General Relativity: An Introduction for Physicists · Problem 14.3

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

Physics · 3-sphere-metric-volume

Problem

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

(w, x, y, z)

, a 3-sphere of radius

R

is defined by

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

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

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

Show that the total volume of the 3-sphere is

V = 2\pi^2 R^3