General Relativity: An Introduction for Physicists · Problem 11.6

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · photon-emission-near-horizon

Problem

A particle in the Schwarzschild geometry emits a radially outgoing photon at coordinates

(t_E, r_E)

, which is received by the distant fixed observer at

(t_R, r_R)

. Show that, if

r_E

lies just outside the horizon

r = 2\mu

, the radial coordinate 'seen' by the distant observer at the time

t_R

is given by

r_E(t_R) \approx 2\mu + 2\mu e^{-c(t_R - r_R)/4\mu}.