General Relativity: An Introduction for Physicists · Problem 8.5

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · charged-particle-geodesic-electromagnetic-field

Problem

The energy–momentum tensor of an electromagnetic field interacting with a source satisfies

\nabla_\mu T^{\mu\nu}_{\rm em} = -F^{\nu\sigma}J_\sigma

, where

J_\sigma

is the 4-current density of the source. Hence show that the worldline of a particle of charge

q

in an electromagnetic field satisfies

\ddot{z}^\nu + \Gamma^\nu_{\sigma\mu}\dot{z}^\mu \dot{z}^\sigma = \frac{q}{m}F^\nu{}_{\sigma}\dot{z}^\sigma,

and interpret this result physically.