Spacetime and Geometry · Problem 6.3

Perturbation Theory and Gravitational Radiation — Sean Carroll

Physics — General Relativity · charged-particle-reissner-nordstrom-penrose

Problem

In the presence of an electromagnetic field, a particle of charge

e

and mass

m

obeys

\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\rho\sigma}\frac{dx^\rho}{d\tau}\frac{dx^\sigma}{d\tau} = \frac{e}{m}F^\mu{}_\nu\frac{dx^\nu}{d\tau}. \tag{6.122}

Imagine that such a particle is moving in the field of a Reissner--Nordstr\"om black hole with charge

Q

and mass

M

. (a) Show that the energy

E = m\left(1 - \frac{2GM}{r} + \frac{GQ^2}{r^2}\right)\frac{dt}{d\tau} + \frac{eQ}{r} \tag{6.123}

is conserved. (b) Will a Penrose-type process work for a charged black hole? What is the change in the black hole mass,

\delta M

, for the maximum physical process?