General Relativity: An Introduction for Physicists · Problem 7.22

The equivalence principle and spacetime curvature — Hobson, Efstathiou & Lasenby

Physics · geodesic-deviation-sphere

Problem

On the surface of a sphere, show that, along the geodesic ϕ=constant\phi = \text{constant}, the geodesic deviation vector ξi\xi^i satisfies D2ξθDs2=0,D2ξϕDs2=−ξϕ(dθds)2.\frac{D^2\xi^\theta}{Ds^2} = 0, \qquad \frac{D^2\xi^\phi}{Ds^2} = -\xi^\phi\left(\frac{d\theta}{ds}\right)^2. Choose a geodesic ϕ=ϕ0\phi = \phi_0 with path length s=θs = \theta measured from θ=0\theta = 0, and a neighbouring geodesic ϕ=ϕ0+δϕ0\phi = \phi_0 + \delta\phi_0, also with s=θs = \theta, and define ξi(θ)\xi^i(\theta) as the vector between s=θs = \theta on one and s=θs = \theta on the other. Show that ξi(θ)=(0,ξϕ)\xi^i(\theta) = (0, \xi^\phi) for all θ\theta. Show in addition that if ξϕ=0\xi^\phi = 0 when θ=0\theta = 0 then ξϕξϕ=l2sin⁡2θ,\xi^\phi \xi_\phi = l^2\sin^2\theta, where l2l^2 is a constant, and that the two geodesics pass through θ=π\theta = \pi.

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