Causality conditions

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 condition: global hyperbolicity. For such spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface.

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

Given are the definitions of these causality conditions for a Lorentzian manifold [math]\displaystyle{ (M,g) }[/math]. Where two or more are given they are equivalent.

Notation:

• [math]\displaystyle{ p \ll q }[/math] denotes the chronological relation.
• [math]\displaystyle{ p \prec q }[/math] denotes the causal relation.

(See causal structure for definitions of [math]\displaystyle{ \,I^+(x) }[/math], [math]\displaystyle{ \,I^-(x) }[/math] and [math]\displaystyle{ \,J^+(x) }[/math], [math]\displaystyle{ \,J^-(x) }[/math].)

Non-totally vicious

• For some points [math]\displaystyle{ p \in M }[/math] we have [math]\displaystyle{ p \not\ll p }[/math].

Chronological

• There are no closed chronological (timelike) curves.
• The chronological relation is irreflexive: [math]\displaystyle{ p \not\ll p }[/math] for all [math]\displaystyle{ p \in M }[/math].

Causal

• There are no closed causal (non-spacelike) curves.
• If both [math]\displaystyle{ p \prec q }[/math] and [math]\displaystyle{ q \prec p }[/math] then [math]\displaystyle{ p = q }[/math]

Distinguishing

Past-distinguishing

• Two points [math]\displaystyle{ p, q \in M }[/math] which share the same chronological past are the same point:

[math]\displaystyle{ I^-(p) = I^-(q) \implies p = q }[/math]

• Equivalently, for any neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ p \in M }[/math] there exists a neighborhood [math]\displaystyle{ V \subset U, p \in V }[/math] such that no past-directed non-spacelike curve from [math]\displaystyle{ p }[/math] intersects [math]\displaystyle{ V }[/math] more than once.

Future-distinguishing

• Two points [math]\displaystyle{ p, q \in M }[/math] which share the same chronological future are the same point:

[math]\displaystyle{ I^+(p) = I^+(q) \implies p = q }[/math]

• Equivalently, for any neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ p \in M }[/math] there exists a neighborhood [math]\displaystyle{ V \subset U, p \in V }[/math] such that no future-directed non-spacelike curve from [math]\displaystyle{ p }[/math] intersects [math]\displaystyle{ V }[/math] more than once.

Strongly causal

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

Stably causal

A manifold satisfying any of the weaker causality conditions defined above may 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 [math]\displaystyle{ M }[/math]. This is a scalar field [math]\displaystyle{ t }[/math] on [math]\displaystyle{ M }[/math] whose gradient [math]\displaystyle{ \nabla^a t }[/math] is everywhere timelike and future-directed. This global time function gives us a stable way to distinguish between future and past for each point of the spacetime (and so we have no causal violations).

Globally hyperbolic

• [math]\displaystyle{ \,M }[/math] is strongly causal and every set [math]\displaystyle{ J^+(x) \cap J^-(y) }[/math] (for points [math]\displaystyle{ x,y \in M }[/math]) is compact.

Robert Geroch showed^[3] that a spacetime is globally hyperbolic if and only if there exists a Cauchy surface for [math]\displaystyle{ M }[/math]. This means that:

• [math]\displaystyle{ M }[/math] is topologically equivalent to [math]\displaystyle{ \mathbb{R} \times\!\, S }[/math] for some Cauchy surface [math]\displaystyle{ S }[/math] (Here [math]\displaystyle{ \mathbb{R} }[/math] denotes the real line).

See also

• Spacetime
• Lorentzian manifold
• Causal structure
• Globally hyperbolic manifold
• Closed timelike curve

References

• S.W. Hawking, G.F.R. Ellis (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-20016-4. 
• S.W. Hawking, W. Israel (1979). General Relativity, an Einstein Centenary Survey. Cambridge University Press. ISBN 0-521-22285-0. 

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