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A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Lusanna, L., author.
Title: Non-inertial frames and Dirac observables in relativity / Luca Lusanna
(National Institute for Nuclear Physics(INFN), Firenze).
Description: Cambridge ; New York, NY : Cambridge University Press, [2019] |
Series: Cambridge monographs on mathematical physics | Includes
bibliographical references and subject index.
Identifiers: LCCN 2019005996 | ISBN 9781108480826 (hardback : alk. paper)
Subjects: LCSH: Dirac equation. | Wave equation. | Quantum field theory. |
General relativity (Physics)
Classification: LCC QC174.45 .L87 2019 | DDC 530.12–dc23
LC record available at https://lccn.loc.gov/2019005996
ISBN 978-1-108-48082-6 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-10-320.jpg)
![Preface xv
view like in SR. Moreover, the inclusion of particle physics requires the possibility
of introducing a notion of Poincaré algebra. These requirements select a family
of Einstein space-times: the globally hyperbolic, asymptotically flat at spatial
infinity, and without super-translations. In them there are the asymptotic ADM
Poincaré generators: When one switches off the Newton constant they tend to
the Poincaré generators of SR.
This class of singularity-free Einstein space-times contains asymptotic inertial
observers to be identified with the fixed stars of astronomy. Moreover, the ADM
energy generates a real temporal evolution modulo the gauge freedom associated
with the first-class constraints. As a consequence there is not a frozen picture
like in loop QG, where the 3-spaces are compact manifolds without boundary,
implying the absence of a Poincaré algebra.
To be able to include fermions, Einstein 4-metric will be decomposed upon
tetrads and the ADM action will be assumed to depend on them. This enlarges
the gauge group and the number of first-class constraints due to the extra gauge
freedom to orientate three gyroscopes and freedom in how they are transported
along time-like trajectories.
The big open problem is to find the DOs of Einstein metric and tetrad gravity,
since no one is able to solve the super-Hamiltonian and super-momentum con-
straints of GR. The best that can be done is to find a Shanmugadhasan canonical
transformation adapted to all the first-class constraints except the four unsolved
constraints.
However, this allows making a preliminary separation before trying to solve
the Hamilton equations between physical canonical variables describing tidal
effects (the gravitational waves of the linearized theory) and generalized gauge
variables describing relativistic inertial effects. The gauge fixing of these inertial
gauge variables (including a clock synchronization convention for the choice of
the instantaneous 3-spaces) fixes a global non-inertial frame with well-defined
relativistic inertial forces, so that the Hamilton equations become deterministic.
As a consequence, one can study post-Minkowskian (PM) and then post-
Newtonian (PN) approximations of Einstein equations in the presence of point
particles and of the electromagnetic field, with implications for dark matter.
Due to the relevance of DB theory of constraints, in the third part of the book
there is a review of what is known on such a theory and on the state of the art
in the search for DOs.
It is assumed that readers of this book of relativistic analytical mechanics
have a good knowledge of SR and GR and of their mathematical, algebraic,
and canonical formalisms, so there is no review of them. See Refs. [1–14] for
the status of SR and GR and Refs. [15–17] for the status of the experiments on
which they are based. See Refs. [18, 19] for some information on the status of
QG. References [20–30] contain the main literature about constraint theory.
This book is the final result of many years of research, mainly together with
Dr. David Alba (now a teacher in high schools) and Professor Horace W. Crater,](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-19-320.jpg)
![xvi Preface
who died last year, meaning I could not ask him to be my co-author. I thank
Professor Massimo Pauri for showing me the relevance of trying to solve foun-
dational problems in relativity and for the pleasure of being his collaborator in
many works.
Index notation: Greek indices μ, ν, . . . have the values 0, 1, 2, 3 (0 denotes
the time axis), while Latin indices i, j, . . . have the values 1, 2, 3. The flat
metric 4
ημν = (1; 0, 0, 0) of the inertial frames in SR has = +1 in the particle
physics convention and = −1 in the general relativity convention. The symbol
≈ means Dirac weak equality, while the symbol
◦
= means evaluated by using
the equations of motion. The symbol ≡ means identically equal. By convention,
repeated indices are summed. αβγδ and ijk are completely antisymmetric tensors
with 0123 = 123 = 123
= 1.
Dimensions of the quantities appearing in this book: [τ] = [xμ
] = [
σ] = [
ηi] =
[l], [
κi] = [Pμ
] = [E/c] = [m l t−1
], [4
g] = [3
g] = [n] = [n(a)] = [3
e(a)r] =
[ ˙
ηi] = [θi] = [0], [G = 6.7 × 10−8
cm3
s−2
g−1
] = [m−1
l3
t−2
], [G/c3
] = [m−1
t] ≈
2.5×10−39
sec/g, [S] = [] = [JAB
] = [m l2
t−1
], [3
R] = [3
Ωrs(a)] = [l−2
], [3
ωr(a)] =
[3
Krs] = [l−1
], [3
πr
(a)] = [3
Π̃rs
] = [m l−1
t−1
], [TAB
] = [M] = [Mr] = [H] =
[H(a)] = [m l−2
t−1
].](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-20-320.jpg)
![Part I
Special Relativity
Minkowski Space-Time
In this first part, after a review of inertial and non-inertial frames in the non-
relativistic Galilei space-time, I will study such frames in the Minkowski space-
time of special relativity (SR).
In Newtonian physics, time and space are absolute notions whose metrological
units are defined by means of standard clocks and rods, whose structure is not
specified. This is satisfactory for the non-relativistic quantum mechanics used
in molecular physics and in quantum information, where gravitation effects are
described by Newtonian gravity.
However, in atomic physics one needs the description of light, whose quantum
nature gives rise to the notion of massless photons whose trajectories do not exist
in Galilei space-time. Moreover particle physics has to face high-speed objects.
As a consequence, the Minkowski space-time of SR has to be introduced and a
new type of metrology has been developed with different standards for length and
time [31]. See references [32]–[38] for updated reviews on relativistic metrology
on Earth, in the Solar System, and in astronomy.
The fundamental theoretical scale for time is the SI (International System of
Units) atomic second, which is the duration of 9 192 631 770 periods of the
radiation corresponding to the transition between two hyperfine levels of the
ground state of the cesium 133 atom at rest at a temperature of 0K. In practice
one uses the International Atomic Time (TAI), defined as a suitable weighted
average of the SI kept by (mainly cesium) atomic clocks in about 70 laboratories
worldwide.
To introduce a convention for the synchronization of distant clocks one uses
the notion of two-way (or round-trip) velocity of light c involving only one clock:
The observer emits a ray of light that is reflected somewhere and then reabsorbed
by the observer, so that only the clock of the observer is used to measure the time
of flight of the ray. It is this velocity that is isotropic and constant in SR (the
light postulate) and replaces the standard of length in relativistic metrology. The
one-way velocity of light from one observer A to an observer B has a meaning](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-21-320.jpg)
![1
Galilei and Minkowski Space-Times
In this chapter we review the properties the non-relativistic Galilei space-time
and of the relativistic Minkowski one. See Refs. [39–43] for a detailed study of
the rotation group and of the kinematical Galilei and Poincaré groups connecting
the inertial frames of the respective space-times.
1.1 The Galilei Space-Time of Non-Relativistic Physics
and Its Inertial and Non-Inertial Frames
In Newtonian physics the notions of time and space are absolute, so that the
chrono-geometrical structure of Galilei space-time is not dynamical. One has at
each instant of the absolute time t, registered by an ideal clock, an instantaneous
Euclidean 3-space R3
t with the standard notion of Euclidean distance, measured
with ideal rods. The clocks in each point of R3
t are synchronized at the time t,
so that Galilei space-time has the structure R × R3
, where R denotes the time
axis and R3
is an abstract Euclidean 3-space associated to the fixed stars of
astronomy. As a consequence, one can parametrize Galilei space-time as the
straight trajectory of an inertial observer (the time axis) endowed with a foliation
of Euclidean 3-spaces orthogonal to the time axis.
The Galilei relativity principle assumes the existence of preferred rigid inertial
frames of reference in uniform translational motion, one with respect to the other
with inertial Cartesian coordinates (t, xi
) centered on an inertial observer, whose
trajectory is the time axis. In these frames, free bodies move along straight lines
(Newton’s first law) and Newton’s equations take the simplest form. The laws of
nature are covariant and there is no preferred observer. The connection between
different inertial frames is realized with the kinematical Galilei transformations:
t
= t + , x
i
= Rij
xj
+ vi
t + i
, where and i
are the time and space rigid
translations, R (R−1
= RT
, the transposed matrix) is the O(3) matrix describing
rigid rotations, and vi
are the parameters of the rigid Galilei boosts.
Due to the absolute nature of Newtonian time, the points on a t = const. sec-
tion of Galilei space-time are all simultaneous (instantaneous absolute 3-space),](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-23-320.jpg)
![4 Galilei and Minkowski Space-Times
whichever inertial system we are using. As a consequence, the causal notions of
before and after a certain event are absolute.
A particle of mass m has the trajectory described by inertial Cartesian
3-coordinates xi
m(t) in Galilei space-time. In the Hamiltonian phase space it
has the momentum pi = m δij
d xi(t)
dt
. For a free particle the Galilei generators
are H =
p2
/2 m (energy), pi (momentum), Ki = m δij xj
m − t pi (boost),
Ji = ijk δjh xh
m pk (rotation). ijk
is the completely antisymmetric tensor.
For a system of mutually interacting N particles of mass mk, trajectory xi
k(t),
momenta pk i(t) = mk δij
d x
j
k
(t)
dt
, k = 1, . . . , N, the Galilei generators are H =
N
k=1
p2
k
2 mk
+ V (
xh(t) −
xk(t)),
P =
N
k=1
pk,
J =
N
k=1
xk(t) ×
pk(t),
K =
N
k=1 (t
pk(t)−mk
xk(t)) = t
P −m
x. Here,
x =
N
k=1
mk
m
xk(t) (m =
n
k=1 mk)
is the Newton center of mass, whose conjugate variable is
P. Therefore, the
conserved Galilei boosts identify the Newtonian center of mass.
Usually the interacting potential depends only on the relative distances of the
particles (and not on their velocities) and appears only in the energy (the Hamil-
tonian) and not in the boosts differently from what happens at the relativistic
level with the Poincaré group.
For isolated N-body systems the ten generators of the Galilei group are
Noether constants of motion. The Abelian nature of the Noether constants
(the 3-momentum) associated to the invariance under translations allow making
a global separation of the center of mass from the relative variables (usually
the Jacobi coordinates, identified by the centers of mass of subsystems, are
preferred): In phase space this can be done with canonical transformations of
the point type both in the coordinates and in the momenta.
Instead, the non-Abelian nature of the Noether constants (the angular momen-
tum) associated with the invariance under rotations implies that there is no
unique separation [44] of the relative variables in six orientational ones (the body
frame in the case of rigid bodies) and in the remaining vibrational (or shape)
ones. As a consequence, an isolated deformable body or a system of particles
may rotate by changing the shape (the falling cat, the diver).
In Refs. [45, 46] there is a kinematical treatment of non-relativistic N-body
systems by means of canonical spin bases and of dynamical body frames, which
can be extended to the relativistic case in which the notions of Jacobi coordinates,
reduced masses, and tensors of inertia are absent and can be recovered only when
extended bodies are simulated with multipolar expansions [47].
Another non-conventional aspect of non-relativistic physics is the many-time
formulation of classical particle dynamics [48] with as many first-class constraints
as particles. Like in the special relativistic case, a distinction arises between
physical positions and canonical configuration variables and a non-relativistic
version of the no-interaction theorem (see Chapter 3) emerges.
See Refs. [49, 50] for Newtonian gravity, where the Newton equivalence prin-
ciple states the equality of inertial and gravitational mass, as a gauge theory of
the Galilei group.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-24-320.jpg)
![1.2 The Minkowski Space-Time 5
To define rigid non-inertial frames, let us consider an arbitrary accelerated
observer whose Cartesian trajectory is yi
(t) and let us introduce the rigid non-
inertial coordinates (t, σi
) by imposing xi
= yi
(t) + Rij
(t) σj
, where R(t) is
a time-dependent rotation matrix, which can be parametrized with three Euler
angles. It describes the rigid rotation of the non-inertial frame. It is convenient to
write the 3-velocity of the accelerated observer in the form vi
(t) = Rij
(t) d yj(t)
dt
.
The angular velocity of the rotating frame is Ωi
(t) = 1
2
ijk
Ωjk
with Ωjk
(t) =
−Ωkj
= (d R(t)
dt
RT
(t))jk
.
A particle of mass m with trajectory given by the Cartesian 3-coordinates
xi
m(t) is described in the rigid non-inertial frames by 3-coordinates ηr
(t) such
that xi
m(t) = yi
(t) + Rij
(t) ηj
(t).
As shown in every book on Newtonian mechanics, a particle moving in
an external potential V (t, xk
m(t)) = Ṽ (t, ηr
(t)) has the equation of motion
m
d2 xi
m(t)
dt2 = −
∂ V (t,xk
m(t))
∂ xi
m
, whose form in the rigid non-inertial frames becomes
m
d2
η(t)
dt2
= −
∂ Ṽ (t, ηk
(t))
∂
η
− m
d
v(t)
dt
+
ω(t) ×
v(t) +
d
ω(t)
dt
×
η(t)
+2
ω(t) ×
d
η(t)
dt
+
ω(t) × [
ω(t) ×
η(t)]
. (1.1)
In this equation there are the standard Euler, Jacobi, Coriolis, and centrifugal
inertial (or fictitious) forces, proportional to the mass of the body, associated
with the acceleration of the non-inertial observer and with the angular velocity
of the rotating rigid non-inertial frame.
The extension to non-rigid non-inertial frames with coordinates (t, σi
)
(σr
are global non-Cartesian 3-coordinates) is done in Ref. [51] by putting
the Cartesian 3-coordinates xi
equal to arbitrary functions Ai
(t, σr
), well
behaved at spatial infinity: xi
= Ai
(t, σr
). This coordinate transformation
must be invertible with inverse σr
= Sr
(t, xi
). The invertibility conditions are
det J(t, σr
) 0, where Ja
r (t, σu
) = ∂ Aa(t,σu)
∂ σr is the three-dimensional Jacobian,
whose inverse is ˜
Jr
a(t, σu
) =
∂ Sr(t,xu)
∂ xa
xb=Ab(t,σu)
(Ja
r(t, σu
) ˜
Jr
b(t, σu
) = δa
b ,
˜
Js
a(t, σu
) Ja
r(t, σu
) = δs
r ). The group of Galilei transformations connecting
inertial frames is replaced by some subgroup of the 3-diffeomorphisms of the
Euclidean 3-space connecting the non-inertial ones. The quantum mechanics of
particles in non-rigid non-inertial frames is studied in Ref. [51].
1.2 The Minkowski Space-Time: Inertial Frames, Cartesian
Coordinates, Matter, Energy–Momentum Tensor,
and Poincaré Generators
The Minkowski space-time of special relativity (SR) is an affine 4-manifold
isomorphic to R4
with Lorentz signature in which neither time nor space are
absolute notions. As a consequence there is no unique notion of instantaneous
3-space and one needs some metrological convention about time and space to be](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-25-320.jpg)
![6 Galilei and Minkowski Space-Times
able to formulate particle physics in the laboratories on Earth in the approxima-
tion of neglecting gravity. The only intrinsic structure of Minkowski space-time
is the conformal one connected with the Lorentz signature: It defines the light-
cone as the locus of incoming and outgoing radiation.
There is no absolute notion of simultaneity: Given an event, all the points
outside the light-cone with vertices in that event are not causally connected
with that event (they have space-like separation from it), so that the notions
of before and after an event become observer-dependent. Therefore there is
no notion of an instantaneous 3-space, of a spatial distance, and of a one-way
velocity of light between two observers (the problem of the synchronization of
distant clocks). Instead, as already said, there is an absolute chrono-geometrical
structure: the light postulate saying that the two-way (or round-trip) velocity
of light c (only one clock is needed for its definition) is (1) constant and (2)
isotropic. Let us remark that the clocks are assumed to be standard atomic
clocks measuring proper time [52–54].
Einstein relativity principle privileges the inertial frames of Minkowski space-
time centered on inertial observers endowed with an atomic clock: Their trajec-
tories are the time axis in Cartesian coordinates xμ
= (xo
= c t; xi
) where the flat
metric tensor with Lorentz signature is 4
ημν = (1; −1, −1, −1). These inertial
frames are in uniform translational motion, one with respect to the other. All
special relativistic physical systems, defined in the inertial frames of Minkowski
space-time, are assumed to be manifestly covariant under the transformations
of the kinematical Poincaré group connecting the inertial frames. The laws of
physics are covariant and there is no preferred observer.
The xo
= const. hyper-planes of inertial frames are usually taken as Euclidean
instantaneous 3-spaces, on which all the clocks are synchronized. They can be
selected with Einstein’s convention for the synchronization of distant clocks to
the clock of an inertial observer. This inertial observer A sends a ray of light at
xo
i to a second accelerated observer B, who reflects it toward A. The reflected ray
is reabsorbed by the inertial observer at xo
f . The convention states that the clock
of B at the reflection point must be synchronized with the clock of A when it
signs 1
2
(xo
i +xo
f ). This convention selects the xo
= const. hyper-planes of inertial
frames as simultaneity 3-spaces and implies that only with this synchronization
the two-way (A–B–A) and one-way (A–B or B–A) velocities of light coincide and
the spatial distance between two simultaneous point is the (3-geodesic) Euclidean
distance. However, if observer A is accelerated, the convention can break down
due to the possible appearance of coordinate singularities.
Relativistic matter is defined in the relativistic inertial frames of Minkowski
space-time centered on inertial observers using Cartesian 4-coordinates. It is in
these frames that one defines the matter Lagrangian when it is known. Once
one has the Lagrangian L(x) of a matter system the energy–momentum tensor
is defined by replacing the flat 4-metric 4
ημν appearing in the Lagrangian with a
4-metric 4
gμν (x) like the one used in general relativity (GR), so that one gets a
new Lagrangian Lg(x) and by using the definition Tμν
(x) = − 2
√
−det4g(x)
δ Sg
δ4gμν (x)
,](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-26-320.jpg)
![1.3 The 1+3 Approach 7
where Sg =
d4
x Lg(x). In inertial frames with Cartesian 4-coordinates xμ
,
the Poincaré generators, assumed finite due to suitable boundary conditions at
spatial infinity, have the following expression: Pμ
=
d3
x Tμo
(xo
,
x), Jμν
=
d3
x [xμ
Tνo
(xo
,
x) − xν
Tμo
(xo
,
x)].
In Appendix A there are some properties of the Poincaré algebra and group. At
the Hamiltonian level the canonical Poincaré generators Pμ
, Jμν
satisfy the Pois-
son algebra {Pμ
, Pν
} = 0, {Pμ
, Jαβ
} = ημα
Pβ
−ημβ
Pα
, {Jαβ
, Jμν
} = Cαβμν
ρσ Jρσ
(Cαβμν
ρσ = ηαμ
δβ
ρ δν
σ + ηβν
δα
ρ δμ
σ − ηαν
δβ
ρ δμ
σ − ηβμ
δα
ρ δν
σ). If Jr
= −1
2
ruv
Juv
is
the generator of space rotations and Kr
= Jro
one of the boosts, the form of
the canonical Poincaré algebra becomes {Jr
, Js
} = rst
Jt
, {Kr
, Ks
} = rsu
Ju
,
{Jr
, Ks
} = {Kr
, Js
} = rsu
Ku
.
To describe point particles with spin, with electric charge and with antiparti-
cles of negative mass in a way that avoids self-reaction divergences at the classical
level and gives the correct quantum theory after quantization, one needs a semi-
classical approach, named pseudo-classical mechanics, in which these degrees
of freedom are described with Grassmann variables. In Appendix B there is a
review of this approach and of the needed mathematical tools.
For the detailed mathematical properties of Minkowski space-time, see any
book on SR, such as the recent ones of Gourgoulhon Ref. [12, 13].
1.3 The 1+3 Approach to Local Non-Inertial
Frames and Its Limitations
Since the actual time-like observers are accelerated, we need some statement
correlating the measurements made by them to those made by inertial observers,
the only ones with a general framework for the interpretation of their experiments
based on Einstein convention for the synchronization of clocks. This statement
is usually the hypothesis of locality, which can be expressed in the following
terms [55–60]: An accelerated observer at each instant along its world-line is
physically equivalent to an otherwise identical momentarily comoving inertial
observer, namely a non-inertial observer passes through a continuous infinity of
hypothetical momentarily comoving inertial observers.
While this hypothesis is verified in Newtonian mechanics and in those rela-
tivistic cases in which a phenomenon can be reduced to point-like coincidences
of classical point particles and light rays (geometrical optic approximation), its
validity is questionable with moving continuous media (for instance the consti-
tutive equations of the electromagnetic field inside them in non-inertial frames
are still unknown) and in the presence of electromagnetic fields when their
wavelength is comparable with the acceleration radii of the observer (the observer
is not “static” enough to be able to measure the frequency of such a wave). See
Refs. [61, 62] for a review of these topics.
The fact that we can describe phenomena only locally near the observer and
that the actual observers are accelerated leads to the 1+3 point of view (or
threading splitting) [63–70], which tries to solve this problem starting from](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-27-320.jpg)
![8 Galilei and Minkowski Space-Times
the local properties of an accelerated observer, whose world-line is assumed
to be the time axis of some frame. Given the world-line γ of the accelerated
observer, we describe it with Lorentzian coordinates xμ
(τ), parametrized with
an affine parameter τ, with respect to a given inertial system. Its unit 4-velocity is
uμ
γ (τ) = ẋμ
(τ)/
ẋ2(τ) [ẋμ
= dxμ
dτ
]. The observer proper time τγ(τ) is defined by
˙
x̃
2
(τγ) = 1 if we use the notations xμ
(τ) = x̃μ
(τγ(τ)) and uμ
(τ) = ũμ
(τγ(τ)) =
d x̃μ
(τγ)/d τγ , and it is indicated by a comoving standard atomic clock.
By a conventional choice of three spatial axes Eμ
(a)(τ) = Ẽμ
(a)(τγ(τ)), a =
1, 2, 3, orthogonal to uμ
(τ) = Eμ
(o)(τ) = ũμ
(τγ(τ)) = Ẽμ
(o)(τγ(τ)), the non-inertial
observer is endowed with an ortho-normal tetrad Eμ
(α)(τ) = Ẽμ
(α)(τγ(τ)), α =
0, 1, 2, 3. This amounts to a choice of three comoving gyroscopes in addition
to the comoving standard atomic clock. Usually the spatial axes are chosen to
be Fermi–Walker transported as a standard of non-rotation, which takes into
account the Thomas precession (see [71]).
Since only the observer 4-velocity is given, this only allows identification of
the tangent plane of the vectors orthogonal to this 4-velocity in each point of
the world-line. Since there is no notion of a 3-space simultaneous with a point
of γ and whose tangent space at that point is Rũ(τγ), this tangent plane is
identified with an instantaneous 3-space both in SR and GR (it is the local
observer rest-frame at that point). This identification is the basic limitation of
this approach because the hyper-planes at different times intersect each other at
a distance from the world-line depending on the acceleration of the observer so
that the approach works only in a world-tube whose radius is this distance. See
Refs. [63–70] for the definition of the (linear and rotational) acceleration radii
of the observer. At each point of γ with proper time τγ(τ), the tangent space to
Minkowski space-time in that point has the 1+3 splitting of vectors in vectors
parallel to ũμ
(τγ) and vectors lying in the three-dimensional (so-called local
observer rest-frame) subspace Rũ(τγ) orthogonal to ũμ
(τγ).
Then Fermi normal coordinates [72–76] are defined on each hyper-plane
orthogonal to the observer unit 4-velocity uμ
(τ) and are used to define a notion
of spatial distance. On each hyper-plane one considers three space-like geodesics
as spatial coordinate lines. However, this produces only local coordinates and
a notion of simultaneity valid only inside the world-tube. See Refs. [77–79] for
variants of this approach, all unable to avoid the coordinate singularity on the
world-tube.
To this type of coordinate singularities we have to add the singularities shown
by all the rotating coordinate systems (the problem of the rotating disk): In
all the proposed uniformly rotating coordinate systems the induced 4-metric
expressed in these coordinates has pathologies (the component 4
goo vanishes)
at the distance R from the rotation axis where ω R = c with ω being the
constant angular velocity of rotation. This is the horizon problem: At R the
time-like 4-velocity of a disk point becomes light-like, even if there is no real
horizon as happens for Schwarzschild black holes. Again, given the unit 4-velocity](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-28-320.jpg)
![1.3 The 1+3 Approach 9
field of the points of the rotating disk, there is no notion of an instantaneous
3-space orthogonal to the associated congruence of time-like observers, due to
the non-zero vorticity of the congruence [71] (see Section 2.1 for the definition
of vorticity). Due to the Frobenius theorem, the congruence is (locally) hyper-
surface orthogonal, i.e., locally synchronizable [71], if and only if the vorticity
vanishes. Moreover, an attempt to use Einstein convention to synchronize the
clocks on the rim of the disk fails and one finds a synchronization gap (see Refs.
[80–84] and the bibliographies of Refs. [61, 62] for these problems).
One does not know how to define the 3-geometry of the rotating disk, how
to measure the length of the circumference, and which time and notion of
simultaneity has to be used to evaluate the velocity of (massive or massless)
particles in uniform motion along the circumference.
The other important phenomenon connected with the rotating disk is the
Sagnac effect (see again Refs. [61, 62, 80–84]), namely the phase difference
generated by the difference in the time needed for a round-trip by two light rays,
emitted at the same point, one co-rotating and the other counter-rotating with
the disk. This effect, which has been tested for light, X-rays, and matter waves
(Cooper pairs, neutrons, electrons, and atoms) and must be taken into account
for the relativistic corrections to space navigation, has again an enormous
number of theoretical interpretations (both in SR and GR). Here the lack of
a good notion of simultaneity leads to problems of time discontinuities or
desynchronization effects when comparing clocks on the rim of the rotating disk.
In conclusion, in SR inertial frames are a limiting theoretical notion since,
also disregarding GR, all the observers on Earth are non-inertial. According
to the IAU 2000 Resolutions [32–35], for the physics in the solar system one
can consider the Solar System Barycentric Celestial Reference System (BCRS)
centered on the barycenter of the Solar System (with the axes identified by fixed
stars (quasars) of the Hypparcos catalog) as a quasi-inertial frame. Instead, the
Geocentric Celestial Reference System (GCRS), with origin in the center of the
geoid, is a non-inertial frame whose axes are non-rotating with respect to the
Solar Frame. Instead, every frame centered on an observer fixed on the surface of
Earth (using the yet non-relativistic International Terrestrial Reference System
[ITRS]) is both non-inertial and rotating. All these frames use notions of time
connected to TAI.
Let us also remark that the physical protocols (think of GPS) can establish
a clock synchronization convention only inside future light-cone of the physical
observer defining the local 3-spaces only inside it, in accord with the 1+3 point
of view.
This state of affairs and the need for predictability (a well-posed Cauchy
problem for field theory) lead to the necessity of abandoning the 1+3 point
of view and shifting to the 3+1 one. In this point of view, besides the world-line
of an arbitrary time-like observer, it is given a global 3+1 splitting of Minkowski
space-time, namely a foliation of it whose leaves are space-like hyper-surfaces.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-29-320.jpg)
![10 Galilei and Minkowski Space-Times
Each leaf is both a Cauchy surface for the description of physical systems and
an instantaneous Riemannian 3-space, namely a notion of simultaneity implied
by a clock synchronization convention different from Einstein’s one. Even if
it is unphysical (i.e., non-factual) to give initial data (the Cauchy problem)
on a non-compact space-like hyper-surface, this is the only way to be able to
use the existing existence and uniqueness theorems for the solutions of partial
differential equations like the Maxwell ones, needed to test the predictions of the
theory.1
Once we have given the Cauchy data on the initial Cauchy surface (an
unphysical process), we can predict the future with every observer receiving the
information only from his/her past light-cone (retarded information from inside
it; electromagnetic signals on it). As emphasized by Havas [86], the 3+1 approach
is based on Møller’s formalization [87, 88] of the notion of simultaneity.
For non-relativistic observers the situation is simpler, but the non-factual need
to give the Cauchy data on a whole initial absolute Euclidean 3-space is present
also in this case for non-relativistic field equations like the Euler equation for
fluids.
Moreover, to study relativistic Hamiltonian dynamics one has to follow its
formulation given by Dirac [89] with the instant, front (or light), and point forms
and the associated canonical realizations of the Poincaré algebra. In the instant
form, the simultaneity hyper-surfaces (Cauchy surfaces) defining a parameter
for the time evolution are space-like hyper-planes xo
= const., in the front form
hyper-planes x−
= 1
2
(xo
− x3
) = const. tangent to future light-cones, while
in the point form the future branch of a two-sheeted hyperboloid x2
0. In
a 6N-dimensional phase space for N scalar particles the ten generators of the
Poincaré algebra are classified into kinematical generators (the generators of
the stability group of the simultaneity hyper-surface) and dynamical generators
(the only ones to be modified with respect to the free case in the presence of
interactions) according to the chosen concept of simultaneity. While in the instant
and point forms there are four dynamical generators (in the former energy and
boosts, in the latter the 4-momentum), the front form has only three of them.
We will see that the 3+1 approach is the natural framework to implement the
instant form of relativistic Hamiltonian dynamics.
Let us add the final remark that both in the 1+3 and in the 3+1 approach we
call observer an idealization by means of a time-like world-line whose tangent
vector in each point is the 4-velocity of the observer. If the 4-velocity is completed
with a spatial triad to form a tetrad in each point of the world-line, we get
an idealized observer with both a clock and a gyroscope. While this notion is
compatible with the absolute metrology of SR, in GR it corresponds to a test
1 As far as we know the theorem on the existence and unicity of solutions has not yet been
extended starting from data given only on the past light-cone. See Ref. [85] for an attempt
to rephrase the instant form of dynamics in a form employing only data from the causal
past light-cone of the observer.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-30-320.jpg)
![1.3 The 1+3 Approach 11
observer. To describe dynamical observers we need a model with dynamical
matter in both cases. Therefore, an observer, or better a mathematical observer,
is an idealization of a measuring apparatus containing an atomic clock and
defining, by means of gyroscopes, a set of spatial axes (and then a, maybe ortho-
normal, tetrad with a convention for its transport) in each point of the world-line.
See Ref. [90] for properties of mathematical and dynamical observers.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-31-320.jpg)
![2
Global Non-Inertial Frames in Special Relativity
In this chapter I will describe the 3+1 approach to global non-inertial frames
in Minkowski space-time and the associated parametrized Minkowski theories
for the description of matter admitting a Lagrangian description in these
frames.
2.1 The 3+1 Approach to Global Non-Inertial Frames
and Radar 4-Coordinates
Assume that the world-line xμ
(τ) of an arbitrary time-like observer carrying a
standard atomic clock is given: τ is an arbitrary monotonically increasing func-
tion of the proper time of this clock. Then one gives an admissible 3+1 splitting
of Minkowski space-time, namely a nice foliation with space-like instantaneous
3-spaces Στ . It is the mathematical idealization of a protocol for clock synchro-
nization allowing the formulation of the Cauchy problem for the equations of
motion of fields: All the clocks in the points of Στ sign the same time of the
atomic clock of the observer. The observer and the foliation define a global
non-inertial reference frame after a choice of 4-coordinates. On each 3-space Στ
one chooses curvilinear 3-coordinates σr
having the observer as the origin. The
quantities σA
= (τ; σr
) are the Lorentz-scalar and observer-dependent “radar
4-coordinates,” first introduced by Bondi [62, 91, 92]. See Ref. [71] for the
mathematical aspects of curved spaces like Στ .
If xμ
→ σA
(x) is the coordinate transformation from the Cartesian
4-coordinates xμ
of an inertial frame centered on a reference inertial observer
to radar coordinates, its inverse σA
→ xμ
= zμ
(τ, σr
) defines the embedding
functions zμ
(τ, σr
) describing the 3-spaces Στ as embedded 3-manifolds into
Minkowski space-time. The induced 4-metric on Στ is the following functional
of the embedding:
4
gAB(τ, σr
) = zμ
A(τ, σr
) 4
ημν zν
B(τ, σr
), (2.1)](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-32-320.jpg)
, one has zμ
τ (τ, σr
) = [N lμ
+
Nr
zμ
r ](τ, σr
), with N(τ, σr
) = [zμ
τ lμ](τ, σr
) = 1 + n(τ, σr
) and Nr(τ, σr
) =
− 4
gτr(τ, σr
) being the lapse and shift functions respectively. The unit
normal lμ
(τ, σu
) (lμ(τ, σu
) zμ
r (τ, σu
) = 0, l2
(τ, σu
) = 1), and the space-
like 4-vectors zμ
r (τ, σu
) identify a (in general non-orthonormal) tetrad in
each point of Minkowski space-time. The tetrad in the origin (lμ
(τ, 0) (in
general non-parallel to the observer 4-velocity), zμ
r (τ, 0)) is a set of axes
carried by the observer; their τ-dependence implies a convention of transport
along the world-line. The lapse function measures the proper time interval
N(τ, σr
)dτ at z(τ, σr
) ∈ Στ between Στ and Στ+dτ . The shift functions
Nr
(τ, σr
) are defined so that Nr
(τ, σr
)dτ describes the horizontal shift on
Στ such that, if zμ
(τ + dτ, σr
+ dσr
) ∈ Στ+dτ , then zμ
(τ + dτ, σr
+ dσr
) ≈
zμ
(τ, σr
) + N(τ, σr
) dτlμ
(τ, σr
) + [dσs
+ Ns
(τ, σr
)dτ]zμ
s (τ, σr
).
In each point of the 3-spaces Στ , one introduces arbitrary triads 3
er
(a)(τ, σu
)
(i.e., three gyroscopes) and the conjugated cotriads 3
e(a)r(τ, σu
) ((3
er
a)
3
e(b)r)
(τ, σu
) = δab, (3
er
(a)
3
e(a)s)(τ, σu
) = δr
s ).
