Parallel transport additional explorations part1&2 sqrd

Parallel Transport: Additional Explorations Part One and PartTwo
Roa, F. J.P.
Let us explore a portion of space-time as viewed in the perspective of Carter-Penrose (CP)
diagram. In this exploration, we are to be concerned with region I of the said CP diagram and
such region has all space-time points contained outside the event horizons 𝑟 = 2𝐺𝑀 𝑞 of a
Schwarzschild blackhole and such region is bounded by four asymptotic curves to be
approximated as straight lines that correspond to the future event horizon 𝐻+
(𝑟 → 2𝐺𝑀 𝑞; 𝑡 →
∞), future null infinity 𝐹+
(𝑟 → ∞; 𝑡 → ∞), past null infinity 𝐹−
(𝑟 → ∞; 𝑡 → − ∞) and
past event horizon 𝐻−
(𝑟 → 2𝐺𝑀 𝑞; 𝑡 → − ∞).
These boundary lines are generated by two distinct families of curves each as parametrized
differently from the other.
For the future null infinity and past event horizon, they are generated by the curve
(1.1)
𝐿−
= 𝐿−( 𝑢̃): 𝜒 + 𝜂 = 2𝑢̃′
tan 𝑢̃′
= 𝑢̃ = 𝑡 + 𝑟 ∗
where 𝑢̃ serves here as parameter whose particular value can generate one of the boundaries.
This is just the in-going Eddington-Finkelstein coordinate and is defined by the time-like
coordinate t and the Regge-Wheeler coordinate 𝑟 ∗, which in turn is just a redefinition of the
spacelike coordinate r via the following expression
(1.2)
𝑟 ∗ = 𝑟 + 2𝐺𝑀 𝑞 𝑙𝑛(
𝑟
2𝐺𝑀 𝑞
− 1)
2𝐺𝑀 𝑞 ≤ 𝑟 ≤ ∞
− ∞ ≤ 𝑟 ∗ ≤ ∞
With the in-going Eddington-Finkelstein coordinate we can also define for its corresponding null
Kruskal coordinate
(1.3)
𝑢 = 𝑒𝑥𝑝 (
𝑢̃
4𝐺𝑀 𝑞
) = √
𝑟
2𝐺𝑀 𝑞
− 1 𝑒𝑥𝑝(
𝑡 + 𝑟
4𝐺𝑀 𝑞
)
The fixed or constant value of this ingoing coordinate holds for a line that all ingoing null paths
follow along
(1.4)
𝐿−( 𝑢̃0) = 𝛾−
∶ 𝜒 + 𝜂 = 2𝑢̃ ′
0

𝑢̃0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑢̃ ′
0 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
We take that this constant value is non-infinite since when the ingoing coordinate goes infinite,
either the past event horizon or future null infinity is generated.
That is, say for instance the said coordinate is at a negative infinite value, this generates the past
event horizon
(1.5)
𝐿−( 𝑢̃ = − ∞) = 𝐻−
∶ 𝜒 + 𝜂 = − 𝜋
𝑢̃′
= −𝜋/2
𝑢 = 0, [𝑟 → 2𝐺𝑀 𝑞; 𝑟 ∗ → − ∞, 𝑡 → − ∞]
while on the opposite end where the same coordinate takes positive infinite value, the future null
infinity is generated.
(1.6)
𝐿−( 𝑢̃ = ∞) = 𝐹+
∶ 𝜒 + 𝜂 = 𝜋
𝑢̃′
= 𝜋/2
𝑢 = ∞, [𝑟 → ∞; 𝑟 ∗ → ∞, 𝑡 → ∞]
The other boundary lines namely, the future event horizon and the past null infinity are in turn
generated by the curve
(2.1)
𝐿+( 𝑣): 𝜒 − 𝜂 = −2𝑣̃′
tan 𝑣̃′
= 𝑣̃ = 𝑡 − 𝑟 ∗
In here, the specific parameter is 𝑣, which is also a component in the null Kruskal coordinate and
this can be defined by the other component (the out-going) of the Eddington-Finkelstein
coordinate 𝑣̃ via
(2.2)
𝑣 = 𝑒𝑥𝑝(−
𝑣̃
4𝐺𝑀 𝑞
) = √
𝑟
2𝐺𝑀 𝑞
− 1 𝑒𝑥𝑝(−
𝑡 − 𝑟
4𝐺𝑀 𝑞
)
and taking for a constant (noninfinite) value of this Kruskal coordinate generates all the out-
going null paths
(2.3)
𝐿+( 𝑣0) = 𝛾+
∶ 𝜒 − 𝜂 = −2𝑣̃ ′
0
With 𝑣 as the parameter we can reach the future event horizon at 𝑣 = 0,
(2.4)
𝐿+( 𝑣 = 0) = 𝐻+
∶ 𝜒 − 𝜂 = − 𝜋
𝑣̃ = ∞ , 𝑣̃′
= 𝜋/2

