General Relativity: An Introduction for Physicists · Problem 9.25

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · alice-bob-circular-orbit-twin-paradox

Problem

Alice and Bob are astronauts in a space capsule, with no engine, in a circular orbit at r=Rr = R (where R>3μR > 3\mu) in the equatorial plane of the Schwarzschild geometry. At some point in the orbit, Bob leaves the capsule, uses his rocket-pack to maintain a hovering position at that fixed point in space and then rejoins the capsule after it has completed one orbit. Show that the proper time interval measured by Alice while Bob is out of the capsule is ΔτA=2π[R2μc2(R−3μ)]1/2.\Delta\tau_A = 2\pi\left[\frac{R^2}{\mu c^2}(R-3\mu)\right]^{1/2}. If ΔτB\Delta\tau_B is the corresponding proper time interval measured by Bob between these two events, show that ΔτBΔτA=(R−2μR−3μ)1/2.\frac{\Delta\tau_B}{\Delta\tau_A} = \left(\frac{R-2\mu}{R-3\mu}\right)^{1/2}. Briefly compare this result with the 'twin paradox' result in special relativity. If Bob chooses not to rejoin the capsule but instead observes it fly past him, show that he will measure the capsule's speed as v=(μc2R−2μ)1/2.v = \left(\frac{\mu c^2}{R-2\mu}\right)^{1/2}.

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