The components of the 4-metric 4
gAB(τ, σr
) and of the inverse 4-metric
4
gAB
(τ, σr
) ((4
gAC 4
gcb)(τ, σr
) = δA
B) can be parametrized in the following way
(a = 1, 2, 3; ā = 1, 2):
4
gττ (τ, σu
) =
N2
− Nr Nr
(τ, σu
),
− 4
gτr(τ, σu
) = Nr(τ, σu
) =
3
grs Ns
(τ, σu
),
3
grs(τ, σu
) = − 4
grs(τ, σu
) =
3
a=1
3
e(a)r
3
e(a)s
(τ, σu
)
=
φ̃2/3
3
a=1
e2
2
b̄=1
γb̄a Rb̄ Vra(θi
) Vsa(θi
)
(τ, σu
),
4
gττ
(τ, σu
) = N−2
(τ, σu
), 4
gτr
(τ, σu
) = −
Nr
N−2
(τ, σu
),
4
grs
(τ, σu
) = −
3
grs
−
Nr
Ns
N2
(τ, σu
),
with 3
grs
(τ, σu
) =
3
er
(a)
3
es
(a)
(τ, σu
). (2.2)
The quantity φ̃2
(τ, σu
) = det 3
grs(τ, σu
) = det [− 4
grs(τ, σu
)] = γ(τ, σu
)
appearing in the 3-metric is the 3-volume element on Στ . We have lμ
(τ, σu
)
= (φ̃−1
μ
αβγ zα
1 zβ
2 zγ
3 )(τ, σu
) = 1
N(τ,σu)
(1, Nr
(τ, σu
)), lμ(τ, σu
) = N(τ, σu
) 4
ημo =
N(τ, σu
) (1; 000), d4
z = φ̃(τ, σu
) dτ d3
σ =
γ(τ, σu) dτd3
σ, and
g(τ, σu) =
−det 4gAB(τ, σu) = N(τ, σu
)
γ(τ, σu).](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-33-320.jpg)
 are the three positive
eigenvalues of the 3-metric 3
grs(τ, σr
) (γāa are suitable numerical constants sat-
isfying
a γāa = 0,
a γāa γb̄a = δāb̄,
ā γāa γāb = δab − 1
3
).
Vra(θi
(τ, σr
)) are the components of a diagonalizing rotation matrix depending
on three Euler angles, θi
(τ, σr
), i = 1, 2, 3. The gauge Euler angles θr
give a
description of the 3-coordinate systems on Στ from a local point of view, because
they give the orientation of the tangents to the three 3-coordinate lines through
each point. However, as shown in appendix A of Ref. [94] and in Ref. [95], it is
more convenient to replace the three Euler angles with the first kind coordinates
θ̃i
(τ, σr
) (−∞ θ̃i
+∞) on the O(3) group manifold: With this choice we
have Vru(θ̃i
) = (e−
i T̂i θ̃i
)ru, where (T̂i)ru = rui.
Therefore, starting from the four independent embedding functions zμ
(τ, σr
),
one obtains the ten components 4
gAB of the 4-metric, which play the role of the
inertial potentials generating the relativistic apparent forces in the non-inertial
frame. For instance, the shift functions Nr(τ, σu
) = − 4
gτr(τ, σu
) describe iner-
tial forces of the gravito-magnetic type induced by the global non-inertial frame.
It can be shown [98–101] that the usual non-relativistic Newtonian inertial poten-
tials are hidden in these functions. The extrinsic curvature tensor 3
Krs(τ, σu
) =
[ 1
2 N
(Nr|s + Ns|r − ∂τ 4
grs)](τ, σu
), describing the shape of the instantaneous 3-
spaces Στ of the non-inertial frame as embedded 3-sub-manifolds of Minkowski
space-time, is a secondary inertial potential, functional of the ten inertial poten-
tials 4
gAB(τ, σr
).
In this framework a relativistic positive-energy scalar particle with world-line
xμ
o (τ) is described by 3-coordinates ηr
(τ) defined by xμ
o (τ) = zμ
(τ, ηr
(τ)), satis-
fying equations of motion containing relativistic inertial forces with the correct
non-relativistic limit as shown in Refs. [98–101]. Fields have to be redefined so
as to know the clock synchronization convention; for instance, a Klein–Gordon
field φ̃(xμ
) has to be replaced with φ(τ, σr
) = φ̃(zμ
(τ, σr
)).
The foliation is nice and admissible if it satisfies the following conditions:
1. N(τ, σr
) 0 in every point of Στ so that the 3-spaces never intersect, avoiding
the coordinate singularity of the Fermi coordinates of the 1+3 approach.
2. 4
gττ (τ, σr
) = (zμ
τ
4
ημν zν
τ )(τ, σr
) = (N2
− Nu Nu
)(τ, σr
) 0, so as to avoid
the coordinate singularity of the rotating disk, and with the positive-definite
3-metric 3
grs(τ, σu
) = − 4
grs(τ, σu
) having three positive eigenvalues (these
are the Møller conditions [87]).
3. All the 3-spaces Στ must tend to the same space-like hyper-plane at spatial
infinity with a unit normal μ
τ , which is the time-like 4-vector of a set of
asymptotic orthonormal tetrads μ
A (μ
A
4
ημν ν
B = 4
ηAB = (+ − −−)). These
tetrads are carried by asymptotic inertial observers and the spatial axes μ
r are
identified by the fixed stars of star catalogues. At spatial infinity the lapse
function tends to 1, the shift functions vanish, and the 4-metric becomes
Euclidean with the same behavior, as will be discussed in relation to general
relativity in Part II.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-34-320.jpg)
![16 Global Non-Inertial Frames in Special Relativity
the expression αi(τ,
σ) = F(σ) α̃i(τ), i = 1, 2, 3. The unit normal is lμ
= μ
τ =
const. and the lapse function is 1+n(τ, σu
) = (zμ
τ lμ)(τ, σu
) = μ
τ ẋμ(τ) 0.
In Eq. (2.5) one uses the notations Ω(r)(τ, σ) = R−1
(τ,
σ) ∂r R(τ, σ) and
R−1
(τ, σ) ∂τ R(τ, σ)
u
v
= δum
mvr
Ωr(τ,σ)
c
, with Ωr
(τ, σ) = F(σ) Ω̃(τ, σ)
n̂r
(τ, σ) being the angular velocity.1
The angular velocity vanishes at spatial
infinity and has an upper bound proportional to the minimum of the linear
velocity vl(τ) = ẋμ(τ) lμ
orthogonal to the space-like hyper-planes. When
the rotation axis is fixed and Ω̃(τ, σ) = ω = const., a simple choice for the
function F(σ) is F(σ) = 1
1+ ω2 σ2
c2
.2
To evaluate the non-relativistic limit for c → ∞, where τ = c t with
t the absolute Newtonian time, one chooses the gauge function F(σ) =
1
1+ ω2 σ2
c2
→c→∞ 1 − ω2 σ2
c2 + O(c−4
). This implies that the corrections to
rigidly rotating non-inertial frames coming from Møller conditions are of order
O(c−2
) and become important at the distance from the rotation axis where
the horizon problem for rigid rotations appears.
As shown in Ref. [62], global rigid rotations are forbidden in relativistic
theories, because, if one uses the embedding zμ
(τ, σu
) = xμ
(τ) + μ
r Rr
s(τ) σs
describing a global rigid rotation with angular velocity Ωr
= Ωr
(τ), then the
resulting gττ (τ, σu
) violates Møller conditions because it vanishes at σ = σR =
1
Ω(τ)
ẋ2(τ) + [ẋμ(τ) μ
r Rr
s(τ) (σ̂ × Ω̂(τ))r]2 −ẋμ(τ) μ
r Rr
s(τ) (σ̂ × Ω̂(τ))r
(σu
= σ σ̂u
, Ωr
= Ω Ω̂r
, σ̂2
= Ω̂2
= 1). At this distance from the rotation
axis the tangential rotational velocity becomes equal to the velocity of light
(the horizon problem of the rotating disk). Let us remark that even if in
the existing theory of rotating relativistic stars [102] one uses differential
rotations, notwithstanding that in the study of the magnetosphere of pulsars
often the notion of light cylinder of radius σR is still used.
3. The search of admissible 3+1 splittings with non-Euclidean 3-spaces is
much more difficult. The simplest case [90] is a parametrization of the
embeddings (Eq. 2.1) in terms of Lorentz matrices ΛA
B(τ, σ) →σ→∞ δA
B
(ΛA
C (τ, σ) 4
ηAB ΛB
D(τ, σ) = 4
ηCD) depending on σ =
r (σr)2 and with
ΛA
B(τ, 0) finite:3
zμ
(τ, σr
) = xμ
(τ) + μ
A ΛA
r (τ, σ) σr
→σ→∞ xμ
(τ) + μ
r σr
,
zμ
(τ, 0) = xμ
(τ) = xμ
o + μ
A fA
(τ),
1 n̂r(τ, σ) defines the instantaneous rotation axis and 0 Ω̃(τ, σ) 2 max
˙
α̃(τ),
˙
β̃(τ), ˙
γ̃(τ)
.
2 Nearly rigid rotating systems, like a rotating disk of radius σo, can be described by using a
function F(σ) approximating the step function θ(σ − σo).
3 It corresponds to the locality hypothesis of Ref. [55–60], according to which at each instant
of time the detectors of an accelerated observer give the same indications as the detectors
of the instantaneously comoving inertial observer.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-36-320.jpg)
![2.1 The 3+1 Approach 17
ẋμ
(τ) = μ
A
˙
fA
(τ) = μ
A α(τ) γx(τ)
1
βr
x(τ)
, ẋ2
(τ) = α2
(τ) 0,
fA
(τ) =
τ
o
dτ1 α(τ1) γx(τ1)
1
βr
x(τ)
, γx(τ) =
1
1 −
β2
x(τ)
. (2.6)
implying zμ
r (τ, σu
) = Λμ
a (τ, σu
) 3
e(a)r(τ, σu
) and 3
grs(τ, σu
) =
3
a=1 (3
e(a)r
3
e(a)s)
(τ, σu
).
The origin of the 3-coordinates σr
in the 3-spaces Στ is a time-like accelerated
observer with world-line xμ
(τ), whose instantaneous 3-velocity, divided by c, is
βx(τ) (|
βx(τ)| 1) and τ is the proper time of the observer when α(τ) = 1. The
4-velocity of the observer is uμ
(τ) = μ
A βA
x (τ)/ 1 −
β2
x(τ), βA
x (τ) = (1; βr
x(τ)).
The Lorentz matrix is written in the form Λ = B R as the product of a
boost B(τ, σ) and a rotation R(τ, σ), like the one in Eq. (2.5), (Rτ
τ = 1,
Rτ
r = 0, Rr
s = Rr
s). The components of the boost are Bτ
τ (τ, σ) = γ(τ, σ) =
1/ 1 −
β2(τ, σ), Bτ
r (τ, σ) = γ(τ, σ) βr(τ, σ), Rr
s(τ, σ) = δr
s + γ2 βr βs
1+γ
(τ, σ), with
βr
(τ, σ) = G(σ) βr
(τ). The Møller conditions are restrictions on G(σ) →σ→∞ 0
with G(0) finite, whose explicit form is given in Ref. [90] for the following two
cases: (1) boosts with small velocities; and (2) time-independent boosts.
See Ref. [99] for the description of the electromagnetic field and of phenomena
like the Sagnac effect and the Faraday rotation in this framework for non-
inertial frames. Moreover, the embedding (Eq. 2.5) has been used in Ref. [103]
on quantum mechanics in non-inertial frames.
Each admissible 3+1 splitting of space-time allows one to define two associated
congruences of time-like observers.
1. The congruence of the Eulerian observers with the unit normal lμ
(τ, σr
) =
zμ
A(τ, σr
) lA
(τ, σr
) to the 3-spaces embedded in Minkowski space-time as unit
4-velocity. The world-lines of these observers are the integral curves of the unit
normal and in general are not geodesics. In adapted radar 4-coordinates the
orthonormal tetrads carried by the Eulerian observers are lA
(τ, σr
), 4
◦
Ē
A
(a)(τ, σr
) =
(0; 3
eu
(a)(τ, σr
)), where 3
eu
(a)(τ, σr
) (a = 1, 2, 3) are triads on the 3-space.
If 4
∇ is the covariant derivative associated with the 4-metric 4
gAB(τ, σr
)
induced by a 3+1 splitting, the equation ((4
∇A)B
C = δB
C ∂A + 4
ΓB
AC ; 3
hAB
(τ, σr
) = 4
gAB(τ, σr
) − lA (τ, σr
)lB(τ, σr
))
4
∇A lB(τ, σr
) =
lA
3
aB + σAB +
1
3
θ 3
hAB − ωAB
(τ, σr
) (2.7)
defines the acceleration 3
aA
(τ, σr
) (3
aA
(τ, σr
) lA(τ, σr
) = 0), the expansion
θ(τ, σr
), the shear σAB(τ, σr
) = σBA(τ, σr
) (σAB(τ, σr
) lB
(τ, σr
) = 0), and
the vorticity or twist ωAB(τ, σr
) = −ωBA(τ, σr
) (ωAB(τ, σr
) lB
(τ, σr
) = 0) of
the Eulerian observers with ωAB(τ, σr
) = 0 since they are surface-forming by
construction.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-37-320.jpg)
![18 Global Non-Inertial Frames in Special Relativity
2. The skew congruence with unit 4-velocity vμ
(τ, σr
) = zμ
A(τ, σr
) vA
(τ, σr
) (in
general it is not surface-forming, i.e., it has a non-vanishing vorticity, like that of
a rotating disk). The skew congruence is defined by requiring that its observers
have the world-lines (integral curves of the 4-velocity) defined by σr
= const. for
every τ, because the unit 4-velocity tangent to the flux lines xμ
σo
(τ) = zμ
(τ, σr
o)
is vμ
σo
(τ) = zμ
τ (τ, σr
o)/
4gττ (τ, σr
o) (there is no horizon problem because it is
everywhere time-like in admissible 3+1 splittings). They carry contro-variant
orthonormal tetrads, given in Ref. [104], not adapted to the foliation, connected
in each point by a Lorentz transformation to the ones of the Eulerian observer
present in this point.
Let us add another motivation for using the 3+1 approach. Let us consider
the standard action for a system of N charged scalar particles plus the electro-
magnetic field:
S =
N
i=1
dτ
− mic
ẋ2
i (τ) − ei ẋμ
i (τ) Aμ(xi(τ))
−
1
4
d4
z Fμν
(z) Fμν (z)
= dτ
Lm + LI
+ d4
z Lem, (2.8)
where mi and ei are the masses and charges of the particles and Fμν (z) =
∂μ Aν (z) − ∂ν Aμ(z). The momenta are
piμ(τ) = −
∂(Lm + LI )(τ)
∂ẋμ
i
= mic
ẋμ
i (τ)
ẋ2
i (τ)
+ ei Aμ(xi(τ)),
πμ
(zo
,
z) = −
∂L(zo
,
z)
∂ ∂o Aμ(zo,
z)
= Foμ
(zo
,
z), (2.9)
and we get the following primary constraints:
φ̄i(τ) =
pi(τ) − ei A(xi(τ))
2
− m2
i c2
≈ 0,
πo
(zo
,
z) ≈ 0. (2.10)
Since the natural time parameter for the field degrees of freedom is zo
, while
the particle world-lines are parametrized by an arbitrary scalar τ, there is no
concept of equal time and it is impossible to evaluate the Poisson bracket of these
constraints. Also, due to the same reason, the Dirac Hamiltonian, which would
be H̄D = H̄c +
N
i=1 λi
(τ) χ̄i(τ) +
d3
zλo
(zo
,
z)πo
(zo
,
z) with H̄c the canonical
Hamiltonian and with λi
(τ), λo
(zo
,
z) Dirac’s multipliers, does not make sense.
This problem is present even at the level of the Euler–Lagrange equations: How
does one formulate a Cauchy problem for a system of coupled equations, some
of which are ordinary differential equations in the affine parameter τ along the
particle world-line, while the others are partial differential equations depending
on Minkowski coordinates zμ
?](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-38-320.jpg)
![2.2 Parametrized Minkowski Theory 19
The standard non-manifestly covariant approach uses zo
as the time parameter
by rewriting the action (Eq. 2.8) in the following form:
S = d4
z
−
N
i=1
dτ δ4
(xi(τ) − z)
mi c
ẋ2
i (τ) + ei ẋμ
i (τ) Aμ(z)
−
1
4
Fμν
(z) Fμν (z)
= d4
z
−
N
i=1
δ3
(
xi(zo
) −
z)
mi c
1 − (
d
xi(zo)
dzo
)2 + ei
Ao(z)
−
d
xi(zo
)
dzo
·
A(z)
−
1
4
Fμν
(z) Fμν (z)
,
(2.11)
where δ(xo
i (τ) − zo
) = δ(τ − fi(zo
)) / |
dxo
i
dτ
| has been used as a gauge fixing,
eliminating the variables xo
i . Since the po
i are determined by the mass-shell
constraints, we remain with six degrees of freedom for particle and no-particle
constraint. The net result of this prescription is to break τ-reparametrization
invariance.
This problem is due to the lack of a covariant concept of equal time between
field and particle variables, so that the 3+1 approach is needed.
2.2 Parametrized Minkowski Theory for Matter Admitting
a Lagrangian Description
In the global non-inertial frames of Minkowski space-time it is possible to describe
isolated systems (particles, strings, fields, fluids) admitting a Lagrangian formu-
lation by means of parametrized Minkowski theories [98, 99, 105]. The existence
of a Lagrangian, which can be coupled to an external gravitational field, makes
possible the determination of the matter energy–momentum tensor and of the ten
conserved Poincaré generators Pμ
and Jμν
(assumed finite) of every configuration
of the isolated system.
First of all, one must replace the matter variables of the isolated system with
new ones knowing the clock synchronization convention defining the 3-spaces
Στ . For instance a Klein–Gordon field φ̃(x) will be replaced with φ(τ, σr
) =
φ̃(z(τ, σr
)); and the same for every other field. Instead, for a relativistic particle
with world-line xμ
(τ), one must make a choice of its energy sign, then the
positive- (or negative-) energy particle will be described by 3-coordinates ηr
(τ)
defined by the intersection of its world-line with Στ : xμ
(τ) = zμ
(τ, ηr
(τ)). Differ-
ent from all the previous approaches to relativistic mechanics, the dynamical con-
figuration variables are the 3-coordinates ηr
(τ) and not the world-lines xμ
(τ) (to
rebuild them in an arbitrary frame one needs the embedding defining that frame).](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-39-320.jpg)
![20 Global Non-Inertial Frames in Special Relativity
Then, one replaces the external gravitational 4-metric in the coupled
Lagrangian with the 4-metric 4
gAB(τ, σr
), which is a functional of the embedding
defining an admissible 3+1 splitting of Minkowski space-time, and the matter
fields with the new ones knowing the instantaneous 3-spaces Στ .
Parametrized Minkowski theories are defined by the resulting Lagrangian
depending on the given matter and on the embedding zμ
(τ, σr
). The resulting
action is invariant under the frame-preserving diffeomorphisms τ → τ
(τ, σu
),
σr
→ σ
r
(σu
) first introduced in Ref. [106]. As a consequence, there are four
first-class constraints with exactly vanishing Poisson brackets (an Abelianized
analogue of the super-Hamiltonian and super-momentum constraints of canonical
gravity) determining the momenta conjugated to the embeddings in terms of the
matter energy–momentum tensor. These constraints ensure the independence
of the description from the choice of the foliation and with their help it is
possible to rebuild some kind of covariance (Wigner covariance, as we shall
see) notwithstanding the 3+1 splitting of flat space-time. This implies that the
embeddings zμ
(τ, σr
) are gauge variables, so that all the admissible non-inertial
or inertial frames are gauge-equivalent, namely physics does not depend on the
clock synchronization convention and on the choice of the 3-coordinates σr
– only
the appearances of phenomena change by changing the notion of instantaneous
3-space. Therefore, the embedding configuration variables zμ
(τ,
σ) will describe
all the possible inertial effects compatible with special relativity.
The matter energy–momentum tensor associated with this Lagrangian allows
the determination of the ten conserved Poincaré generators Pμ
and Jμν
of
every configuration of the system (in non-inertial frames they are asymptotic
generators at spatial infinity, like the ADM ones in general relativity). The
behavior of matter at spatial infinity on each 3-space Στ must be such that
the Poincaré generators are finite with a 4-momentum not space-like.
As an example, one may consider N free scalar particles with masses mi.
Usually the time-like world-lines of the particles are described by Cartesian
4-coordinates xμ
i (τi), depending on the particle proper time in an inertial frame.
At the Hamiltonian level one has that the 4-momenta piμ(τi) satisfy the mass-
shell first-class constraints p2
i (τi) − m2
i = 0. Therefore, there are two solutions
with opposite sign for the energies (particles and antiparticles), so that the time
components xo
i (τ) are gauge variables, sources of the problem of how to eliminate
the relative times in relativistic bound states.
In the 3+1 approach the time-like world-lines of the N particles are Cartesian
4-coordinates xμ
i (τ) parametrized with the time τ of Bondi radar 4-coordinates
so that we have xμ
i (τ) = zμ
(τ, ηr
i (τ)), i = 1, . . . , N. Therefore, each particle is
identified by the three numbers σr
= ηr
i (τ) individuating the intersection of each
world-line with the 3-space Στ , and not by four. As a consequence, each particle
must have a well-defined sign of the energy, ηi = sign po
i = ±. Each particle
with a definite sign of the energy will be described by the six Lorentz-scalar](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-40-320.jpg)
![2.2 Parametrized Minkowski Theory 21
canonical coordinates ηr
i (τ), κir(τ) at the Hamiltonian level, like in Newtonian
mechanics.
There are no longer mass-shell constraints, because we cannot describe the
two topologically disjoint branches of the mass hyperboloid simultaneously as
in the standard manifestly Lorentz-covariant approach. By using Eq. (2.1),
the particle 4-velocities can be written in the form ẋμ
i (τ) = N(τ, ηu
i (τ)) lμ
(τ, ηu
i (τ)) + [η̇r
i (τ) + Nr
(τ, ηu
i (τ))] zμ
r (τ, ηu
i (τ)), with ẋ2
i (τ) =
N2
− 3
grs
(Nr
+η̇r
i (τ)) (Ns
+η̇s
i )
(τ, ηu
i (τ)). Therefore the derived usual particle momenta,
satisfying p2
i − m2
i c2
≈ 0, are pμ
i (τ) = mi c ẋμ
i (τ)/
ẋ2
i (τ) =
ηi
m2
i c2 − 3grs(τ, ηu
i (τ)) κir(τ) κis(τ) lμ
(τ, ηu
i (τ)) + κir(τ) 3
grs
(τ, ηu
i (τ))
zμ
s (τ, ηu
i (τ)).
In parametrized Minkowski theories the free particles are described by the
following action depending on the configurational variables ηr
i (τ) of the particles
with energy sign ηi and on the embedding variables zμ
(τ, σr
) of an arbitrary
admissible 3+1 splitting of Minkowski space-time:
S = dτ d3
σ L(τ, σu
) = dτ L(τ),
L(τ, σu
) = −
N
i=1
δ3
(σu
− ηu
i (τ))
mic ηi [4gττ (τ, σu) + 2 4gτr(τ, σu) η̇r
i (τ) + 4grs(τ, σu) η̇r
i (τ) η̇s
i (τ)]
= −
N
i=1
δ3
(σu
− ηu
i (τ))
ηi mi c N2(τ, σu)+ 4grs(τ, σu) [η̇r
i (τ)+Nr(τ, σu)] [η̇s
i (τ)+Ns(τ, σu)].
(2.12)
This action is invariant under separate τ- and σr
-reparametrizations.
The resulting canonical momenta and their Poisson brackets are
ρμ(τ, σu
) = −
∂L(τ, σu
)
∂zμ
τ (τ, σu)
=
N
i=1
δ3
(σu
− ηu
i (τ)) ηi mi c
zτμ(τ, σu
) + zrμ(τ, σu
) η̇r
i (τ)
[4gττ (τ, σu) + 2 4gτr(τ, σu) η̇r
i (τ) + 4grs(τ, σu) η̇r
i (τ) η̇s
i (τ)]
= [(ρν lν
) lμ + (ρν zν
r ) 3
grs
zsμ](τ, σu
),
κir(τ) = −
∂L(τ)
∂η̇r
i (τ)
= 3
grs(τ, ηu
i (τ)) κi
s
(τ)
= ηi mi c
4gτr(τ, ηu
i (τ)) + 4grs(τ, ηu
i (τ)) η̇s
i (τ)
[4gττ (τ, ηu
i (τ))+2 4gτr(τ, ηu
i (τ)) η̇r
i (τ)+4grs(τ, ηu
i (τ)) η̇r
i (τ) η̇s
i (τ)]
,
{zμ
(τ, σu
), ρν (τ, σ
u
)} = − 4
ημ
ν δ3
(σu
− σ
u
),
{ηr
i (τ), κjs(τ)} = −δij δr
s . (2.13)](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-41-320.jpg)
 =
N
i=1
δ3
(σu
− ηu
i (τ))
φ̃(τ, σu)
κir(τ),
Trs(τ, σu
) =
zrμ zsν Tμν
(τ, σu
)
= [Nr Ns Tττ
+ (Nr
3
gsm + Ns
3
grm) Trm
+ 3
grm
3
gsn Tmn
](τ, σu
)
=
N
i=1
δ3
(σu
− ηu
i (τ))
φ̃(τ, σu)
ηi
κir(τ) κis(τ)
m2
i c2 + 3gvw(τ, σu) κiv(τ) κiw(τ)
.
(2.14)
The four first-class constraints implying the gauge nature of the embedding
and the gauge equivalence of the description in different non-inertial frames are:4
Hμ(τ, σu
) = ρμ(τ, σu
) − φ̃(τ, σu)
lμ T⊥⊥ − zrμ
3
grs
T⊥s
(τ, σu
) ≈ 0,
{Hμ(τ, σu
), Hμ(τ, σv
)} = 0,
d3
σ Hμ
(τ, σu
) = Pμ
−
N
i=1
lμ
(τ, ηu
i (τ)) ηi
m2
i c2 + 3grs(τ, ηu
i (τ)) κir(τ) κis(τ)
−
N
i=1
zμ
r (τ, ηu
i (τ)) 3
grs
(τ, ηu
i (τ)) κis(τ) ≈ 0. (2.15)
Since the canonical Hamiltonian vanishes, one has the Dirac Hamiltonian
(λμ
(τ, σu
) are Dirac’s multipliers):
4 Since the constraints (Eq. 2.15) are distributions concentrated at the position of particles,
in evaluating their Poisson brackets we must use the embeddings zμ(τ, σu) at a generic
point and not zμ(τ, ηu
i (τ)).](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-42-320.jpg)
![2.2 Parametrized Minkowski Theory 23
HD = d3
σ λμ
(τ, σu
) Hμ(τ, σu
)
= d3
σ
λμ(τ, σu
) lμ
(τ, σu
) Hλ(τ, σu
)
+λμ(τ, σu
) zμ
s (τ, σu
) 3
gsr
(τ, σu
) Hλ r(τ, σu
)
,
Hλ(τ, σu
) = (lμ Hμ
)(τ, σu
) ≈ 0,
Hλ r(τ, σu
) = (zr μHμ
)(τ, σu
) ≈ 0,
{Hλ r(τ, σu
), Hλ s(τ, σ
u
)} = Hλ r(τ, σ
u
)
∂δ3
(σu
, σ
u
)
∂σs
+ Hλ s(τ, σu
)
∂δ3
(σu
, σ
u
)
∂σr
,
{Hλ(τ, σu
), Hλ r(τ, σ
u
)} = Hλ(τ, σu
)
∂δ3
(σu
, σ
u
)
∂σr
,
{Hλ(τ, σu
), Hλ(τ, σ
u
)} =
3
grs
(τ, σu
) Hλ s(τ, σu
)
+3
grs
(τ, σ
u
) Hλ s(τ, σ
u
)
∂δ3
(σu
, σ
u
)
∂σr
, (2.16)
and one finds that {Hμ(τ, σu
), HD} = 0. Therefore, there are only the four
first-class constraints of Eq. (2.16). The constraints Hμ(τ, σu
) ≈ 0 describe
the arbitrariness of the foliation: Physical results do not depend on its choice.
In Eq. (2.16) we have also shown the non-holonomic form Hλ(τ, σu
) ≈ 0,
Hλ r(τ, σu
) ≈ 0 of the constraints, which satisfies the universal Dirac algebra
of the super-momentum and super-Hamiltonian constraints of ADM canonical
metric gravity (see Chapter 5).
The same description can be given when the matter are not particles but the
Klein–Gordon [107] and Dirac [108] fields and for the electromagnetic field [30],
starting by their usual description given, for instance, in Ref. [109].
To describe the physics in a given admissible non-inertial frame described by
an embedding zμ
F (τ, σu
), one must add the gauge fixings:
ζμ
(τ, σu
) = zμ
(τ, σu
) − zμ
F (τ, σu
) ≈ 0. (2.17)
If we put zμ
F (τ, σu
) = xμ
(τ) + bμ
r (τ) σr
, with xμ
(τ) a non-inertial observer
(origin of the 3-coordinates σr
) and with bμ
A(τ) an orthonormal tetrad such
that the constant (future-pointing) unit normal to the 3-spaces is lμ
= bμ
τ =
μ
αβγ bα
1 (τ) bβ
2 (τ) bγ
3 (τ) = const., we get a foliation of Minkowski space-time with
space-like hyper-planes.
If xμ
(τ) is the world-line of a time-like inertial observer, the gauge fixings with
zμ
F (τ, σu
) = xμ
(τ) + bμ
r (τ) σr
(2.18)
describe inertial frames in Minkowski space-time with zμ
τ (τ, σr
) ≈ ẋμ
(τ) +
ḃμ
r (τ) σr
and zμ
r (τ) ≈ bμ
r (τ). Usually the inertial frames have bμ
r (τ) = μ
r with](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-43-320.jpg)
![24 Global Non-Inertial Frames in Special Relativity
μ
r constant asymptotic triads. Moreover, we have 3
grs(τ) = bμ
r (τ) 4
ημν bν
s (τ) =
a
3
e(a)r(τ) 3
e(a)s(τ) = δrs
(in terms of cotriads inside Στ contained in the
chosen parametrization of bμ
r (τ)) with inverse 3
grs
(τ) =
a
3
er
(a)(τ) 3
es
(a)(τ)
(in terms of the dual triads). The shift and lapse functions have the expres-
sion Nr(τ, σu
) = − 4
gτr(τ, σu
) = (ẋμ
(τ) + ḃμ
s (τ) σs
) 4
ημν bν
r (τ), N2
(τ, σu
) =
(Nr
Nr)(τ, σ) + (ẋμ
(τ) + ḃμ
r (τ) σr
) 4
ημν (ẋν
(τ) + ḃν
s (τ) σs
).
The second-class constraints implied by the gauge fixings (Eq. 2.17) together
with the constraints Hμ(τ, σu
) ≈ 0 lead to the following Dirac brackets:
{f, g}∗
= {f, g} − d3
σ
{f, ζμ
(τ, σu
)} {Hμ(τ, σu
), g}
−{f, Hμ(τ, σu
)} {ζμ
(τ, σu
), g}
. (2.19)
The preservation in time of the gauge fixings ( d
dτ
ζμ
(τ, σu
) = {ζμ
(τ, σu
),
HD} ≈ 0) implies the following form of the Dirac multipliers: λμ
(τ, σu
) = λ̃μ
(τ)+
λ̃μ
ν (τ) bν
r (τ) σr
, with λ̃μ
(τ) = −ẋμ
(τ), λ̃μν
(τ) = 1
2
[ḃμ
r (τ) bν
r (τ) − bμ
r (τ) ḃν
r (τ)].
The space-like hyper-planes still depend on ten residual gauge degrees of
freedom: (1) the world-line xμ
(τ) of the inertial observer chosen as the origin of
the 3-coordinates σr
; and (2) six variables parametrizing an orthonormal tetrad
bμ
A(τ). Their ten conjugate variables are contained in the canonical momenta
ρμ(τ, σu
) conjugated to the embedding and are (1) the total 4-momentum Pμ
canonically conjugate to xμ
(τ), {xμ
, Pν
}∗
= − 4
ημν
; and (2) six momentum
variables canonically conjugate to the tetrads bμ
A(τ). They are determined by
the constraints (Eq. 2.15). Therefore, after this gauge fixing the Dirac Hamilto-
nian depends only on the following ten surviving first-class constraints implied
by Eq. (2.15):
HD = λ̃μ
(τ) H̃μ(τ) −
1
2
λ̃μν
(τ) H̃μν (τ),
H̃μ
(τ) = d3
σ Hμ
(τ,
σ) = Pμ
− lμ
N
i=1
ηi
m2
i c2 + 3grs(τ) κir(τ) κis(τ)
−bμ
r (τ)
N
i=1
3
grs
(τ) κis(τ) ≈ 0,
H̃μν
(τ) = d3
σ σr
bμ
r (τ) Hν
(τ,
σ) − bν
r (τ) Hμ
(τ,
σ)
= Sμν
− (bμ
r (τ) lν
− bν
(τ) lμ
)
N
i=1
ηr
i (τ))
ηi
m2
i c2 + 3grs(τ, ηu
i (τ)) κir(τ) κis(τ)
−(bμ
r (τ) bν
s (τ) − bν
r (τ) bμ
s (τ))
N
i=1
ηr
i (τ)) κis(τ) ≈ 0,](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-44-320.jpg)
![2.2 Parametrized Minkowski Theory 25
{H̃μ
(τ), H̃ν
(τ)} = {H̃α
(τ), H̃μν
(τ)} = 0,
{H̃μν
(τ), H̃αβ
(τ)} = Cμναβ
γδ H̃γδ
(τ), (2.20)
with Sμν
the spin part of the Lorentz generators whose form, implied by
Eqs. (2.14) and (2.18), is
Jμν
= xμ
(τ) Pν
− xν
(τ) Pμ
+ Sμν
,
Sμν
= d3
σ σr
bμ
r (τ) ρν
(τ,
σ) − bν
r (τ) ρμ
(τ,
σ)
, (2.21)
and with Cμναβ
γδ = δν
γ δα
δ
4
ημβ
+ δμ
γ δβ
δ
4
ηνα
− δν
γ δβ
δ
4
ημα
− δμ
γ δα
δ
4
ηνβ
being the
structure constants of the Lorentz algebra.
