General Relativity: An Introduction for Physicists · Problem 8.3

The gravitational field equations — Hobson, Efstathiou & Lasenby

Physics · electromagnetic-energy-momentum-tensor

Problem

The electromagnetic field in vacuo has an energy–momentum tensor

T^{\mu\nu}_{\rm em}

. By analogy with the energy–momentum tensor for dust, we require that (i)

T^{\mu\nu}_{\rm em}

is symmetric; (ii)

\nabla_\mu T^{\mu\nu}_{\rm em} = 0

; (iii)

T^{\mu\nu}_{\rm em}

must be quadratic in the dynamical variable

F^{\mu\nu}

. Hence show that

T^{\mu\nu}_{\rm em} = \alpha(F^\mu{}_\sigma F^{\nu\sigma} - \tfrac{1}{4}g^{\mu\nu}F_{\sigma\rho}F^{\sigma\rho}),

where

\alpha

is a constant. By examining the component

T^{00}_{\rm em}

in local Cartesian inertial coordinates, show that the constant

\alpha = -1/\mu_0