General Relativity: An Introduction for Physicists · Problem 13.10

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-equatorial-contravariant-metric

Problem

Show that, in the equatorial plane

\theta = \pi/2

of the Kerr geometry, the contravariant metric components in Boyer–Lindquist coordinates are

g^{tt} = \frac{1}{c^2 \Delta}\left(r^2 + a^2 + \frac{2\mu a^2}{r}\right), \qquad g^{t\phi} = \frac{2\mu a}{cr\Delta},

g^{rr} = -\frac{\Delta}{r^2}, \qquad g^{\phi\phi} = -\frac{1}{\Delta}\left(1 - \frac{2\mu}{r}\right).