Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 34D / c / Solution

For Alice's circular geodesic, the conserved quantities or radial geodesic equation give

Ω = d ϕ / d t = r_{s} / (2 r^{3})

. Consequently

(\frac{d τ _{A}}{d t})^{2} = f - r^{2} Ω^{2} = 1 - \frac{3 r _{s}}{2 r} .

Bob at rest far away has

d τ_{B} / d t ≃ 1

. Thus the ratio of ageing rates is

k^{2} = \frac{1}{1 - 3 r _{s} / ( 2 r )} = \frac{2 r}{2 r - 3 r _{s}} .

For Alice's stable orbit

r > 3 r_{s}

, the denominator factor exceeds

1/2

, proving $k < 2$ . Both gravitational time dilation and Alice's orbital motion enter this proper-time ratio.

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