[𝑟 → 2𝐺𝑀 𝑞 ; 𝑟 ∗ → − ∞, 𝑡 → ∞]
and we can travel back in time towards an infinite past to reach the past null infinity where 𝑣 =
∞
(2.5)
𝐿+( 𝑣 = ∞) = 𝐹−
∶ 𝜒 − 𝜂 = 𝜋
𝑣̃ = −∞ , 𝑣̃′
= − 𝜋/2
[ 𝑟 ∗ → ∞, 𝑡 → − ∞]
The spacetime of region I is where events or specifically motions take place outside the BH
event horizons and such motions are fundamentally happening in a given geometry of the
spacetime and such geometry is encoded in the fundamental line element that is given with a
particular metric solution to Einstein’ field equation. In our present case, we have the
Schwarzschild metric solution and using the Eddington-Finkelstein coordinates we write the said
fundamental line element in the form given by
(2.6)
𝑑𝑆2( 𝑢̃, 𝑣̃) = −𝜂 𝑚 𝑑𝑢̃ 𝑑𝑣̃ + 𝑟2
𝑑Ω2
= −𝜂 𝑚 𝑑𝑢̃ 𝑑𝑣̃
in which for convenience, we suppress, setting 𝑑Ω2
= 0 the fundamental line element of the unit
two-sphere
(2.7)
𝑑Ω2
= 𝑑θ2
+ 𝑠𝑖𝑛2
θ 𝑑𝜙2
(2.8)
𝜂 𝑚 = 1 −
2𝐺𝑀 𝑞
𝑟
In the null Kruskal coordinates, we write (2.6) into an alternative form
(2.9)
𝑑𝑆2( 𝑢, 𝑣) =
32𝐺3
𝑀 𝑞
3
𝑟
𝑒𝑥𝑝(−
𝑟
2𝐺𝑀 𝑞
) 𝑑𝑢 𝑑𝑣
Briefly, we will examine the asymptotic behavior of each of these fundamental line elements,
specifically the responses very near the future event horizon and also very near the past null
infinity.
As we approach very near the future event horizon 𝐻+
(𝑟∗ → − ∞; 𝑡 → ∞), where 𝑣 = 0, 𝑣̃ =
∞, 𝑢 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 and 𝑢̃ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, the fundamental line element (2.6) is vanishing since the
metric solution 𝜂 𝑚 approaches to zero as the said horizon is approached. However, according to
(2.9) the fundamental line element stays nonvanishing, and very near the future event horizon
this takes the following approximate form
(3.1)

