Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 37E / a

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- Solution a

Solution

In the static limit, the supplied linearized Riemann tensor gives

R_{00} = δ^{ij} R_{i 0 j 0} = - \frac{1}{2} \nabla^{2} h_{00} .

The trace of the Einstein field equations is

R = - κ T

. Here

T ≃ - ρ

, so

R = κ ρ

. The

00

field equation is therefore

R_{00} - \frac{1}{2} η_{00} R = R_{00} + \frac{1}{2} R = κ ρ,

which yields

R_{00} = κ ρ /2

. Combining the two expressions for

R_{00}

gives

- \nabla^{2} h_{00} = κ ρ .

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