Since Pμ
is the total conserved 4-momentum of the isolated system, the
conjugate variable xμ
(τ) describes an inertial observer taking into account its
global properties, namely having the role of some kind of relativistic center of
mass of the isolated system (see next chapter).
From now on we will use the notation Pμ
= (Po
= E/c;
P) = M c uμ
(P) =
Mc (
1 +
h2;
h)
def
= Mc hμ
, where
h =
v/c is an a-dimensional 3-velocity and
with M c =
√
P2.
For an isolated system with total conserved time-like 4-momentum Pμ
, the
“inertial rest-frame” is the inertial 3+1 splitting with lμ
= Pμ
/
√
P2 and with
bμ
r (τ) = μ
r , where lμ
= μ
o and μ
r are constant orthonormal tetrads.
Instead, the “non-inertial rest-frames” of an isolated system with total con-
served time-like 4-momentum Pμ
are those admissible non-inertial 3+1 splittings
whose 3-spaces Στ tend to space-like hyper-planes perpendicular to Pμ
at spatial
infinity. In these non-inertial rest-frames the Poincaré generators are asymptotic
(constant of the motion) symmetry generators like the asymptotic ADM ones in
the asymptotically Minkowskian space-times of general relativity (see Part II).
The non-relativistic limit of the parametrized Minkowski theories gives the
parametrized Galilei theories, which are defined and studied in Ref. [51] using
also the results of Refs. [49, 50]. Also the inertial and non-inertial frames in
Galilei space-time of Section 1.1 are gauge-equivalent in this formulation. In
this approach, a non-relativistic particle of 3-coordinates xa
(t) in an inertial
frame is described in the non-inertial frames by 3-coordinates ηu
(t) defined
by the equation xa
(t) = Aa
(t, ηu
(t)), where Aa
is the function describing the
non-relativistic non-inertial frame (see after Eq. (1.1)). Therefore the standard
velocity takes the form
ẋa
(t) =
dAa
(t, ηu
(t))
dt
=
∂ Aa
(t, ηu
(t))
∂t
+ Ja
r (t, ηu
(t))
d ηu
(t)
dt
. (2.22)
In the case of N particles xa
i (t), i = 1, . . . , N, interacting with arbitrary poten-
tial V (t,
x1(t), . . . ,
xN (t)) the equation of motion mi
d2xa
i (t)
dt2 = − ∂ V
∂ xa
i
(t,
x1(t), . . . ,
xN (t)) take the form ( ˜
Jr
a is the inverse of Ja
r )](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-45-320.jpg)
![26 Global Non-Inertial Frames in Special Relativity
mi
d
dt
∂ Aa
(t, ηu
i (t))
∂t
+ Ja
r (t, ηu
i (t))
d ηr
i (t)
dt
= −
∂ V̂
∂ xa
i
|xu
i (t)=Au(t,ηv
i (t)) = − ˜
Jr
a (t, ηu
i (t))
j
∂ V̂
∂ ηr
j
, (2.23)
with V̂ = V
xi(t)=
A(t,
ηi(t)).
The Lagrangian of parametrized Galilei theory is
L(t) = d3
σ
ia
δ3
(σu
− ηu
i (t))
1
2mi
∂ Aa
(t, σu
)
∂ t
+ Ja
r (t, σu
)
dηr
i (t)
dt
2
− V̂ .
(2.24)
It depends on the particles and on the non-inertial frame variable Aa
(t, σu
).
As shown in Ref. [51] the action S =
dt L(t) is invariant under local Noether
transformations δ ηk
i (t) = Fa
(t, ηv
i (t)) ˜
Jr
a (t, ηv
i (t)), δ Aa
(t, σk
) = Fa
(t, σk
) with
Fa
(t, σk
) arbitrary functions.
Therefore, these are first-class constraints implying that the functions Aa
(t, σr
)
are gauge variables. As a consequence, the description of physics in inertial
and non-inertial frames is connected by gauge transformations, like in the
relativistic case.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-46-320.jpg)
![3
Relativistic Dynamics and the Relativistic
Center of Mass
In this chapter I face the problem of how to get a consistent description of
relativistic particle dynamics, eliminating the problem of relative times in rel-
ativistic bound states and clarifying the endless problem of which definition of
the relativistic center of mass has to be used.
After a review of the attempts to describe interacting relativistic particles
taking into account the no-interaction theorem, I will show that the dynamics
must be formulated in terms of suitable covariant relative variables after the
separation of an external canonical but not covariant center of mass.
The world-tube in Minkowski space-time, where the non-covariance effects are
concentrated, has an intrinsic radius, the Møller radius, determined by the value
of the Poincaré Casimir invariants associated to the given configuration of the
isolated system. It exists due to the Lorentz signature of Minkowski space-time.
It is a classical unit of length determined by the system itself, which exists also for
classical fields and is a natural candidate for a ultraviolet cutoff in quantization.
To describe the relative motions of an isolated system of interacting particles in
a covariant way in special relativity (SR) I use the inertial foliation of Minkowski
space-time in which the 3-spaces are space-like hyper-planes orthogonal to the
time-like (assumed finite) 4-momentum of the isolated system, namely its inertial
rest-frame.
This allows defining canonical bases of Wigner-covariant relative variables
inside the 3-spaces (named Wigner hyper-planes) of these rest-frames and defin-
ing the Wigner-covariant rest-frame instant form of dynamics. As a consequence,
I can develop a new Hamiltonian kinematics for positive-energy scalar and spin-
ning particles.
Before introducing these new developments, let us explore a brief historical
review of the theories involving interacting relativistic point particles.
The theory of relativistic bound states and the interpretational problems with
the Bethe–Salpeter equation [109] require the understanding of the instantaneous
approximations to quantum field theory so as to arrive at an effective relativistic](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-47-320.jpg)
![28 Relativistic Dynamics and Center of Mass
wave equation and to an acceptable scalar product. In turn, the wave equation
must also result from the quantization of a relativistic action-at-a-distance two-
body problem in relativistic mechanics (but with the particles interpreted as
asymptotic states of quantum field theory), since only in this way can we get a
solution to the interpretational problems connected to the gauge nature of the
relative times.
Usually the particle world-lines qμ
i (τi), i = 1, . . . , N, are parametrized with
independent affine parameters τi and the action principles describing them are
invariant under separate reparametrizations of each world-line, since this is geo-
metrically possible even in the presence of interactions with a finite time delay
to avoid instantaneous action at a distance. Since the dynamical correlation
among the points on the particle’s world-lines is not in general one-to-one in
these approaches, it is impossible to develop a Hamiltonian formulation starting
from the Euler–Lagrange integro-differential equations of motion implied by
the delay. The natural development of these approaches was field theory – for
instance the study of the coupled system of relativistic charged particles plus the
electromagnetic field.
As an alternative to field theory, there was the development of relativistic
mechanics with action-at-a-distance interactions described by suitable potentials
implying a one-to-one correlation among the world-lines [110–113]. Geometrically
each particle has its world-line described by a 4-vector (the 4-position) qμ
i (τi),
i = 1, . . . , N, parametrized with an independent arbitrary affine scalar parameter
τi
1
By inverting qo
i (τi) to get τi = τi(qo
i ), we can identify the world-line in a
non-manifestly covariant way with
qi =
qi(qo
i ): In this form they are named
predictive coordinates. The instant form amounts to putting qo
1 = . . . = qo
N = xo
and to describing the world-lines with the functions
qi(xo
). Each one of these
configuration variables has a different associated notion of velocity:
dq
μ
i (τi)
dτi
(or
dq
μ
i (τ)
dτ
),
d
qi(qo
i )
dqo
i
(predictive velocities),
d
qi(xo)
dxo ; and of acceleration:
d2q
μ
i (τi)
(dτi)2 (or
d2q
μ
i (τ)
dτ2 ),
d2
qi(qo
i )
(dqo
i )2 (predictive accelerations),
d2
qi(xo)
(dxo)2 .
Bel’s non-manifestly covariant predictive mechanics [114–117] is the attempt
to describe relativistic mechanics with N-time predictive equations of motion for
the predictive coordinates
qi(qo
i ) in Newtonian form: mi
d2
qi(qo
i )
(dqo
i )2
◦
=
Fi(qo
k,
qk(qo
k),
d
qk(qo
k)
dqo
k
). Since the left-hand side of these equations depends only
on qo
i , the predictive forces must satisfy the predictive conditions
d
Fi
dqo
k
= 0 for
k = i. Moreover they must be invariant under space translations and behave
like space three-vectors under spatial rotations. Finally, they must satisfy
1 The standard choice in the manifestly covariant approach with a 4N-dimensional
configuration space is τ1 = . . . = τN = τ. Another possibility is the choice of proper times
τi = τiP T .](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-48-320.jpg)
![Relativistic Dynamics and Center of Mass 29
the Currie–Hill equations [118, 119] (or Currie–Hill world-line conditions),
whose satisfaction implies that the predictive positions
qi(xo
) behave under
Lorentz boosts like the spatial components of 4-vectors. Bel [114–117] proved
that these equations constitute the necessary and sufficient conditions that
guarantee that the dynamics is Lorentz invariant with respect to finite Lorentz
transformations. However, the Currie–Hill equations are so non-linear that it is
practically impossible to find consistent predictive forces and develop this point
of view.
The first well-posed Hamiltonian formulation of relativistic mechanics was
given by Dirac [89] with the instant, front (or light), and point forms of rela-
tivistic Hamiltonian dynamics and the associated canonical realizations of the
Poincaré algebra. His non-manifestly covariant Hamiltonian instant form has a
well-defined non-relativistic limit and 1/c expansions containing the deviations
(potentials) from the free case.
However, the development of Hamiltonian models was blocked by the no-
interaction theorem of Currie, Jordan, and Sudarshan [120–122] (see Ref. [123]
for a review). Its original form was formulated in the Hamiltonian Dirac instant
form in the 6N-dimensional phase space
qi(xo
),
pi(xo
)
of N particles. The no-
interaction theorem states that in the hypotheses: (1) the configuration variables
qi(xo
) are canonical, i.e., {
qi(xo
),
qj(xo
)} = 0; (2) the Lorentz boosts can be
implemented as canonical transformations (existence of a canonical realization
of the Poincaré group) and the
qi(xo
) are the space components of 4-vectors; and
(3) the system is non-singular (the transformation from positions and velocity
to canonical coordinates is non-singular; the existence of a Lagrangian it is not
assumed). This implies only free motion.
As a consequence of the theorem, if we denote xμ
i (τ), piμ(τ) the canonical
coordinates of the manifestly covariant approach and
xi(xo
),
pi(xo
) their equal
time restriction in the instant form, we have
xi(xo
) =
qi(xo
) except for free
motion. Let us remark that, since the manifestly covariant approach gives the
classical basis for the theory of covariant wave equations, the 4-coordinates xμ
i (τ)
(and not the geometrical 4-positions qμ
i (τ)) are the coordinates locally minimally
coupled to external fields.
Many attempts were made to avoid this theorem by relaxing one of its hypothe-
ses or by renouncing the concept of the world-line till when Droz Vincent’s many-
time Hamiltonian formalism [124–128] (a refinement of the manifestly covariant
non-manifestly predictive approach; it is the origin of the multi-temporal equa-
tions of Ref. [48]), Todorov’s quasi-potential approach to bound states [130–133],
and Komar’s study of toy models for general relativity (GR) [134–137] converged
toward manifestly covariant models based on singular Lagrangians and/or the
Dirac–Bergmann theory of constraints [138–140].
Since the Lagrangian formulation is usually not known, a system of N
relativistic scalar particles is usually described in a manifestly covariant](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-49-320.jpg)
![30 Relativistic Dynamics and Center of Mass
8N-dimensional phase space with coordinates
xμ
i (τ), piμ(τ)
[{xμ
i (τ), pjν (τ)} =
−δij δμ
ν , {xμ
i (τ), xν
j (τ)} = {piμ(τ), pjν (τ)} = 0], where τ is a scalar evolution
parameter. The description is independent of the choice of τ: The Lagrangian
(even if usually not explicitly known) is assumed τ-reparametrization invariant,
so that at the Hamiltonian level the canonical Hamiltonian vanishes identically,
Hc ≡ 0. Since the physical degrees of freedom for N scalar particles are 6N,
there are constraints, which, in the case of N free scalar particles of mass mi
are just the mass-shell conditions φi(q, p) = p2
i − m2
i ≈ 0, i = 1, . . . , N. These
constraints say that the time variables xo
i (τ) are the gauge variables of a
τ-reparametrization invariant theory with canonical Hamiltonian Hc ≡ 0. The
Dirac Hamiltonian is HD =
N
i=1 λi
(τ) φi if all the first-class constraints are
primary. The final constraint sub-manifold is the union of 2N
(for generic masses
mi) disjoint sub-manifolds corresponding to the choice of either the positive-
or negative-energy branch of each two-sheeted mass-shell hyperboloid. Each
branch is a non-compact sub-manifold of phase space on which each particle
has a well-defined sign of the energy and 2N
is a topological number (the zeroth
homotopy class of the constraint sub-manifold).2
However, only in the case of two-body systems it is known how to introduce
interactions (due to the Droz Vincent–Todorov–Komar model) with an arbitrary
action-at-a-distance interaction instantaneous in the rest-frame described by the
two first-class constraints φi = p2
i − m2
i c2
+ V (r2
⊥) ≈ 0, i = 1, 2, with rμ
⊥ =
(ημν
− pμ
pν
/ p2
)rν , rμ
= xμ
1 − xμ
2 , pμ = p1μ + p2μ. For N 2 a closed form of
the N first-class constraints is not known explicitly (there is only an existence
proof): Only versions of the model with explicit gauge fixings, so that all the
constraints except one are second class, are known.
This model has been completely understood both at the classical and quantum
level [139]. The no-interaction theorem is initially avoided due to the singular
nature of the Lagrangian: There is a canonical realization of the Poincaré group
and the canonical coordinates xμ
i are 4-vectors. However, when we restrict our-
selves to the constraint sub-manifold and look for canonical coordinates adapted
to it and to the Poincaré group, it turns out that among the final canonical
coordinates will always appear the canonical non-covariant center of mass of
the particle system. Therefore, all these models have the following properties:
(1) the canonical and predictive 4-positions do not coincide (except in the free
case); and (2) the decoupled canonical center of mass is not covariant.
Let us see how to overcome these problems.
2 Let us remark that when the particles are coupled to weak external fields the 2N
sub-manifolds are deformed but remain disjoint. But when the strength of the external
fields increases, the various sub-manifolds may intersect each other and this topological
discontinuity is the signal that we are entering a non-classical regime where quantum pair
production becomes relevant due to the disappearance of mass gaps.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-50-320.jpg)
![3.1 The Wigner-Covariant Rest-Frame Instant Form 31
3.1 The Wigner-Covariant Rest-Frame Instant Form
of Dynamics for Isolated Systems
Given an inertial observer described by a 4-coordinate xμ
(τ) assumed canoni-
cally conjugated to the finite time-like 4-momentum Pμ
, P2
0 ({xμ
, Pν
} =
− 4
ημν
) of an isolated system to get the inertial frame of the rest-frame instant
form, we must add the following gauge fixings on the orthonormal tetrads bμ
A(τ)
appearing in Eq. (2.18) and use the following new Dirac brackets:
δμ
A(τ) = bμ
A(τ) − μ
A(u(P)) ≈ 0, ⇒ zμ
F (τ, σu
) = xμ
(τ) + μ
r (u(P)) σr
,
{f, g}∗∗
= {f, g}∗
−
1
4
{f, H̃μν
}∗
4
ημσ A
ν (u(P))−4
ηνσ D
μ (u(P))
{δσ
D, g}∗
+{f, δσ
D}∗
4
ησν D
μ (u(P))−4
ησμ D
ν (u(P))
{H̃μν
, g}∗
,
{xμ
, Pν
}∗∗
= − 4
ημν
, {ηr
i , κjs}∗∗
= δij δr
s , (3.1)
where μ
A(u(P)) = μ
A(
h) = Lμ
A(P,
◦
P) (A
μ (u(P)) ν
A(u(P)) = δν
μ) are the columns
of the standard Wigner boost sending the time-like 4-momentum Pμ
to its rest-
frame form Pμ
= Lμ
ν (P,
◦
P)
◦
P
ν
,
◦
P
μ
= M c (1;
0), which are given in Eqs. (A.7)–
(A.10) of Appendix A. In the Euclidean 3-spaces of the rest-frame, the Lorentz
scalar 3-coordinates σr
can be chosen as Cartesian 3-vectors
σ.
As a consequence of the Pμ
-dependence of the gauge fixings (Eq. 3.1), as shown
in Ref. [105], the Lorentz-scalar 3-vectors ηr
i (τ), κir(τ) giving the Hamiltonian
description of particles in the rest-frame (xμ
i (τ) = xμ
(τ) + μ
r (u(P)) ηr
i (τ); κir(τ)
is given in Eq. (2.13), restricted to the rest-frame) become Wigner spin-1 3-
vectors. They transform under the Wigner rotations of Eqs. (A.11) and (A.12)
of Appendix A. If a Lorentz transformation x
μ
i (τ) = Λμ
ν xν
i (τ) is done, we
get ηr
i (τ) → η
r
i (τ) = Rr
s(Λ, P) ηs
i (τ), μ
r (u(P)) ηr
i (τ) → Λμ
ν ν
r (u(Λ P)) η
r
i (τ).
Therefore the scalar product of two such vectors is a Lorentz scalar. The same
happens for all the 3-vectors living inside these instantaneous 3-spaces. Therefore
the instantaneous 3-spaces Στ are orthogonal to Pμ
: They are named Wigner
hyper-planes.
From now on, i, j . . . will denote Euclidean indices, while r, s . . . will denote
Wigner spin-1 indices. With the notation introduced after Eq. (2.20) with
h =
v/c =
P/Mc we have μ
τ (u(P)) = uμ
(P) = Pμ
/Mc = μ
τ (
h) = hμ
,
μ
r (u(P)) =
− hr; δi
r − hi hr
1+
√
1+
h2
= μ
r (
h) = (
1 +
h2;
h), 4
ημν
= μ
A(
h) 4
ηAB
ν
B(
h) = (hμ
hν
− μ
r (
h) ν
r (
h)).
The Pμ
-dependent gauge fixing (Eq. 3.1) changes the interpretation of the
residual gauge variables, because the constraints (Eq. 2.15) are reduced to the
following remaining four first-class constraints (being in an inertial frame we
have Tττ
= T⊥⊥ and Tτr
= −3
grs
T⊥s):](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-51-320.jpg)
![32 Relativistic Dynamics and Center of Mass
H̃μ
(τ) = d3
σ Hμ
(τ,
σ)
= Pμ
− uμ
(P) d3
σ Tττ
(τ,
σ) − μ
r (u(P)) d3
σ Tτr
(τ,
σ) ≈ 0,
⇓
Mc =
√
P2 ≈ d3
σ Tττ
(τ,
σ),
Pr
(int) = d3
σ Tτr
(τ,
σ) ≈ 0, (3.2)
which implies that M is the mass of the isolated system and that its total
3-momentum vanishes in the global rest-frame.
At this stage, only the four degrees of freedom xμ
(τ) of the original embedding
zμ
(τ, σr
) are still free parameters. However, due to the dependence of the gauge
fixings (Eq. 3.1) upon Pμ
, the final canonical gauge variable conjugated to Pμ
is not the 4-vector xμ
, but the following non-covariant position variable x̃μ
(τ)
[105] (see Section A.5 of Appendix A; Ts is the Lorentz-scalar rest time):
x̃μ
= xμ
−
1
M c (Po + M c)
Pν Sνμ
+ M c
Soμ
− Soν Pν Pμ
M2 c2
,
u(P) · x̃ = u(P) · x = c Ts, {x̃μ
(τ), Pν
} = −4
ημν
,
{Ts, M} = {u(P) · x̃,
√
P2 = −. (3.3)
Let us remark that, since the Poincaré generators Pμ
, Jμν
are global quantities
(they know the whole instantaneous 3-space), collective variables like x̃μ
(τ),
being defined in terms of them, are also global non-local quantities: As a con-
sequence, they are non-measurable with local means [98–101, 141–143] (see also
the review papers in Refs. [144–147]). This is a fundamental difference from the
non-relativistic 3-center of mass.
As shown in Ref. [105] after the gauge fixing (Eq. 3.1), the final form of the
external Poincaré generators (Eq. 2.14) of an arbitrary isolated system in the
rest-frame instant form is:
Pμ
, Jμν
= x̃μ
Pν
− x̃ν
Pμ
+ S̃μν
,
Po
= M2 c2 +
P2 = Mc 1 +
h2,
P = Mc
h,
S̃r
= S̄r
=
1
2
ruv
S̄uv
, S̃0r
=
rsu
Ps
S̄u
Mc + Po
,
Jij
= x̃i
Pj
− x̃j
Pi
+ iju
S̄u
= zi
hj
− zj
hi
+ iju
S̄u
,
Ki
= Joi
= x̃o
Pi
− x̃i
M2 c2 +
P2 −
isu
Ps
S̄u
M c +
M2 c2 +
P2
= − 1 +
h2 zi
+
(
S̄ ×
h)i
1 +
1 +
h2
, (3.4)](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-52-320.jpg)
![3.1 The Wigner-Covariant Rest-Frame Instant Form 33
with S̃μν
function only of the 3-spin S̄r
of the rest spin tensor S̄AB =
μ
A(u(P)) ν
B(u(P)) Sμν (both the spin tensors S̃μν
and S̄AB
satisfy the Lorentz
algebra). It is this external realization that implements the Wigner rotations on
the Wigner hyper-planes through the last term in the Lorentz boosts.
Note that both L̃μν
= x̃μ
Pν
−x̃ν
Pμ
and S̃μν
= Jμν
−L̃μν
are conserved. Since
we assume P2
0, the Pauli–Lubanski invariant is W2
= − P2
S̄
2
.
The Lorentz boosts are differently interaction-dependent from the Galilei ones.
Let us remark that this realization is universal in the sense that it depends
on the nature of the isolated system only through a U(2) algebra [142], whose
generators are the invariant mass M (which in turn depends on the relative
variables and on the type of interaction) and the internal spin
S̄, which is
interaction-independent, being in an instant form of dynamics.
The Dirac Hamiltonian is now HD = λ(τ)
Mc−
d3
σ Tττ
(τ,
σ)
+
λ(τ)·
P(int)
and the embedding of Wigner hyper-planes is
zμ
W (τ,
σ) = xμ
(τ) + μ
r (u(P)) σr
, (3.5)
with xμ
a function of x̃μ
, Pμ
, and Sμν
(or S̄AB
) according to Eq. (3.3). The
gauge freedom in the choice of xμ
is connected with the arbitrariness of the spin
boosts S̄τr
(or Soi
).
Eq. (3.2) implies that the eight variables x̃μ
(τ) and Pμ
are restricted only
by a first-class constraint identifying Mc =
√
P2 with the invariant mass of
the isolated system, evaluated by using its energy–momentum tensor: Therefore,
these eight variables are to be reduced to six physical variables describing the
external decoupled relativistic center of mass of the isolated system (
√
P2 is
determined by the constraint and its conjugate variable, the rest time Ts =
u(P) · x̃(τ) is a gauge variable). The choice of a gauge fixing for the rest time Ts
is equivalent to the identification of a clock carried by the inertial observer. The
natural inertial observer for this description is the external Fokker–Pryce center
of inertia (the only covariant collective variable), which is also a function of τ,
z, and
h (see Eq. (3.8)). Therefore, there are three collective position degrees
of freedom hidden in the embedding field zμ
(τ,
σ), which become non-local non-
measurable physical variables.
As a consequence, every isolated system (i.e., a closed universe) can be visu-
alized as a decoupled non-covariant collective (non-local) pseudo-particle (the
external center of mass), described by canonical variables identifying the frozen
Jacobi data
z,
h (see Eq. (3.8)), carrying a “pole–dipole structure,” namely
the invariant mass M c (the Lorentz-scalar monopole) and the rest spin
S̄ (the
dipole, a Wigner spin-1 3-vector) of the system, and with an associated external
realization of the Poincaré group:3
3 The last term in the Lorentz boosts induces the Wigner rotation of the 3-vectors inside the
Wigner 3-spaces.](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-53-320.jpg)
![34 Relativistic Dynamics and Center of Mass
Pμ
= M c hμ
= M c
1 +
h2;
h
,
Jij
= zi
hj
− zj
hi
+ ijk
Sk
, Ki
= Joi
= − 1 +
h2 zi
+
(
S ×
h)i
1 +
1 +
h2
.
(3.6)
The universal breaking of Lorentz covariance is connected to this decoupled
non-local collective variable and is irrelevant because all the dynamics of the
isolated system lives inside the Wigner 3-spaces and is Wigner-covariant. The
invariant mass and the rest spin are built in terms of the Wigner-covariant
variables of the given isolated system (the 6N variables
ηi(τ) and
κi(τ) for a
system of N particles) living inside the Wigner 3-spaces [98–101, 140].
The presence of the three first-class constraints
P(int) ≈ 0 (the rest-frame
conditions) on these variables implies that each Wigner 3-space is a rest-frame
of the isolated system whose Wigner spin-1 3-vector describing the internal
3-center of mass, is a gauge variable. In this way we avoid a double counting of
the center of mass and the dynamics inside the Wigner hyper-planes is described
only by Wigner spin-1 internal relative variables.
Let us now add the τ-dependent gauge fixing,
c Ts − τ ≈ 0, (3.7)
where c Ts = u(P) · x̃ = u(P) · x is the Lorentz-scalar rest time, to the first-class
constraint Mc−
d3
σ Tττ
(τ,
σ) ≈ 0 of Eq. (3.2). After this imposition (from now
on τ/c is the rest time Ts, which satisfies {Ts, M} = {u(P) · x̃,
√
P2} = −), we
have the following properties [100, 101]:
1. Due to the τ-dependence of the gauge fixing c Ts − τ ≈ 0, the Dirac Hamil-
tonian given before Eq. (3.5) is replaced by the Hamiltonian HD =
λ(τ) ·
P(int) ≈ 0: There is a frozen description of dynamics in the external space
of the frozen external center of mass modulo the three first-class constraints
P(int) ≈ 0, as in the standard Hamilton–Jacobi description.
2. Inside the Wigner 3-space the evolution in τ = cTs of the relative variables
after the elimination of the internal center of mass is governed by the
total mass M of the isolated system, i.e., by the energy generator of
the internal Poincaré group, as the natural physical Hamiltonian. The
4-momentum Pμ
becomes the 4-momentum of the isolated system: Pμ
=
Mc (
1 +
h2;
h) = Mc hμ
with
h arbitrary a-dimensional 3-velocity and
Mc ≡
d3
σ Tττ
(τ,
σ).
3. The six physical degrees of freedom describing the external decoupled
relativistic center of mass are the non-evolving Jacobi data:
z = Mc
x̃ −
P
Po
x̃o
,
h =
P
Mc
, {zi
, hj
} = δij
. (3.8)](https://image.slidesharecdn.com/5214299-250603040406-eeea2e0f/85/Noninertial-Frames-And-Dirac-Observables-In-Relativity-Lusanna-54-320.jpg)
Noninertial Frames And Dirac Observables In Relativity Lusanna
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- 5. NON-INERTIAL FRAMES AND DIRAC OBSERVABLES IN RELATIVITY Interpreting general relativity relies on a proper description of non-inertial frames and Dirac observables. This book describes global non-inertial frames in special and general relativity. The first part covers special relativity and Minkowski space-time, before covering general relativity, globally hyperbolic Einstein space- time, and the application of the 3+1 splitting method to general relativity. It uses a Hamiltonian description and the Dirac–Bergmann theory of constraints to show the transition between one non-inertial frame and another is a gauge transformation, extra variables describing the frame are gauge variables, and the measureable matter quantities are gauge-invariant Dirac observables. Point particles, fluids, and fields are also discussed, including how to treat the problems of relative times in the description of relativistic bound states, and the problem of relativistic center of mass. Providing a detailed description of mathematical methods, it is perfect for theoretical physicists, researchers, and students working in special and general relativity. L u c a L u s a n n a is a retired research director of the Firenze section of the National Institute for Nuclear Physics (INFN). He is a fellow of the General Relativity and Gravitation Society (SIGRAV), the Italian Physical Society, of which he has been president for five years. Since 2009 he has been the director of the Theoretical Topical Team of the ACES mission of ESA, tasked with putting an atomic clock on the International Space Station.
- 6. CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS S. J. Aarseth Gravitational N-Body Simulations: Tools and Algorithms† D. Ahluwalia Mass Dimension One Fermions J. Ambjørn, B. Durhuus and T. Jonsson Quantum Geometry: A Statistical Field Theory Approach† A. M. Anile Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics J. A. de Azcárraga and J. M. Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics† O. Babelon, D. Bernard and M. Talon Introduction to Classical Integrable Systems† F. Bastianelli and P. van Nieuwenhuizen Path Integrals and Anomalies in Curved Space† D. Baumann and L. McAllister Inflation and String Theory V. Belinski and M. Henneaux The Cosmological Singularity† V. Belinski and E. Verdaguer Gravitational Solitons† J. Bernstein Kinetic Theory in the Expanding Universe† G. F. Bertsch and R. A. Broglia Oscillations in Finite Quantum Systems† N. D. Birrell and P. C. W. Davies Quantum Fields in Curved Space† K. Bolejko, A. Krasiński, C. Hellaby and M-N. Célérier Structures in the Universe by Exact Methods: Formation, Evolution, Interactions D. M. Brink Semi-Classical Methods for Nucleus-Nucleus Scattering† M. Burgess Classical Covariant Fields† E. A. Calzetta and B.-L. B. Hu Nonequilibrium Quantum Field Theory S. Carlip Quantum Gravity in 2+1 Dimensions† P. Cartier and C. DeWitt-Morette Functional Integration: Action and Symmetries† J. C. Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion† P. D. B. Collins An Introduction to Regge Theory and High Energy Physics† M. Creutz Quarks, Gluons and Lattices† P. D. D’Eath Supersymmetric Quantum Cosmology† J. Dereziński and C. Gérard Mathematics of Quantization and Quantum Fields F. de Felice and D. Bini Classical Measurements in Curved Space-Times F. de Felice and C. J. S Clarke Relativity on Curved Manifolds† B. DeWitt Supermanifolds, 2nd edition† P. G. O. Freund Introduction to Supersymmetry† F. G. Friedlander The Wave Equation on a Curved Space-Time† J. L. Friedman and N. Stergioulas Rotating Relativistic Stars Y. Frishman and J. Sonnenschein Non-Perturbative Field Theory: From Two Dimensional Conformal Field Theory to QCD in Four Dimensions J. A. Fuchs Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory† J. Fuchs and C. Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists† Y. Fujii and K. Maeda The Scalar-Tensor Theory of Gravitation† J. A. H. Futterman, F. A. Handler, R. A. Matzner Scattering from Black Holes† A. S. Galperin, E. A. Ivanov, V. I. Ogievetsky and E. S. Sokatchev Harmonic Superspace† R. Gambini and J. Pullin Loops, Knots, Gauge Theories and Quantum Gravity† T. Gannon Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics† A. Garcı́a-Dı́az Exact Solutions in Three-Dimensional Gravity M. Göckeler and T. Schücker Differential Geometry, Gauge Theories, and Gravity† C. Gómez, M. Ruiz-Altaba and G. Sierra Quantum Groups in Two-Dimensional Physics† M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 1: Introduction M. B. Green, J. H. Schwarz and E. Witten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and Phenomenology V. N. Gribov The Theory of Complex Angular Momenta: Gribov Lectures on Theoretical Physics† J. B. Griffiths and J. Podolský Exact Space-Times in Einstein’s General Relativity† T. Harko and F. Lobo Extensions of f(R) Gravity: Curvature-Matter Couplings and Hybrid Metric-Palatini Gravity S. W. Hawking and G. F. R. Ellis The Large Scale Structure of Space-Time† F. Iachello and A. Arima The Interacting Boson Model† F. Iachello and P. van Isacker The Interacting Boson-Fermion Model† C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† G. Jaroszkiewicz Principles of Discrete Time Mechanics G. Jaroszkiewicz Quantized Detector Networks
- 7. C. V. Johnson D-Branes† P. S. Joshi Gravitational Collapse and Spacetime Singularities† J. I. Kapusta and C. Gale Finite-Temperature Field Theory: Principles and Applications, 2nd edition† V. E. Korepin, N. M. Bogoliubov and A. G. Izergin Quantum Inverse Scattering Method and Correlation Functions† J. Kroon Conformal Methods in General Relativity M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory† S. Mallik and S. Sarkar Hadrons at Finite Temperature A. Malyarenko and M. Ostoja-Starzewski Tensor-Valued Random Fields for Continuum Physics N. Manton and P. Sutcliffe Topological Solitons† N. H. March Liquid Metals: Concepts and Theory† I. Montvay and G. Münster Quantum Fields on a Lattice† P. Nath Supersymmetry, Supergravity, and Unification L. O’Raifeartaigh Group Structure of Gauge Theories† T. Ortı́n Gravity and Strings, 2nd edition A. M. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† M. Paranjape The Theory and Applications of Instanton Calculations L. Parker and D. Toms Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity R. Penrose and W. Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time Geometry† S. Pokorski Gauge Field Theories, 2nd edition† J. Polchinski String Theory Volume 1: An Introduction to the Bosonic String† J. Polchinski String Theory Volume 2: Superstring Theory and Beyond† J. C. Polkinghorne Models of High Energy Processes† V. N. Popov Functional Integrals and Collective Excitations† L. V. Prokhorov and S. V. Shabanov Hamiltonian Mechanics of Gauge Systems S. Raychaudhuri and K. Sridhar Particle Physics of Brane Worlds and Extra Dimensions A. Recknagel and V. Schiomerus Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes M. Reuter and F. Saueressig Quantum Gravity and the Functional Renormalization Group R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton: Deep Inelastic Scattering† P. Romatschke and U. Romatschke Relativistic Fluid Dynamics In and Out of Equilibrium: And Applications to Relativistic Nuclear Collisions C. Rovelli Quantum Gravity† W. C. Saslaw Gravitational Physics of Stellar and Galactic Systems† R. N. Sen Causality, Measurement Theory and the Differentiable Structure of Space-Time M. Shifman and A. Yung Supersymmetric Solitons Y. M. Shnir Topological and Non-Topological Solitons in Scalar Field Theories H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition† J. Stewart Advanced General Relativity† J. C. Taylor Gauge Theories of Weak Interactions† T. Thiemann Modern Canonical Quantum General Relativity† D. J. Toms The Schwinger Action Principle and Effective Action† A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells, Jr Twistor Geometry and Field Theory† E. J. Weinberg Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics† † Available in paperback
- 9. Non-Inertial Frames and Dirac Observables in Relativity L U C A L U S A N N A National Institute for Nuclear Physics (INFN)
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108480826 DOI: 10.1017/9781108691239 © Cambridge University Press 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Lusanna, L., author. Title: Non-inertial frames and Dirac observables in relativity / Luca Lusanna (National Institute for Nuclear Physics(INFN), Firenze). Description: Cambridge ; New York, NY : Cambridge University Press, [2019] | Series: Cambridge monographs on mathematical physics | Includes bibliographical references and subject index. Identifiers: LCCN 2019005996 | ISBN 9781108480826 (hardback : alk. paper) Subjects: LCSH: Dirac equation. | Wave equation. | Quantum field theory. | General relativity (Physics) Classification: LCC QC174.45 .L87 2019 | DDC 530.12–dc23 LC record available at https://lccn.loc.gov/2019005996 ISBN 978-1-108-48082-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. To Enrica, Haja and Leo: thank you for being always with me!