𝑑𝑆2( 𝑢̃, 𝑣̃) → 0 as 𝜂 𝑚 (𝑟 → 2𝐺𝑀 𝑞 ) → 0
𝑑𝑆2( 𝑢, 𝑣) ≈
16𝐺2
𝑀 𝑞
2
𝑒
𝑑𝑢 𝑑𝑣
where here 𝑣 is the parameter.
What we have just observed is that the vanishing of the fundamental line element written in the
Eddington-Finkelstein coordinates is just due to the coordinate singularity of the given metric
solution and such singularity was removed as we switched to the null Kruskal form (2.9).
Given that 𝑣 is the parameter in this situation, we can think of 𝑣̃ as being also parametrized by 𝑣
and this is written (see (2.2)) as
(3.2)
𝑣̃ = −4𝐺𝑀 𝑞 ln 𝑣
Proceeding, let us say we go for the point of intersection 𝑃𝐼𝐴 between 𝐿−( 𝑢̃ = 0) and 𝐿+( 𝑣) and
such 𝑃𝐼𝐴 is given by
(3.3)
𝑃𝐼𝐴 ∶ (𝜒 = −𝑣̃′( 𝑣); 𝜂 = 𝑣̃′( 𝑣))
By inspection these coordinates appear to be parametrized by 𝑣 via 𝑣̃(𝑣) and these can be
thought of as parametric curves along 𝐿−( 𝑢̃ = 0) with 𝑣 as the parameter. These curves can
reach both the future event horizon and past null infinity.
(3.4)
𝐻+
: 𝜒( 𝑣 = 0) = −𝜂( 𝑣 = 0) = − 𝜋/2
𝐹−
: 𝜒( 𝑣 = ∞) = −𝜂( 𝑣 = ∞) = 𝜋/2
At the far end that is, very near the past null infinity 𝐹−
(𝑟 ∗ → ∞; 𝑡 → − ∞), where 𝑢̃ =
𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, 𝑢 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, while 𝑣̃ = −∞ and 𝑣 = ∞, the fundamental line element as
expressed in the Eddington-Finkelstein form is non-vanishing and is approximately flat.
(3.5)
𝑑𝑆2( 𝑢̃, 𝑣̃) = − 𝑑𝑢̃ 𝑑𝑣̃
𝑑Ω2
= 0
𝜂 𝑚(𝑟 → ∞) = 1
Proceeding in the same manner, we obtain for the point of intersection 𝑃𝐼 𝐵 between 𝐿−( 𝑢̃) and
𝐹−
and this is given by

(3.6)
𝑃𝐼 𝐵 : (𝜒( 𝑢̃) = 𝑢̃′( 𝑢̃) +
𝜋
2
; 𝜂( 𝑢̃) = 𝑢̃′( 𝑢̃) − 𝜋/2)
The component coordinates in this point can be thought of as parametric curves along 𝐹−
with 𝑢̃
as the parameter. These curves can reach 𝐻−
and 𝐹+
, respectively
(3.7)
𝐻−
: 𝜒( 𝑢̃ = −∞ ) = 0; 𝜂( 𝑢̃ = −∞) = − 𝜋
𝐹+
: 𝜒( 𝑢̃ = ∞) = 𝜋; 𝜂( 𝑢̃ = ∞) = 0
In here, we also note that
(3.8)
𝜒( 𝑢̃ = 0) =
𝜋
2
; 𝜂( 𝑢̃ = 0) = −𝜋/2
and make some remarks that we restrict 𝐿−( 𝑢̃) up to the value of tilde 𝑢̃ = 0 so that 𝐿−( 𝑢̃)
cannot reach 𝐹+
. We must also note that 𝐿−( 𝑢̃ = 0) is where 𝐿+( 𝑣) can span along, going from
future event horizon back to past null infinity.
Given these coordinate curves, we can obtain for their corresponding tangent vectors. Of special
interest are the tangent vectors to the coordinate curves 𝜒( 𝑣) and 𝜂( 𝑣) along 𝐿−( 𝑢̃ = 0).
(4.1)
𝑑𝜒
𝑑𝑣
= −
𝑑𝜂
𝑑𝑣
=
4𝐺𝑀 𝑞
𝑣
𝑐𝑜𝑠2
(arctan(4𝐺𝑀 𝑞 𝑙𝑛𝑣))
These tangent vectors vanish on the surface of 𝐻+
(4.2)
𝑑𝜂
𝑑𝑣
|
𝑣 = 0
= −
𝑑𝜒
𝑑𝑣
|
𝑣 =0
= 0
and also on the surface of 𝐹−
(4.3)
𝑑𝜒
𝑑𝑣
|
𝑣 = ∞
= −
𝑑𝜂
𝑑𝑣
|
𝑣 = ∞
= 0

