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}.

@ __General references__: Nolan et al a1812 [arbitrary character, first order perturbations].

@ __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 PRD(15)-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*).

@ __References__: Akers et al PRD(18)-a1711 [boundary of the future].

**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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