Causality Conditions

In General > s.a. causality; causality violations [including chronology protection].
* 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.
$ Chronology: There are no closed timelike curves, collection of points {p_i} M, such that p₁ p₂ ... p_n p₁.
$ Causality: There are no closed causal curves, collection of points {p_i} M, such that p₁< p₂ < ... < p_n< p₁.
@ References: Hubeny et al IJMPD(05)gq [stringy motivation for spacetimes with almost-closed timelike curves]; Minguzzi CQG(07)gq/06 [for warped products], JMP(08)-a0712 [non-imprisonment conditions]; JGP(09) [and continuity of Lorentzian distance].
> Related topics: see spacetime subsets [lines].

Future / Past Distinguishing Condition
* 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 V U 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 U p, there is another neighborhood V U of x, such that no causal curve intersects V more than once; Alternatively, if p has arbitrarily small causally convex neighborhoods.
* Property: If K M 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 C^k open topology in which all metrics satisfy are causal).
$ Def: There exists a continuous nonzero timelike vector field t^a such that the metric g'_ab := g_ab – t_a t_b has no closed timelike curves.
* Relationships: Equivalent to the existence of a global time function f : M → R with past-directed timelike _a t.
* 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.
@ References: in Wald 84; Rácz GRG(87), GRG(88); Sánchez gq/04-in [t functions and Cauchy surfaces]; Minguzzi CQG(08), CQG(08)gq/07 [and K-causality]; Minguzzi CMP(09) [and lightlike lines]; Minguzzi & Rinaldelli CQG(09)-a0904 [compact stable causality]; Minguzzi a0905, a0909 [and time functions].

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-in [t 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.

Global Hyperbolicity
$ 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 J ⁺(p) J ^–(q) is 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 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].
@ Splitting, time functions: Bernal & Sánchez CMP(03)gq, CMP(05)gq/04, gq/04-in; Cotsakis GRG(04) [sliced spaces]; Sánchez gq/04-in.

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