Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 311 / 2 / iv

Table of contents
- Solution iv

Solution

Contract the Einstein field equations

G_{ab} = 8 π T_{ab}

twice with

n^{a}

. Since

n^{a} n_{a} = - 1

and

ρ = T_{ab} n^{a} n^{b}

,

R + 2 R_{ab} n^{a} n^{b} = 16 π ρ .

If

K_{ab}

is proportional to

h_{ab}

in three dimensions, tracing gives

K_{ab} = (K /3) h_{ab}

and

K_{ab} K^{ab} = K^{2} /3

. The Scalar Gauss equation gives the Hamiltonian constraint

^{(3)} R = 16 π ρ - \frac{2}{3} K^{2} = \frac{2}{3} (24 π ρ - K^{2}) .

Therefore

24 π ρ - K^{2} \geq 0 ⟹^{(3)} R \geq 0 .

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