Causality Conditions| Causality Conditions |
In General > s.a. causality; causality
violations [including chronology protection]; non-causal spacetimes.
* Remark: It is usually assumed that
classical spacetimes satisfy the strong causality condition; This is important for
the singularity theorems and for Minkowski quantum field theory, but in curved
space quantum field theory the Green functions may have acausal poles.
@ General references: Hubeny et al IJMPD(05)gq
[stringy motivation for spacetimes with almost-closed timelike curves];
Minguzzi & Sánchez gq/06-proc [causal hierarchy of spacetimes, rev];
Minguzzi JMP(08)-a0712 [non-imprisonment conditions],
JGP(09) [and continuity of the Lorentzian distance];
Pourkhandani & Bahrampour CQG(12) [and the topology of the space of causal curves];
Aké Hau et al CQG(20)-a2003 [for Lorentzian length spaces];
Minguzzi & Costa e Silva CQG-a2005 [and smooth spacetime coverings];
Carballo-Rubio et al a2005 [in modified gravity].
@ Specific types of spacetimes: Minguzzi CQG(07)gq/06 [for warped products];
Blanco & Moreira ACI-a1507 [Carter spacetimes].
> Related topics: see spacetime subsets [lines].
> Online resources:
see Wikipedia page.
Chronology / Causality Condition
$ Chronology: There are no closed timelike
curves, collection of points {pi}
∈ M, such that p1 \(\ll\)
p2 \(\ll\) ...
\(\ll\) pn
\(\ll\) p1.
$ Causality: There are no closed causal
curves, collection of points {pi}
∈ M, such that p1<
p2 < ... <
pn
< p1.
Future / Past Distinguishing Condition > s.a. Horismos.
* Idea: Any two points
with the same chronological future (past) coincide.
$ Def: It holds at p
in M if for all U neighborhoods of p, there is
another neighborhood V ⊂ U of p, such that
every future- (or past-) directed non-spatial curve through
p (cf. strong causality) only meets V once.
* Relationships: If either
is satisfied, the Alexandrov topology can be defined.
Strong Causality Condition
* Idea: There are no almost closed timelike curves.
$ Def: It holds at p in M if,
for all U ∋ p, there is another neighborhood V ⊂ U
of x, such that no causal curve intersects V more than once; Alternatively,
if p has arbitrarily small causally convex neighborhoods.
* Property: If K ⊂ M is
compact, every causal curve confined to K has future and past endpoints in K
[@ in Wald 84].
* Relationships: It implies that
the Alexandrov topology is equivalent to the manifold one.
@ References: Minguzzi JGP(09)-a0810.
Stable Causality Condition
* Idea: (M, g)
is not "on the verge" of having a bad causal structure, in the
sense that the light cones can be widened everywhere without violating
the causality condition (there is a neighborhood of g in the
Ck open topology in which
all metrics satisfy are causal).
$ Def: There exists a continuous
non-zero timelike vector field ta
such that the metric g'ab
:= gab − ta
tb has no closed timelike curves.
* Relationships: Equivalent
to the global existence of a time function f: M →
\(\mathbb R\) (with timelike gradient dt).
* Compact stable causality:
The light cones can be widened outside any arbitrarily large compact set, i.e.
in a neighborhood of infinity, without spoiling causality; The condition can
be obtained as the antisymmetry condition of a new causal relation, but not
as a causal stability condition with respect to a topology on metrics.
@ General references: in Wald 84;
Rácz GRG(87),
GRG(88);
Sánchez gq/04-proc [time functions and Cauchy surfaces];
Minguzzi CMP(09) [and lightlike lines];
Minguzzi & Rinaldelli CQG(09)-a0904 [compact stable causality];
Minguzzi a0905-wd,
CMP(10)-a0909 [and time functions];
Howard AIP(10)-a1601 [almost stable causality].
@ K-causality: Minguzzi CQG(08),
CQG(08)gq/07;
Ebrahimi a1404 [and domain theory].
@ Specific spacetimes: Chruściel & Szybka ATMP(11)-a1010 [Pomeransky-Senkov black rings]
Causal Continuity > s.a. metric types [degenerate].
$ Def: (M, g) is past
and future distinguishing, and I +(p)
and I −(p) vary continuously
with p.
@ References: in Geroch JMP(70);
Hawking & Sachs CMP(74);
Vyas & Akolia GRG(86);
Borde et al CQG(99)gq [and topology change];
Sánchez gq/04-proc [time functions and Cauchy surfaces];
Minguzzi CQG(08)-a0712.
Causal Simplicity
$ Def: (M, g) is past
and future distinguishing, and J +(p)
and J −(p) are closed for all
p in M.
@ References:
Sánchez gq/06-wd [sufficient condition];
Minguzzi JGP(09)-a0810;
Minguzzi JMP(12)-a1204 [causal simplicity removes holes from spacetime];
Chernov CQG(18)-a1712 [linking and causality].
Global Hyperbolicity
> s.a. determinism; differentiable manifolds
[and inequivalent smooth structures]; types of spacetimes.
$ Def 1: (Leray) The
collection of causal curves joining p and q is compact
for all p, q in M (in a suitable topology).
$ Def 2: (M, g) is time-orientable
and the Alexandrov sets / causal diamonds J +(p)
∩ J −(q) are compact for all p,
q in M.
$ Def 3:
(Geroch) (M, g) admits a Cauchy surface.
* Properties: It is always causally
simple, strongly causal, and topologically Σ × \(\mathbb R\).
@ General references: Lichnerowicz in(68);
Choquet-Bruhat in(68);
Geroch JMP(70);
Matori JMP(88) [spatially closed spacetimes];
Clarke CQG(98)gq/97 [generalization];
Choquet-Bruhat & Cotsakis JGP(02) [and completeness];
Martin & Panangaden CMP(06)gq/04 [dense subsets];
Bernal & Sánchez CQG(07)gq/06 [conditions];
Benavides Navarro & Minguzzi JMP(11)-a1108 [stability in the interval topology];
Sämann AHP(16)-a1412 [spacetimes with continuous metrics];
Hounnonkpe & Minguzzi CQG(19)-a1908 [without causality condition].
@ Splitting, time functions: Bernal & Sánchez CMP(03)gq,
CMP(05)gq/04,
gq/04-proc;
Cotsakis GRG(04) [sliced spaces];
Sánchez gq/04-proc;
Minguzzi CQG(16)-a1601 [existence of smooth Cauchy steep time functions];
Bleybel a2103 [using temporal foliations of causal sets].
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