General Relativity: An Introduction for Physicists · Problem 11.16

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · kruskal-line-element

Problem

Show that, in terms of the Kruskal coordinates

u

and

v

defined in Exercise 11.15, the Schwarzschild line element takes the form

ds^2 = \frac{32\mu^3}{r}\exp\left(-\frac{r}{2\mu}\right)\left(dv^2 - du^2\right) - r^2\left(d\theta^2 + \sin^2\theta\,d\phi^2\right),

where

r

is considered as a function of

v

and

u

and is defined implicitly by

u^2 - v^2 = \left(\frac{r}{2\mu}-1\right)\exp\left(\frac{r}{2\mu}\right).

Show further that

v

is timelike and

u

is spacelike throughout the Schwarzschild geometry.