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 Incoming links: Stable causality

Causal hierarchy 2026-10-05

The causal hierarchy orders restrictions on the causal structure of a spacetime, from exclusion of closed timelike curves through stable causality and global hyperbolicity.

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Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 3 / ii / Solution Created 2026-10-03 Updated 2026-10-05

Let

t^{a}

exhibit the stable causality of

g

, so

g - t_{a} t_{b}

has no closed timelike curves. Set

r^{a} = \frac{t ^{a}}{2}, g_{ab}^{'} = g_{ab} - r_{a} r_{b} = g_{ab} - \frac{1}{2} t_{a} t_{b} .

r^{a}

remains timelike for

g^{'}

. Widening the cones of

g^{'}

once more using

r

gives

g_{ab}^{'} - r_{a} r_{b} = g_{ab} - t_{a} t_{b},

which is causal by assumption. Hence

g - rr is itself stably causal .

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Strong causality 2026-10-05

Strong causality means that every event has arbitrarily small neighborhoods which no causal curve can leave and then re-enter. Stable causality implies strong causality.

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