- 13. Contents Preface page xiii Part I Special Relativity: Minkowski Space-Time 1 Galilei and Minkowski Space-Times 3 1.1 The Galilei Space-Time of Non-Relativistic Physics and Its Inertial and Non-Inertial Frames 3 1.2 The Minkowski Space-Time: Inertial Frames, Cartesian Coordinates, Matter, Energy–Momentum Tensor, and Poincaré Generators 5 1.3 The 1+3 Approach to Local Non-Inertial Frames and Its Limitations 7 2 Global Non-Inertial Frames in Special Relativity 12 2.1 The 3+1 Approach to Global Non-Inertial Frames and Radar 4-Coordinates 12 2.2 Parametrized Minkowski Theory for Matter Admitting a Lagrangian Description 19 3 Relativistic Dynamics and the Relativistic Center of Mass 27 3.1 The Wigner-Covariant Rest-Frame Instant Form of Dynamics for Isolated Systems 31 3.2 The Relativistic Center-of-Mass Problem 36 3.3 The Elimination of Relative Times in Relativistic Systems of Particles and in Relativistic Bound States 39 3.4 Wigner-Covariant Quantum Mechanics of Point Particles 44 3.5 The Non-Inertial Rest-Frames 47 4 Matter in the Rest-Frame Instant Form of Dynamics 54 4.1 The Klein–Gordon Field 54 4.2 The Electromagnetic Field and Its Dirac Observables 65 4.3 Relativistic Atomic Physics 71 4.4 The Dirac Field 79 4.5 Yang–Mills Fields 96 4.6 Relativistic Fluids, Relativistic Micro-Canonical Ensemble, and Steps toward Relativistic Statistical Mechanics 105
- 14. x Contents Part II General Relativity: Globally Hyperbolic Einstein Space-Times 5 Hamiltonian Gravity in Einstein Space-Times 123 5.1 Global 3+1 Splittings of Globally Hyperbolic Space-Times without Super-Translations and Asymptotically Minkowskian at Spatial Infinity Admitting a Hamiltonian Formulationof Gravity 123 5.2 The ADM Hamiltonian Formulation of Einstein Gravity and the Asymptotic ADM Poincaré Generators in the Non-Inertial Rest-Frames 127 6 ADM Tetrad Gravity and Its Constraints 138 6.1 ADM Tetrad Gravity, Its Hamiltonian Formulation, and Its First-Class Constraints 138 6.2 The Shanmugadhasan Canonical Transformation to the York Canonical Basis for the Search of the Dirac Observables of the Gravitational Field 145 6.3 The Non-Harmonic 3-Orthogonal Schwinger Time Gauges and the Metrological Interpretation of the Inertial Gauge Variables 149 6.4 Point Particles and the Electromagnetic Field as Matter 155 7 Post-Minkowskian and Post-Newtonian Approximations 167 7.1 The Post-Minkowskian Approximation in the 3-Orthogonal Gauges 167 7.2 The Post-Newtonian Expansion of the Post-Minkowskian Linearization 178 7.3 Dark Matter as a Relativistic Inertial Effect and Relativistic Celestial Metrology 181 Part III Dirac–Bergmann Theory of Constraints 8 Singular Lagrangians and Constraint Theory 189 8.1 Regular Lagrangians and the First Noether Theorem 189 8.2 Singular Lagrangians and Dirac–Bergmann Theory of Hamiltonian First- and Second-Class Constraints 194 8.3 The Gauge Transformations in Field Theory and General Relativity 204 9 Dirac Observables Invariant under the Hamiltonian Gauge Transformations Generated by First-Class Constraints 223 9.1 Shanmugadhasan Canonical Bases Adapted to the Constraints and the Dirac Observables 228
- 15. Contents xi 9.2 The Null Eigenvalues of the Hessian Matrix of Singular Lagrangians and Degenerate Cases 232 9.3 The Second Noether Theorem 235 9.4 Constraints in Field Theory 236 10 Concluding Remarks and Open Problems 243 Appendix A Canonical Realizations of Lie Algebras, Poincaré Group, Poincaré Orbits, and Wigner Boosts 248 Appendix B Grassmann Variables and Pseudo-Classical Lagrangians 265 Appendix C Relativistic Perfect Fluids and Covariant Thermodynamics 276 References 293 Index 320
- 17. Preface While in Newtonian physics (NP) both global inertial and rigid non-inertial frames of the Galilei space-time are under control due to the absolute notions of time and Euclidean 3-space, in the Minkowski space-time of special relativity (SR) only global inertial frames centered on inertial observers are defined by using Einstein convention for the synchronization of the clocks present in each point to the one of the inertial observer. Only in this way can instantaneous Euclidean 3-spaces be selected. The Lorentz signature of Minkowski space-time implies that the only intrinsic notion in SR is the conformal structure, namely the light-cone, identifying the allowed paths of the light rays sent by an observer. However, all physical observers are accelerated and till now there is only the so- called 1+3 point of view for identifying a local instantaneous 3-space in a region around the observer whose radius depends upon the observer’s acceleration. With this description it is not possible to define a Cauchy problem in non-inertial frames for classical fields like the Maxwell one. Therefore, an open theoretical challenge is to find a consistent description of classical particles, fluids, and fields in the non-inertial reference frames of the Minkowski space-time of SR. This accomplishment would allow extension of the existent formulations of these systems in the global inertial frames centered on inertial observers, which are connected in a manifestly covariant way by the transformations of the kinematical Poincaré group (the relativity principle), to a formulation covariant under some family of space-time diffeomorphisms. As a consequence, this description would allow a smooth transition to Einstein space-times of general relativity (GR) with the relativity principle replaced by the equivalence principle forbidding the existence of global inertial frames. In GR only a free-falling observer will recover SR as a local approximation in a neighborhood where tidal effects are negligible. In SR these steps are preliminary to the transition to the quantum the- ory of matter, because they allow finding the solution of problems like the elimination of relative times in relativistic bound states, the clarification of the notion of relativistic center of mass (a non-local non-measurable quantity), and a consistent treatment of relativistic atomic physics, relativistic fluids, and relativistic statistical mechanics. The final open challenge will be to find a consistent formulation of quantum gravity (QG). In this book the classical SR and GR space-times are assumed to be nice differentiable 4-manifolds with trivial topology, a notion criticized in many approaches to QG.
- 18. xiv Preface The only known mathematical method to define global non-inertial frames with well-defined global instantaneous non-Euclidean 3-spaces is the 3+1 point of view described in the first part of the book, where also the implications for relativistic metrology will be examined. In it one considers an arbitrary observer endowed with a nice foliation of Minkowski space-time whose leaves are the instantaneous (in general non-Euclidean) 3-spaces. Moreover, the 3+1 point of view allows extension of the Lagrangian description of matter to a formulation (the parametrized Minkowski theory) in which the Lagrangian depends upon the variables identifying the non-inertial frames. The Lagrangian is singular, being invariant under the so-called frame-preserving diffeomorphisms, so that the study of its properties requires the second Noether theorem. As a consequence, one needs Dirac–Bergmann (DB) theory of con- straints to study its Hamiltonian formulation and to show that the transition from a non-inertial frame to another (either inertial or non-inertial) frame is a gauge transformation and that the extra variables describing the frame are gauge variables, like those present in Maxwell and Yang–Mills theories, while the measurable matter quantities are the gauge-invariant Dirac observables (DOs). In the 3+1 point of view it is possible to describe matter in SR separating the unobservable relativistic center of mass and by defining physical relative variables. The ten Poincaré generators can be defined in a consistent way, so that it is possible to define a relativistic quantum mechanics of point particles belonging to irreducible representations of the Poincaré group, as required by every model of particle physics. While the first part of the book is dedicated to SR, the second part considers GR, where the geometrical view of Einstein implies that the 4-metric of the space-time with Lorentz signature not only determines the chrono-geometrical structure of the space-time by means of the line element, but is also the dynamical field mediating the gravitational interaction. Now the allowed paths of light rays (the conformal structure) are point-dependent and the 4-metric teaches relativistic causality to all the other fields. Behind Einstein GR there is the principle of general covariance. The Hilbert action is invariant under passive diffeomorphisms (ordinary coordinate trans- formations), while Einstein equations are form-invariant in every 4-coordinate system (invariance under active diffeomorphisms), so that all the matter fields must have a tensorial character. The Hamiltonian formulation of GR requires shifting from the Hilbert action to the ADM one, in which the passive diffeomorphisms are replaced by local Noether transformations in the framework of the second Noether theorem, so that in phase space one needs to use the DB theory of constraints. In an arbitrary Einstein space-time one can define only local non-inertial frames around a non-inertial time-like observer. However, the need for a good formulation of the Cauchy problem for Einstein and matter fields requires the notion of instantaneous 3-space, namely the possibility of using the 3+1 point of
- 19. Preface xv view like in SR. Moreover, the inclusion of particle physics requires the possibility of introducing a notion of Poincaré algebra. These requirements select a family of Einstein space-times: the globally hyperbolic, asymptotically flat at spatial infinity, and without super-translations. In them there are the asymptotic ADM Poincaré generators: When one switches off the Newton constant they tend to the Poincaré generators of SR. This class of singularity-free Einstein space-times contains asymptotic inertial observers to be identified with the fixed stars of astronomy. Moreover, the ADM energy generates a real temporal evolution modulo the gauge freedom associated with the first-class constraints. As a consequence there is not a frozen picture like in loop QG, where the 3-spaces are compact manifolds without boundary, implying the absence of a Poincaré algebra. To be able to include fermions, Einstein 4-metric will be decomposed upon tetrads and the ADM action will be assumed to depend on them. This enlarges the gauge group and the number of first-class constraints due to the extra gauge freedom to orientate three gyroscopes and freedom in how they are transported along time-like trajectories. The big open problem is to find the DOs of Einstein metric and tetrad gravity, since no one is able to solve the super-Hamiltonian and super-momentum con- straints of GR. The best that can be done is to find a Shanmugadhasan canonical transformation adapted to all the first-class constraints except the four unsolved constraints. However, this allows making a preliminary separation before trying to solve the Hamilton equations between physical canonical variables describing tidal effects (the gravitational waves of the linearized theory) and generalized gauge variables describing relativistic inertial effects. The gauge fixing of these inertial gauge variables (including a clock synchronization convention for the choice of the instantaneous 3-spaces) fixes a global non-inertial frame with well-defined relativistic inertial forces, so that the Hamilton equations become deterministic. As a consequence, one can study post-Minkowskian (PM) and then post- Newtonian (PN) approximations of Einstein equations in the presence of point particles and of the electromagnetic field, with implications for dark matter. Due to the relevance of DB theory of constraints, in the third part of the book there is a review of what is known on such a theory and on the state of the art in the search for DOs. It is assumed that readers of this book of relativistic analytical mechanics have a good knowledge of SR and GR and of their mathematical, algebraic, and canonical formalisms, so there is no review of them. See Refs. [1–14] for the status of SR and GR and Refs. [15–17] for the status of the experiments on which they are based. See Refs. [18, 19] for some information on the status of QG. References [20–30] contain the main literature about constraint theory. This book is the final result of many years of research, mainly together with Dr. David Alba (now a teacher in high schools) and Professor Horace W. Crater,
- 20. xvi Preface who died last year, meaning I could not ask him to be my co-author. I thank Professor Massimo Pauri for showing me the relevance of trying to solve foun- dational problems in relativity and for the pleasure of being his collaborator in many works. Index notation: Greek indices μ, ν, . . . have the values 0, 1, 2, 3 (0 denotes the time axis), while Latin indices i, j, . . . have the values 1, 2, 3. The flat metric 4 ημν = (1; 0, 0, 0) of the inertial frames in SR has = +1 in the particle physics convention and = −1 in the general relativity convention. The symbol ≈ means Dirac weak equality, while the symbol ◦ = means evaluated by using the equations of motion. The symbol ≡ means identically equal. By convention, repeated indices are summed. αβγδ and ijk are completely antisymmetric tensors with 0123 = 123 = 123 = 1. Dimensions of the quantities appearing in this book: [τ] = [xμ ] = [ σ] = [ ηi] = [l], [ κi] = [Pμ ] = [E/c] = [m l t−1 ], [4 g] = [3 g] = [n] = [n(a)] = [3 e(a)r] = [ ˙ ηi] = [θi] = [0], [G = 6.7 × 10−8 cm3 s−2 g−1 ] = [m−1 l3 t−2 ], [G/c3 ] = [m−1 t] ≈ 2.5×10−39 sec/g, [S] = [] = [JAB ] = [m l2 t−1 ], [3 R] = [3 Ωrs(a)] = [l−2 ], [3 ωr(a)] = [3 Krs] = [l−1 ], [3 πr (a)] = [3 Π̃rs ] = [m l−1 t−1 ], [TAB ] = [M] = [Mr] = [H] = [H(a)] = [m l−2 t−1 ].
- 21. Part I Special Relativity Minkowski Space-Time In this first part, after a review of inertial and non-inertial frames in the non- relativistic Galilei space-time, I will study such frames in the Minkowski space- time of special relativity (SR). In Newtonian physics, time and space are absolute notions whose metrological units are defined by means of standard clocks and rods, whose structure is not specified. This is satisfactory for the non-relativistic quantum mechanics used in molecular physics and in quantum information, where gravitation effects are described by Newtonian gravity. However, in atomic physics one needs the description of light, whose quantum nature gives rise to the notion of massless photons whose trajectories do not exist in Galilei space-time. Moreover particle physics has to face high-speed objects. As a consequence, the Minkowski space-time of SR has to be introduced and a new type of metrology has been developed with different standards for length and time [31]. See references [32]–[38] for updated reviews on relativistic metrology on Earth, in the Solar System, and in astronomy. The fundamental theoretical scale for time is the SI (International System of Units) atomic second, which is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium 133 atom at rest at a temperature of 0K. In practice one uses the International Atomic Time (TAI), defined as a suitable weighted average of the SI kept by (mainly cesium) atomic clocks in about 70 laboratories worldwide. To introduce a convention for the synchronization of distant clocks one uses the notion of two-way (or round-trip) velocity of light c involving only one clock: The observer emits a ray of light that is reflected somewhere and then reabsorbed by the observer, so that only the clock of the observer is used to measure the time of flight of the ray. It is this velocity that is isotropic and constant in SR (the light postulate) and replaces the standard of length in relativistic metrology. The one-way velocity of light from one observer A to an observer B has a meaning
- 22. 2 Special Relativity only after a choice of a convention for synchronizing the clock in A with the one in B. One uses the conventional value c = 299 792 458 m s−1 for the two-way velocity of light. To measure the 3-distance between two objects in an inertial frame, one puts an atomic clock in the first object, then sends a ray of light to the second object, where it is reflected and then reabsorbed by the first object, whose measure of the flight time allows finding the 3-distance. As a consequence, the meter is the length of the path traveled by light in a vacuum in an inertial frame during a time interval of 1/c of a second.
- 23. 1 Galilei and Minkowski Space-Times In this chapter we review the properties the non-relativistic Galilei space-time and of the relativistic Minkowski one. See Refs. [39–43] for a detailed study of the rotation group and of the kinematical Galilei and Poincaré groups connecting the inertial frames of the respective space-times. 1.1 The Galilei Space-Time of Non-Relativistic Physics and Its Inertial and Non-Inertial Frames In Newtonian physics the notions of time and space are absolute, so that the chrono-geometrical structure of Galilei space-time is not dynamical. One has at each instant of the absolute time t, registered by an ideal clock, an instantaneous Euclidean 3-space R3 t with the standard notion of Euclidean distance, measured with ideal rods. The clocks in each point of R3 t are synchronized at the time t, so that Galilei space-time has the structure R × R3 , where R denotes the time axis and R3 is an abstract Euclidean 3-space associated to the fixed stars of astronomy. As a consequence, one can parametrize Galilei space-time as the straight trajectory of an inertial observer (the time axis) endowed with a foliation of Euclidean 3-spaces orthogonal to the time axis. The Galilei relativity principle assumes the existence of preferred rigid inertial frames of reference in uniform translational motion, one with respect to the other with inertial Cartesian coordinates (t, xi ) centered on an inertial observer, whose trajectory is the time axis. In these frames, free bodies move along straight lines (Newton’s first law) and Newton’s equations take the simplest form. The laws of nature are covariant and there is no preferred observer. The connection between different inertial frames is realized with the kinematical Galilei transformations: t = t + , x i = Rij xj + vi t + i , where and i are the time and space rigid translations, R (R−1 = RT , the transposed matrix) is the O(3) matrix describing rigid rotations, and vi are the parameters of the rigid Galilei boosts. Due to the absolute nature of Newtonian time, the points on a t = const. sec- tion of Galilei space-time are all simultaneous (instantaneous absolute 3-space),
- 24. 4 Galilei and Minkowski Space-Times whichever inertial system we are using. As a consequence, the causal notions of before and after a certain event are absolute. A particle of mass m has the trajectory described by inertial Cartesian 3-coordinates xi m(t) in Galilei space-time. In the Hamiltonian phase space it has the momentum pi = m δij d xi(t) dt . For a free particle the Galilei generators are H = p2 /2 m (energy), pi (momentum), Ki = m δij xj m − t pi (boost), Ji = ijk δjh xh m pk (rotation). ijk is the completely antisymmetric tensor. For a system of mutually interacting N particles of mass mk, trajectory xi k(t), momenta pk i(t) = mk δij d x j k (t) dt , k = 1, . . . , N, the Galilei generators are H = N k=1 p2 k 2 mk + V ( xh(t) − xk(t)), P = N k=1 pk, J = N k=1 xk(t) × pk(t), K = N k=1 (t pk(t)−mk xk(t)) = t P −m x. Here, x = N k=1 mk m xk(t) (m = n k=1 mk) is the Newton center of mass, whose conjugate variable is P. Therefore, the conserved Galilei boosts identify the Newtonian center of mass. Usually the interacting potential depends only on the relative distances of the particles (and not on their velocities) and appears only in the energy (the Hamil- tonian) and not in the boosts differently from what happens at the relativistic level with the Poincaré group. For isolated N-body systems the ten generators of the Galilei group are Noether constants of motion. The Abelian nature of the Noether constants (the 3-momentum) associated to the invariance under translations allow making a global separation of the center of mass from the relative variables (usually the Jacobi coordinates, identified by the centers of mass of subsystems, are preferred): In phase space this can be done with canonical transformations of the point type both in the coordinates and in the momenta. Instead, the non-Abelian nature of the Noether constants (the angular momen- tum) associated with the invariance under rotations implies that there is no unique separation [44] of the relative variables in six orientational ones (the body frame in the case of rigid bodies) and in the remaining vibrational (or shape) ones. As a consequence, an isolated deformable body or a system of particles may rotate by changing the shape (the falling cat, the diver). In Refs. [45, 46] there is a kinematical treatment of non-relativistic N-body systems by means of canonical spin bases and of dynamical body frames, which can be extended to the relativistic case in which the notions of Jacobi coordinates, reduced masses, and tensors of inertia are absent and can be recovered only when extended bodies are simulated with multipolar expansions [47]. Another non-conventional aspect of non-relativistic physics is the many-time formulation of classical particle dynamics [48] with as many first-class constraints as particles. Like in the special relativistic case, a distinction arises between physical positions and canonical configuration variables and a non-relativistic version of the no-interaction theorem (see Chapter 3) emerges. See Refs. [49, 50] for Newtonian gravity, where the Newton equivalence prin- ciple states the equality of inertial and gravitational mass, as a gauge theory of the Galilei group.
- 25. 1.2 The Minkowski Space-Time 5 To define rigid non-inertial frames, let us consider an arbitrary accelerated observer whose Cartesian trajectory is yi (t) and let us introduce the rigid non- inertial coordinates (t, σi ) by imposing xi = yi (t) + Rij (t) σj , where R(t) is a time-dependent rotation matrix, which can be parametrized with three Euler angles. It describes the rigid rotation of the non-inertial frame. It is convenient to write the 3-velocity of the accelerated observer in the form vi (t) = Rij (t) d yj(t) dt . The angular velocity of the rotating frame is Ωi (t) = 1 2 ijk Ωjk with Ωjk (t) = −Ωkj = (d R(t) dt RT (t))jk . A particle of mass m with trajectory given by the Cartesian 3-coordinates xi m(t) is described in the rigid non-inertial frames by 3-coordinates ηr (t) such that xi m(t) = yi (t) + Rij (t) ηj (t). As shown in every book on Newtonian mechanics, a particle moving in an external potential V (t, xk m(t)) = Ṽ (t, ηr (t)) has the equation of motion m d2 xi m(t) dt2 = − ∂ V (t,xk m(t)) ∂ xi m , whose form in the rigid non-inertial frames becomes m d2 η(t) dt2 = − ∂ Ṽ (t, ηk (t)) ∂ η − m d v(t) dt + ω(t) × v(t) + d ω(t) dt × η(t) +2 ω(t) × d η(t) dt + ω(t) × [ ω(t) × η(t)] . (1.1) In this equation there are the standard Euler, Jacobi, Coriolis, and centrifugal inertial (or fictitious) forces, proportional to the mass of the body, associated with the acceleration of the non-inertial observer and with the angular velocity of the rotating rigid non-inertial frame. The extension to non-rigid non-inertial frames with coordinates (t, σi ) (σr are global non-Cartesian 3-coordinates) is done in Ref. [51] by putting the Cartesian 3-coordinates xi equal to arbitrary functions Ai (t, σr ), well behaved at spatial infinity: xi = Ai (t, σr ). This coordinate transformation must be invertible with inverse σr = Sr (t, xi ). The invertibility conditions are det J(t, σr ) 0, where Ja r (t, σu ) = ∂ Aa(t,σu) ∂ σr is the three-dimensional Jacobian, whose inverse is ˜ Jr a(t, σu ) = ∂ Sr(t,xu) ∂ xa xb=Ab(t,σu) (Ja r(t, σu ) ˜ Jr b(t, σu ) = δa b , ˜ Js a(t, σu ) Ja r(t, σu ) = δs r ). The group of Galilei transformations connecting inertial frames is replaced by some subgroup of the 3-diffeomorphisms of the Euclidean 3-space connecting the non-inertial ones. The quantum mechanics of particles in non-rigid non-inertial frames is studied in Ref. [51]. 1.2 The Minkowski Space-Time: Inertial Frames, Cartesian Coordinates, Matter, Energy–Momentum Tensor, and Poincaré Generators The Minkowski space-time of special relativity (SR) is an affine 4-manifold isomorphic to R4 with Lorentz signature in which neither time nor space are absolute notions. As a consequence there is no unique notion of instantaneous 3-space and one needs some metrological convention about time and space to be
- 26. 6 Galilei and Minkowski Space-Times able to formulate particle physics in the laboratories on Earth in the approxima- tion of neglecting gravity. The only intrinsic structure of Minkowski space-time is the conformal one connected with the Lorentz signature: It defines the light- cone as the locus of incoming and outgoing radiation. There is no absolute notion of simultaneity: Given an event, all the points outside the light-cone with vertices in that event are not causally connected with that event (they have space-like separation from it), so that the notions of before and after an event become observer-dependent. Therefore there is no notion of an instantaneous 3-space, of a spatial distance, and of a one-way velocity of light between two observers (the problem of the synchronization of distant clocks). Instead, as already said, there is an absolute chrono-geometrical structure: the light postulate saying that the two-way (or round-trip) velocity of light c (only one clock is needed for its definition) is (1) constant and (2) isotropic. Let us remark that the clocks are assumed to be standard atomic clocks measuring proper time [52–54]. Einstein relativity principle privileges the inertial frames of Minkowski space- time centered on inertial observers endowed with an atomic clock: Their trajec- tories are the time axis in Cartesian coordinates xμ = (xo = c t; xi ) where the flat metric tensor with Lorentz signature is 4 ημν = (1; −1, −1, −1). These inertial frames are in uniform translational motion, one with respect to the other. All special relativistic physical systems, defined in the inertial frames of Minkowski space-time, are assumed to be manifestly covariant under the transformations of the kinematical Poincaré group connecting the inertial frames. The laws of physics are covariant and there is no preferred observer. The xo = const. hyper-planes of inertial frames are usually taken as Euclidean instantaneous 3-spaces, on which all the clocks are synchronized. They can be selected with Einstein’s convention for the synchronization of distant clocks to the clock of an inertial observer. This inertial observer A sends a ray of light at xo i to a second accelerated observer B, who reflects it toward A. The reflected ray is reabsorbed by the inertial observer at xo f . The convention states that the clock of B at the reflection point must be synchronized with the clock of A when it signs 1 2 (xo i +xo f ). This convention selects the xo = const. hyper-planes of inertial frames as simultaneity 3-spaces and implies that only with this synchronization the two-way (A–B–A) and one-way (A–B or B–A) velocities of light coincide and the spatial distance between two simultaneous point is the (3-geodesic) Euclidean distance. However, if observer A is accelerated, the convention can break down due to the possible appearance of coordinate singularities. Relativistic matter is defined in the relativistic inertial frames of Minkowski space-time centered on inertial observers using Cartesian 4-coordinates. It is in these frames that one defines the matter Lagrangian when it is known. Once one has the Lagrangian L(x) of a matter system the energy–momentum tensor is defined by replacing the flat 4-metric 4 ημν appearing in the Lagrangian with a 4-metric 4 gμν (x) like the one used in general relativity (GR), so that one gets a new Lagrangian Lg(x) and by using the definition Tμν (x) = − 2 √ −det4g(x) δ Sg δ4gμν (x) ,
- 27. 1.3 The 1+3 Approach 7 where Sg = d4 x Lg(x). In inertial frames with Cartesian 4-coordinates xμ , the Poincaré generators, assumed finite due to suitable boundary conditions at spatial infinity, have the following expression: Pμ = d3 x Tμo (xo , x), Jμν = d3 x [xμ Tνo (xo , x) − xν Tμo (xo , x)]. In Appendix A there are some properties of the Poincaré algebra and group. At the Hamiltonian level the canonical Poincaré generators Pμ , Jμν satisfy the Pois- son algebra {Pμ , Pν } = 0, {Pμ , Jαβ } = ημα Pβ −ημβ Pα , {Jαβ , Jμν } = Cαβμν ρσ Jρσ (Cαβμν ρσ = ηαμ δβ ρ δν σ + ηβν δα ρ δμ σ − ηαν δβ ρ δμ σ − ηβμ δα ρ δν σ). If Jr = −1 2 ruv Juv is the generator of space rotations and Kr = Jro one of the boosts, the form of the canonical Poincaré algebra becomes {Jr , Js } = rst Jt , {Kr , Ks } = rsu Ju , {Jr , Ks } = {Kr , Js } = rsu Ku . To describe point particles with spin, with electric charge and with antiparti- cles of negative mass in a way that avoids self-reaction divergences at the classical level and gives the correct quantum theory after quantization, one needs a semi- classical approach, named pseudo-classical mechanics, in which these degrees of freedom are described with Grassmann variables. In Appendix B there is a review of this approach and of the needed mathematical tools. For the detailed mathematical properties of Minkowski space-time, see any book on SR, such as the recent ones of Gourgoulhon Ref. [12, 13]. 1.3 The 1+3 Approach to Local Non-Inertial Frames and Its Limitations Since the actual time-like observers are accelerated, we need some statement correlating the measurements made by them to those made by inertial observers, the only ones with a general framework for the interpretation of their experiments based on Einstein convention for the synchronization of clocks. This statement is usually the hypothesis of locality, which can be expressed in the following terms [55–60]: An accelerated observer at each instant along its world-line is physically equivalent to an otherwise identical momentarily comoving inertial observer, namely a non-inertial observer passes through a continuous infinity of hypothetical momentarily comoving inertial observers. While this hypothesis is verified in Newtonian mechanics and in those rela- tivistic cases in which a phenomenon can be reduced to point-like coincidences of classical point particles and light rays (geometrical optic approximation), its validity is questionable with moving continuous media (for instance the consti- tutive equations of the electromagnetic field inside them in non-inertial frames are still unknown) and in the presence of electromagnetic fields when their wavelength is comparable with the acceleration radii of the observer (the observer is not “static” enough to be able to measure the frequency of such a wave). See Refs. [61, 62] for a review of these topics. The fact that we can describe phenomena only locally near the observer and that the actual observers are accelerated leads to the 1+3 point of view (or threading splitting) [63–70], which tries to solve this problem starting from
- 28. 8 Galilei and Minkowski Space-Times the local properties of an accelerated observer, whose world-line is assumed to be the time axis of some frame. Given the world-line γ of the accelerated observer, we describe it with Lorentzian coordinates xμ (τ), parametrized with an affine parameter τ, with respect to a given inertial system. Its unit 4-velocity is uμ γ (τ) = ẋμ (τ)/ ẋ2(τ) [ẋμ = dxμ dτ ]. The observer proper time τγ(τ) is defined by ˙ x̃ 2 (τγ) = 1 if we use the notations xμ (τ) = x̃μ (τγ(τ)) and uμ (τ) = ũμ (τγ(τ)) = d x̃μ (τγ)/d τγ , and it is indicated by a comoving standard atomic clock. By a conventional choice of three spatial axes Eμ (a)(τ) = Ẽμ (a)(τγ(τ)), a = 1, 2, 3, orthogonal to uμ (τ) = Eμ (o)(τ) = ũμ (τγ(τ)) = Ẽμ (o)(τγ(τ)), the non-inertial observer is endowed with an ortho-normal tetrad Eμ (α)(τ) = Ẽμ (α)(τγ(τ)), α = 0, 1, 2, 3. This amounts to a choice of three comoving gyroscopes in addition to the comoving standard atomic clock. Usually the spatial axes are chosen to be Fermi–Walker transported as a standard of non-rotation, which takes into account the Thomas precession (see [71]). Since only the observer 4-velocity is given, this only allows identification of the tangent plane of the vectors orthogonal to this 4-velocity in each point of the world-line. Since there is no notion of a 3-space simultaneous with a point of γ and whose tangent space at that point is Rũ(τγ), this tangent plane is identified with an instantaneous 3-space both in SR and GR (it is the local observer rest-frame at that point). This identification is the basic limitation of this approach because the hyper-planes at different times intersect each other at a distance from the world-line depending on the acceleration of the observer so that the approach works only in a world-tube whose radius is this distance. See Refs. [63–70] for the definition of the (linear and rotational) acceleration radii of the observer. At each point of γ with proper time τγ(τ), the tangent space to Minkowski space-time in that point has the 1+3 splitting of vectors in vectors parallel to ũμ (τγ) and vectors lying in the three-dimensional (so-called local observer rest-frame) subspace Rũ(τγ) orthogonal to ũμ (τγ). Then Fermi normal coordinates [72–76] are defined on each hyper-plane orthogonal to the observer unit 4-velocity uμ (τ) and are used to define a notion of spatial distance. On each hyper-plane one considers three space-like geodesics as spatial coordinate lines. However, this produces only local coordinates and a notion of simultaneity valid only inside the world-tube. See Refs. [77–79] for variants of this approach, all unable to avoid the coordinate singularity on the world-tube. To this type of coordinate singularities we have to add the singularities shown by all the rotating coordinate systems (the problem of the rotating disk): In all the proposed uniformly rotating coordinate systems the induced 4-metric expressed in these coordinates has pathologies (the component 4 goo vanishes) at the distance R from the rotation axis where ω R = c with ω being the constant angular velocity of rotation. This is the horizon problem: At R the time-like 4-velocity of a disk point becomes light-like, even if there is no real horizon as happens for Schwarzschild black holes. Again, given the unit 4-velocity
- 29. 1.3 The 1+3 Approach 9 field of the points of the rotating disk, there is no notion of an instantaneous 3-space orthogonal to the associated congruence of time-like observers, due to the non-zero vorticity of the congruence [71] (see Section 2.1 for the definition of vorticity). Due to the Frobenius theorem, the congruence is (locally) hyper- surface orthogonal, i.e., locally synchronizable [71], if and only if the vorticity vanishes. Moreover, an attempt to use Einstein convention to synchronize the clocks on the rim of the disk fails and one finds a synchronization gap (see Refs. [80–84] and the bibliographies of Refs. [61, 62] for these problems). One does not know how to define the 3-geometry of the rotating disk, how to measure the length of the circumference, and which time and notion of simultaneity has to be used to evaluate the velocity of (massive or massless) particles in uniform motion along the circumference. The other important phenomenon connected with the rotating disk is the Sagnac effect (see again Refs. [61, 62, 80–84]), namely the phase difference generated by the difference in the time needed for a round-trip by two light rays, emitted at the same point, one co-rotating and the other counter-rotating with the disk. This effect, which has been tested for light, X-rays, and matter waves (Cooper pairs, neutrons, electrons, and atoms) and must be taken into account for the relativistic corrections to space navigation, has again an enormous number of theoretical interpretations (both in SR and GR). Here the lack of a good notion of simultaneity leads to problems of time discontinuities or desynchronization effects when comparing clocks on the rim of the rotating disk. In conclusion, in SR inertial frames are a limiting theoretical notion since, also disregarding GR, all the observers on Earth are non-inertial. According to the IAU 2000 Resolutions [32–35], for the physics in the solar system one can consider the Solar System Barycentric Celestial Reference System (BCRS) centered on the barycenter of the Solar System (with the axes identified by fixed stars (quasars) of the Hypparcos catalog) as a quasi-inertial frame. Instead, the Geocentric Celestial Reference System (GCRS), with origin in the center of the geoid, is a non-inertial frame whose axes are non-rotating with respect to the Solar Frame. Instead, every frame centered on an observer fixed on the surface of Earth (using the yet non-relativistic International Terrestrial Reference System [ITRS]) is both non-inertial and rotating. All these frames use notions of time connected to TAI. Let us also remark that the physical protocols (think of GPS) can establish a clock synchronization convention only inside future light-cone of the physical observer defining the local 3-spaces only inside it, in accord with the 1+3 point of view. This state of affairs and the need for predictability (a well-posed Cauchy problem for field theory) lead to the necessity of abandoning the 1+3 point of view and shifting to the 3+1 one. In this point of view, besides the world-line of an arbitrary time-like observer, it is given a global 3+1 splitting of Minkowski space-time, namely a foliation of it whose leaves are space-like hyper-surfaces.
