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 *x*_{0 }< *x*_{1} < *x*_{2} < ...
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]; Kissinger et al a1708 [and process terminality].

@ __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

@

>

**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 *T** _{pq}* of
timelike paths from

*

*

*

*

@

**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 [S^{1} 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, a1801-proc [extension]; > 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].

@ __Causal structure and gravity__: Friedman et al PRD(13)-a1305 [shared causal pasts and futures in cosmology]; Gomes a1603 [quantum gravity and superpositions of causal structures]; Suvorov & Melatos PRD-a1709 [gravitational waves in *f*(*R*) gravity].

@ __Generalization__: Yurtsever JMP(90);
Bois & Trelut RQS-ap/03
[and temporal order]; Minguzzi a1709-conf [for general closed cone structures].

@ __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]; Sormani & Vega CQG(16)-a1508 [null distance function]; > s.a. arrow of time; Link
Theory; Paneitz Operator.

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18 jan 2018