General Relativity: An Introduction for Physicists · Problem 11.10

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · orthonormal-frame-tidal-forces

Problem

A massive particle is released from rest at coordinate radius rr in the Schwarzschild geometry. Show that a frame of orthonormal basis vectors defining the inertial instantaneous rest frame of the particle may be taken as (e^0)μ=1cuμ=1c(1−2μr)−1/2δ0μ,(e^1)μ=(1−2μr)1/2δ1μ,\left(\hat{e}_0\right)^\mu = \frac{1}{c}u^\mu = \frac{1}{c}\left(1-\frac{2\mu}{r}\right)^{-1/2}\delta_0^\mu, \qquad \left(\hat{e}_1\right)^\mu = \left(1-\frac{2\mu}{r}\right)^{1/2}\delta_1^\mu, (e^2)μ=1rδ2μ,(e^3)μ=1rsin⁡θδ3μ.\left(\hat{e}_2\right)^\mu = \frac{1}{r}\delta_2^\mu, \qquad \left(\hat{e}_3\right)^\mu = \frac{1}{r\sin\theta}\delta_3^\mu. Hence show that the spatial components of the orthogonal connecting vector between two such nearby particles satisfy d2ζr^dτ2=+2μc2r3ζr^,d2ζθ^dτ2=−μc2r3ζθ^,d2ζϕ^dτ2=−μc2r3ζϕ^.\frac{d^2\zeta^{\hat{r}}}{d\tau^2} = +\frac{2\mu c^2}{r^3}\zeta^{\hat{r}}, \qquad \frac{d^2\zeta^{\hat{\theta}}}{d\tau^2} = -\frac{\mu c^2}{r^3}\zeta^{\hat{\theta}}, \qquad \frac{d^2\zeta^{\hat{\phi}}}{d\tau^2} = -\frac{\mu c^2}{r^3}\zeta^{\hat{\phi}}.

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