General Relativity: An Introduction for Physicists · Problem 3.18

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · freely-falling-particles-geodesic-equations

Problem

By transforming from a local inertial coordinate system ξμ\xi^\mu in which ds2=c2dτ2=ημν dξμ dξν,ds^2 = c^2 d\tau^2 = \eta_{\mu\nu}\, d\xi^\mu\, d\xi^\nu, to a general coordinate system xμx^\mu, show that freely falling particles obey the geodesic equations of motion d2xλdτ2+Γλμνdxμdτdxνdτ=0,\frac{d^2 x^\lambda}{d\tau^2} + \Gamma^\lambda{}_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0, where Γλμν=∂xλ∂ξα∂2ξα∂xμ∂xν.\Gamma^\lambda{}_{\mu\nu} = \frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial x^\mu \partial x^\nu}.

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