Causal Structures in Spacetime  

In General > s.a. causality; spacetime / Chronological Space; lorentzian geometry [including analogs]; singularities; types of metrics.
* Idea: The causal structure of a spacetime is a global property, and contains almost all the information about the metric (9/10 in 4D, all except for the conformal factor); It can be considered as the fundamental structure in quantum gravity.
$ Def: A partial ordering on a set of points (poset), indicated by p < q (possibly with additional conditions).
$ Causal completeness: A spacetime is causally complete if every bounded, increasing sequence x0 < x1 < x2 < ... in M converges.
@ General references: Joshi 93; García-Parrado & Senovilla CQG(05)gq [rev]; Minguzzi & Sánchez in(08)gq/06 [hierarchy of conditions]; Howard AIP(10)-a1601 [and singularities]; Chruściel a1110 [elements of causality theory]; Foldes a1206 [maximal chains and antichain cutsets]; Ried et al nPhys(15)-a1406 [inferring causal relations]; Stoica a1504 [as fundamental].
@ Abstract causal structure: Kronheimer & Penrose PCPS(67); Pimenov 68; Kronheimer GRG(71); Carter GRG(71); Lerner in(72); Penrose 72; in Hawking & Ellis 73; Woodhouse PhD(73); in Beem et al 96; Rainer JMP(99)gq [topological manifolds]; Jaroszkiewicz gq/00 [discrete spacetime]; García-Parrado & Sánchez CQG(05)mp; Cegła & Jancewicz JMP(13) [lattice structure approach].
@ Causal fs, maps: Vyas & Joshi GRG(83); Joshi GRG(89); García & Senovilla mp/02-proc, CQG(03)gq/02 [between manifolds], CQG(03)gq, CQG(04)gq/03 [symmetries]; Janardhan & Saraykar Pra(08)gq/05 [using K-causality], GRG(13)-a1208 [causal-cone-preserving transformations in spacetime].
@ Causal boundaries: Harris JMP(98) [universality]; > s.a. spacetime boundaries.
> Online resources: see Wikipedia page.

Chronological Homotopy Theory > s.a. spacetime subsets [lines].
* Idea: Paths which are close also have close parametrizations, but smoothness is irrelevant.
* Topology on paths: The space Tpq of timelike paths from p to q has the compact-open topology generated by UK,U := {γTpq | γ(K) ⊂ U, K compact in [0,1], U open in M}.
* Homotopy: Two paths are chronologically homotopic if they lie in the same path-connected component of Tpq.
* Homotopy of manifolds: (M, g) and (M', g') have the same homotopy type if there is a homeomorphism MM' preserving the homotopy structures of all Tpq.
* Euler number: χ(Tpq):= χ(K), where K is a finite cell complex with the same homotopy type as Tpq.
* Applications: May lead to a way of defining black holes in closed universes.
@ References: Smith AJM(60), PNAS(60); Kronheimer GRG(71); Morales & Sánchez CQG(15)-a1505 [globally hyperbolic spacetimes with infinitely many causal homotopy classes of curves].

Types of Spacetimes > s.a. causality violations; non-commutative geometry; types of lorentzian geometries.
@ Special metrics: Lester JMP(84) [de Sitter and Einstein cylinder]; Calvão et al JMP(88) [Gödel-type]; Levichev GRG(89) [homogeneous]; Matschull CQG(96)gq/95 [degenerate]; Gratus & Tucker JMP(96)gq [degenerate 2D]; Singh & Sahdev gq/01 [S1 time topology]; Chruściel & Grant CQG(12)-a1111 [continuous metrics, systematic study]; Harris CQG(15)-a1412 + CQG+ [static and stationary spacetimes].
@ Non-Hausdorff spacetime: Hájíček CMP(71); Sharlow AP(98).
> Specific types: see gödel spacetime; gravitational waves [pp-waves]; minkowski space; schwarzschild and Kruskal Extension.

Various Causality-Type Relations > s.a. spacetime subsets [causal and chronological futures/pasts].
$ K-causality: K+ is the smallest relation containing I+ that is transitive and (topologically) closed.
* At singular points: The light cone structure at degenerate points might be different but well-defined; A point p has a single past (future) light cone if for all neighborhoods U of p, not containing other singular points, I(p, N) (I+(p, N)) is connected.
@ K-causality: Sorkin & Woolgar CQG(96)gq/95; Dowker et al CQG(00)gq/99 [degenerate metrics]; Miller a1702-proc; > s.a. causality conditions.

Related Concepts > s.a. causality violations; Horismos; initial-value form; null infinity [causal completion]; spacetime subsets; variational principles [causal].
* Recovery of spacetime structure: (Hawking-Malament theorem) The causal relations among points in a sufficiently causal spacetime (or among points in a countable, dense subset) determine uniquely the topology, differentiable structure and metric (up to a conformal factor which is constant if the points are uniformly embedded) of the manifold.
@ And curvature: Woodhouse CMP(76); Szabados GRG(82); Gibbons & Solodukhin PLB(07)ht [Alexandrov sets and curvature], PLB(07)-a0706 [asymptotically de Sitter case]; > s.a. Alexandrov Sets; wave phenomena.
@ And spacetime topology: Fuller & Wheeler PR(62); Konstantinov IJMPD(95)gq/94, gq/97-MG8, G&C(97)gq/98 [non-trivial]; Chamblin gq/95-conf; Lobo & Crawford gq/02-conf; Nielsen Flagga & Antonsen IJTP(04) [Stiefel-Whitney class]; Borchers & Sen 06; BenDaniel a0806 [denumerable spacetime]; Parrikar & Surya CQG(11)-a1102 [dimensionality]; Kovár a1112 [de Groot dual]; Saraykar & Janardhan G&C-a1411 [rev]; > s.a. spacetime topology.
@ Recovery of spacetime structure: Hawking et al JMP(76); Malament JMP(77); Briginshaw IJTP(80), IJTP(80) [and conformal group]; Martin & Panangaden CMP(06)gq/04; in Malament gq/05-ch; Kim CQG(08)-a0801 [from Cauchy surface]; > s.a. causal sets.
@ And initial data: Klainerman & Rodnianski IM(05)m.AP/03 [vacuum].
@ Generalization: Yurtsever JMP(90); Bois & Trelut RQS-ap/03 [and temporal order].
@ Other topics: Szabados GRG(87) [and measurability]; Kreinovich IJTP(94) [approximate causality]; Casini CQG(02)gq [logic]; Harris CQG(04)gq/03 [and discrete group actions]; Diethert et al IJMPA(08)-a0710 [causal structure as emergent from symmetry breaking]; Chernov & Nemirovski GFA(10)-a0810 [Legendrian links and Low conjecture]; Friedman et al PRD(13)-a1305 [shared causal pasts and futures in cosmology]; Sormani & Vega CQG(16)-a1508 [null distance function]; Gomes a1603 [quantum gravity and superpositions of causal structures]; > s.a. arrow of time; Link Theory; Paneitz Operator.

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