General Relativity: An Introduction for Physicists · Problem 12.16

Further spherically symmetric geometries — Hobson, Efstathiou & Lasenby

Physics · reissner-nordstrom-advanced-eddington-finkelstein

Problem

Show that, by introducing the advanced Eddington–Finkelstein timelike coordinate

ct' = ct - \frac{r_-^2}{r_+ - r_-}\ln\left|\frac{r}{r_-} - 1\right| + \frac{r_+^2}{r_+ - r_-}\ln\left|\frac{r}{r_+} - 1\right|,

the Reissner–Nordström line element takes the form

ds^2 = c^2\Delta\,dt'^2 - 2(1 - \Delta)\,dt'\,dr - (2 - \Delta)\,dr^2 - r^2(d\theta^2 + \sin^2\theta\,d\phi^2),

where

\Delta \equiv \Delta(r) = 1 - 2\mu/r + q^2/r^2

. Hence show that the worldlines of radially moving photons in advanced Eddington–Finkelstein coordinates are given by

ct' + r = \text{constant} \quad (\text{incoming}), \qquad c\frac{dt'}{dr} = \frac{2 - \Delta}{\Delta} \quad (\text{outgoing}).

What is the significance, if any, of the fact that

c\,dt'/dr = 0

\Delta(r) = 2

for outgoing radially moving photons?