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*^{ ±}[*γ*],
with *γ* a
future/past-endless causal curve; It can be used to define boundary points or singularities.

* __Submanifolds__: The submanifolds of a given (Riemannian or) Lorentzian manifold can be classified into some interesting subfamilies, such as minimal (maximal), totally geodesic, Einstein, isotropic submanifolds, etc, most of which have been extensively studied; > s.a. Extremal Surface; Totally Geodesic Submanifold.

@ __References__: Thomas & Wichmann JMP(97);
Casini CQG(02)
[lattice structure]; Perlick gq/05-conf [totally
umbilical submanifolds]; Cabrerizo et al JGP(12) [isotropic submanifolds]; Sanders CQG(13)-a1211 [spacelike and timelike compactness].

**Lines** > s.a. causality violations [closed
timelike curves]; geodesics;
lines; minkowski space; Worldline.

* __Null curve__: A curve *γ*: \(\mathbb R\) →
*M* with an everywhere null tangent vector, *g*_{ab} *γ*^{·a} *γ*^{·b} =
0 [@ Urbantke JMP(89)].

@ __References__: Martin CQG(06)gq/04 [compactness
of the space of causal curves];
Minguzzi JMP(08)-a0712 [limit
curve theorems]; Duggal & Jin 07, JGP(10) [null
curves and hypersurfaces]; Adamo & Newman CQG(10)-a0911 [complex
world-lines and shear-free null geodesics in Minkowski space]; > s.a. topology.

**Surfaces** > s.a. extrinsic
curvature [extremal surfaces]; geometry
of FLRW 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

*σ*:= –\(1\over2\)*ρ*^{ab} *ρ*_{ab} = *VX* + *W*^{ 2}
= 0 ,

with *ρ*_{ab}:=
2*k*_{[a}* m*_{b]}, *k* =
stationary Killing vector field; *m* =
axisymmetric Killing vector field; *X*:= *m*_{a}* m*^{a}; *V*:= –*k*_{a}* k*^{a}; *W*:= *k*_{a}* m*^{a}.

@ __Null surfaces__: Silva-Ortigoza GRG(00)
[in ^{3}Minkowski, singularities]; Duggal & Jin
07, JGP(10)
[null curves and hypersurfaces]; Gorkavyy
DG&A(08)
[in ^{n}Minkowski, minimal]; Inoguchi
& Lee IJGMP(09) [in ^{3}Minkowski]; Grant AHP(11)-a1008 [properties of the area of slices of the null cone of a point]; Padmanabhan PRD(11)-a1012 [Navier-Stokes fluid dynamics of null surfaces]; Adamo & Newman PRD(11)-a1101 [shear-free null geodesic congruences, and future null infinity]; Chakraborty & Padmanabhan a1508 [thermodynamical interpretation of geometrical variables];
> s.a. gravitational thermodynamics.

@ __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-en
[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* | ∃ *λ*: [0,1] → *M* timelike or null, future/past-directed,
such that *λ*(0) = *p* and *λ*(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 ∈ 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* | ∃ *λ*:
[0,1] → *M* timelike, future/past-directed, such that *λ*(0)
= *p*, *λ*(1) = *q*} .

$ __For a subset__: The
union of the chronological futures/pasts of its points, *I*^{ ±}(*S*):= ∪_{p ∈ S} *I*^{ ±}(*p*).

* __Remark__: This 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 of *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__: A subset *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 *Q*_{1} and *Q*_{2} are
two past sets and *P* = *Q*_{1} ∪ *Q*_{2},
then either *P* = *Q*_{1} or
*P* = *Q*_{2}.

$ __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 a subset *S* ⊂ *M*,
the future/past domain of dependence (also called causal development) of *S* are

*D*^{±}(*S*):=
{*x* ∈ *M* | every past/future-directed endless timelike curve from *x* meets *S*} .

@ __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*^{–}(\(\cal I\)^{+}) ∩ *I*^{+}(\(\cal I\)^{}^{–}).

* __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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