Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 35D / a

Table of contents
- Solution a

Solution

For

k = 1

, put

r = sin χ

. The spatial metric tensor becomes

a^{2} [d χ^{2} + sin^{2} χ (d θ^{2} + sin^{2} θ d ϕ^{2})] .

The global closed spatial slice is a 3-sphere with

0 \leq χ \leq π

; the original

r

chart covers only a hemisphere, so two charts are needed. A circle at fixed

χ

in an equatorial angular plane has circumference

2 πa sin χ

. The full spatial volume is

4 π a^{3} \int_{0}^{π} sin^{2} χ d χ = 2 π^{2} a^{3} .

Integrating a single

r \in [0, 1]

chart without its second copy would give only half this volume.

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