- 30. 10 Galilei and Minkowski Space-Times Each leaf is both a Cauchy surface for the description of physical systems and an instantaneous Riemannian 3-space, namely a notion of simultaneity implied by a clock synchronization convention different from Einstein’s one. Even if it is unphysical (i.e., non-factual) to give initial data (the Cauchy problem) on a non-compact space-like hyper-surface, this is the only way to be able to use the existing existence and uniqueness theorems for the solutions of partial differential equations like the Maxwell ones, needed to test the predictions of the theory.1 Once we have given the Cauchy data on the initial Cauchy surface (an unphysical process), we can predict the future with every observer receiving the information only from his/her past light-cone (retarded information from inside it; electromagnetic signals on it). As emphasized by Havas [86], the 3+1 approach is based on Møller’s formalization [87, 88] of the notion of simultaneity. For non-relativistic observers the situation is simpler, but the non-factual need to give the Cauchy data on a whole initial absolute Euclidean 3-space is present also in this case for non-relativistic field equations like the Euler equation for fluids. Moreover, to study relativistic Hamiltonian dynamics one has to follow its formulation given by Dirac [89] with the instant, front (or light), and point forms and the associated canonical realizations of the Poincaré algebra. In the instant form, the simultaneity hyper-surfaces (Cauchy surfaces) defining a parameter for the time evolution are space-like hyper-planes xo = const., in the front form hyper-planes x− = 1 2 (xo − x3 ) = const. tangent to future light-cones, while in the point form the future branch of a two-sheeted hyperboloid x2 0. In a 6N-dimensional phase space for N scalar particles the ten generators of the Poincaré algebra are classified into kinematical generators (the generators of the stability group of the simultaneity hyper-surface) and dynamical generators (the only ones to be modified with respect to the free case in the presence of interactions) according to the chosen concept of simultaneity. While in the instant and point forms there are four dynamical generators (in the former energy and boosts, in the latter the 4-momentum), the front form has only three of them. We will see that the 3+1 approach is the natural framework to implement the instant form of relativistic Hamiltonian dynamics. Let us add the final remark that both in the 1+3 and in the 3+1 approach we call observer an idealization by means of a time-like world-line whose tangent vector in each point is the 4-velocity of the observer. If the 4-velocity is completed with a spatial triad to form a tetrad in each point of the world-line, we get an idealized observer with both a clock and a gyroscope. While this notion is compatible with the absolute metrology of SR, in GR it corresponds to a test 1 As far as we know the theorem on the existence and unicity of solutions has not yet been extended starting from data given only on the past light-cone. See Ref. [85] for an attempt to rephrase the instant form of dynamics in a form employing only data from the causal past light-cone of the observer.
- 31. 1.3 The 1+3 Approach 11 observer. To describe dynamical observers we need a model with dynamical matter in both cases. Therefore, an observer, or better a mathematical observer, is an idealization of a measuring apparatus containing an atomic clock and defining, by means of gyroscopes, a set of spatial axes (and then a, maybe ortho- normal, tetrad with a convention for its transport) in each point of the world-line. See Ref. [90] for properties of mathematical and dynamical observers.
- 32. 2 Global Non-Inertial Frames in Special Relativity In this chapter I will describe the 3+1 approach to global non-inertial frames in Minkowski space-time and the associated parametrized Minkowski theories for the description of matter admitting a Lagrangian description in these frames. 2.1 The 3+1 Approach to Global Non-Inertial Frames and Radar 4-Coordinates Assume that the world-line xμ (τ) of an arbitrary time-like observer carrying a standard atomic clock is given: τ is an arbitrary monotonically increasing func- tion of the proper time of this clock. Then one gives an admissible 3+1 splitting of Minkowski space-time, namely a nice foliation with space-like instantaneous 3-spaces Στ . It is the mathematical idealization of a protocol for clock synchro- nization allowing the formulation of the Cauchy problem for the equations of motion of fields: All the clocks in the points of Στ sign the same time of the atomic clock of the observer. The observer and the foliation define a global non-inertial reference frame after a choice of 4-coordinates. On each 3-space Στ one chooses curvilinear 3-coordinates σr having the observer as the origin. The quantities σA = (τ; σr ) are the Lorentz-scalar and observer-dependent “radar 4-coordinates,” first introduced by Bondi [62, 91, 92]. See Ref. [71] for the mathematical aspects of curved spaces like Στ . If xμ → σA (x) is the coordinate transformation from the Cartesian 4-coordinates xμ of an inertial frame centered on a reference inertial observer to radar coordinates, its inverse σA → xμ = zμ (τ, σr ) defines the embedding functions zμ (τ, σr ) describing the 3-spaces Στ as embedded 3-manifolds into Minkowski space-time. The induced 4-metric on Στ is the following functional of the embedding: 4 gAB(τ, σr ) = zμ A(τ, σr ) 4 ημν zν B(τ, σr ), (2.1)
- 33. 2.1 The 3+1 Approach 13 where zμ A(τ, σr ) = ∂ zμ (τ, σr )/∂ σA , zA μ (τ, σr ) = 4 gAB (τ, σr ) 4 ημν zν B(τ, σr ), and 4 ημν = (+ − −−) is the flat metric. While the 4-vectors zμ r (τ, σu ) are tangent to Στ , so that the unit normal lμ (τ, σu ) is proportional to μ αβγ [zα 1 zβ 2 zγ 3 ](τ, σu ), one has zμ τ (τ, σr ) = [N lμ + Nr zμ r ](τ, σr ), with N(τ, σr ) = [zμ τ lμ](τ, σr ) = 1 + n(τ, σr ) and Nr(τ, σr ) = − 4 gτr(τ, σr ) being the lapse and shift functions respectively. The unit normal lμ (τ, σu ) (lμ(τ, σu ) zμ r (τ, σu ) = 0, l2 (τ, σu ) = 1), and the space- like 4-vectors zμ r (τ, σu ) identify a (in general non-orthonormal) tetrad in each point of Minkowski space-time. The tetrad in the origin (lμ (τ, 0) (in general non-parallel to the observer 4-velocity), zμ r (τ, 0)) is a set of axes carried by the observer; their τ-dependence implies a convention of transport along the world-line. The lapse function measures the proper time interval N(τ, σr )dτ at z(τ, σr ) ∈ Στ between Στ and Στ+dτ . The shift functions Nr (τ, σr ) are defined so that Nr (τ, σr )dτ describes the horizontal shift on Στ such that, if zμ (τ + dτ, σr + dσr ) ∈ Στ+dτ , then zμ (τ + dτ, σr + dσr ) ≈ zμ (τ, σr ) + N(τ, σr ) dτlμ (τ, σr ) + [dσs + Ns (τ, σr )dτ]zμ s (τ, σr ). In each point of the 3-spaces Στ , one introduces arbitrary triads 3 er (a)(τ, σu ) (i.e., three gyroscopes) and the conjugated cotriads 3 e(a)r(τ, σu ) ((3 er a) 3 e(b)r) (τ, σu ) = δab, (3 er (a) 3 e(a)s)(τ, σu ) = δr s ). The components of the 4-metric 4 gAB(τ, σr ) and of the inverse 4-metric 4 gAB (τ, σr ) ((4 gAC 4 gcb)(τ, σr ) = δA B) can be parametrized in the following way (a = 1, 2, 3; ā = 1, 2): 4 gττ (τ, σu ) = N2 − Nr Nr (τ, σu ), − 4 gτr(τ, σu ) = Nr(τ, σu ) = 3 grs Ns (τ, σu ), 3 grs(τ, σu ) = − 4 grs(τ, σu ) = 3 a=1 3 e(a)r 3 e(a)s (τ, σu ) = φ̃2/3 3 a=1 e2 2 b̄=1 γb̄a Rb̄ Vra(θi ) Vsa(θi ) (τ, σu ), 4 gττ (τ, σu ) = N−2 (τ, σu ), 4 gτr (τ, σu ) = − Nr N−2 (τ, σu ), 4 grs (τ, σu ) = − 3 grs − Nr Ns N2 (τ, σu ), with 3 grs (τ, σu ) = 3 er (a) 3 es (a) (τ, σu ). (2.2) The quantity φ̃2 (τ, σu ) = det 3 grs(τ, σu ) = det [− 4 grs(τ, σu )] = γ(τ, σu ) appearing in the 3-metric is the 3-volume element on Στ . We have lμ (τ, σu ) = (φ̃−1 μ αβγ zα 1 zβ 2 zγ 3 )(τ, σu ) = 1 N(τ,σu) (1, Nr (τ, σu )), lμ(τ, σu ) = N(τ, σu ) 4 ημo = N(τ, σu ) (1; 000), d4 z = φ̃(τ, σu ) dτ d3 σ = γ(τ, σu) dτd3 σ, and g(τ, σu) = −det 4gAB(τ, σu) = N(τ, σu ) γ(τ, σu).
- 34. 14 Global Non-Inertial Frames in Special Relativity The quantities λa(τ, σr ) = [φ̃1/3 e 2 b̄=1 γb̄a Rb̄ ](τ, σr ) are the three positive eigenvalues of the 3-metric 3 grs(τ, σr ) (γāa are suitable numerical constants sat- isfying a γāa = 0, a γāa γb̄a = δāb̄, ā γāa γāb = δab − 1 3 ). Vra(θi (τ, σr )) are the components of a diagonalizing rotation matrix depending on three Euler angles, θi (τ, σr ), i = 1, 2, 3. The gauge Euler angles θr give a description of the 3-coordinate systems on Στ from a local point of view, because they give the orientation of the tangents to the three 3-coordinate lines through each point. However, as shown in appendix A of Ref. [94] and in Ref. [95], it is more convenient to replace the three Euler angles with the first kind coordinates θ̃i (τ, σr ) (−∞ θ̃i +∞) on the O(3) group manifold: With this choice we have Vru(θ̃i ) = (e− i T̂i θ̃i )ru, where (T̂i)ru = rui. Therefore, starting from the four independent embedding functions zμ (τ, σr ), one obtains the ten components 4 gAB of the 4-metric, which play the role of the inertial potentials generating the relativistic apparent forces in the non-inertial frame. For instance, the shift functions Nr(τ, σu ) = − 4 gτr(τ, σu ) describe iner- tial forces of the gravito-magnetic type induced by the global non-inertial frame. It can be shown [98–101] that the usual non-relativistic Newtonian inertial poten- tials are hidden in these functions. The extrinsic curvature tensor 3 Krs(τ, σu ) = [ 1 2 N (Nr|s + Ns|r − ∂τ 4 grs)](τ, σu ), describing the shape of the instantaneous 3- spaces Στ of the non-inertial frame as embedded 3-sub-manifolds of Minkowski space-time, is a secondary inertial potential, functional of the ten inertial poten- tials 4 gAB(τ, σr ). In this framework a relativistic positive-energy scalar particle with world-line xμ o (τ) is described by 3-coordinates ηr (τ) defined by xμ o (τ) = zμ (τ, ηr (τ)), satis- fying equations of motion containing relativistic inertial forces with the correct non-relativistic limit as shown in Refs. [98–101]. Fields have to be redefined so as to know the clock synchronization convention; for instance, a Klein–Gordon field φ̃(xμ ) has to be replaced with φ(τ, σr ) = φ̃(zμ (τ, σr )). The foliation is nice and admissible if it satisfies the following conditions: 1. N(τ, σr ) 0 in every point of Στ so that the 3-spaces never intersect, avoiding the coordinate singularity of the Fermi coordinates of the 1+3 approach. 2. 4 gττ (τ, σr ) = (zμ τ 4 ημν zν τ )(τ, σr ) = (N2 − Nu Nu )(τ, σr ) 0, so as to avoid the coordinate singularity of the rotating disk, and with the positive-definite 3-metric 3 grs(τ, σu ) = − 4 grs(τ, σu ) having three positive eigenvalues (these are the Møller conditions [87]). 3. All the 3-spaces Στ must tend to the same space-like hyper-plane at spatial infinity with a unit normal μ τ , which is the time-like 4-vector of a set of asymptotic orthonormal tetrads μ A (μ A 4 ημν ν B = 4 ηAB = (+ − −−)). These tetrads are carried by asymptotic inertial observers and the spatial axes μ r are identified by the fixed stars of star catalogues. At spatial infinity the lapse function tends to 1, the shift functions vanish, and the 4-metric becomes Euclidean with the same behavior, as will be discussed in relation to general relativity in Part II.
- 35. 2.1 The 3+1 Approach 15 By using the asymptotic tetrads μ A we can give the following parametrization of the embedding functions: zμ (τ, σr ) = xμ (τ) + μ A FA (τ, σr ), FA (τ, 0) = 0, xμ (τ) = xμ o + μ A fA (τ), (2.3) where xμ (τ) is the world-line of the observer. The functions fA (τ) determine the 4-velocity uμ (τ) = ẋμ (τ)/ ẋ2(τ) (ẋμ (τ) = dxμ(τ) dτ ) and the 4-acceleration aμ (τ) = duμ(τ) dτ of the observer. For an inertial frame centered on the inertial observer xμ (τ) = xμ o + μ τ τ, the embedding is zμ (τ, σr ) = xμ (τ) + μ r σr , with μ A being an orthonormal tetrad identifying the Cartesian axes. It is difficult to construct explicit examples of admissible 3+1 splittings because the Møller conditions are non-linear differential conditions on the functions fA (τ) and FA (τ, σr ). When these conditions are satisfied, Eq. (2.2) describes a global non-inertial frame in Minkowski space-time. Till now the solution of Møller conditions is known in the following two cases in which the instantaneous 3-spaces are parallel Euclidean space-like hyper-planes not equally spaced due to a linear acceleration: 1. Rigid non-inertial reference frames with translational acceleration. An exam- ple are the following embeddings: zμ (τ, σu ) = xμ o + μ τ f(τ) + μ r σr , 4 gττ (τ, σu ) = df(τ) dτ 2 , 4 gτr(τ, σu ) = 0, 4 grs(τ, σu ) = − δrs. (2.4) This is a foliation with parallel hyper-planes with normal lμ = μ τ = const. and with the time-like observer xμ (τ) = xμ o + μ τ f(τ) as the origin of the 3-coordinates. The hyper-planes have translational acceleration ẍμ (τ) = μ τ ¨ f(τ), so that they are not uniformly distributed like in the inertial case f(τ) = τ. 2. Differentially rotating non-inertial frames without the coordinate singularity of the rotating disk. The embedding defining these frames is as follows: zμ (τ, σu ) = xμ (τ) + μ r Rr s(τ, σ) σs →σ→∞ xμ (τ) + μ r σr , Rr s(τ, σ) = Rr s(αi(τ, σ)) = Rr s(F(σ) α̃i(τ)), 0 F(σ) 1 A σ , d F(σ) dσ = 0 (Moller conditions), zμ τ (τ, σu ) = ẋμ (τ) − μ r Rr s(τ, σ) δsw wuv σu Ωv (τ, σ) c , zμ r (τ, σu ) = μ k Rk v (τ, σ) δv r + Ωv (r)u(τ, σ) σu , (2.5) where σ = | σ| and Rr s(αi(τ, σ)) is a rotation matrix satisfying the asymptotic conditions Rr s(τ, σ) →σ→∞δr s , ∂A Rr s(τ, σ) →σ→∞ 0, whose Euler angles have
- 36. 16 Global Non-Inertial Frames in Special Relativity the expression αi(τ, σ) = F(σ) α̃i(τ), i = 1, 2, 3. The unit normal is lμ = μ τ = const. and the lapse function is 1+n(τ, σu ) = (zμ τ lμ)(τ, σu ) = μ τ ẋμ(τ) 0. In Eq. (2.5) one uses the notations Ω(r)(τ, σ) = R−1 (τ, σ) ∂r R(τ, σ) and R−1 (τ, σ) ∂τ R(τ, σ) u v = δum mvr Ωr(τ,σ) c , with Ωr (τ, σ) = F(σ) Ω̃(τ, σ) n̂r (τ, σ) being the angular velocity.1 The angular velocity vanishes at spatial infinity and has an upper bound proportional to the minimum of the linear velocity vl(τ) = ẋμ(τ) lμ orthogonal to the space-like hyper-planes. When the rotation axis is fixed and Ω̃(τ, σ) = ω = const., a simple choice for the function F(σ) is F(σ) = 1 1+ ω2 σ2 c2 .2 To evaluate the non-relativistic limit for c → ∞, where τ = c t with t the absolute Newtonian time, one chooses the gauge function F(σ) = 1 1+ ω2 σ2 c2 →c→∞ 1 − ω2 σ2 c2 + O(c−4 ). This implies that the corrections to rigidly rotating non-inertial frames coming from Møller conditions are of order O(c−2 ) and become important at the distance from the rotation axis where the horizon problem for rigid rotations appears. As shown in Ref. [62], global rigid rotations are forbidden in relativistic theories, because, if one uses the embedding zμ (τ, σu ) = xμ (τ) + μ r Rr s(τ) σs describing a global rigid rotation with angular velocity Ωr = Ωr (τ), then the resulting gττ (τ, σu ) violates Møller conditions because it vanishes at σ = σR = 1 Ω(τ) ẋ2(τ) + [ẋμ(τ) μ r Rr s(τ) (σ̂ × Ω̂(τ))r]2 −ẋμ(τ) μ r Rr s(τ) (σ̂ × Ω̂(τ))r (σu = σ σ̂u , Ωr = Ω Ω̂r , σ̂2 = Ω̂2 = 1). At this distance from the rotation axis the tangential rotational velocity becomes equal to the velocity of light (the horizon problem of the rotating disk). Let us remark that even if in the existing theory of rotating relativistic stars [102] one uses differential rotations, notwithstanding that in the study of the magnetosphere of pulsars often the notion of light cylinder of radius σR is still used. 3. The search of admissible 3+1 splittings with non-Euclidean 3-spaces is much more difficult. The simplest case [90] is a parametrization of the embeddings (Eq. 2.1) in terms of Lorentz matrices ΛA B(τ, σ) →σ→∞ δA B (ΛA C (τ, σ) 4 ηAB ΛB D(τ, σ) = 4 ηCD) depending on σ = r (σr)2 and with ΛA B(τ, 0) finite:3 zμ (τ, σr ) = xμ (τ) + μ A ΛA r (τ, σ) σr →σ→∞ xμ (τ) + μ r σr , zμ (τ, 0) = xμ (τ) = xμ o + μ A fA (τ), 1 n̂r(τ, σ) defines the instantaneous rotation axis and 0 Ω̃(τ, σ) 2 max ˙ α̃(τ), ˙ β̃(τ), ˙ γ̃(τ) . 2 Nearly rigid rotating systems, like a rotating disk of radius σo, can be described by using a function F(σ) approximating the step function θ(σ − σo). 3 It corresponds to the locality hypothesis of Ref. [55–60], according to which at each instant of time the detectors of an accelerated observer give the same indications as the detectors of the instantaneously comoving inertial observer.
- 37. 2.1 The 3+1 Approach 17 ẋμ (τ) = μ A ˙ fA (τ) = μ A α(τ) γx(τ) 1 βr x(τ) , ẋ2 (τ) = α2 (τ) 0, fA (τ) = τ o dτ1 α(τ1) γx(τ1) 1 βr x(τ) , γx(τ) = 1 1 − β2 x(τ) . (2.6) implying zμ r (τ, σu ) = Λμ a (τ, σu ) 3 e(a)r(τ, σu ) and 3 grs(τ, σu ) = 3 a=1 (3 e(a)r 3 e(a)s) (τ, σu ). The origin of the 3-coordinates σr in the 3-spaces Στ is a time-like accelerated observer with world-line xμ (τ), whose instantaneous 3-velocity, divided by c, is βx(τ) (| βx(τ)| 1) and τ is the proper time of the observer when α(τ) = 1. The 4-velocity of the observer is uμ (τ) = μ A βA x (τ)/ 1 − β2 x(τ), βA x (τ) = (1; βr x(τ)). The Lorentz matrix is written in the form Λ = B R as the product of a boost B(τ, σ) and a rotation R(τ, σ), like the one in Eq. (2.5), (Rτ τ = 1, Rτ r = 0, Rr s = Rr s). The components of the boost are Bτ τ (τ, σ) = γ(τ, σ) = 1/ 1 − β2(τ, σ), Bτ r (τ, σ) = γ(τ, σ) βr(τ, σ), Rr s(τ, σ) = δr s + γ2 βr βs 1+γ (τ, σ), with βr (τ, σ) = G(σ) βr (τ). The Møller conditions are restrictions on G(σ) →σ→∞ 0 with G(0) finite, whose explicit form is given in Ref. [90] for the following two cases: (1) boosts with small velocities; and (2) time-independent boosts. See Ref. [99] for the description of the electromagnetic field and of phenomena like the Sagnac effect and the Faraday rotation in this framework for non- inertial frames. Moreover, the embedding (Eq. 2.5) has been used in Ref. [103] on quantum mechanics in non-inertial frames. Each admissible 3+1 splitting of space-time allows one to define two associated congruences of time-like observers. 1. The congruence of the Eulerian observers with the unit normal lμ (τ, σr ) = zμ A(τ, σr ) lA (τ, σr ) to the 3-spaces embedded in Minkowski space-time as unit 4-velocity. The world-lines of these observers are the integral curves of the unit normal and in general are not geodesics. In adapted radar 4-coordinates the orthonormal tetrads carried by the Eulerian observers are lA (τ, σr ), 4 ◦ Ē A (a)(τ, σr ) = (0; 3 eu (a)(τ, σr )), where 3 eu (a)(τ, σr ) (a = 1, 2, 3) are triads on the 3-space. If 4 ∇ is the covariant derivative associated with the 4-metric 4 gAB(τ, σr ) induced by a 3+1 splitting, the equation ((4 ∇A)B C = δB C ∂A + 4 ΓB AC ; 3 hAB (τ, σr ) = 4 gAB(τ, σr ) − lA (τ, σr )lB(τ, σr )) 4 ∇A lB(τ, σr ) = lA 3 aB + σAB + 1 3 θ 3 hAB − ωAB (τ, σr ) (2.7) defines the acceleration 3 aA (τ, σr ) (3 aA (τ, σr ) lA(τ, σr ) = 0), the expansion θ(τ, σr ), the shear σAB(τ, σr ) = σBA(τ, σr ) (σAB(τ, σr ) lB (τ, σr ) = 0), and the vorticity or twist ωAB(τ, σr ) = −ωBA(τ, σr ) (ωAB(τ, σr ) lB (τ, σr ) = 0) of the Eulerian observers with ωAB(τ, σr ) = 0 since they are surface-forming by construction.
- 38. 18 Global Non-Inertial Frames in Special Relativity 2. The skew congruence with unit 4-velocity vμ (τ, σr ) = zμ A(τ, σr ) vA (τ, σr ) (in general it is not surface-forming, i.e., it has a non-vanishing vorticity, like that of a rotating disk). The skew congruence is defined by requiring that its observers have the world-lines (integral curves of the 4-velocity) defined by σr = const. for every τ, because the unit 4-velocity tangent to the flux lines xμ σo (τ) = zμ (τ, σr o) is vμ σo (τ) = zμ τ (τ, σr o)/ 4gττ (τ, σr o) (there is no horizon problem because it is everywhere time-like in admissible 3+1 splittings). They carry contro-variant orthonormal tetrads, given in Ref. [104], not adapted to the foliation, connected in each point by a Lorentz transformation to the ones of the Eulerian observer present in this point. Let us add another motivation for using the 3+1 approach. Let us consider the standard action for a system of N charged scalar particles plus the electro- magnetic field: S = N i=1 dτ − mic ẋ2 i (τ) − ei ẋμ i (τ) Aμ(xi(τ)) − 1 4 d4 z Fμν (z) Fμν (z) = dτ Lm + LI + d4 z Lem, (2.8) where mi and ei are the masses and charges of the particles and Fμν (z) = ∂μ Aν (z) − ∂ν Aμ(z). The momenta are piμ(τ) = − ∂(Lm + LI )(τ) ∂ẋμ i = mic ẋμ i (τ) ẋ2 i (τ) + ei Aμ(xi(τ)), πμ (zo , z) = − ∂L(zo , z) ∂ ∂o Aμ(zo, z) = Foμ (zo , z), (2.9) and we get the following primary constraints: φ̄i(τ) = pi(τ) − ei A(xi(τ)) 2 − m2 i c2 ≈ 0, πo (zo , z) ≈ 0. (2.10) Since the natural time parameter for the field degrees of freedom is zo , while the particle world-lines are parametrized by an arbitrary scalar τ, there is no concept of equal time and it is impossible to evaluate the Poisson bracket of these constraints. Also, due to the same reason, the Dirac Hamiltonian, which would be H̄D = H̄c + N i=1 λi (τ) χ̄i(τ) + d3 zλo (zo , z)πo (zo , z) with H̄c the canonical Hamiltonian and with λi (τ), λo (zo , z) Dirac’s multipliers, does not make sense. This problem is present even at the level of the Euler–Lagrange equations: How does one formulate a Cauchy problem for a system of coupled equations, some of which are ordinary differential equations in the affine parameter τ along the particle world-line, while the others are partial differential equations depending on Minkowski coordinates zμ ?
- 39. 2.2 Parametrized Minkowski Theory 19 The standard non-manifestly covariant approach uses zo as the time parameter by rewriting the action (Eq. 2.8) in the following form: S = d4 z − N i=1 dτ δ4 (xi(τ) − z) mi c ẋ2 i (τ) + ei ẋμ i (τ) Aμ(z) − 1 4 Fμν (z) Fμν (z) = d4 z − N i=1 δ3 ( xi(zo ) − z) mi c 1 − ( d xi(zo) dzo )2 + ei Ao(z) − d xi(zo ) dzo · A(z) − 1 4 Fμν (z) Fμν (z) , (2.11) where δ(xo i (τ) − zo ) = δ(τ − fi(zo )) / | dxo i dτ | has been used as a gauge fixing, eliminating the variables xo i . Since the po i are determined by the mass-shell constraints, we remain with six degrees of freedom for particle and no-particle constraint. The net result of this prescription is to break τ-reparametrization invariance. This problem is due to the lack of a covariant concept of equal time between field and particle variables, so that the 3+1 approach is needed. 2.2 Parametrized Minkowski Theory for Matter Admitting a Lagrangian Description In the global non-inertial frames of Minkowski space-time it is possible to describe isolated systems (particles, strings, fields, fluids) admitting a Lagrangian formu- lation by means of parametrized Minkowski theories [98, 99, 105]. The existence of a Lagrangian, which can be coupled to an external gravitational field, makes possible the determination of the matter energy–momentum tensor and of the ten conserved Poincaré generators Pμ and Jμν (assumed finite) of every configuration of the isolated system. First of all, one must replace the matter variables of the isolated system with new ones knowing the clock synchronization convention defining the 3-spaces Στ . For instance a Klein–Gordon field φ̃(x) will be replaced with φ(τ, σr ) = φ̃(z(τ, σr )); and the same for every other field. Instead, for a relativistic particle with world-line xμ (τ), one must make a choice of its energy sign, then the positive- (or negative-) energy particle will be described by 3-coordinates ηr (τ) defined by the intersection of its world-line with Στ : xμ (τ) = zμ (τ, ηr (τ)). Differ- ent from all the previous approaches to relativistic mechanics, the dynamical con- figuration variables are the 3-coordinates ηr (τ) and not the world-lines xμ (τ) (to rebuild them in an arbitrary frame one needs the embedding defining that frame).
- 40. 20 Global Non-Inertial Frames in Special Relativity Then, one replaces the external gravitational 4-metric in the coupled Lagrangian with the 4-metric 4 gAB(τ, σr ), which is a functional of the embedding defining an admissible 3+1 splitting of Minkowski space-time, and the matter fields with the new ones knowing the instantaneous 3-spaces Στ . Parametrized Minkowski theories are defined by the resulting Lagrangian depending on the given matter and on the embedding zμ (τ, σr ). The resulting action is invariant under the frame-preserving diffeomorphisms τ → τ (τ, σu ), σr → σ r (σu ) first introduced in Ref. [106]. As a consequence, there are four first-class constraints with exactly vanishing Poisson brackets (an Abelianized analogue of the super-Hamiltonian and super-momentum constraints of canonical gravity) determining the momenta conjugated to the embeddings in terms of the matter energy–momentum tensor. These constraints ensure the independence of the description from the choice of the foliation and with their help it is possible to rebuild some kind of covariance (Wigner covariance, as we shall see) notwithstanding the 3+1 splitting of flat space-time. This implies that the embeddings zμ (τ, σr ) are gauge variables, so that all the admissible non-inertial or inertial frames are gauge-equivalent, namely physics does not depend on the clock synchronization convention and on the choice of the 3-coordinates σr – only the appearances of phenomena change by changing the notion of instantaneous 3-space. Therefore, the embedding configuration variables zμ (τ, σ) will describe all the possible inertial effects compatible with special relativity. The matter energy–momentum tensor associated with this Lagrangian allows the determination of the ten conserved Poincaré generators Pμ and Jμν of every configuration of the system (in non-inertial frames they are asymptotic generators at spatial infinity, like the ADM ones in general relativity). The behavior of matter at spatial infinity on each 3-space Στ must be such that the Poincaré generators are finite with a 4-momentum not space-like. As an example, one may consider N free scalar particles with masses mi. Usually the time-like world-lines of the particles are described by Cartesian 4-coordinates xμ i (τi), depending on the particle proper time in an inertial frame. At the Hamiltonian level one has that the 4-momenta piμ(τi) satisfy the mass- shell first-class constraints p2 i (τi) − m2 i = 0. Therefore, there are two solutions with opposite sign for the energies (particles and antiparticles), so that the time components xo i (τ) are gauge variables, sources of the problem of how to eliminate the relative times in relativistic bound states. In the 3+1 approach the time-like world-lines of the N particles are Cartesian 4-coordinates xμ i (τ) parametrized with the time τ of Bondi radar 4-coordinates so that we have xμ i (τ) = zμ (τ, ηr i (τ)), i = 1, . . . , N. Therefore, each particle is identified by the three numbers σr = ηr i (τ) individuating the intersection of each world-line with the 3-space Στ , and not by four. As a consequence, each particle must have a well-defined sign of the energy, ηi = sign po i = ±. Each particle with a definite sign of the energy will be described by the six Lorentz-scalar
- 41. 2.2 Parametrized Minkowski Theory 21 canonical coordinates ηr i (τ), κir(τ) at the Hamiltonian level, like in Newtonian mechanics. There are no longer mass-shell constraints, because we cannot describe the two topologically disjoint branches of the mass hyperboloid simultaneously as in the standard manifestly Lorentz-covariant approach. By using Eq. (2.1), the particle 4-velocities can be written in the form ẋμ i (τ) = N(τ, ηu i (τ)) lμ (τ, ηu i (τ)) + [η̇r i (τ) + Nr (τ, ηu i (τ))] zμ r (τ, ηu i (τ)), with ẋ2 i (τ) = N2 − 3 grs (Nr +η̇r i (τ)) (Ns +η̇s i ) (τ, ηu i (τ)). Therefore the derived usual particle momenta, satisfying p2 i − m2 i c2 ≈ 0, are pμ i (τ) = mi c ẋμ i (τ)/ ẋ2 i (τ) = ηi m2 i c2 − 3grs(τ, ηu i (τ)) κir(τ) κis(τ) lμ (τ, ηu i (τ)) + κir(τ) 3 grs (τ, ηu i (τ)) zμ s (τ, ηu i (τ)). In parametrized Minkowski theories the free particles are described by the following action depending on the configurational variables ηr i (τ) of the particles with energy sign ηi and on the embedding variables zμ (τ, σr ) of an arbitrary admissible 3+1 splitting of Minkowski space-time: S = dτ d3 σ L(τ, σu ) = dτ L(τ), L(τ, σu ) = − N i=1 δ3 (σu − ηu i (τ)) mic ηi [4gττ (τ, σu) + 2 4gτr(τ, σu) η̇r i (τ) + 4grs(τ, σu) η̇r i (τ) η̇s i (τ)] = − N i=1 δ3 (σu − ηu i (τ)) ηi mi c N2(τ, σu)+ 4grs(τ, σu) [η̇r i (τ)+Nr(τ, σu)] [η̇s i (τ)+Ns(τ, σu)]. (2.12) This action is invariant under separate τ- and σr -reparametrizations. The resulting canonical momenta and their Poisson brackets are ρμ(τ, σu ) = − ∂L(τ, σu ) ∂zμ τ (τ, σu) = N i=1 δ3 (σu − ηu i (τ)) ηi mi c zτμ(τ, σu ) + zrμ(τ, σu ) η̇r i (τ) [4gττ (τ, σu) + 2 4gτr(τ, σu) η̇r i (τ) + 4grs(τ, σu) η̇r i (τ) η̇s i (τ)] = [(ρν lν ) lμ + (ρν zν r ) 3 grs zsμ](τ, σu ), κir(τ) = − ∂L(τ) ∂η̇r i (τ) = 3 grs(τ, ηu i (τ)) κi s (τ) = ηi mi c 4gτr(τ, ηu i (τ)) + 4grs(τ, ηu i (τ)) η̇s i (τ) [4gττ (τ, ηu i (τ))+2 4gτr(τ, ηu i (τ)) η̇r i (τ)+4grs(τ, ηu i (τ)) η̇r i (τ) η̇s i (τ)] , {zμ (τ, σu ), ρν (τ, σ u )} = − 4 ημ ν δ3 (σu − σ u ), {ηr i (τ), κjs(τ)} = −δij δr s . (2.13)
- 42. 22 Global Non-Inertial Frames in Special Relativity The Poincaré generators and the energy–momentum tensor of this system are ({zμ (τ, σu ), Pν } = − 4 ημν ; in inertial frames we have T⊥⊥ = Tττ and T⊥r = −3 grs Tτs ): Pμ = d3 σρμ (τ, σu ), Jμν = d3 σ(zμ ρν − zν ρμ )(τ, σu ), TAB (τ, σu ) = − 2 − det 4gCD(τ, σu) δ S δ 4gAB(τ, σu) , Tμν = zμ A zν B TAB , T⊥⊥(τ, σu ) = lμ lν Tμν (τ, σu ) = (N Tττ )(τ, σu ) = N i=1 δ3 (σu − ηu i (τ)) φ̃(τ, σu) ηi m2 i c2 + 3grs(τ, σu) κir(τ) κis(τ), T⊥r(τ, σu ) = lμ zrν Tμν (τ, σu ) = −[N 3 grs (Tττ Ns + Tτs ](τ, σu ) = N i=1 δ3 (σu − ηu i (τ)) φ̃(τ, σu) κir(τ), Trs(τ, σu ) = zrμ zsν Tμν (τ, σu ) = [Nr Ns Tττ + (Nr 3 gsm + Ns 3 grm) Trm + 3 grm 3 gsn Tmn ](τ, σu ) = N i=1 δ3 (σu − ηu i (τ)) φ̃(τ, σu) ηi κir(τ) κis(τ) m2 i c2 + 3gvw(τ, σu) κiv(τ) κiw(τ) . (2.14) The four first-class constraints implying the gauge nature of the embedding and the gauge equivalence of the description in different non-inertial frames are:4 Hμ(τ, σu ) = ρμ(τ, σu ) − φ̃(τ, σu) lμ T⊥⊥ − zrμ 3 grs T⊥s (τ, σu ) ≈ 0, {Hμ(τ, σu ), Hμ(τ, σv )} = 0, d3 σ Hμ (τ, σu ) = Pμ − N i=1 lμ (τ, ηu i (τ)) ηi m2 i c2 + 3grs(τ, ηu i (τ)) κir(τ) κis(τ) − N i=1 zμ r (τ, ηu i (τ)) 3 grs (τ, ηu i (τ)) κis(τ) ≈ 0. (2.15) Since the canonical Hamiltonian vanishes, one has the Dirac Hamiltonian (λμ (τ, σu ) are Dirac’s multipliers): 4 Since the constraints (Eq. 2.15) are distributions concentrated at the position of particles, in evaluating their Poisson brackets we must use the embeddings zμ(τ, σu) at a generic point and not zμ(τ, ηu i (τ)).
