General Relativity: An Introduction for Physicists · Problem 11.8

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · dust-collapse-proper-time

Problem

A spherical distribution of dust of coordinate radius

R

and total mass

M

collapses from rest under its own gravity. Show that, as the collapse progresses, the coordinate radius

r

of the star's surface and the elapsed proper time

\tau

of an observer sitting on the surface are related by

\tau(r) = -\frac{1}{(2GM)^{1/2}}\int_R^r \left(\frac{r}{1-r/R}\right)^{1/2}dr.

By making the substitution

r = R\cos^2(\psi/2)

, or otherwise, show that the solution can be expressed parametrically as

r = \frac{R}{2}(1+\cos\phi), \qquad \tau = \frac{R}{2}\left(\frac{R}{2GM}\right)^{1/2}(\psi+\sin\psi).

Calculate the proper time experienced by the observer before the star collapses to a point.