Causality
Conditions| Causality
Conditions |
In General > s.a. causality; causality
violations [including chronology protection].
* 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.
$ Chronology: There are
no closed timelike curves, collection of points {pi}
M, such that p1 
p2 ... pn p1.
$ Causality: There are
no closed causal curves, collection of points {pi}
M, such that p1< p2 < ... < pn< p1.
@ References: Hubeny et al IJMPD(05)gq [stringy
motivation for spacetimes with almost ctc's]; Minguzzi CQG(07)gq/06 [for
warped products], a0712 [non-imprisonment
conditions].
> Related topics: see spacetime subsets [lines].
Future / Past Distinguishing Condition
* 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.
Stable Causality Condition
* Idea: (M, g)
is not "on the verge" of having
a bad causal structure (there is a neighborhood of g in the Ck open
topology ...).
$ Def: There exists a
continuous nonzero timelike vector field ta such
that the metric g'ab
:= gab – ta tb has
no closed timelike curves.
* Relationships: Equivalent
to the existence of a global time function f : M → R with
past-directed timelike 
a t.
@ References: in Wald 84; Rácz GRG(87), GRG(88);
Sánchez gq/04-in
[t functions and Cauchy surfaces]; Minguzzi CQG(08), CQG(08)gq/07 [and
K-causality].
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-in
[t functions and Cauchy surfaces]; Minguzzi 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 [sufficient
condition].
Global Hyperbolicity
$ 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 J+(p) 
J–(q)
is 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
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].
@ Splitting, time functions: Bernal & Sánchez CMP(03)gq,
CMP(05)gq/04, gq/04-in;
Cotsakis GRG(04) [sliced spaces]; Sánchez gq/04-in.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008