- 43. 2.2 Parametrized Minkowski Theory 23 HD = d3 σ λμ (τ, σu ) Hμ(τ, σu ) = d3 σ λμ(τ, σu ) lμ (τ, σu ) Hλ(τ, σu ) +λμ(τ, σu ) zμ s (τ, σu ) 3 gsr (τ, σu ) Hλ r(τ, σu ) , Hλ(τ, σu ) = (lμ Hμ )(τ, σu ) ≈ 0, Hλ r(τ, σu ) = (zr μHμ )(τ, σu ) ≈ 0, {Hλ r(τ, σu ), Hλ s(τ, σ u )} = Hλ r(τ, σ u ) ∂δ3 (σu , σ u ) ∂σs + Hλ s(τ, σu ) ∂δ3 (σu , σ u ) ∂σr , {Hλ(τ, σu ), Hλ r(τ, σ u )} = Hλ(τ, σu ) ∂δ3 (σu , σ u ) ∂σr , {Hλ(τ, σu ), Hλ(τ, σ u )} = 3 grs (τ, σu ) Hλ s(τ, σu ) +3 grs (τ, σ u ) Hλ s(τ, σ u ) ∂δ3 (σu , σ u ) ∂σr , (2.16) and one finds that {Hμ(τ, σu ), HD} = 0. Therefore, there are only the four first-class constraints of Eq. (2.16). The constraints Hμ(τ, σu ) ≈ 0 describe the arbitrariness of the foliation: Physical results do not depend on its choice. In Eq. (2.16) we have also shown the non-holonomic form Hλ(τ, σu ) ≈ 0, Hλ r(τ, σu ) ≈ 0 of the constraints, which satisfies the universal Dirac algebra of the super-momentum and super-Hamiltonian constraints of ADM canonical metric gravity (see Chapter 5). The same description can be given when the matter are not particles but the Klein–Gordon [107] and Dirac [108] fields and for the electromagnetic field [30], starting by their usual description given, for instance, in Ref. [109]. To describe the physics in a given admissible non-inertial frame described by an embedding zμ F (τ, σu ), one must add the gauge fixings: ζμ (τ, σu ) = zμ (τ, σu ) − zμ F (τ, σu ) ≈ 0. (2.17) If we put zμ F (τ, σu ) = xμ (τ) + bμ r (τ) σr , with xμ (τ) a non-inertial observer (origin of the 3-coordinates σr ) and with bμ A(τ) an orthonormal tetrad such that the constant (future-pointing) unit normal to the 3-spaces is lμ = bμ τ = μ αβγ bα 1 (τ) bβ 2 (τ) bγ 3 (τ) = const., we get a foliation of Minkowski space-time with space-like hyper-planes. If xμ (τ) is the world-line of a time-like inertial observer, the gauge fixings with zμ F (τ, σu ) = xμ (τ) + bμ r (τ) σr (2.18) describe inertial frames in Minkowski space-time with zμ τ (τ, σr ) ≈ ẋμ (τ) + ḃμ r (τ) σr and zμ r (τ) ≈ bμ r (τ). Usually the inertial frames have bμ r (τ) = μ r with
- 44. 24 Global Non-Inertial Frames in Special Relativity μ r constant asymptotic triads. Moreover, we have 3 grs(τ) = bμ r (τ) 4 ημν bν s (τ) = a 3 e(a)r(τ) 3 e(a)s(τ) = δrs (in terms of cotriads inside Στ contained in the chosen parametrization of bμ r (τ)) with inverse 3 grs (τ) = a 3 er (a)(τ) 3 es (a)(τ) (in terms of the dual triads). The shift and lapse functions have the expres- sion Nr(τ, σu ) = − 4 gτr(τ, σu ) = (ẋμ (τ) + ḃμ s (τ) σs ) 4 ημν bν r (τ), N2 (τ, σu ) = (Nr Nr)(τ, σ) + (ẋμ (τ) + ḃμ r (τ) σr ) 4 ημν (ẋν (τ) + ḃν s (τ) σs ). The second-class constraints implied by the gauge fixings (Eq. 2.17) together with the constraints Hμ(τ, σu ) ≈ 0 lead to the following Dirac brackets: {f, g}∗ = {f, g} − d3 σ {f, ζμ (τ, σu )} {Hμ(τ, σu ), g} −{f, Hμ(τ, σu )} {ζμ (τ, σu ), g} . (2.19) The preservation in time of the gauge fixings ( d dτ ζμ (τ, σu ) = {ζμ (τ, σu ), HD} ≈ 0) implies the following form of the Dirac multipliers: λμ (τ, σu ) = λ̃μ (τ)+ λ̃μ ν (τ) bν r (τ) σr , with λ̃μ (τ) = −ẋμ (τ), λ̃μν (τ) = 1 2 [ḃμ r (τ) bν r (τ) − bμ r (τ) ḃν r (τ)]. The space-like hyper-planes still depend on ten residual gauge degrees of freedom: (1) the world-line xμ (τ) of the inertial observer chosen as the origin of the 3-coordinates σr ; and (2) six variables parametrizing an orthonormal tetrad bμ A(τ). Their ten conjugate variables are contained in the canonical momenta ρμ(τ, σu ) conjugated to the embedding and are (1) the total 4-momentum Pμ canonically conjugate to xμ (τ), {xμ , Pν }∗ = − 4 ημν ; and (2) six momentum variables canonically conjugate to the tetrads bμ A(τ). They are determined by the constraints (Eq. 2.15). Therefore, after this gauge fixing the Dirac Hamilto- nian depends only on the following ten surviving first-class constraints implied by Eq. (2.15): HD = λ̃μ (τ) H̃μ(τ) − 1 2 λ̃μν (τ) H̃μν (τ), H̃μ (τ) = d3 σ Hμ (τ, σ) = Pμ − lμ N i=1 ηi m2 i c2 + 3grs(τ) κir(τ) κis(τ) −bμ r (τ) N i=1 3 grs (τ) κis(τ) ≈ 0, H̃μν (τ) = d3 σ σr bμ r (τ) Hν (τ, σ) − bν r (τ) Hμ (τ, σ) = Sμν − (bμ r (τ) lν − bν (τ) lμ ) N i=1 ηr i (τ)) ηi m2 i c2 + 3grs(τ, ηu i (τ)) κir(τ) κis(τ) −(bμ r (τ) bν s (τ) − bν r (τ) bμ s (τ)) N i=1 ηr i (τ)) κis(τ) ≈ 0,
- 45. 2.2 Parametrized Minkowski Theory 25 {H̃μ (τ), H̃ν (τ)} = {H̃α (τ), H̃μν (τ)} = 0, {H̃μν (τ), H̃αβ (τ)} = Cμναβ γδ H̃γδ (τ), (2.20) with Sμν the spin part of the Lorentz generators whose form, implied by Eqs. (2.14) and (2.18), is Jμν = xμ (τ) Pν − xν (τ) Pμ + Sμν , Sμν = d3 σ σr bμ r (τ) ρν (τ, σ) − bν r (τ) ρμ (τ, σ) , (2.21) and with Cμναβ γδ = δν γ δα δ 4 ημβ + δμ γ δβ δ 4 ηνα − δν γ δβ δ 4 ημα − δμ γ δα δ 4 ηνβ being the structure constants of the Lorentz algebra. Since Pμ is the total conserved 4-momentum of the isolated system, the conjugate variable xμ (τ) describes an inertial observer taking into account its global properties, namely having the role of some kind of relativistic center of mass of the isolated system (see next chapter). From now on we will use the notation Pμ = (Po = E/c; P) = M c uμ (P) = Mc ( 1 + h2; h) def = Mc hμ , where h = v/c is an a-dimensional 3-velocity and with M c = √ P2. For an isolated system with total conserved time-like 4-momentum Pμ , the “inertial rest-frame” is the inertial 3+1 splitting with lμ = Pμ / √ P2 and with bμ r (τ) = μ r , where lμ = μ o and μ r are constant orthonormal tetrads. Instead, the “non-inertial rest-frames” of an isolated system with total con- served time-like 4-momentum Pμ are those admissible non-inertial 3+1 splittings whose 3-spaces Στ tend to space-like hyper-planes perpendicular to Pμ at spatial infinity. In these non-inertial rest-frames the Poincaré generators are asymptotic (constant of the motion) symmetry generators like the asymptotic ADM ones in the asymptotically Minkowskian space-times of general relativity (see Part II). The non-relativistic limit of the parametrized Minkowski theories gives the parametrized Galilei theories, which are defined and studied in Ref. [51] using also the results of Refs. [49, 50]. Also the inertial and non-inertial frames in Galilei space-time of Section 1.1 are gauge-equivalent in this formulation. In this approach, a non-relativistic particle of 3-coordinates xa (t) in an inertial frame is described in the non-inertial frames by 3-coordinates ηu (t) defined by the equation xa (t) = Aa (t, ηu (t)), where Aa is the function describing the non-relativistic non-inertial frame (see after Eq. (1.1)). Therefore the standard velocity takes the form ẋa (t) = dAa (t, ηu (t)) dt = ∂ Aa (t, ηu (t)) ∂t + Ja r (t, ηu (t)) d ηu (t) dt . (2.22) In the case of N particles xa i (t), i = 1, . . . , N, interacting with arbitrary poten- tial V (t, x1(t), . . . , xN (t)) the equation of motion mi d2xa i (t) dt2 = − ∂ V ∂ xa i (t, x1(t), . . . , xN (t)) take the form ( ˜ Jr a is the inverse of Ja r )
- 46. 26 Global Non-Inertial Frames in Special Relativity mi d dt ∂ Aa (t, ηu i (t)) ∂t + Ja r (t, ηu i (t)) d ηr i (t) dt = − ∂ V̂ ∂ xa i |xu i (t)=Au(t,ηv i (t)) = − ˜ Jr a (t, ηu i (t)) j ∂ V̂ ∂ ηr j , (2.23) with V̂ = V xi(t)= A(t, ηi(t)). The Lagrangian of parametrized Galilei theory is L(t) = d3 σ ia δ3 (σu − ηu i (t)) 1 2mi ∂ Aa (t, σu ) ∂ t + Ja r (t, σu ) dηr i (t) dt 2 − V̂ . (2.24) It depends on the particles and on the non-inertial frame variable Aa (t, σu ). As shown in Ref. [51] the action S = dt L(t) is invariant under local Noether transformations δ ηk i (t) = Fa (t, ηv i (t)) ˜ Jr a (t, ηv i (t)), δ Aa (t, σk ) = Fa (t, σk ) with Fa (t, σk ) arbitrary functions. Therefore, these are first-class constraints implying that the functions Aa (t, σr ) are gauge variables. As a consequence, the description of physics in inertial and non-inertial frames is connected by gauge transformations, like in the relativistic case.
- 47. 3 Relativistic Dynamics and the Relativistic Center of Mass In this chapter I face the problem of how to get a consistent description of relativistic particle dynamics, eliminating the problem of relative times in rel- ativistic bound states and clarifying the endless problem of which definition of the relativistic center of mass has to be used. After a review of the attempts to describe interacting relativistic particles taking into account the no-interaction theorem, I will show that the dynamics must be formulated in terms of suitable covariant relative variables after the separation of an external canonical but not covariant center of mass. The world-tube in Minkowski space-time, where the non-covariance effects are concentrated, has an intrinsic radius, the Møller radius, determined by the value of the Poincaré Casimir invariants associated to the given configuration of the isolated system. It exists due to the Lorentz signature of Minkowski space-time. It is a classical unit of length determined by the system itself, which exists also for classical fields and is a natural candidate for a ultraviolet cutoff in quantization. To describe the relative motions of an isolated system of interacting particles in a covariant way in special relativity (SR) I use the inertial foliation of Minkowski space-time in which the 3-spaces are space-like hyper-planes orthogonal to the time-like (assumed finite) 4-momentum of the isolated system, namely its inertial rest-frame. This allows defining canonical bases of Wigner-covariant relative variables inside the 3-spaces (named Wigner hyper-planes) of these rest-frames and defin- ing the Wigner-covariant rest-frame instant form of dynamics. As a consequence, I can develop a new Hamiltonian kinematics for positive-energy scalar and spin- ning particles. Before introducing these new developments, let us explore a brief historical review of the theories involving interacting relativistic point particles. The theory of relativistic bound states and the interpretational problems with the Bethe–Salpeter equation [109] require the understanding of the instantaneous approximations to quantum field theory so as to arrive at an effective relativistic
- 48. 28 Relativistic Dynamics and Center of Mass wave equation and to an acceptable scalar product. In turn, the wave equation must also result from the quantization of a relativistic action-at-a-distance two- body problem in relativistic mechanics (but with the particles interpreted as asymptotic states of quantum field theory), since only in this way can we get a solution to the interpretational problems connected to the gauge nature of the relative times. Usually the particle world-lines qμ i (τi), i = 1, . . . , N, are parametrized with independent affine parameters τi and the action principles describing them are invariant under separate reparametrizations of each world-line, since this is geo- metrically possible even in the presence of interactions with a finite time delay to avoid instantaneous action at a distance. Since the dynamical correlation among the points on the particle’s world-lines is not in general one-to-one in these approaches, it is impossible to develop a Hamiltonian formulation starting from the Euler–Lagrange integro-differential equations of motion implied by the delay. The natural development of these approaches was field theory – for instance the study of the coupled system of relativistic charged particles plus the electromagnetic field. As an alternative to field theory, there was the development of relativistic mechanics with action-at-a-distance interactions described by suitable potentials implying a one-to-one correlation among the world-lines [110–113]. Geometrically each particle has its world-line described by a 4-vector (the 4-position) qμ i (τi), i = 1, . . . , N, parametrized with an independent arbitrary affine scalar parameter τi 1 By inverting qo i (τi) to get τi = τi(qo i ), we can identify the world-line in a non-manifestly covariant way with qi = qi(qo i ): In this form they are named predictive coordinates. The instant form amounts to putting qo 1 = . . . = qo N = xo and to describing the world-lines with the functions qi(xo ). Each one of these configuration variables has a different associated notion of velocity: dq μ i (τi) dτi (or dq μ i (τ) dτ ), d qi(qo i ) dqo i (predictive velocities), d qi(xo) dxo ; and of acceleration: d2q μ i (τi) (dτi)2 (or d2q μ i (τ) dτ2 ), d2 qi(qo i ) (dqo i )2 (predictive accelerations), d2 qi(xo) (dxo)2 . Bel’s non-manifestly covariant predictive mechanics [114–117] is the attempt to describe relativistic mechanics with N-time predictive equations of motion for the predictive coordinates qi(qo i ) in Newtonian form: mi d2 qi(qo i ) (dqo i )2 ◦ = Fi(qo k, qk(qo k), d qk(qo k) dqo k ). Since the left-hand side of these equations depends only on qo i , the predictive forces must satisfy the predictive conditions d Fi dqo k = 0 for k = i. Moreover they must be invariant under space translations and behave like space three-vectors under spatial rotations. Finally, they must satisfy 1 The standard choice in the manifestly covariant approach with a 4N-dimensional configuration space is τ1 = . . . = τN = τ. Another possibility is the choice of proper times τi = τiP T .
- 49. Relativistic Dynamics and Center of Mass 29 the Currie–Hill equations [118, 119] (or Currie–Hill world-line conditions), whose satisfaction implies that the predictive positions qi(xo ) behave under Lorentz boosts like the spatial components of 4-vectors. Bel [114–117] proved that these equations constitute the necessary and sufficient conditions that guarantee that the dynamics is Lorentz invariant with respect to finite Lorentz transformations. However, the Currie–Hill equations are so non-linear that it is practically impossible to find consistent predictive forces and develop this point of view. The first well-posed Hamiltonian formulation of relativistic mechanics was given by Dirac [89] with the instant, front (or light), and point forms of rela- tivistic Hamiltonian dynamics and the associated canonical realizations of the Poincaré algebra. His non-manifestly covariant Hamiltonian instant form has a well-defined non-relativistic limit and 1/c expansions containing the deviations (potentials) from the free case. However, the development of Hamiltonian models was blocked by the no- interaction theorem of Currie, Jordan, and Sudarshan [120–122] (see Ref. [123] for a review). Its original form was formulated in the Hamiltonian Dirac instant form in the 6N-dimensional phase space qi(xo ), pi(xo ) of N particles. The no- interaction theorem states that in the hypotheses: (1) the configuration variables qi(xo ) are canonical, i.e., { qi(xo ), qj(xo )} = 0; (2) the Lorentz boosts can be implemented as canonical transformations (existence of a canonical realization of the Poincaré group) and the qi(xo ) are the space components of 4-vectors; and (3) the system is non-singular (the transformation from positions and velocity to canonical coordinates is non-singular; the existence of a Lagrangian it is not assumed). This implies only free motion. As a consequence of the theorem, if we denote xμ i (τ), piμ(τ) the canonical coordinates of the manifestly covariant approach and xi(xo ), pi(xo ) their equal time restriction in the instant form, we have xi(xo ) = qi(xo ) except for free motion. Let us remark that, since the manifestly covariant approach gives the classical basis for the theory of covariant wave equations, the 4-coordinates xμ i (τ) (and not the geometrical 4-positions qμ i (τ)) are the coordinates locally minimally coupled to external fields. Many attempts were made to avoid this theorem by relaxing one of its hypothe- ses or by renouncing the concept of the world-line till when Droz Vincent’s many- time Hamiltonian formalism [124–128] (a refinement of the manifestly covariant non-manifestly predictive approach; it is the origin of the multi-temporal equa- tions of Ref. [48]), Todorov’s quasi-potential approach to bound states [130–133], and Komar’s study of toy models for general relativity (GR) [134–137] converged toward manifestly covariant models based on singular Lagrangians and/or the Dirac–Bergmann theory of constraints [138–140]. Since the Lagrangian formulation is usually not known, a system of N relativistic scalar particles is usually described in a manifestly covariant
- 50. 30 Relativistic Dynamics and Center of Mass 8N-dimensional phase space with coordinates xμ i (τ), piμ(τ) [{xμ i (τ), pjν (τ)} = −δij δμ ν , {xμ i (τ), xν j (τ)} = {piμ(τ), pjν (τ)} = 0], where τ is a scalar evolution parameter. The description is independent of the choice of τ: The Lagrangian (even if usually not explicitly known) is assumed τ-reparametrization invariant, so that at the Hamiltonian level the canonical Hamiltonian vanishes identically, Hc ≡ 0. Since the physical degrees of freedom for N scalar particles are 6N, there are constraints, which, in the case of N free scalar particles of mass mi are just the mass-shell conditions φi(q, p) = p2 i − m2 i ≈ 0, i = 1, . . . , N. These constraints say that the time variables xo i (τ) are the gauge variables of a τ-reparametrization invariant theory with canonical Hamiltonian Hc ≡ 0. The Dirac Hamiltonian is HD = N i=1 λi (τ) φi if all the first-class constraints are primary. The final constraint sub-manifold is the union of 2N (for generic masses mi) disjoint sub-manifolds corresponding to the choice of either the positive- or negative-energy branch of each two-sheeted mass-shell hyperboloid. Each branch is a non-compact sub-manifold of phase space on which each particle has a well-defined sign of the energy and 2N is a topological number (the zeroth homotopy class of the constraint sub-manifold).2 However, only in the case of two-body systems it is known how to introduce interactions (due to the Droz Vincent–Todorov–Komar model) with an arbitrary action-at-a-distance interaction instantaneous in the rest-frame described by the two first-class constraints φi = p2 i − m2 i c2 + V (r2 ⊥) ≈ 0, i = 1, 2, with rμ ⊥ = (ημν − pμ pν / p2 )rν , rμ = xμ 1 − xμ 2 , pμ = p1μ + p2μ. For N 2 a closed form of the N first-class constraints is not known explicitly (there is only an existence proof): Only versions of the model with explicit gauge fixings, so that all the constraints except one are second class, are known. This model has been completely understood both at the classical and quantum level [139]. The no-interaction theorem is initially avoided due to the singular nature of the Lagrangian: There is a canonical realization of the Poincaré group and the canonical coordinates xμ i are 4-vectors. However, when we restrict our- selves to the constraint sub-manifold and look for canonical coordinates adapted to it and to the Poincaré group, it turns out that among the final canonical coordinates will always appear the canonical non-covariant center of mass of the particle system. Therefore, all these models have the following properties: (1) the canonical and predictive 4-positions do not coincide (except in the free case); and (2) the decoupled canonical center of mass is not covariant. Let us see how to overcome these problems. 2 Let us remark that when the particles are coupled to weak external fields the 2N sub-manifolds are deformed but remain disjoint. But when the strength of the external fields increases, the various sub-manifolds may intersect each other and this topological discontinuity is the signal that we are entering a non-classical regime where quantum pair production becomes relevant due to the disappearance of mass gaps.
- 51. 3.1 The Wigner-Covariant Rest-Frame Instant Form 31 3.1 The Wigner-Covariant Rest-Frame Instant Form of Dynamics for Isolated Systems Given an inertial observer described by a 4-coordinate xμ (τ) assumed canoni- cally conjugated to the finite time-like 4-momentum Pμ , P2 0 ({xμ , Pν } = − 4 ημν ) of an isolated system to get the inertial frame of the rest-frame instant form, we must add the following gauge fixings on the orthonormal tetrads bμ A(τ) appearing in Eq. (2.18) and use the following new Dirac brackets: δμ A(τ) = bμ A(τ) − μ A(u(P)) ≈ 0, ⇒ zμ F (τ, σu ) = xμ (τ) + μ r (u(P)) σr , {f, g}∗∗ = {f, g}∗ − 1 4 {f, H̃μν }∗ 4 ημσ A ν (u(P))−4 ηνσ D μ (u(P)) {δσ D, g}∗ +{f, δσ D}∗ 4 ησν D μ (u(P))−4 ησμ D ν (u(P)) {H̃μν , g}∗ , {xμ , Pν }∗∗ = − 4 ημν , {ηr i , κjs}∗∗ = δij δr s , (3.1) where μ A(u(P)) = μ A( h) = Lμ A(P, ◦ P) (A μ (u(P)) ν A(u(P)) = δν μ) are the columns of the standard Wigner boost sending the time-like 4-momentum Pμ to its rest- frame form Pμ = Lμ ν (P, ◦ P) ◦ P ν , ◦ P μ = M c (1; 0), which are given in Eqs. (A.7)– (A.10) of Appendix A. In the Euclidean 3-spaces of the rest-frame, the Lorentz scalar 3-coordinates σr can be chosen as Cartesian 3-vectors σ. As a consequence of the Pμ -dependence of the gauge fixings (Eq. 3.1), as shown in Ref. [105], the Lorentz-scalar 3-vectors ηr i (τ), κir(τ) giving the Hamiltonian description of particles in the rest-frame (xμ i (τ) = xμ (τ) + μ r (u(P)) ηr i (τ); κir(τ) is given in Eq. (2.13), restricted to the rest-frame) become Wigner spin-1 3- vectors. They transform under the Wigner rotations of Eqs. (A.11) and (A.12) of Appendix A. If a Lorentz transformation x μ i (τ) = Λμ ν xν i (τ) is done, we get ηr i (τ) → η r i (τ) = Rr s(Λ, P) ηs i (τ), μ r (u(P)) ηr i (τ) → Λμ ν ν r (u(Λ P)) η r i (τ). Therefore the scalar product of two such vectors is a Lorentz scalar. The same happens for all the 3-vectors living inside these instantaneous 3-spaces. Therefore the instantaneous 3-spaces Στ are orthogonal to Pμ : They are named Wigner hyper-planes. From now on, i, j . . . will denote Euclidean indices, while r, s . . . will denote Wigner spin-1 indices. With the notation introduced after Eq. (2.20) with h = v/c = P/Mc we have μ τ (u(P)) = uμ (P) = Pμ /Mc = μ τ ( h) = hμ , μ r (u(P)) = − hr; δi r − hi hr 1+ √ 1+ h2 = μ r ( h) = ( 1 + h2; h), 4 ημν = μ A( h) 4 ηAB ν B( h) = (hμ hν − μ r ( h) ν r ( h)). The Pμ -dependent gauge fixing (Eq. 3.1) changes the interpretation of the residual gauge variables, because the constraints (Eq. 2.15) are reduced to the following remaining four first-class constraints (being in an inertial frame we have Tττ = T⊥⊥ and Tτr = −3 grs T⊥s):
- 52. 32 Relativistic Dynamics and Center of Mass H̃μ (τ) = d3 σ Hμ (τ, σ) = Pμ − uμ (P) d3 σ Tττ (τ, σ) − μ r (u(P)) d3 σ Tτr (τ, σ) ≈ 0, ⇓ Mc = √ P2 ≈ d3 σ Tττ (τ, σ), Pr (int) = d3 σ Tτr (τ, σ) ≈ 0, (3.2) which implies that M is the mass of the isolated system and that its total 3-momentum vanishes in the global rest-frame. At this stage, only the four degrees of freedom xμ (τ) of the original embedding zμ (τ, σr ) are still free parameters. However, due to the dependence of the gauge fixings (Eq. 3.1) upon Pμ , the final canonical gauge variable conjugated to Pμ is not the 4-vector xμ , but the following non-covariant position variable x̃μ (τ) [105] (see Section A.5 of Appendix A; Ts is the Lorentz-scalar rest time): x̃μ = xμ − 1 M c (Po + M c) Pν Sνμ + M c Soμ − Soν Pν Pμ M2 c2 , u(P) · x̃ = u(P) · x = c Ts, {x̃μ (τ), Pν } = −4 ημν , {Ts, M} = {u(P) · x̃, √ P2 = −. (3.3) Let us remark that, since the Poincaré generators Pμ , Jμν are global quantities (they know the whole instantaneous 3-space), collective variables like x̃μ (τ), being defined in terms of them, are also global non-local quantities: As a con- sequence, they are non-measurable with local means [98–101, 141–143] (see also the review papers in Refs. [144–147]). This is a fundamental difference from the non-relativistic 3-center of mass. As shown in Ref. [105] after the gauge fixing (Eq. 3.1), the final form of the external Poincaré generators (Eq. 2.14) of an arbitrary isolated system in the rest-frame instant form is: Pμ , Jμν = x̃μ Pν − x̃ν Pμ + S̃μν , Po = M2 c2 + P2 = Mc 1 + h2, P = Mc h, S̃r = S̄r = 1 2 ruv S̄uv , S̃0r = rsu Ps S̄u Mc + Po , Jij = x̃i Pj − x̃j Pi + iju S̄u = zi hj − zj hi + iju S̄u , Ki = Joi = x̃o Pi − x̃i M2 c2 + P2 − isu Ps S̄u M c + M2 c2 + P2 = − 1 + h2 zi + ( S̄ × h)i 1 + 1 + h2 , (3.4)
- 53. 3.1 The Wigner-Covariant Rest-Frame Instant Form 33 with S̃μν function only of the 3-spin S̄r of the rest spin tensor S̄AB = μ A(u(P)) ν B(u(P)) Sμν (both the spin tensors S̃μν and S̄AB satisfy the Lorentz algebra). It is this external realization that implements the Wigner rotations on the Wigner hyper-planes through the last term in the Lorentz boosts. Note that both L̃μν = x̃μ Pν −x̃ν Pμ and S̃μν = Jμν −L̃μν are conserved. Since we assume P2 0, the Pauli–Lubanski invariant is W2 = − P2 S̄ 2 . The Lorentz boosts are differently interaction-dependent from the Galilei ones. Let us remark that this realization is universal in the sense that it depends on the nature of the isolated system only through a U(2) algebra [142], whose generators are the invariant mass M (which in turn depends on the relative variables and on the type of interaction) and the internal spin S̄, which is interaction-independent, being in an instant form of dynamics. The Dirac Hamiltonian is now HD = λ(τ) Mc− d3 σ Tττ (τ, σ) + λ(τ)· P(int) and the embedding of Wigner hyper-planes is zμ W (τ, σ) = xμ (τ) + μ r (u(P)) σr , (3.5) with xμ a function of x̃μ , Pμ , and Sμν (or S̄AB ) according to Eq. (3.3). The gauge freedom in the choice of xμ is connected with the arbitrariness of the spin boosts S̄τr (or Soi ). Eq. (3.2) implies that the eight variables x̃μ (τ) and Pμ are restricted only by a first-class constraint identifying Mc = √ P2 with the invariant mass of the isolated system, evaluated by using its energy–momentum tensor: Therefore, these eight variables are to be reduced to six physical variables describing the external decoupled relativistic center of mass of the isolated system ( √ P2 is determined by the constraint and its conjugate variable, the rest time Ts = u(P) · x̃(τ) is a gauge variable). The choice of a gauge fixing for the rest time Ts is equivalent to the identification of a clock carried by the inertial observer. The natural inertial observer for this description is the external Fokker–Pryce center of inertia (the only covariant collective variable), which is also a function of τ, z, and h (see Eq. (3.8)). Therefore, there are three collective position degrees of freedom hidden in the embedding field zμ (τ, σ), which become non-local non- measurable physical variables. As a consequence, every isolated system (i.e., a closed universe) can be visu- alized as a decoupled non-covariant collective (non-local) pseudo-particle (the external center of mass), described by canonical variables identifying the frozen Jacobi data z, h (see Eq. (3.8)), carrying a “pole–dipole structure,” namely the invariant mass M c (the Lorentz-scalar monopole) and the rest spin S̄ (the dipole, a Wigner spin-1 3-vector) of the system, and with an associated external realization of the Poincaré group:3 3 The last term in the Lorentz boosts induces the Wigner rotation of the 3-vectors inside the Wigner 3-spaces.
