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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-58/1/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 58 / 1 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

A static observer has four-velocity

u = F^{- 1/2} \partial_{t}

. Its only nonzero acceleration component is

a^{r} = Γ^{r}_{tt} / F = F^{'} /2 = - r / R^{2}

. Using

g_{rr} = F^{- 1}

, the magnitude is

a = \frac{∣ F ^{'} ∣}{2 F} = \frac{r}{R ^{2} 1 - r ^{2} / R ^{2}} .

It vanishes only at

r = 0

, where

F = 1

and

d τ = F d t = d t

. Thus $t$ is proper time for the central inertial static observer. Other static observers need inward acceleration to resist de Sitter separation and have a redshifted proper time.

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