Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 37A / c

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- Solution c

Solution

A radial timelike geodesic has conserved

E = f \dot{t}

and normalization

\overset{r}{˙}^{2} = E^{2} - f

. Choose

E > 1

and the outward sign. Then

Δ τ = \int_{0}^{b} \frac{d r}{E ^{2} - 1 + r ^{2} / b ^{2}} < \infty,

so the observer reaches

r = b

in finite proper time even though static coordinate

t

diverges.

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