# 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 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 ${\displaystyle (M,g)}$. Where two or more are given they are equivalent.

Notation:

## Distinguishing

### Past-distinguishing

${\displaystyle I^{-}(p)=I^{-}(q)\implies p=q}$

## 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:

## Globally hyperbolic

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