General Relativity: An Introduction for Physicists · Problem 9.3

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · schwarzschild-geodesic-connection-coefficients

Problem

The Schwarzschild line element is

ds^2 = c^2\left(1-\frac{2\mu}{r}\right)dt^2 - \left(1-\frac{2\mu}{r}\right)^{-1}dr^2 - r^2\,d\theta^2 - r^2\sin^2\theta\,d\phi^2.

By considering the 'Lagrangian'

L = g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu

, where the dots denote differentiation with respect to an affine parameter

\lambda

, calculate the connection coefficients

\Gamma^\mu_{\nu\sigma}

. Hence verify that the geodesic equations are given by (9.17--9.20).