General Relativity: An Introduction for Physicists · Problem 11.5

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · retarded-eddington-finkelstein-coordinates

Problem

Show that, on introduction of the retarded 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 find the equations for the worldlines of radially moving photons in retarded Eddington–Finkelstein coordinates. Use this result to sketch the spacetime diagram showing the light-cone structure in this coordinate system.

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