Causality conditions

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In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.^[1]

The weaker the causality condition on a spacetime, the more unphysical the spacetime is. Spacetimes with closed timelike curves, for example, present severe interpretational difficulties. See the grandfather paradox.

It is reasonable to believe that any physical spacetime will satisfy the strongest causality condtion: global hyperbolicity. For such spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface.

1 The hierarchy
2 Non-totally vicious
3 Chronological
4 Causal
5 Distinguishing
- 5.1 Past-distinguishing
- 5.2 Future-distinguishing
6 Strongly causal
7 Stably causal
8 Globally hyperbolic
9 See also
10 References

[edit] The hierarchy

There is a hierarchy of causality conditions, each one of which is strictly stronger than the previous. This is sometimes called the causal ladder. The conditions, from weakest to strongest, are:

Non-totally vicious
Chronological
Causal
Distinguishing
Strongly causal
Stably causal
Causally continuous
Causally simple
Globally hyperbolic

We now give definitions of these causality conditions for a Lorentzian manifold $(M, g)$ . Where two or more are given they are equivalent.

Notation:

$p \ll q$ denotes the chronological relation.
$p \prec q$ denotes the causal relation.

(See causal structure for definitions.)

[edit] Non-totally vicious

For some points $p \in M$ we have $p \not\ll p$ .

[edit] Chronological

There are no closed chronological (timelike) curves.
The chronological relation is irreflexive: $p \not\ll p$ for all $p \in M$ .

[edit] Causal

There are no closed causal (non-spacelike) curves.
If both $p \prec q$ and $q \prec p$ then $p = q$

[edit] Distinguishing

[edit] Past-distinguishing

Two points $p, q \in M$ which share the same chronological past are the same point:

$I^-(p) = I^-(q) \implies p = q$

For any neighborhood $U$ of $p \in M$ there exists a neighborhood $V \subset U, p \in V$ such that no past-directed non-spacelike curve from $p$ intersects $V$ more than once.

[edit] Future-distinguishing

Two points $p, q \in M$ which share the same chronological future are the same point:

$I^+(p) = I^+(q) \implies p = q$

For any neighborhood $U$ of $p \in M$ there exists a neighborhood $V \subset U, p \in V$ such that no future-directed non-spacelike curve from $p$ intersects $V$ more than once.

[edit] Strongly causal

For any neighborhood $U$ of $p \in M$ there exists no timelike curve that passes through $U$ more than once.
For any neighborhood $U$ of $p \in M$ there exists a neighborhood $V \subset U, p \in V$ such that $V$ is causally convex in $M$ (and thus in $U$ ).
The Alexandrov topology agrees with the manifold topology.

[edit] Stably causal

A manifold satisfying any of the weaker causality conditions defined above will fail to do so if the metric is given a small perturbation. A spacetime is stably causal if it cannot be made to contain closed causal curves by arbitrarily small perturbations of the metric. Stephen Hawking showed^[2] that this is equivalent to:

There exists a global time function on $M$ . This is a scalar field $t$ on $M$ whose gradient $\nabla^a t$ is everywhere timelike and future-directed.

[edit] Globally hyperbolic

$\,M$ is strongly causal and every set $J^+(x) \cap J^-(y)$ (for points $x,y \in M$ ) is compact.

Robert Geroch showed^[3] that a spacetime is globally hyperbolic if and only if there existences a Cauchy surface for $M$ . This means that:

$M$ is topologically equivalent to $\mathbb{R} \times S$ for some Cauchy surface $S$ (Here $\mathbb{R}$ denotes the real line).

[edit] See also

[edit] References

^ E. Minguzzi and M. Sanchez, The causal hierarchy of spacetimes, arXiv:gr-qc/0609119v2
^ S.W. Hawking, The existence of cosmic time functions Proc. R. Soc. Lond. (1969), A308, 433
^ R. Geroch, Domain of Dependence J. Math. Phys. (1970) 11, 437-449

S.W. Hawking, G.F.R. Ellis, (1973). The Large Scale Structure of Spacetime. Cambridge: Cambridge University Press. ISBN 0-521-20016-4.

S.W. Hawking, W. Israel,. General Relativity, an Einstein Centenary Survey. Cambridge University Press. ISBN 0-521-22285-0.