General Relativity: An Introduction for Physicists · Problem 11.15

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · kruskal-coordinates-derivation

Problem

Suppose that

p

and

q

are respectively the advanced and retarded Eddington–Finkelstein time parameters, defined in terms of Schwarzschild coordinates by

p = ct + r + 2\mu\ln\left|\frac{r}{2\mu}-1\right|, \qquad q = ct - r - 2\mu\ln\left|\frac{r}{2\mu}-1\right|.

A new set of (Kruskal) coordinates is defined by

v = \tfrac{1}{2}\left(e^{p/4\mu} - e^{-q/4\mu}\right) \qquad \text{and} \qquad u = \tfrac{1}{2}\left(e^{p/4\mu} + e^{-q/4\mu}\right).

Show that these new coordinates are related to Schwarzschild coordinates for

r > 2\mu

v = \left(\frac{r}{2\mu}-1\right)^{1/2}\exp\left(\frac{r}{4\mu}\right)\sinh\left(\frac{ct}{4\mu}\right),

u = \left(\frac{r}{2\mu}-1\right)^{1/2}\exp\left(\frac{r}{4\mu}\right)\cosh\left(\frac{ct}{4\mu}\right),

and for

r < 2\mu

v = \left(1-\frac{r}{2\mu}\right)^{1/2}\exp\left(\frac{r}{4\mu}\right)\cosh\left(\frac{ct}{4\mu}\right),

u = \left(1-\frac{r}{2\mu}\right)^{1/2}\exp\left(\frac{r}{4\mu}\right)\sinh\left(\frac{ct}{4\mu}\right).