The other set of tangent vectors is for the coordinate curves 𝜒( 𝑢̃) and 𝜂( 𝑢̃) along 𝐹−
.
(Cautionary remark: 𝜂 here is not to be confused with component metric appearing in (2.7). 𝜂
here is one component of the coordinate curves.)
(4.4)
𝑑𝜒
𝑑𝑢̃
=
𝑑𝜂
𝑑𝑢̃
= 𝑐𝑜𝑠2(arctan 𝑢̃)
These vanish on the surface of 𝐻−
(4.5)
𝑑𝜒
𝑑𝑢̃
|
𝑢̃ = −∞
=
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = −∞
= 0
However, on the surface bounded by the line 𝐿−( 𝑢̃ = 0), these are non-vanishing
(4.6)
𝑑𝜒
𝑑𝑢̃
|
𝑢̃ = 0
=
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = 0
= 1
Also of note here is
(4.7)
−
1
4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑣
|
𝑣 = 1
= 1
Now let us explore what can result from matching up one piece of (4.6) with (4.7). That is, by
matching up we mean
(4.8)
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = 0
= −
1
4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑣
|
𝑣 = 1
= 1
leading to a linear relation between the two different parameters
(4.9)
𝑢̃ = −4𝐺𝑀 𝑞 𝑣 + 4𝐺𝑀 𝑞
𝑢̃ 0𝑞 = 4𝐺𝑀 𝑞, 𝑤ℎ𝑒𝑛 𝑣 = 0

where
(4.10)
𝑑𝜂
𝑑𝑣
=
𝜕𝑢̃
𝜕𝑣
𝑑𝜂
𝑑𝑢̃
= −4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑢̃
Note in this that 𝑢̃0𝑞 = 4𝐺𝑀 𝑞 (when 𝑣 = 0) can be thought of as constant of integration upon
integration of
𝜕𝑢̃
𝜕𝑣
= −4𝐺𝑀 𝑞 with respect to 𝑣, given −∞ ≤ 𝑢̃ ≤ 0, 1 ≤ 𝑣 ≤ ∞.
On the other hand we can choose this constant of integration to be zero, 𝑢̃0𝑞 = 0 so that
(4.11)
𝑢̃ = −4𝐺𝑀 𝑞 𝑣
−∞ ≤ 𝑢̃ ≤ 0, 0 ≤ 𝑣 ≤ ∞
𝑣 = 1, 𝑢̃ = −4𝐺𝑀 𝑞
𝑣 = 0, 𝑢̃ = 0
and note that by chain rule we have
(4.12.1)
𝑑𝜂
𝑑𝑣
|
𝑣 = 1
= −4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = −4𝐺 𝑀 𝑞
and substituting this in (4.8) to yield
(4.12.2)
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = 0
=
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = −4𝐺 𝑀 𝑞
= 1
This seems to suggest that the tangent vector is constant along the parameter as evaluated at two
different values of this parameter 𝑢̃.
Also by chain rule we note that
(4.13)
𝑑𝜂
𝑑𝑢̃
=
𝜕𝑣
𝜕𝑢̃
𝑑𝜂
𝑑𝑣
= (
𝜕𝑢̃
𝜕𝑣
)
−1
𝑑𝜂
𝑑𝑣
= −
1
4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑣
and evaluate

(4.14)
𝑑𝜂
𝑑𝑢̃
|
𝑢̃ = 0
= −
1
4𝐺𝑀 𝑞
𝑑𝜂
𝑑𝑣
|
𝑣 = 0
Then substitute this in (4.8) to find
(4.15)
𝑑𝜂
𝑑𝑣
|
𝑣 = 0
=
𝑑𝜂
𝑑𝑣
|
𝑣 = 1
= −4𝐺𝑀 𝑞
Also, this is suggestive that this tangent vector is constant along the given parameter as evaluated
at two different values of this parameter 𝑣.
Ref’s
[1]Ohanian, H. C., GRAVITATION AND SPACETIME, New York: W. W. Norton and
Company Inc., copyright 1976
[2]Townsend, P. K., Blackholes – Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012
[3]Carroll, S. M., Lecture notes On General Relativity, http://www.arxiv.org/abs/gr-qc/9712019

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