Causal
Structures in Spacetime| Causal
Structures in Spacetime |
In General > s.a. [causality, spacetime]; 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, all except the conformal
factor); It can be considered as the fundamental structure in quantum gravity.
$ 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 gq/06-in
[hierarchy of conditions].
@ Causal f 's, maps: Vyas & Joshi GRG(83);
Joshi GRG(89);
García & Senovilla mp/02-in, CQG(03)gq/02 [between
manifolds], CQG(03)gq,
CQG(04)gq/03 [symmetries];
Janardhan & Saraykar Pra(08)gq/05 [using
K-causality].
@ Causal boundaries: Harris JMP(98) [universality]; > s.a. spacetime
boundaries.
Abstract Causal Structure > s.a. Chronological
Space.
$ Def: A partial ordering
on a set of points (poset), indicated by p < q (possibly with additional conditions).
@ References: 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.
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 :=
{
Tpq |
(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: 
(Tpq):=
(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).
Specific Spacetimes > s.a. gödel
spacetime; gravitational
waves [pp-waves]; minkowski; schwarzschild and Kruskal
Extension.
@ 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].
@ Non-Hausdorff spacetime: Hájícek CMP(71);
Sharlow AP(98).
> Related topics: see
causality violations; non-commutative
geometry.
Various Causality-Type Relations
* 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]; > s.a. causality conditions.
Related Concepts > s.a. causality
violations; Horismos; initial-value form; Link
Theory; null
infinity [causal
completion];
spacetime subsets.
* Recovery of spacetime
structure:
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-in,
G&C(97)gq/98 [nontrivial];
Chamblin gq/95-in;
Lobo & Crawford gq/02-in;
Nielsen Flagga
& Antonsen IJTP(04)
[Stiefel-Whitney class]; > 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-in;
Kim CQG(08)-a0801 [from
Cauchy surface]; > s.a. causal
sets.
@ And initial data: Klainerman & Rodnianski 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 a0810 [Legendrian
links and Low
conjecture].
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15 jul 2009