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, qS such that qI +(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), pM; 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, gab 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:= –\(1\over2\)ρab ρab = VX + W 2 = 0 ,

with ρab:= 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]; Duggal & Jin 07, JGP(10) [null curves and hypersurfaces]; Gorkavyy DG&A(08) [in nMinkowski, minimal]; Inoguchi & Lee IJGMP(09) [in 3Minkowski]; 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 pM, J+ is the set of points that can be reached from p by a future-directed causal curve, J ±(p):= {qM | ∃ 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):= ∪pS J ±(p).

Chronological Future / Past of a Subset of Spacetime
$ For a point: The chronological future/past of pM is the set of points that can be reached from p by a future/past-directed timelike curve,

I ±(p):= {qM | ∃ 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):= ∪pS 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 PM is a past set if there is an SM 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 = Q1Q2, then either P = Q1 or P = Q2.
$ PIP, proper indecomposable past set: An IP of the form P = I(x) for some xM; 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 xM; 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 SM, the future/past domain of dependence (also called causal development) of S are

D±(S):= {xM | 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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