General Relativity: An Introduction for Physicists · Problem 12.13

Further spherically symmetric geometries — Hobson, Efstathiou & Lasenby

Physics · electromagnetic-field-tensor-spherical-symmetry

Problem

Show that the electromagnetic field tensor outside a static spherically symmetric charged matter distribution has the form

[F_{\mu\nu}] = E(r)\begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},

where

E(r)

is some arbitrary function. Hence show that, if the line element outside the matter distribution has the form

ds^2 = A(r)\,dt^2 - B(r)\,dr^2 - r^2(d\theta^2 + \sin^2\theta\,d\phi^2),

the energy–momentum tensor of the electromagnetic field in this region is given by

[T_{\mu\nu}] = \frac{1}{2}c^2\epsilon_0 E^2\,\mathrm{diag}\!\left(\frac{1}{B},\,-\frac{1}{A},\,\frac{r^2 E^2}{AB},\,\frac{r^2 E^2\sin^2\theta}{AB}\right).