Causality Conditions  

In General > s.a. causality; causality violations [including chronology protection]; non-causal spacetimes.
* Remark: It is usually assumed that classical spacetimes satisfy the strong causality condition; This is important for the singularity theorems and for Minkowski quantum field theory, but in curved space quantum field theory the Green functions may have acausal poles.
@ General references: Hubeny et al IJMPD(05)gq [stringy motivation for spacetimes with almost-closed timelike curves]; Minguzzi & Sánchez gq/06-proc [causal hierarchy of spacetimes, rev]; Minguzzi JMP(08)-a0712 [non-imprisonment conditions], JGP(09) [and continuity of the Lorentzian distance]; Pourkhandani & Bahrampour CQG(12) [and the topology of the space of causal curves].
@ Specific types of spacetimes: Minguzzi CQG(07)gq/06 [for warped products]; Blanco & Moreira a1507 [Carter spacetimes].
> Related topics: see spacetime subsets [lines].
> Online resources: see Wikipedia page.

Chronology / Causality Condition
$ Chronology: There are no closed timelike curves, collection of points {pi} ∈ M, such that p1 \(\ll\) p2 \(\ll\) ... \(\ll\) pn\(\ll\) p1.
$ Causality: There are no closed causal curves, collection of points {pi} ∈ M, such that p1< p2 < ... < pn< p1.

Future / Past Distinguishing Condition > s.a. Horismos.
* Idea: Any two points with the same chronological future (past) coincide.
$ Def: It holds at p in M if for all U neighborhoods of p, there is another neighborhood VU of p, such that every future- (or past-) directed non-spatial curve through p (cf. strong causality) only meets V once.
* Relationships: If either is satisfied, the Alexandrov topology can be defined.

Strong Causality Condition
* Idea: There are no almost closed timelike curves.
$ Def: It holds at p in M if, for all Up, there is another neighborhood VU of x, such that no causal curve intersects V more than once; Alternatively, if p has arbitrarily small causally convex neighborhoods.
* Property: If KM is compact, every causal curve confined to K has future and past endpoints in K [@ in Wald 84].
* Relationships: It implies that the Alexandrov topology is equivalent to the manifold one.
@ References: Minguzzi JGP(09)-a0810.

Stable Causality Condition
* Idea: (M, g) is not "on the verge" of having a bad causal structure, in the sense that the light cones can be widened everywhere without violating the causality condition (there is a neighborhood of g in the Ck open topology in which all metrics satisfy are causal).
$ Def: There exists a continuous non-zero timelike vector field ta such that the metric g'ab := gabta tb has no closed timelike curves.
* Relationships: Equivalent to the global existence of a time function f : M → \(\mathbb R\) (with timelike gradient dt).
* Compact stable causality: The light cones can be widened outside any arbitrarily large compact set, i.e. in a neighborhood of infinity, without spoiling causality; The condition can be obtained as the antisymmetry condition of a new causal relation, but not as a causal stability condition with respect to a topology on metrics.
@ General references: in Wald 84; Rácz GRG(87), GRG(88); Sánchez gq/04-proc [time functions and Cauchy surfaces]; Minguzzi CMP(09) [and lightlike lines]; Minguzzi & Rinaldelli CQG(09)-a0904 [compact stable causality]; Minguzzi a0905-wd, CMP(10)-a0909 [and time functions]; Howard AIP(10)-a1601 [almost stable causality].
@ K-causality: Minguzzi CQG(08), CQG(08)gq/07; Ebrahimi a1404 [and domain theory].
@ Specific spacetimes: Chruściel & Szybka ATMP(11)-a1010 [Pomeransky-Senkov black rings]

Causal Continuity > s.a. metric types [degenerate].
$ Def: (M, g) is past and future distinguishing, and I +(p) and I(p) vary continuously with p.
@ References: in Geroch JMP(70); Hawking & Sachs CMP(74); Vyas & Akolia GRG(86); Borde et al CQG(99)gq [and topology change]; Sánchez gq/04-proc [time functions and Cauchy surfaces]; Minguzzi CQG(08)-a0712.

Causal Simplicity
$ Def: (M, g) is past and future distinguishing, and J +(p) and J(p) are closed for all p in M.
@ References: Sánchez gq/06-wd [sufficient condition]; Minguzzi JGP(09)-a0810; Minguzzi JMP(12)-a1204 [causal simplicity removes holes from spacetime].

Global Hyperbolicity > s.a. differentiable manifolds [and inequivalent smooth structures].
$ Def 1: (Leray) The collection of causal curves joining p and q is compact for all p, q in M (in a suitable topology).
$ Def 2: (M, g) is time-orientable and the Alexandrov sets / causal diamonds J +(p) ∩ J(q) are compact for all p, q in M.
$ Def 3: (Geroch) (M, g) admits a Cauchy surface.
* Properties: It is always causally simple, strongly causal, and topologically Σ × \(\mathbb R\).
@ General references: Lichnerowicz in(68); Choquet-Bruhat in(68); Geroch JMP(70); Matori JMP(88) [spatially closed spacetimes]; Clarke CQG(98)gq/97 [generalization]; Choquet-Bruhat & Cotsakis JGP(02) [and completeness]; Martin & Panangaden CMP(06)gq/04 [dense subsets]; Bernal & Sánchez CQG(07)gq/06 [conditions]; Benavides Navarro & Minguzzi JMP(11)-a1108 [stability in the interval topology]; Sämann a1412 [spacetimes with continuous metrics].
@ Splitting, time functions: Bernal & Sánchez CMP(03)gq, CMP(05)gq/04, gq/04-proc; Cotsakis GRG(04) [sliced spaces]; Sánchez gq/04-proc; Minguzzi a1601 [existence of smooth Cauchy steep time functions].

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