong relation, consistently keeping in view the lesson which he desired to teach. It can be frankly admitted that a certain mannerism resulted from his way of working, but this mannerism was by no means the dull formality into which many artists descend when they substitute a convention for inspiration; it was rather a revelation of his personality and of that belief in the rightness of his own judgment which counts for so much in the development of the really strong man. Except for the short time in which he was influenced by Rossetti, his life was spent in illustrating an entirely independent view of artistic responsibilities; and it would be difficult now to question this independence with the wonderful series of his paintings available to prove how earnestly and how seriously he strove to realise his ideals in art.
- 68. The plates are printed by Bemrose & Sons, Ltd., Derby and London The text at the Ballantyne Press, Edinburgh
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