General Relativity: An Introduction for Physicists · Problem 9.14

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · energy-release-radial-infall

Problem

Suppose that the particle in Exercise 9.13 has rest mass

m_0

and that it stopped at

r = r_1

. If its excess energy was converted to radiation that is observed at infinity, show that the energy released as seen by a stationary observer at

r_1

E = m_0 c^2 \left(\frac{\gamma_0}{\sqrt{1-2GM/(c^2 r_1)}} - 1\right).

What is the energy released as observed at infinity? Show that this tends to

\gamma_0 m_0 c^2

r_1

tends to

2GM/c^2