Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 3 / e

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- Solution e

Solution

Put

S_{1} = \sum_{i = 1}^{3} p_{i}, S_{2} = \sum_{i = 1}^{3} p_{i}^{2} .

Contracting the curvature components gives

R_{00} = \frac{S _{1} - S _{2}}{t ^{2}}, R_{ii} = \frac{p _{i} ( S _{1} - 1 )}{t ^{2}}

in the orthonormal frame, with no sum on

i

. Vacuum requires

S_{1} = S_{2}

and

p_{i} (S_{1} - 1) = 0

for every

i

. Since the

p_{i}

are not all zero, some

p_{i} \neq = 0

S_{1} = 1

and then

S_{2} = 1

. Conversely these conditions make every Ricci component vanish. Thus the Kasner metric is vacuum exactly when

i \sum p_{i} = 1, i \sum p_{i}^{2} = 1,

so

A = B = 1

.

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