- 54. 34 Relativistic Dynamics and Center of Mass Pμ = M c hμ = M c 1 + h2; h , Jij = zi hj − zj hi + ijk Sk , Ki = Joi = − 1 + h2 zi + ( S × h)i 1 + 1 + h2 . (3.6) The universal breaking of Lorentz covariance is connected to this decoupled non-local collective variable and is irrelevant because all the dynamics of the isolated system lives inside the Wigner 3-spaces and is Wigner-covariant. The invariant mass and the rest spin are built in terms of the Wigner-covariant variables of the given isolated system (the 6N variables ηi(τ) and κi(τ) for a system of N particles) living inside the Wigner 3-spaces [98–101, 140]. The presence of the three first-class constraints P(int) ≈ 0 (the rest-frame conditions) on these variables implies that each Wigner 3-space is a rest-frame of the isolated system whose Wigner spin-1 3-vector describing the internal 3-center of mass, is a gauge variable. In this way we avoid a double counting of the center of mass and the dynamics inside the Wigner hyper-planes is described only by Wigner spin-1 internal relative variables. Let us now add the τ-dependent gauge fixing, c Ts − τ ≈ 0, (3.7) where c Ts = u(P) · x̃ = u(P) · x is the Lorentz-scalar rest time, to the first-class constraint Mc− d3 σ Tττ (τ, σ) ≈ 0 of Eq. (3.2). After this imposition (from now on τ/c is the rest time Ts, which satisfies {Ts, M} = {u(P) · x̃, √ P2} = −), we have the following properties [100, 101]: 1. Due to the τ-dependence of the gauge fixing c Ts − τ ≈ 0, the Dirac Hamil- tonian given before Eq. (3.5) is replaced by the Hamiltonian HD = λ(τ) · P(int) ≈ 0: There is a frozen description of dynamics in the external space of the frozen external center of mass modulo the three first-class constraints P(int) ≈ 0, as in the standard Hamilton–Jacobi description. 2. Inside the Wigner 3-space the evolution in τ = cTs of the relative variables after the elimination of the internal center of mass is governed by the total mass M of the isolated system, i.e., by the energy generator of the internal Poincaré group, as the natural physical Hamiltonian. The 4-momentum Pμ becomes the 4-momentum of the isolated system: Pμ = Mc ( 1 + h2; h) = Mc hμ with h arbitrary a-dimensional 3-velocity and Mc ≡ d3 σ Tττ (τ, σ). 3. The six physical degrees of freedom describing the external decoupled relativistic center of mass are the non-evolving Jacobi data: z = Mc x̃ − P Po x̃o , h = P Mc , {zi , hj } = δij . (3.8)
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- 56. room with cloths, his head was covered with a strip of pure linen, his raiment was put on, and he performed his homage to the gods; and when he entered the perfuming-room, there approached him the court women attendants, appointed by the grand chamberlain and sent by the king, slaves of Vilāsavatī, with Kulavardhanā, and zenana women sent from the whole zenana, bearing in baskets different ornaments, wreaths, unguents, and robes, which they presented to him. Having taken them in due order from the women, he first himself anointed Vaiçampāyana. When his own anointing was done, and giving to those around him flowers, perfumes, robes, and jewels, as was meet, (202) he went to the banquet-hall, rich in a thousand jewelled vessels, like the autumn sky gleaming with stars. He there sat on a doubled rug, with Vaiçampāyana next him, eagerly employed, as was fitting, in praising his virtues, and the host of princes, placed each in order of seniority on the ground, felt the pleasure of their service increased by seeing the great courtesy with which the prince said: ‘Let this be given to him, and that to him!’And so he duly partook of his morning meal. ‘“After rinsing his mouth and taking betel, he stayed there a short time, and then went to Indrāyudha, and there, without sitting down, while his attendants stood behind him, with upraised faces, awaiting his commands, and talking mostly about Indrāyudha’s points, he himself, with heart uplifted by Indrāyudha’s merits, scattered the fodder before him, and departing, visited the court; and in the same order of routine he saw the king, and, returning home, spent the night there. Next day, at dawn, he beheld approaching a chamberlain, by name Kailāsa, the chief of the zenana, greatly trusted by the king, accompanied by a maiden of noble form, in her first youth, from her life at court self-possessed, yet not devoid of modesty, (203) growing to maidenhood, and in her veil of silk red with cochineal, resembling the Eastern quarter clothed in early sunshine. (204) And Kailāsa, bowing and approaching, with his right hand placed on the ground, spoke as follows: ‘“‘Prince, Queen Vilāsavatī bids me say: “This maiden, by name Patralekhā, daughter of the King of Kulūta, was brought with the captives by the great king on his conquest of the royal city of Kulūta while she was yet a little child, and was placed among the zenana women. And tenderness
- 57. grew up in me towards her, seeing she was a king’s daughter and without a protector, and she was long cared for and brought up by me just like a daughter. Therefore, I now send her to thee, thinking her fit to be thy betel- bearer; but she must not be looked on by thee, great prince of many days, as thine other attendants. She must be cared for as a young maiden; she must be shielded from the thoughtless like thine own nature; she must be looked on as a pupil. (205) Like a friend, she must be admitted to all thy confidences. By reason of the love that has long grown up in me, my heart rests on her as on my own daughter; and being sprung from a great race, she is fitted for such duties; in truth, she herself will in a few days charm the prince by her perfect gentleness. My love for her is of long growth, and therefore strong; but as the prince does not yet know her character, this is told to him. Thou must in all ways strive, happy prince, that she may long be thy fitting companion.”’ When Kailāsa had thus spoken and was silent, Candrāpīḍa looked long and steadily at Patralekhā as she made a courteous obeisance, and with the words, ‘As my mother wishes,’ dismissed the chamberlain. And Patralekhā, from her first sight of him, was filled with devotion to him, and never left the prince’s side either by night or day, whether he was sleeping, or sitting, or standing, or walking, or going to the court, just as if she were his shadow; while he felt for her a great affection, beginning from his first glance at her, and constantly growing; he daily showed more favour to her, and counted her in all his secrets as part of his own heart. ‘“As the days thus passed on, the king, eager for the anointing of Candrāpīḍa as crown prince, (206) appointed chamberlains to gather together all things needful for it; and when it was at hand, Çukanāsa, desirous of increasing the prince’s modesty, great as it already was, spoke to him at length during one of his visits: ‘Dear Candrāpīḍa, though thou hast learnt what is to be known, and read all the çāstras, no little remains for thee to learn. For truly the darkness arising from youth is by nature very thick, nor can it be pierced by the sun, nor cleft by the radiance of jewels, nor dispelled by the brightness of lamps. The intoxication of Lakshmī is terrible, and does not cease even in old age. There is, too, another blindness of power, evil, not to be cured by any salve. The fever of pride runs very high, and no cooling appliances can allay it. The madness that rises from
- 58. tasting the poison of the senses is violent, and not to be counteracted by roots or charms. The defilement of the stain of passion is never destroyed by bathing or purification. The sleep of the multitude of royal pleasures is ever terrible, and the end of night brings no waking. Thus thou must often be told at length. Lordship inherited even from birth, fresh youth, peerless beauty, superhuman talent, all this is a long succession of ills. (207) Each of these separately is a home of insolence; how much more the assemblage of them! For in early youth the mind often loses its purity, though it be cleansed with the pure waters of the çāstras. The eyes of the young become inflamed, though their clearness is not quite lost. Nature, too, when the whirlwind of passion arises, carries a man far in youth at its own will, like a dry leaf borne on the wind. This mirage of pleasure, which captivates the senses as if they were deer, always ends in sorrow. When the mind has its consciousness dulled by early youth, the characteristics of the outer world fall on it like water, all the more sweetly for being but just tasted. Extreme clinging to the things of sense destroys a man, misleading him like ignorance of his bearings. But men such as thou art the fitting vessels for instruction. For on a mind free from stain the virtue of good counsel enters easily, as the moon’s rays on a moon crystal. The words of a guru, though pure, yet cause great pain when they enter the ears of the bad, as water does; (208) while in others they produce a nobler beauty, like the ear-jewel on an elephant. They remove the thick darkness of many sins, like the moon in the gloaming.190 The teaching of a guru is calming, and brings to an end the faults of youth by turning them to virtue, just as old age takes away the dark stain of the locks by turning them to gray. This is the time to teach thee, while thou hast not yet tasted the pleasures of sense. For teaching pours away like water in a heart shattered by the stroke of love’s arrow. Family and sacred tradition are unavailing to the froward and undisciplined. Does a fire not burn when fed on sandal-wood? Is not the submarine fire the fiercer in the water that is wont to quench fire? But the words of a guru are a bathing without water, able to cleanse all the stains of man; they are a maturity that changes not the locks to gray; they give weight without increase of bulk; though not wrought of gold, they are an ear-jewel of no common order; without light they shine; without startling they awaken. They are specially needed for kings, for the admonishers of kings are few.
- 59. (209) For from fear, men follow like an echo the words of kings, and so, being unbridled in their pride, and having the cavity of their ears wholly stopped, they do not hear good advice even when offered; and when they do hear, by closing their eyes like an elephant, they show their contempt, and pain the teachers who offer them good counsel. For the nature of kings, being darkened by the madness of pride’s fever, is perturbed; their wealth causes arrogance and false self-esteem; their royal glory causes the torpor brought about by the poison of kingly power. First, let one who strives after happiness look at Lakshmī. For this Lakshmī, who now rests like a bee on the lotus-grove of a circle of naked swords, has risen from the milk ocean, has taken her glow from the buds of the coral-tree, her crookedness from the moon’s digit, her restlessness from the steed Uccaiḥçrava, her witchery from Kālakūṭa poison, her intoxication from nectar, and from the Kaustubha gem her hardness. (210) All these she has taken as keepsakes to relieve her longing with memory of her companions’ friendship. There is nothing so little understood here in the world as this base Lakshmī. When won, she is hard to keep; when bound fast by the firm cords of heroism, she vanishes; when held by a cage of swords brandished by a thousand fierce champions, she yet escapes; when guarded by a thick band of elephants, dark with a storm of ichor, she yet flees away. She keeps not friendships; she regards not race; she recks not of beauty; she follows not the fortunes of a family; she looks not on character; she counts not cleverness; she hears not sacred learning; she courts not righteousness; she honours not liberality; she values not discrimination; she guards not conduct; she understands not truth; she makes not auspicious marks her guide; like the outline of an aërial city, she vanishes even as we look on her. She is still dizzy with the feeling produced by the eddying of the whirlpool made by Mount Mandara. As if she were the tip of a lotus-stalk bound to the varying motion of a lotus-bed, she gives no firm foothold anywhere. Even when held fast with great effort in palaces, she totters as if drunk with the ichor of their many wild elephants. (211) She dwells on the sword’s edge as if to learn cruelty. She clings to the form of Nārāyaṇa as if to learn constant change of form. Full of fickleness, she leaves even a king, richly endowed with friends, judicial power, treasure, and territory, as she leaves a lotus at the end of day, though it have root, stalk, bud, and wide-spreading petals. Like a creeper, she is ever a
- 60. parasite.191 Like Gangā, though producing wealth, she is all astir with bubbles; like the sun’s ray, she alights on one thing after another; like the cavity of hell, she is full of dense darkness. Like the demon Hiḍambā, her heart is only won by the courage of a Bhīma; like the rainy season, she sends forth but a momentary flash; like an evil demon, she, with the height of many men,192 crazes the feeble mind. As if jealous, she embraces not him whom learning has favoured; she touches not the virtuous man, as being impure; she despises a lofty nature as unpropitious; she regards not the gently-born, as useless. She leaps over a courteous man as a snake; (212) she avoids a hero as a thorn; she forgets a giver as a nightmare; she keeps far from a temperate man as a villain; she mocks at the wise as a fool; she manifests her ways in the world as if in a jugglery that unites contradictions. For, though creating constant fever,193 she produces a chill;194 though exalting men, she shows lowness of soul; though rising from water, she augments thirst; though bestowing lordship,195 she shows an unlordly196 nature; though loading men with power, she deprives them of weight;197 though sister of nectar, she leaves a bitter taste; though of earthly mould,198 she is invisible; though attached to the highest,199 she loves the base; like a creature of dust, she soils even the pure. Moreover, let this wavering one shine as she may, she yet, like lamplight, only sends forth lamp-black. For she is the fostering rain of the poison-plants of desire, the hunter’s luring song to the deer of the senses, the polluting smoke to the pictures of virtue, the luxurious couch of infatuation’s long sleep, the ancient watch-tower of the demons of pride and wealth. (213) She is the cataract gathering over eyes lighted by the çāstras, the banner of the reckless, the native stream of the alligators of wrath, the tavern of the mead of the senses, the music-hall of alluring dances, the lair of the serpents of sin, the rod to drive out good practices. She is the untimely rain to the kalahaṃsas200 of the virtues, the hotbed of the pustules of scandal, the prologue of the drama of fraud, the roar of the elephant of passion, the slaughter-house of goodness, the tongue of Rāhu for the moon of holiness. Nor see I any who has not been violently embraced by her while she was yet unknown to him, and whom she has not deceived. Truly, even in a picture she moves; even in a book she practises magic; even cut in a gem
- 61. she deceives; even when heard she misleads; even when thought on she betrays. ‘“‘When this wretched evil creature wins kings after great toil by the will of destiny, they become helpless, and the abode of every shameful deed. For at the very moment of coronation their graciousness is washed away as if by the auspicious water-jars; (214) their heart is darkened as by the smoke of the sacrificial fire; their patience is swept away as by the kuça brooms of the priest; their remembrance of advancing age is concealed as by the donning of the turban; the sight of the next world is kept afar as by the umbrella’s circle; truth is removed as by the wind of the cowries; virtue is driven out as by the wands of office; the voices of the good are drowned as by cries of “All hail!” and glory is flouted as by the streamers of the banners. ‘“‘For some kings are deceived by successes which are uncertain as the tremulous beaks of birds when loose from weariness, and which, though pleasant for a moment as a firefly’s flash, are contemned by the wise; they forget their origin in the pride of amassing a little wealth, and are troubled by the onrush of passion as by a blood-poisoning brought on by accumulated diseases; they are tortured by the senses, which though but five, in their eagerness to taste every pleasure, turn to a thousand; they are bewildered by the mind, which, in native fickleness, follows its own impulses, and, being but one, gets the force of a hundred thousand in its changes. Thus they fall into utter helplessness. They are seized by demons, conquered by imps, (215) possessed by enchantments, held by monsters, mocked by the wind, swallowed by ogres. Pierced by the arrows of Kāma, they make a thousand contortions; scorched by covetousness, they writhe; struck down by fierce blows, they sink down.201 Like crabs, they sidle; like cripples, with steps broken by sin, they are led helpless by others; like stammerers from former sins of falsehood, they can scarce babble; like saptacchada202 trees, they produce headache in those near them; like dying men, they know not even their kin; like purblind203 men, they cannot see the brightest virtue; like men bitten in a fatal hour, they are not waked even by mighty charms; like lac-ornaments, they cannot endure strong heat;204
- 62. like rogue elephants, being firmly fixed to the pillar of self-conceit, they refuse teaching; bewildered by the poison of covetousness, they see everything as golden; like arrows sharpened by polishing,205 when in the hands of others they cause destruction; (216) with their rods206 they strike down great families, like high-growing fruit; like untimely blossoms, though fair outwardly, they cause destruction; they are terrible of nature, like the ashes of a funeral pyre; like men with cataract, they can see no distance; like men possessed, they have their houses ruled by court jesters; when but heard of, they terrify, like funeral drums; when but thought of, like a resolve to commit mortal sin, they bring about great calamity; being daily filled with sin, they become wholly puffed up. In this state, having allied themselves to a hundred sins, they are like drops of water hanging on the tip of the grass on an anthill, and have fallen without perceiving it. ‘“‘But others are deceived by rogues intent on their own ends, greedy of the flesh-pots of wealth, cranes of the palace lotus-beds! “Gambling,” say these, “is a relaxation; adultery a sign of cleverness; hunting, exercise; drinking, delight; recklessness, heroism; neglect of a wife, freedom from infatuation; (217) contempt of a guru’s words, a claim to others’ submission; unruliness of servants, the ensuring of pleasant service; devotion to dance, song, music, and bad company, is knowledge of the world; hearkening to shameful crimes is greatness of mind; tame endurance of contempt is patience; self-will is lordship; disregard of the gods is high spirit; the praise of bards is glory; restlessness is enterprise; lack of discernment is impartiality.” Thus are kings deceived with more than mortal praises by men ready to raise faults to the grade of virtues, practised in deception, laughing in their hearts, utterly villainous; and thus these monarchs, by reason of their senselessness, have their minds intoxicated by the pride of wealth, and have a settled false conceit in them that these things are really so; though subject to mortal conditions, they look on themselves as having alighted on earth as divine beings with a superhuman destiny; they employ a pomp in their undertakings only fit for gods (218) and win the contempt of all mankind. They welcome this deception of themselves by their followers. From the delusion as to their own divinity established in their minds, they are overthrown by false ideas, and they think their own
- 63. pair of arms have received another pair;207 they imagine their forehead has a third eye buried in the skin.208 They consider the sight of themselves a favour; they esteem their glance a benefit; they regard their words as a present; they hold their command a glorious boon; they deem their touch a purification. Weighed down by the pride of their false greatness, they neither do homage to the gods, nor reverence Brahmans, nor honour the honourable, nor salute those to whom salutes are due, nor address those who should be addressed, nor rise to greet their gurus. They laugh at the learned as losing in useless labour all the enjoyment of pleasure; they look on the teaching of the old as the wandering talk of dotage; they abuse the advice of their councillors as an insult to their own wisdom; they are wroth with the giver of good counsel. ‘“‘At all events, the man they welcome, with whom they converse, whom they place by their side, advance, (219) take as companion of their pleasure and recipient of their gifts, choose as a friend, the man to whose voice they listen, on whom they rain favours, of whom they think highly, in whom they trust, is he who does nothing day and night but ceaselessly salute them, praise them as divine, and exalt their greatness. ‘“‘What can we expect of those kings whose standard is a law of deceit, pitiless in the cruelty of its maxims; whose gurus are family priests, with natures made merciless by magic rites; whose teachers are councillors skilled to deceive others; whose hearts are set on a power that hundreds of kings before them have gained and lost; whose skill in weapons is only to inflict death; whose brothers, tender as their hearts may be with natural affection, are only to be slaughtered. ‘“‘Therefore, my Prince, in this post of empire which is terrible in the hundreds of evil and perverse impulses which attend it, and in this season of youth which leads to utter infatuation, thou must strive earnestly not to be scorned by thy people, nor blamed by the good, nor cursed by thy gurus, nor reproached by thy friends, nor grieved over by the wise. Strive, too, that thou be not exposed by knaves, (220) deceived by sharpers, preyed upon by villains, torn to pieces by wolvish courtiers, misled by rascals, deluded by women, cheated by fortune, led a wild dance by pride, maddened by desire,
- 64. assailed by the things of sense, dragged headlong by passion, carried away by pleasure. ‘“‘Granted that by nature thou art steadfast, and that by thy father’s care thou art trained in goodness, and moreover, that wealth only intoxicates the light of nature, and the thoughtless, yet my very delight in thy virtues makes me speak thus at length. ‘“‘Let this saying be ever ringing in thine ears: There is none so wise, so prudent, so magnanimous, so gracious, so steadfast, and so earnest, that the shameless wretch Fortune cannot grind him to powder. Yet now mayest thou enjoy the consecration of thy youth to kinghood by thy father under happy auspices. Bear the yoke handed down to thee that thy forefathers have borne. Bow the heads of thy foes; raise the host of thy friends; after thy coronation wander round the world for conquest; and bring under thy sway the earth with its seven continents subdued of yore by thy father. ‘“‘This is the time to crown thyself with glory. (221) A glorious king has his commands fulfilled as swiftly as a great ascetic.’ ‘“Having said thus much, he was silent, and by his words Candrāpīḍa was, as it were, washed, wakened, purified, brightened, bedewed, anointed, adorned, cleansed, and made radiant, and with glad heart he returned after a short time to his own palace. ‘“Some days later, on an auspicious day, the king, surrounded by a thousand chiefs, raised aloft, with Çukanāsa’s help, the vessel of consecration, and himself anointed his son, while the rest of the rites were performed by the family priest. The water of consecration was brought from every sacred pool, river and ocean, encircled by every plant, fruit, earth, and gem, mingled with tears of joy, and purified by mantras. At that very moment, while the prince was yet wet with the water of consecration, royal glory passed on to him without leaving Tārāpīḍa, as a creeper still clasping its own tree passes to another. (222) Straightway he was anointed from head to foot by Vilāsavatī, attended by all the zenana, and full of tender love, with sweet sandal white as moonbeams. He was garlanded with fresh white
- 65. flowers; decked209 with lines of gorocanā; adorned with an earring of dūrvā grass; clad in two new silken robes with long fringes, white as the moon; bound with an amulet round his hand, tied by the family priest; and had his breast encircled by a pearl-necklace, like the circle of the Seven Ṛishis come down to see his coronation, strung on filaments from the lotus-pool of the royal fortune of young royalty. ‘“From the complete concealment of his body by wreaths of white flowers interwoven and hanging to his knees, soft as moonbeams, and from his wearing snowy robes he was like Narasiṃha, shaking his thick mane,210 or like Kailāsa, with its flowing streams, or Airāvata, rough with the tangled lotus-fibres of the heavenly Ganges, or the Milky Ocean, all covered with flakes of bright foam. (223) ‘“Then his father himself for that time took the chamberlain’s wand to make way for him, and he went to the hall of assembly and mounted the royal throne, like the moon on Meru’s peak. Then, when he had received due homage from the kings, after a short pause the great drum that heralded his setting out on his triumphal course resounded deeply, under the stroke of golden drum-sticks. Its sound was as the noise of clouds gathering at the day of doom; or the ocean struck by Mandara; or the foundations of earth by the earthquakes that close an aeon; or a portent-cloud, with its flashes of lightning; or the hollow of hell by the blows of the snout of the Great Boar. And by its sound the spaces of the world were inflated, opened, separated, outspread, filled, turned sunwise, and deepened, and the bonds that held the sky were unloosed. The echo of it wandered through the three worlds; for it was embraced in the lower world by Çesha, with his thousand hoods raised and bristling in fear; it was challenged in space by the elephants of the quarters tossing their tusks in opposition; it was honoured with sunwise turns in the sky by the sun’s steeds, tossing211 their heads in their snort of terror; (224) it was wondrously answered on Kailāsa’s peak by Çiva’s bull, with a roar of joy in the belief that it was his master’s loudest laugh; it was met in Meru by Airāvata, with deep trumpeting; it was reverenced in the hall of the gods by Yama’s bull, with his curved horns turned sideways in wrath at so strange a sound; and it was heard in terror by the guardian gods of the world.
- 66. ‘“Then, at the roar of the drum, followed by an outcry of ‘All hail!’ from all sides, Candrāpīḍa came down from the throne, and with him went the glory of his foes. He left the hall of assembly, followed by a thousand chiefs, who rose hastily around him, strewing on all sides the large pearls that fell from the strings of their necklaces as they struck against each other, like rice sportively thrown as a good omen for their setting off to conquer the world. He showed like the coral-tree amid the white buds of the kalpa-trees;212 or Airāvata amid the elephants of the quarters bedewing him with water from their trunks; or heaven, with the firmament showering stars; or the rainy season with clouds ever pouring heavy drops. (225) ‘“Then an elephant was hastily brought by the mahout, adorned with all auspicious signs for the journey, and on the inner seat Patralekhā was placed. The prince then mounted, and under the shade of an umbrella with a hundred wires enmeshed with pearls, beauteous as Kailāsa standing on the arms of Rāvaṇa, and white as the whirlpools of the Milky Ocean under the tossing of the mountain, he started on his journey. And as he paused in his departure he saw the ten quarters tawny with the rich sunlight, surpassing molten lac, of the flashing crest-jewels of the kings who watched him with faces hidden behind the ramparts, as if the light were the fire of his own majesty, flashing forth after his coronation. He saw the earth bright as if with his own glow of loyalty when anointed as heir-apparent, and the sky crimson as with the flame that heralded the swift destruction of his foes, and daylight roseate as with lac-juice from the feet of the Lakshmī of earth coming to greet him. ‘“On the way hosts of kings, with their thousand elephants swaying in confusion, their umbrellas broken by the pressure of the crowd, their crest- jewels falling low as their diadems bent in homage, (226) their earrings hanging down, and the jewels falling on their cheeks, bowed low before him, as a trusted general recited their names. The elephant Gandhamādana followed the prince, pink with much red lead, dangling to the ground his ear-ornaments of pearls, having his head outlined with many a wreath of white flowers, like Meru with evening sunlight resting on it, the white stream of Ganges falling across it, and the spangled roughness of a bevy of stars on its peak. Before Candrāpīḍa went Indrāyudha, led by his groom,
- 67. perfumed with saffron and many-hued, with the flash of golden trappings on his limbs. And so the expedition slowly started towards the Eastern Quarter.213 ‘“Then the whole army set forth with wondrous turmoil, with its forest of umbrellas stirred by the elephants’ movements, like an ocean of destruction reflecting on its advancing waves a thousand moons, flooding the earth. (227) ‘“When the prince left his palace Vaiçampāyana performed every auspicious rite, and then, clothed in white, anointed with an ointment of white flowers, accompanied by a great host of powerful kings, shaded by a white umbrella, followed close on the prince, mounted on a swift elephant, like a second Crown Prince, and drew near to him like the moon to the sun. Straightway the earth heard on all sides the cry: ‘The Crown Prince has started!’ and shook with the weight of the advancing army. (228) ‘“In an instant the earth seemed as it were made of horses; the horizon, of elephants; the atmosphere, of umbrellas; the sky, of forests of pennons; the wind, of the scent of ichor; the human race, of kings; the eye, of the rays of jewels; the day, of crests; the universe, of cries of ‘All hail!’ (228–234 condensed) ‘“The dust rose at the advance of the army like a herd of elephants to tear up the lotuses of the sunbeams, or a veil to cover the Lakshmī of the three worlds. Day became earthy; the quarters were modelled in clay; the sky was, as it were, resolved in dust, and the whole universe appeared to consist of but one element. (234) ‘“When the horizon became clear again, Vaiçampāyana, looking at the mighty host which seemed to rise from the ocean, was filled with wonder, and, turning his glance on every side, said to Candrāpīḍa: ‘What, prince, has been left unconquered by the mighty King Tārāpīḍa, for thee to conquer? What regions unsubdued, for thee to subdue? (235) What fortresses untaken, for thee to take? What continents unappropriated, for thee to appropriate? What treasures ungained, for thee to gain? What kings have not been humbled? By whom have the raised hands of salutation, soft as young lotuses, not been placed on the head? By whose brows, encircled
- 68. with golden bands, have the floors of his halls not been polished? Whose crest-jewels have not scraped his footstool? Who have not accepted his staff of office? Who have not waved his cowries? Who have not raised the cry of “Hail!”? Who have not drunk in with the crocodiles of their crests, the radiance of his feet, like pure streams? For all these princes, though they are imbued with the pride of armies, ready in their rough play to plunge into the four oceans; though they are the peers of the great kings Daçaratha, Bhagīratha, Bharata, Dilīpa, Alarka, and Māndhātṛi; though they are anointed princes, soma-drinkers, haughty in the pride of birth, yet they bear on the sprays of crests purified with the shower of the water of consecration the dust of thy feet of happy omen, like an amulet of ashes. By them as by fresh noble mountains, the earth is upheld. These their armies that have entered the heart of the ten regions follow thee alone. (236) For lo! wherever thy glance is cast, hell seems to vomit forth armies, the earth to bear them, the quarters to discharge them, the sky to rain them, the day to create them. And methinks the earth, trampled by the weight of boundless hosts, recalls to-day the confusion of the battles of the Mahābhārata. ‘“‘Here the sun wanders in the groves of pennons, with his orb stumbling over their tops, as if he were trying, out of curiosity, to count the banners. The earth is ceaselessly submerged under ichor sweet as cardamons, and flowing like a plait of hair, from the elephants who scatter it all round, and thick, too, with the murmur of the bees settling on it, so that it shines as if filled with the waves of Yamunā. The lines of moon-white flags hide the horizon, like rivers that in fear of being made turbid by the heavy host have fled to the sky. It is a wonder that the earth has not to-day been split into a thousand pieces by the weight of the army; and that the bonds of its joints, the noble mountains, are not burst asunder; and that the hoods of Çesha, the lord of serpents, in distress at the burden of earth pressed down under the load of troops, do not give way.’ (237) ‘“While he was thus speaking, the prince reached his palace. It was adorned with many lofty triumphal arches; dotted with a thousand pavilions enclosed in grassy ramparts, and bright with many a tent of shining white cloth. Here he dismounted, and performed in kingly wise all due rites; and though the kings and ministers who had come together sought to divert him
- 69. with various tales, he spent the rest of the day in sorrow, for his heart was tortured with bitter grief for his fresh separation from his father. When day was brought to a close he passed the night, too, mostly in sleeplessness, with Vaiçampāyana resting on a couch not far from his own, and Patralekhā sleeping hard by on a blanket placed on the ground; his talk was now of his father, now of his mother, now of Çukanāsa, and he rested but little. At dawn he arose, and with an army that grew at every march, as it advanced in unchanged order, he hollowed the earth, shook the mountains, dried the rivers, emptied the lakes, (238) crushed the woods to powder, levelled the crooked places, tore down the fortresses, filled up the hollows, and hollowed the solid ground. ‘“By degrees, as he wandered at will, he bowed the haughty, exalted the humble, encouraged the fearful, protected the suppliant, rooted out the vicious, and drove out the hostile. He anointed princes in different places, gathered treasures, accepted gifts, took tribute, taught local regulations, established monuments of his visit, made hymns of worship, and inscribed edicts. He honoured Brahmans, reverenced saints, protected hermitages, and showed a prowess that won his people’s love. He exalted his majesty, heaped up his glory, showed his virtues far and wide, and won renown for his good deeds. Thus trampling down the woods on the shore, and turning the whole expanse of ocean to gray with the dust of his army, he wandered over the earth. ‘“The East was his first conquest, then the Southern Quarter, marked by Triçanku, then the Western Quarter, which has Varuṇa for its sign, and immediately afterwards the Northern Quarter adorned by the Seven Ṛishis. Within the three years that he roamed over the world he had subdued the whole earth, with its continents, bounded only by the moat of four oceans. (239) ‘“He then, wandering sunwise, conquered and occupied Suvarṇapura, not far from the Eastern Ocean, the abode of those Kirātas who dwell near Kailāsa, and are called Hemajakūṭas, and as his army was weary from its worldwide wandering, he encamped there for a few days to rest.
