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].

@ __Specific types of spacetimes__: Minguzzi CQG(07)gq/06 [for warped products];
Blanco & Moreira 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 {*p*_{i}}
∈ *M*, such that *p*_{1} \(\ll\)
*p*_{2} \(\ll\) ...
\(\ll\) *p*_{n}
\(\ll\) *p*_{1}.

$ __Causality__: There are no closed
causal curves, collection of points {*p*_{i}}
∈ *M*, such that *p*_{1}<
*p*_{2} < ... <
*p*_{n}
< *p*_{1}.

**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 C^{k} open
topology in which all metrics satisfy are causal).

$ __Def__: There exists a
continuous non-zero timelike vector field *t*^{a}
such that the metric *g'*_{ab}
:= *g*_{ab} − *t*_{a}
*t*_{b} has no closed timelike curves.

* __Relationships__: Equivalent
to the global existence of a time function *f*: *M* →
\(\mathbb R\) (with timelike gradient d*t*).

* __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. 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*.

w
$ __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].

@ __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].

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