Spacetime
Subsets| Spacetime
Subsets |
In General > s.a. causal
structures and causality
conditions; Edge; singularities.
* Achronal set: A subset S of
a Lorentzian manifold M is called an achronal set if none of its points
is in the chronological future of any
other,
or there are no p, q 
S such
that q I+(p),
or I+(S) 
S = Ø.
* Causally closed set: A set S such
that S'' = S, where S' is the causal complement
of S (S' is always closed).
* Terminal Irreducible Past
/ Future set (TIP / TIF): One which is not the union of two past/future
sets (unless one is contained in the other), nor
of the form I–/+(p), p M;
It is then of the form I–/+[
],
with a
future/past-endless causal curve; Can be used to define boundary points or
singularities.
@ References: Thomas & Wichmann JMP(97);
Casini CQG(02)
[lattice structure]; Perlick
gq/05-in [totally
umbilical submanifolds].
Lines > s.a. geodesics;
lines; minkowski space; Timelike
Curve; Worldline.
* Null curve: A curve :
R →
M with an everywhere null tangent vector, gab ·a ·b =
0 [@ Urbantke JMP(89)].
@ References: Martin CQG(06)gq/04 [compactness
of space of causal curves];
Minguzzi JMP(08)-a0712 [limit
curve theorems]; Duggal & Jiin 08 [null
curves and hypersurfaces]; > s.a. topology.
Surfaces > s.a. geometry of FRW
models; Light Cones.
* Outer rotosurface:
The circularity limit, boundary of the outer connected region outside a black
hole where a timelike orbit (with tangent vector a
linear combination
of k and m) can represent uniform circular motion; It
must lie outside or on both the outer ergosurface and the horizon; Defined
as the locus of points in a stationary axisymmetric spacetime where
:= –
ab ab = VX + W 2
= 0 ,
with ab:=
2k[a mb], k =
stationary Killing vector field; m =
axisymmetric Killing vector field; X:= ma ma; V:= –ka ka; W:= ka ma.
@ Null surfaces: Silva-Ortigoza GRG(00)
[in 3Minkowski, singularities]; Gorkavyy
DG&A(08)
[in nMinkowski, minimal]; Inoguchi
& Lee IJGMP(09) [in 3Minkowski].
@ Photon surface: Claudel et al JMP(01)gq/00 [arbitrary
spacetime].
@ Other types: Catoni et al NCB(05)mp [constant
curvature, in Minkowski]; Andersson gq/05-in
[critical surfaces]; Senovilla CQG(07)gq [2-surfaces,
classification]; Hasse & Perlick a0806-in
[2D
timelike, classification]; > s.a. Cauchy
Surface; foliations; horizons; Horismos; Hypersurface; Slice; Trapped
Surface.
Causal Future / Past of a Subset of Spacetime
$ For a point: For p M, J+
is the set of points that can be reached
from p by a future-directed causal curve, J+/–(p):=
{q M | 
:
[0,1] →
M timelike or null, future/past-directed,
such that (0) = p and (1)
= q}.
* Properties: The causal future/past of a point is always a closed
set in
flat spacetime or in any globally hyperbolic spacetime, but not in general.
$ For a subset: The union
of the causal futures/pasts of its points, J+/–(S):= 
p in S J+/–(p).
Chronological Future / Past of a Subset of Spacetime
$ For a point: The chronological
future/past of p M is
the set of
points that can be reached from p by a future/past-directed timelike
curve,
I+/–(p):=
{q M |
:
[0,1] → M timelike,
future/past-directed, such that (0)
= p, (1)
= q} .
$ For a subset: The
union of the chronological futures/pasts of its points, I+/–(S):= p in S I+/–(p).
* Remark: Can be generalized
to I+/–(p, N),
if we require the timelike curve in
question to lie entirely in the neighborhood N of p.
* Properties: (i) I+/–[I+/–(S)]
= I+/–(S);
(ii) I+/–(closure S)=
I+/–(S);
(iii) I+/–(S) is
always open, if the manifold is everywhere Lorentzian (no singular
points).
Past and Future Sets in General > s.a. Alexandrov
Set; spacetime
boundaries.
$ Past set: P 
M is
a past set if there is an S M such
that P = I–(S);
Such a set is always open.
$ IP, indecomposable past set:
One such that if Q1 and Q2 are
two past sets and P = Q1 Q2,
then either P = Q1 or
P = Q2.
$ PIP, proper indecomposable
past
set:
An IP of the form P = I–(x)
for some x M;
Equivalent
to P = I–(c)
for some timelike curve c with endpoint x.
$ TIP, terminal indecomposable
past
set:
An IP not of the form P = I–(x)
for some x M;
Equivalent to P = I–(c)
for some inextendible timelike curve c.
@ References: Geroch et al PRS(72); Budic & Sachs JMP(74).
Domain of Dependence
$ Def: Given S M,
the future/past domain of dependence of S are
D+/–(S):=
{x M |
every past/future-directed endless timelike curve from x meets S}
.
Also called causal development.
@ References: Geroch JMP(70).
Domain of Outer Communications
* Idea: The part of a
spacetime manifold that can be connected to an asymptotic region by both future
and past-directed
timelike curves;
Equivalently, I–(
+) I+(–).
* Result: It is the
maximal connected asymptotically flat region of spacetime such that the
integral curves of the stationary Killing vector field through any p will,
if extended sufficiently far forward, enter and remain in I+(p)
[@ Carter in(73)].
* Properties: If the
causality condition holds, it cannot contain fixed points of the stationary
Killing vector field.
@ Properties: Galloway CQG(95)
[simply connected].
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jul
2009