General Relativity: An Introduction for Physicists · Problem 9.19

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · alice-bob-schwarzschild-twin-paradox

Problem

Alice is situated at a fixed position on the equator of the Earth (which is assumed to be spherical). In Schwarzschild coordinates (t,r,θ,ϕ)(t, r, \theta, \phi), her worldline is described in terms of a parameter τ\tau by t=γτ,r=R,θ=π/2,ϕ=ωτ,t = \gamma\tau, \quad r = R, \quad \theta = \pi/2, \quad \phi = \omega\tau, where γ\gamma and ω\omega are constants and RR is the coordinate radius of the Earth's surface. Bob is a distant stationary observer in space. Show that he will measure the orbital speed of Alice to be v=Rω/γv = R\omega/\gamma. By considering the magnitude of Alice's 4-velocity, show that γ=[1−(v2c2+GMc2r)]−1/2,\gamma = \left[1 - \left(\frac{v^2}{c^2} + \frac{GM}{c^2 r}\right)\right]^{-1/2}, where MM is the mass of the Earth. Interpret this result physically.

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