Causal Structures in Spacetime| 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];
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 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
:= {γ ∈ Tpq
| 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 M → M'
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.
@ 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 [S1 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).
@ Lor-regularity metrics:
Chruściel & Grant CQG(12)-a1111 [continuous, systematic study];
Grant et al LMP(20)-a1901 [topology];
Ling GRG(20)-a1911 [continuous].
> 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,
JPCS(18)-a1801 [extension];
> s.a. causality conditions.
@ Other types of relations:
García-Heveling a2101
[k+ relation, and spacetimes with continuous metrics].
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 GJPAM-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(17)-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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