- 70. ‘“One day during his sojourn there he mounted Indrāyudha to hunt, and as he roamed through the wood he beheld a pair of Kinnaras wandering down at will from the mountains. Wondering at the strange sight, and eager to take them, he brought up his horse respectfully near them and approached them. But they hurried on, fearing the unknown sight of a man, and fleeing from him, while he pursued them, doubling Indrāyudha’s speed by frequent pats on his neck, and went on alone, leaving his army far behind. Led on by the idea that he was just catching them, he was borne in an instant fifteen leagues from his own quarters by Indrāyudha’s speed as it were at one bound, and was left companionless. (240) The pair of Kinnaras he was pursuing were climbing a steep hill in front of him. He at length turned away his glance, which was following their progress, and, checked by the steepness of the ascent, reined in Indrāyudha. Then, seeing that both his horse and himself were tired and heated by their toils, he considered for a moment, and laughed at himself as he thought: ‘Why have I thus wearied myself for nothing, like a child? What matters it whether I catch the pair of Kinnaras or not? If caught, what is the good? if missed, what is the harm? What a folly this is of mine! What a love of busying myself in any trifle! What a passion for aimless toil! What a clinging to childish pleasure! The good work I was doing has been begun in vain. The needful rite I had begun has been rendered fruitless. The duty of friendship I undertook has not been performed. The royal office I was employed in has not been fulfilled. The great task I had entered on has not been completed. My earnest labour in a worthy ambition has been brought to nought. Why have I been so mad as to leave my followers behind and come so far? (241) and why have I earned for myself the ridicule I should bestow on another, when I think how aimlessly I have followed these monsters with their horses’ heads? I know not how far off is the army that follows me. For the swiftness of Indrāyudha traverses a vast space in a moment, and his speed prevented my noticing as I came by what path I should turn back, for my eyes were fixed on the Kinnaras; and now I am in a great forest, spread underfoot with dry leaves, with a dense growth of creepers, underwood, and branching trees. Roam as I may here I cannot light on any mortal who can show me the way to Suvarṇapura. I have often heard that Suvarṇapura is the farthest bound of earth to the north, and that beyond it lies a supernatural forest, and beyond that again is Kailāsa. This then is Kailāsa; so I must turn back now, and
- 71. resolutely seek to make my way unaided to the south. For a man must bear the fruit of his own faults.’ ‘“With this purpose he shook the reins in his left hand, and turned the horse’s head. Then he again reflected: (242) ‘The blessed sun with glowing light now adorns the south, as if he were the zone-gem of the glory of day. Indrāyudha is tired; I will just let him eat a few mouthfuls of grass, and then let him bathe and drink in some mountain rill or river; and when he is refreshed I will myself drink some water, and after resting a short time under the shade of a tree, I will set out again.’ ‘“So thinking, constantly turning his eyes on every side for water, he wandered till at length he saw a track wet with masses of mud raised by the feet of a large troop of mountain elephants, who had lately come up from bathing in a lotus-pool. (243) Inferring thence that there was water near, he went straight on along the slope of Kailāsa, the trees of which, closely crowded as they were, seemed, from their lack of boughs, to be far apart, for they were mostly pines, çāl, and gum olibanum trees, and were lofty, and like a circle of umbrellas, to be gazed at with upraised head. There was thick yellow sand, and by reason of the stony soil the grass and shrubs were but scanty. (244) ‘“At length he beheld, on the north-east of Kailāsa, a very lofty clump of trees, rising like a mass of clouds, heavy with its weight of rain, and massed as if with the darkness of a night in the dark fortnight. ‘“The wind from the waves, soft as sandal, dewy, cool from passing over the water, aromatic with flowers, met him, and seemed to woo him; and the cries of kalahaṃsas drunk with lotus-honey, charming his ear, summoned him to enter. So he went into that clump, and in its midst beheld the Acchoda Lake, as if it were the mirror of the Lakshmī of the three worlds, the crystal chamber of the goddess of earth, the path by which the waters of ocean escape, the oozing of the quarters, the avatar of part of the sky, Kailāsa taught to flow, Himavat liquefied, moonlight melted, Çiva’s smile turned to water, (245) the merit of the three worlds abiding in the shape of a lake, a range of hills of lapis lazuli changed into water, or a mass of autumn
- 72. clouds poured down in one spot. From its clearness it might be Varuṇa’s mirror; it seemed to be fashioned of the hearts of ascetics, the virtues of good men, the bright eyes of deer, or the rays of pearls. (247) ‘“Like the person of a great man, it showed clearly the signs of fish, crocodile, tortoise, and cakṛa;214 like the story of Kārtikeya, the lamentations of the wives of Krauñca215 resounded in it; it was shaken by the wings of white Dhārtarāshṭras, as the Mahābhārata by the rivalry of Pāṇḍavas and Dhārtarāshṭras; and the drinking of poison by Çiva was represented by the drinking of its water by peacocks, as if it were the time of the churning of ocean. It was fair, like a god, with a gaze that never wavers. (248) Like a futile argument, it seemed to have no end; and was a lake most fair and gladdening to the eyes. ‘“The very sight of it seemed to remove Candrāpīḍa’s weariness, and as he gazed he thought: ‘“‘Though my pursuit of the horse-faced pair was fruitless, yet now that I see this lake it has gained its reward. My eyes’ reward in beholding all that is to be seen has now been won, the furthest point of all fair things seen, the limit of all that gladdens us gazed upon, the boundary line of all that charms us descried, the perfection of all that causes joy made manifest, and the vanishing-point of all worthy of sight beheld. (249) By creating this lake water, sweet as nectar, the Creator has made his own labour of creation superfluous. For this, too, like the nectar that gladdens all the senses, produces joy to the eye by its purity, offers the pleasure of touch by its coolness, gladdens the sense of smell by the fragrance of its lotuses, pleases the ear with the ceaseless murmur of its haṃsas, and delights the taste with its sweetness. Truly it is from eagerness to behold this that Çiva leaves not his infatuation for dwelling on Kailāsa. Surely Kṛishṇa no longer follows his own natural desire as to a watery couch, for he sleeps on the ocean, with its water bitter with salt, and leaves this water sweet as nectar! Nor is this, in sooth, the primæval lake; for the earth, when fearing the blows of the tusks of the boar of destruction, entered the ocean, all the waters of which were designed but to be a draught for Agastya; whereas, if it had plunged into this mighty lake, deep as many deep hells, it could not have been
- 73. reached, I say not by one, but not even by a thousand boars. (250) Verily it is from this lake that the clouds of doom at the seasons of final destruction draw little by little their water when they overwhelm the interstices of the universe, and darken all the quarters with their destroying storm. And methinks that the world, Brahmā’s egg, which in the beginning of creation was made of water, was massed together and placed here under the guise of a lake.’ So thinking, he reached the south bank, dismounted and took off Indrāyudha’s harness; (251) and the latter rolled on the ground, arose, ate some mouthfuls of grass, and then the prince took him down to the lake, and let him drink and bathe at will. After that, the prince took off his bridle, bound two of his feet by a golden chain to the lower bough of a tree hard by, and, cutting off with his dagger some dūrvā grass from the bank of the lake, threw it before the horse, and went back himself to the water. He washed his hands, and feasted, like the cātaka, on water; like the cakravāka, he tasted pieces of lotus-fibre; like the moon with its beams, he touched the moon-lotuses with his finger-tips; like a snake, he welcomed the breeze of the waves;216 like one wounded with Love’s arrows, he placed a covering of lotus-leaves on his breast; like a mountain elephant, when the tip of his trunk is wet with spray, he adorned his hands with spray-washed lotuses. Then with dewy lotus-leaves, with freshly-broken fibres, he made a couch on a rock embowered in creepers, and rolling up his cloak for a pillow, lay down to sleep. After a short rest, he heard on the north bank of the lake a sweet sound of unearthly music, borne on the ear, and blent with the chords of the vīnā. (252) Indrāyudha heard it first, and letting fall the grass he was eating, with ears fixed and neck arched, turned towards the voice. The prince, as he heard it, rose from his lotus-couch in curiosity to see whence this song could arise in a place deserted by men, and cast his glance towards the region; but, from the great distance, he was unable, though he strained his eyes to the utmost, to discern anything, although he ceaselessly heard the sound. Desiring in his eagerness to know its source, he determined to depart, and saddling and mounting Indrāyudha, he set forth by the western forest path, making the song his goal; the deer, albeit unasked, were his guides, as they rushed on in front, delighting in the music.217
- 74. (253–256 condensed) ‘“Welcomed by the breezes of Kailāsa, he went towards that spot, which was surrounded by trees on all sides, and at the foot of the slope of Kailāsa, on the left bank of the lake, called Candraprabhā, which whitened the whole region with a splendour as of moonlight, he beheld an empty temple of Çiva. (257) ‘“As he entered the temple he was whitened by the falling on him of ketakī pollen, tossed by the wind, as if for the sake of seeing Çiva he had been forcibly made to perform a vow of putting on ashes, or as if he were robed in the pure merits of entering the temple; and, in a crystal shrine resting on four pillars, he beheld Çiva, the four-faced, teacher of the world, the god whose feet are honoured by the universe, with his emblem, the linga, made of pure pearl. Homage had been paid to the deity by shining lotuses of the heavenly Ganges, that might be mistaken for crests of pearls, freshly-plucked and wet, with drops falling from the ends of their leaves, like fragments of the moon’s disc split and set upright, or like parts of Çiva’s own smile, or scraps of Çesha’s hood, or brothers of Kṛishṇa’s conch, or the heart of the Milky Ocean. (258) ‘“But, seated in a posture of meditation, to the right of the god, facing him, Candrāpīḍa beheld a maiden vowed to the service of Çiva, who turned the region with its mountains and woods to ivory by the brightness of her beauty. For its lustre shone far, spreading through space, white as the tide of the Milky Ocean, overwhelming all things at the day of doom, or like a store of penance gathered in long years and flowing out, streaming forth massed together like Ganges between the trees, giving a fresh whiteness to Kailāsa, and purifying the gazer’s soul, though it but entered his eye. The exceeding whiteness of her form concealed her limbs as though she had entered a crystal shrine, or had plunged into a sea of milk, or were hidden in spotless silk, or were caught on the surface of a mirror, or were veiled in autumn clouds. She seemed to be fashioned from the quintessence of
- 75. whiteness, without the bevy of helps for the creation of the body that consist of matter formed of the five gross elements. (259) She was like sacrifice impersonate, come to worship Çiva, in fear of being seized by the unworthy; or Rati, undertaking a rite of propitiation to conciliate him, for the sake of Kāma’s body; or Lakshmī, goddess of the Milky Ocean, longing for a digit of Çiva’s moon, her familiar friend of yore when they dwelt together in the deep; or the embodied moon seeking Çiva’s protection from Rāhu; or the beauty of Airāvata,218 come to fulfil Çiva’s wish to wear an elephant’s skin; or the brightness of the smile on the right face of Çiva become manifest and taking a separate abode; or the white ash with which Çiva besprinkles himself, in bodily shape; or moonlight made manifest to dispel the darkness of Çiva’s neck; or the embodied purity of Gaurī’s mind; or the impersonate chastity of Kārtikeya; or the brightness of Çiva’s bull, dwelling apart from his body; (260) or the wealth of flowers on the temple trees come of themselves to worship Çiva; or the fulness of Brahmā’s penance come down to earth; or the glory of the Prajāpatis of the Golden Age, resting after the fatigue of wandering through the seven worlds; or the Three Vedas, dwelling in the woods in grief at the overthrow of righteousness in the Kali Age; or the germ of a future Golden Age, in the form of a maiden; or the fulness of a muni’s contemplation, in human shape; or a troop of heavenly elephants, falling into confusion on reaching the heavenly Ganges; or the beauty of Kailāsa, fallen in dread of being uprooted by Rāvaṇa; or the Lakshmī of the Çvetadvīpa219 come to behold another continent; or the grace of an opening kāça-blossom looking for the autumn; or the brightness of Çesha’s body leaving hell and come to earth; or the brilliance of Balarāma, which had left him in weariness of his intoxication; or a succession of bright fortnights massed together. ‘“She seemed from her whiteness to have taken a share from all the haṃsas; (261) or to have come from the heart of righteousness; or to have been fashioned from a shell; or drawn from a pearl; or formed from lotus-fibres; or made of flakes of ivory; or purified by brushes of moonbeams; or inlaid with lime; or whitened with foam-balls of ambrosia; or laved in streams of quicksilver; or rubbed with melted silver; or dug out from the moon’s orb; or decked with the hues of kuṭaja, jasmine, and sinduvāra flowers. She
- 76. seemed, in truth, to be the very furthest bound of whiteness. Her head was bright with matted locks hanging on her shoulders, made, as it were, of the brightness of morning rays taken from the sun on the Eastern Mountain, tawny like the quivering splendour of flashing lightning, and, being wet from recent bathing, marked with the dust of Çiva’s feet clasped in her devotion; she bore Çiva’s feet marked with his name in jewels on her head, fastened with a band of hair; (262) and her brow had a sectarial mark of ashes pure as the dust of stars ground by the heels of the sun’s horses. (266) She was a goddess, and her age could not be known by earthly reckoning, but she resembled a maiden of eighteen summers. ‘“Having beheld her, Candrāpīḍa dismounted, tied his horse to a bough, and then, reverently bowing before the blessed Çiva, gazed again on that heavenly maiden with a steady unswerving glance. And as her beauty, grace, and serenity stirred his wonder, the thought arose in him: ‘How in this world each matter in its turn becomes of no value! For when I was pursuing the pair of Kinnaras wantonly and vainly I beheld this most beautiful place, inaccessible to men, and haunted by the immortals. (267) Then in my search for water I saw this delightful lake sought by the Siddhas. While I rested on its bank I heard a divine song; and as I followed the sound, this divine maiden, too fair for mortal sight, met my eyes. For I cannot doubt her divinity. Her very beauty proclaims her a goddess. And whence in the world of men could there arise such harmonies of heavenly minstrelsy? If, therefore, she vanishes not from my sight, nor mounts the summit of Kailāsa, nor flies to the sky, I will draw near and ask her, “Who art thou, and what is thy name, and why hast thou in the dawn of life undertaken this vow?” This is all full of wonder.’ With this resolve he approached another pillar of the crystal shrine, and sat there, awaiting the end of the song. ‘“Then when she had stilled her lute, like a moon-lotus bed when the pleasant hum of the bees is silenced, (268) the maiden rose, made a sunwise turn and an obeisance to Çiva, and then turning round, with a glance by nature clear, and by the power of penance confident, she, as it were, gave courage to Candrāpīḍa, as if thereby she were sprinkling him with merits,
- 77. laving him with holy water, purifying him with penance, freeing him from stain, giving him his heart’s desire, and leading him to purity. ‘“‘Hail to my guest!’ said she. ‘How has my lord reached this place? Rise, draw near, and receive a guest’s due welcome.’ So she spake; and he, deeming himself honoured even by her deigning to speak with him, reverently arose and bowed before her. ‘As thou biddest, lady,’ he replied, and showed his courtesy by following in her steps like a pupil. And on the way he thought: ‘Lo, even when she beheld me she did not vanish! Truly a hope of asking her questions has taken hold of my heart. And when I see the courteous welcome, rich in kindness, of this maiden, fair though she be with a beauty rare in ascetics, I surely trust that at my petition she will tell me all her story.’ (269) ‘“Having gone about a hundred paces, he beheld a cave, with its entrance veiled by dense tamālas, showing even by day a night of their own; its edge was vocal with the glad bees’ deep murmur on the bowers of creepers with their opening blossoms; it was bedewed with torrents that in their sheer descent fell in foam, dashing against the white rock, and cleft by the axe-like points of the jagged cliff, with a shrill crash as the cold spray rose up and broke; it was like a mass of waving cowries hanging from a door, from the cascades streaming down on either side, white as Çiva’s smile, or as pearly frost. Within was a circle of jewelled pitchers; on one side hung a veil worn in sacred meditation; a clean pair of shoes made of cocoanut matting hung on a peg; one corner held a bark bed gray with dust scattered by the ashes the maiden wore; the place of honour was filled by a bowl of shell carved with a chisel, like the orb of the moon; and close by there stood a gourd of ashes. ‘“On the rock at the entrance Candrāpīḍa took his seat, and when the maiden, having laid her lute on the pillow of the bark bed, took in a leafy cup some water from the cascade to offer to her guest, and he said as she approached (270): ‘Enough of these thy great toils. Cease this excess of grace. Be persuaded, lady. Let this too great honour be abandoned. The very sight of thee, like the aghamarshaṇa hymn, stills all evil and sufficeth for purification. Deign to take thy seat!’Yet being urged by her, he reverently,
- 78. with head bent low, accepted all the homage she gave to her guest. When her cares for her guest were over, she sat down on another rock, and after a short silence he told, at her request, the whole story of his coming in pursuit of the pair of Kinnaras, beginning with his expedition of conquest. The maiden then rose, and, taking a begging bowl, wandered among the trees round the temple; and ere long her bowl was filled with fruits that had fallen of their own accord. As she invited Candrāpīḍa to the enjoyment of them, the thought arose in his heart: ‘Of a truth, there is nought beyond the power of penance. For it is a great marvel how the lords of the forest, albeit devoid of sense, yet, like beings endowed with sense, gain honour for themselves by casting down their fruits for this maiden. A wondrous sight is this, and one never seen before.’ ‘“So, marvelling yet more, he brought Indrāyudha to that spot, unsaddled him, and tied him up hard by. (271) Then, having bathed in the torrent, he partook of the fruits, sweet as ambrosia, and drank the cool water of the cascade, and having rinsed his mouth, he waited apart while the maiden enjoyed her repast of water, roots, and fruit. ‘“When her meal was ended and she had said her evening prayer, and taken her seat fearlessly on the rock, the Prince quietly approached her, and sitting down near her, paused awhile and then respectfully said: ‘“‘Lady, the folly that besets mankind impels me even against my will to question thee, for I am bewildered by a curiosity that has taken courage from thy kindness. For even the slightest grace of a lord emboldens a weak nature: even a short time spent together creates intimacy. Even a slight acceptance of homage produces affection. Therefore, if it weary thee not, I pray thee to honour me with thy story. For from my first sight of thee a great eagerness has possessed me as to this matter. Is the race honoured by thy birth, lady, that of the Maruts, or Ṛishis, or Gandharvas, or Guhyakas, or Apsarases? And wherefore in thy fresh youth, tender as a flower, has this vow been taken? (272) For how far apart would seem thy youth, thy beauty, and thine exceeding grace, from this thy peace from all thoughts of earth! This is marvellous in mine eyes! And wherefore hast thou left the heavenly hermitages that gods may win, and that hold all things needful for the
- 79. highest saints, to dwell alone in this deserted wood? And whereby hath thy body, though formed of the five gross elements, put on this pure whiteness? Never have I heard or seen aught such as this. I pray thee dispel my curiosity, and tell me all I ask.’ ‘“For a little time she pondered his request in silence, and then she began to weep noiselessly, and her eyes were blinded by tears which fell in large drops, carrying with them the purity of her heart, showering down the innocence of her senses, distilling the essence of asceticism, dropping in a liquid form the brightness of her eyes, most pure, falling on her white cheeks like a broken string of pearls, unceasing, splashing on her bosom covered by the bark robe. (273) ‘“And as he beheld her weeping Candrāpīḍa reflected: ‘How hardly can misfortune be warded off, if it takes for its own a beauty like this, which one might have deemed beyond its might! Of a truth there is none whom the sorrows of life in the body leave untouched. Strong indeed is the working of the opposed powers of pleasure and pain.220 These her tears have created in me a further curiosity, even greater than before. It is no slight grief that can take its abode in a form like hers. For it is not a feeble blow that causes the earth to tremble.’ ‘“While his curiosity was thus increased he felt himself guilty of recalling her grief, and rising, brought in his folded hand from the torrent some water to bathe her face. But she, though the torrent of her tears was in nowise checked by his gentleness, yet bathed her reddened eyes, and drying her face with the edge of her bark robe, slowly said with a long and bitter sigh: (274) ‘“‘Wherefore, Prince, wilt thou hear the story of my ascetic life, all unfit for thy ears? for cruel has been my heart, hard my destiny, and evil my condition, even from my birth. Still, if thy desire to know be great, hearken. It has come within the range of our hearing, usually directed to auspicious knowledge, that there are in the abode of the gods maidens called Apsarases. Of these there are fourteen families: one sprung from the mind of Brahmā, another from the Vedas, another from fire, another from the wind, another from nectar when it was churned, another from water, another
- 80. from the sun’s rays, another from the moon’s beams, another from earth, and another from lightning; one was fashioned by Death, and another created by Love; besides, Daksha, father of all, had among his many daughters two, Muni and Arishṭā, and from their union with the Gandharvas were sprung the other two families. These are, in sum, the fourteen races. But from the Gandharvas and the daughters of Daksha sprang these two families. Here Muni bore a sixteenth son, by name Citraratha, who excelled in virtues Sena and all the rest of his fifteen brothers. For his heroism was famed through the three worlds; his dignity was increased by the name of Friend, bestowed by Indra, whose lotus feet are caressed by the crests of the gods cast down before him; and even in childhood he gained the sovereignty of all the Gandharvas by a right arm tinged with the flashing of his sword. (275) Not far hence, north of the land of Bharata, is his dwelling, Hemakūṭa, a boundary mountain in the Kimpurusha country. There, protected by his arm, dwell innumerable Gandharvas. By him this pleasant wood, Caitraratha, was made, this great lake Acchoda was dug out, and this image of Çiva was fashioned. But the son of Arishṭā, in the second Gandharva family, was as a child anointed king by Citraratha, lord of the Gandharvas, and now holds royal rank, and with a countless retinue of Gandharvas dwells likewise on this mountain. Now, from that family of Apsarases which sprang from the moon’s nectar was born a maiden, fashioned as though by the grace of all the moon’s digits poured in one stream, gladdening the eyes of the universe, moonbeam-fair, in name and nature a second Gaurī.221 (276) Her Haṃsa, lord of the second family, wooed, as the Milky Ocean the Ganges; with him she was united, as Rati with Kāma, or the lotus-bed with the autumn; and enjoying the great happiness of such a union she became the queen of his zenana. To this noble pair I was born as only daughter, ill-omened, a prey for grief, and a vessel for countless sorrows; my father, however, having no other child, greeted my birth with a great festival, surpassing that for a son, and on the tenth day, with the customary rites he gave me the fitting name of Mahāçvetā. In his palace I spent my childhood, passed from lap to lap of the Gandharva dames, like a lute, as I murmured the prattle of babyhood, ignorant as yet of the sorrows of love; but in time fresh youth came to me as
- 81. the honey-month to the spring, fresh shoots to the honey-month, flowers to the fresh shoots, bees to the flowers, and honey to the bees. ‘“‘222And one day in the month of honey I went down with my mother to the Acchoda lake to bathe, when its beauties were spread wide in the spring, and all its lotuses were in flower. (278) ‘“‘I worshipped the pictures of Çiva, attended by Bṛingiriṭi, which were carved on the rocks of the bank by Pārvatī when she came down to bathe, and which had the reverential attendance of ascetics portrayed by the thin footprints left in the dust. “How beautiful!” I cried, “is this bower of creepers, with its clusters of flowers of which the bees’ weight has broken the centre and bowed the filaments; this mango is fully in flower, and the honey pours through the holes in the stalks of its buds, which the cuckoo’s sharp claws have pierced; how cool this sandal avenue, which the serpents, terrified at the murmur of hosts of wild peacocks, have deserted; how delightful the waving creepers, which betray by their fallen blossoms the swinging of the wood-nymphs upon them; how pleasant the foot of the trees on the bank where the kalahaṃsas have left the line of their steps imprinted in the pollen of many a flower!” Drawn on thus by the ever-growing charms of the wood, I wandered with my companions. (279) And at a certain spot I smelt the fragrance of a flower strongly borne on the wind, overpowering that of all the rest, though the wood was in full blossom; it drew near, and by its great sweetness seemed to anoint, to delight, and to fill the sense of smell. Bees followed it, seeking to make it their own: it was truly a perfume unknown heretofore, and fit for the gods. I, too, eager to learn whence it came, with eyes turned into buds, and drawn on like a bee by that scent, and attracting to me the kalahaṃsas of the lake by the jangling of my anklets loudly clashed in the tremulous speed of my curiosity, advanced a few steps and beheld a graceful youthful ascetic coming down to bathe. He was like Spring doing penance in grief for Love made the fuel of Çiva’s fire, or the crescent on Çiva’s brow performing a vow to win a full orb, or Love restrained in his eagerness to conquer Çiva: by his great splendour he appeared to be girt by a cage of quivering lightning, embosomed in the globe of the summer sun, or encircled in the flames of a furnace: (280) by the brightness of his form, flashing forth ever more and more, yellow as
- 82. lamplight, he made the grove a tawny gold; his locks were yellow and soft like an amulet dyed in gorocanā. The line of ashes on his brow made him like Ganges with the line of a fresh sandbank, as though it were a sandal- mark to win Sarasvatī,223 and played the part of a banner of holiness; his eyebrows were an arch rising high over the abode of men’s curses; his eyes were so long that he seemed to wear them as a chaplet; he shared with the deer the beauty of their glance; his nose was long and aquiline; the citron of his lower lip was rosy as with the glow of youth, which was refused an entrance to his heart; with his beardless cheek he was like a fresh lotus, the filaments of which have not yet been tossed by the bees in their sport; he was adorned with a sacrificial thread like the bent string of Love’s bow, or a filament from the lotus grove of the pool of penance; in one hand he bore a pitcher like a kesara fruit with its stalk; in the other a crystal rosary, strung as it were with the tears of Rati wailing in grief for Love’s death. (281) His loins were girt with a muñja-grass girdle, as though he had assumed a halo, having outvied the sun by his innate splendour; the office of vesture was performed by the bark of the heavenly coral-tree,224 bright as the pink eyelid of an old partridge, and washed in the waves of the heavenly Ganges; he was the ornament of ascetic life, the youthful grace of holiness, the delight of Sarasvatī, the chosen lord of all the sciences, and the meeting- place of all divine tradition. He had, like the summer season,225 his āshāḍha226; he had, like a winter wood, the brightness of opening millet, and he had like the month of honey, a face adorned with white tilaka.227 With him there was a youthful ascetic gathering flowers to worship the gods, his equal in age and a friend worthy of himself. (282) ‘“‘Then I saw a wondrous spray of flowers which decked his ear, like the bright smile of woodland Çrī joying in the sight of spring, or the grain- offering of the honey-month welcoming the Malaya winds, or the youth of the Lakshmī of flowers, or the cowrie that adorns Love’s elephant; it was wooed by the bees; the Pleiads lent it their grace; and its honey was nectar. “Surely,” I decided, “this is the fragrance which makes all other flowers scentless,” and gazing at the youthful ascetic, the thought arose in my mind: “Ah, how lavish is the Creator who has skill228 to produce the highest perfection of form, for he has compounded Kāma of all miraculous beauty,
- 83. excelling the universe, and yet has created this ascetic even more fair, surpassing him, like a second love-god, born of enchantment. (283) Methinks that when Brahmā229 made the moon’s orb to gladden the world, and the lotuses to be Lakshmī’s palace of delight, he was but practising to gain skill for the creation of this ascetic’s face; why else should such things be created? Surely it is false that the sun with its ray Sushumnā230 drinks all the digits of the moon as it wanes in the dark fortnight, for their beams are cast down to enter this fair form. How otherwise could there be such grace in one who lives in weary penance, beauty’s destroyer?” As I thus thought, Love, beauty’s firm adherent, who knows not good from ill, and who is ever at hand to the young, enthralled me, together with my sighs, as the madness of spring takes captive the bee. Then with a right eye gazing steadily, the eyelashes half closed, the iris darkened by the pupil’s tremulous sidelong glance, I looked long on him. With this glance I, as it were, drank him in, besought him, told him I was wholly his, offered my heart, tried to enter into him with my whole soul, sought to be absorbed in him, implored his protection to save Love’s victim, showed my suppliant state that asked for a place in his heart; (284) and though I asked myself, “What is this shameful feeling that has arisen in me, unseemly and unworthy a noble maiden?” yet knowing this, I could not master myself, but with great difficulty stood firm, gazing at him. For I seemed to be paralyzed, or in a picture, or scattered abroad, or bound, or in a trance, and yet in wondrous wise upheld, as though when my limbs were failing, support was at the same moment given; for I know not how one can be certain in a matter that can neither be told nor taught, and that is not capable of being told, for it is only learnt from within. Can it be ascertained as presented by his beauty, or by my own mind, or by love, or by youth or affection, or by any other causes? I cannot tell. Lifted up and dragged towards him by my senses, led forward by my heart, urged from behind by Love, I yet by a strong effort restrained my impulse. (285) Straightway a storm of sighs went forth unceasingly, prompted by Love as he strove to find a place within me; and my bosom heaved as longing to speak earnestly to my heart, and then I thought to myself: “What an unworthy action is this of vile Kāma, who surrenders me to this cold ascetic free from all thoughts of love! Truly, the heart of woman is foolish exceedingly, since it cannot weigh the fitness of that which it
- 84. loves. For what has this bright home of glory and penance to do with the stirrings of love that meaner men welcome? Surely in his heart he scorns me for being thus deceived by Kāma! Strange it is that I who know this cannot restrain my feeling! (286) Other maidens, indeed, laying shame aside, have of their own accord gone to their lords; others have been maddened by that reckless love-god; but not as I am here alone! How in that one moment has my heart been thrown into turmoil by the mere sight of his form, and passed from my control! for time for knowledge and good qualities always make Love invincible. It is best for me to leave this place while I yet have my senses, and while he does not clearly see this my hateful folly of love. Perchance if he sees in me the effects of a love he cannot approve, he will in wrath make me feel his curse. For ascetics are ever prone to wrath.” Thus having resolved, I was eager to depart, but, remembering that holy men should be reverenced by all, I made an obeisance to him with eyes turned to his face, eyelashes motionless, not glancing downwards, my cheek uncaressed by the flowers dancing in my ears, my garland tossing on my waving hair, and my jewelled earrings swinging on my shoulders. ‘“‘As I thus bent, the irresistible command of love, the inspiration of the spring, the charm of the place, the frowardness of youth, the unsteadiness of the senses, (287) the impatient longing for earthly goods, the fickleness of the mind, the destiny that rules events—in a word, my own cruel fate, and the fact that all my trouble was caused by him, were the means by which Love destroyed his firmness by the sight of my feeling, and made him waver towards me like a flame in the wind. He too was visibly thrilled, as if to welcome the newly-entering Love; his sighs went before him to show the way to his mind which was hastening towards me; the rosary in his hand trembled and shook, fearing the breaking of his vow; drops rose on his cheek, like a second garland hanging from his ear; his eyes, as his pupils dilated and his glance widened in the joy of beholding me, turned the spot to a very lotus-grove, so that the ten regions were filled by the long rays coming forth like masses of open lotuses that had of their own accord left the Acchoda lake and were rising to the sky.
- 85. ‘“‘By the manifest change in him my love was redoubled, and I fell that moment into a state I cannot describe, all unworthy of my caste. “Surely,” I reflected, “Kāma himself teaches this play of the eye, though generally after a long happy love, else whence comes this ascetic’s gaze? (288) For his mind is unversed in the mingled feelings of earthly joys, and yet his eyes, though they have never learnt the art, pour forth the stream of love’s sweetness, rain nectar, are half closed by joy, are slow with distress, heavy with sleep, roaming with pupils tremulous and languid with the weight of gladness, and yet bright with the play of his eyebrows. Whence comes this exceeding skill that tells the heart’s longing wordlessly by a glance alone?” ‘“‘Impelled by these thoughts I advanced, and bowing to the second young ascetic, his companion, I asked: “What is the name of his Reverence? Of what ascetic is he the son? From what tree is this garland woven? For its scent, hitherto unknown, and of rare sweetness, kindles great curiosity in me.” ‘“‘With a slight smile, he replied: “Maiden, what needs this question? But I will enlighten thy curiosity. Listen! ‘“‘“There dwells in the world of gods a great sage, Çvetaketu; his noble character is famed through the universe; his feet are honoured by bands of siddhas, gods, and demons; (289) his beauty, exceeding that of Nalakūbara,231 is dear to the three worlds, and gladdens the hearts of goddesses. Once upon a time, when seeking lotuses for the worship of the gods, he went down to the Heavenly Ganges, which lay white as Çiva’s smile, while its water was studded as with peacocks’ eyes by the ichor of Airāvata. Straightway Lakshmī, enthroned on a thousand-petalled white lotus close by, beheld him coming down among the flowers, and looking on him, she drank in his beauty with eyes half closed by love, and quivering with weight of joyous tears, and with her slender fingers laid on her softly- opening lips; and her heart was disturbed by Love; by her glance alone she won his affection. A son was born, and taking him in her arms with the words, ‘Take him, for he is thine,’ she gave him to Çvetaketu, who performed all the rites of a son’s birth, and called him Puṇḍarīka, because he was born in a puṇḍarīka lotus. Moreover, after initiation, he led him
- 86. through the whole circle of the arts. (290) This is Puṇḍarīka whom you see. And this spray comes from the pārijāta tree,232 which rose when the Milky Ocean was churned by gods and demons. How it gained a place in his ear contrary to his vow, I will now tell. This being the fourteenth day of the month, he started with me from heaven to worship Çiva, who had gone to Kailāsa. On the way, near the Nandana Wood, a nymph, drunk with the juice of flowers, wearing fresh mango shoots in her ear, veiled completely by garlands falling to the knees, girt with kesara flowers, and resting on the fair hand lent her by the Lakshmī of spring, took this spray of pārijāta, and bending low, thus addressed Puṇḍarīka: ‘Sir, let, I pray, this thy form, that gladdens the eyes of the universe, have this spray as its fitting adornment; let it be placed on the tip of thy ear, for it has but the playfulness that belongs to a garland; let the birth of the pārijāta now reap its full blessing!’ At her words, his eyes were cast down in modesty at the praise he so well deserved, and he turned to depart without regarding her; but as I saw her following us, I said, ‘What is the harm, friend. Let her courteous gift be accepted!’ and so by force, against his will, the spray adorns his ear. Now all has been told: who he is, whose son, and what this flower is, and how it has been raised to his ear.” (291) When he had thus spoken, Puṇḍarīka said to me with a slight smile: “Ah, curious maiden, why didst thou take the trouble to ask this? If the flower, with its sweet scent, please thee, do thou accept it,” and advancing, he took it from his own ear and placed it in mine, as though, with the soft murmur of the bees on it, it were a prayer for love. At once, in my eagerness to touch his hand, a thrill arose in me, like a second pārijāta flower, where the garland lay; while he, in the pleasure of touching my cheek, did not see that from his tremulous fingers he had dropped his rosary at the same time as his timidity; but before it reached the ground I seized it, and playfully placed it on my neck, where it wore the grace of a necklace unlike all others, while I learnt the joy of having my neck clasped, as it were, by his arm. ‘“‘As our hearts were thus occupied with each other, my umbrella-bearer addressed me: “Princess, the Queen has bathed. It is nearly time to go home. Do thou, therefore, also bathe.” At her words, like a newly-caught elephant, rebellious at the first touch of the new hook, I was unwillingly dragged away, and as I went down to bathe, I could hardly withdraw my
- 87. eyes, for they seemed to be drowned in the ambrosial beauty of his face, or caught in the thicket of my thrilling cheek, or pinned down by Love’s shafts, or sewn fast by the cords233 of his charms. (292) ‘“‘Meanwhile, the second young ascetic, seeing that he was losing his self-control, gently upbraided him: “Dear Puṇḍarīka, this is unworthy of thee. This is the way trodden by common men. For the good are rich in self- control. Why dost thou, like a man of low caste, fail to restrain the turmoil of thy soul? Whence comes this hitherto unknown assault of the senses, which so transforms thee? Where is thine old firmness? Where thy conquest of the senses? Where thy self-control? Where thy calm of mind, thine inherited holiness, thy carelessness of earthly things? Where the teaching of thy guru, thy learning of the Vedas, thy resolves of asceticism, thy hatred of pleasure, thine aversion to vain delights, thy passion for penance, thy distaste for enjoyments, thy rule over the impulses of youth? Verily all knowledge is fruitless, study of holy books is useless, initiation has lost its meaning, pondering the teaching of gurus avails not, proficiency is worthless, learning leads to nought, since even men like thee are stained by the touch of passion, and overcome by folly. (293) Thou dost not even see that thy rosary has fallen from thy hand, and has been carried away. Alas! how good sense fails in men thus struck down. Hold back this heart of thine, for this worthless girl is seeking to carry it away.” ‘“‘To these words he replied, with some shame: “Dear Kapiñjala, why dost thou thus misunderstand me? I am not one to endure this reckless girl’s offence in taking my rosary!” and with his moonlike face beautiful in its feigned wrath, and adorned the more by the dread frown he tried to assume, while his lip trembled with longing to kiss me, he said to me, “Playful maiden, thou shalt not move a step from this place without giving back my rosary.” Thereupon I loosed from my neck a single row of pearls as the flower-offering that begins a dance in Kāma’s honour, and placed it in his outstretched hand, while his eyes were fixed on my face, and his mind was far away. I started to bathe, but how I started I know not, for my mother and my companions could hardly lead me away by force, like a river driven backwards, and I went home thinking only of him.
- 88. (294) ‘“‘And entering the maidens’ dwelling, I began straightway to ask myself in my grief at his loss: “Am I really back, or still there? Am I alone, or with my maidens? Am I silent, or beginning to speak? Am I awake or asleep? Do I weep or hold back my tears? Is this joy or sorrow, longing or despair, misfortune or gladness, day or night? Are these things pleasures or pains?” All this I understood not. In my ignorance of Love’s course, I knew not whither to go, what to do, hear, see, or speak, whom to tell, nor what remedy to seek. Entering the maidens’ palace, I dismissed my friends at the door, and shut out my attendants, and then, putting aside all my occupations, I stood alone with my face against the jewelled window. I gazed at the region which, in its possession of him, was richly decked, endowed with great treasure, overflowed by the ocean of nectar, adorned with the rising of the full moon, and most fair to behold, I longed to ask his doings even of the breeze wafted from thence, or of the scent of the woodland flowers, or of the song of the birds. (295) I envied even the toils of penance for his devotion to them. For his sake, in the blind adherence of love, I took a vow of silence. I attributed grace to the ascetic garb, because he accepted it, beauty to youth because he owned it, charm to the pārijāta flower because it touched his ear, delight to heaven because he dwelt there, and invincible power to love because he was so fair. Though far away, I turned towards him as the lotus-bed to the sun, the tide to the moon, or the peacock to the cloud. I bore on my neck his rosary, like a charm against the loss of the life stricken by his absence. I stood motionless, though a thrill made the down on my cheek like a kadamba flower ear-ring, as it rose from the joy of being touched by his hand, and from the pārijāta spray in my ear, which spoke sweetly to me of him. ‘“‘Now my betel-bearer, Taralikā, had been with me to bathe; she came back after me rather late, and softly addressed me in my sadness: “Princess, one of those godlike ascetics we saw on the bank of Lake Acchoda—(296) he by whom this spray of the heavenly tree was placed in thy ear—as I was following thee, eluded the glance of his other self, and approaching me with soft steps between the branches of a flowering creeper, asked me concerning thee, saying, ‘Damsel, who is this maiden? Whose daughter is she? What is her name? And whither goes she?’ I replied: ‘She is sprung from Gaurī, an Apsaras of the moon race, and her father Haṃsa is king of
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