General Relativity: An Introduction for Physicists · Problem 14.4

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · 3-hyperboloid-metric-volume

Problem

In a four-dimensional Minkowski space with 'Cartesian' coordinates

(w, x, y, z)

, a 3-hyperboloid is defined by

w^2 - x^2 - y^2 + -z^2 = R^2

. Show that the metric on the surface of the 3-hyperboloid can be written in the form

d\sigma^2 = R^2\left[d\chi^2 + \sinh^2\chi\,(d\theta^2 + \sin^2\theta\,d\phi^2)\right].

Show that the total volume of the 3-hyperboloid is infinite.