General Relativity: An Introduction for Physicists · Problem 11.4

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · advanced-eddington-finkelstein-coordinates

Problem

Show that, on introduction of the advanced Eddington–Finkelstein timelike coordinate t′=ct+2μln⁡∣r/(2μ)−1∣t' = ct + 2\mu\ln|r/(2\mu)-1|, the Schwarzschild line element takes the form ds2=c2(1−2μr)dt′2−4μcr dt′ dr−(1+2μr)dr2−r2(dθ2+sin⁡2θ dϕ2).ds^2 = c^2\left(1-\frac{2\mu}{r}\right)dt'^2 - \frac{4\mu c}{r}\,dt'\,dr - \left(1+\frac{2\mu}{r}\right)dr^2 - r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right). Hence show that the worldlines of radially moving photons in advanced Eddington–Finkelstein coordinates are given by ct′=r+4μln⁡∣r2μ−1∣+constant(outgoing photon),ct' = r + 4\mu\ln\left|\frac{r}{2\mu}-1\right| + \text{constant} \qquad \text{(outgoing photon)}, ct′=−r+constant(incoming photon).ct' = -r + \text{constant} \qquad \text{(incoming photon)}.

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