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];
Witten a1905-ln [intro:
Raychaudhuri equation, singularity theorems, black hole area theorem, topological censorship, ...];
Minguzzi LRR(19) [self-contained rev].

@ __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];
> s.a. causality in quantum theory.

@ __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.

@ __Examples, symmetries__:
Lester JMP(84) [de Sitter and Einstein cylinder];
Calvão et al JMP(88) [Gödel-type];
Levichev GRG(89) [homogeneous];
Singh & Sahdev gq/01 [S^{1} time topology];
Harris CQG(15)-a1412
+ CQG+ [static and stationary spacetimes].

@ __Degenerate metrics__: Matschull CQG(96)gq/95;
Gratus & Tucker JMP(96)gq [2D].

@ __Non-Hausdorff spacetime__:
Hájíček CMP(71);
Sharlow AP(98).

@ __Other types__: Chruściel & Grant CQG(12)-a1111 [continuous metrics, systematic study];
Grant et al a1901 [low-regularity Lorentzian geometry, topology].

> __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 RVMP(18)-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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