General Relativity: An Introduction for Physicists · Problem 13.9

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-horizon-area-penrose-process

Problem

Use your answer to Exercise 13.7 to show that the area of the event horizon

r = r_+

in the Kerr geometry may be written as

A = \frac{8\pi G}{c^4}\left[M^2 + \sqrt{M^4 - \left(\frac{cJ}{G}\right)^2}\right],

where

M

and

J

are the mass and angular momentum of the black hole. Hence show that if the mass and angular momentum change by

\delta M

and

\delta J

respectively, then the corresponding change in the proper area of the horizon is given by

\delta A = \frac{8\pi G}{c} \frac{a}{\Omega_{\rm H}\sqrt{\mu^2 - a^2}}\left(\delta M - \Omega_{\rm H}\frac{\delta J}{c^2}\right),

where

\Omega_{\rm H}

is the 'angular velocity of the horizon', defined in (13.29). Thus show that the area of the event horizon must increase in the Penrose process.