Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 37D / ii

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- Solution ii

Solution

In the Newtonian limit of general relativity, time derivatives are negligible and

T_{00} = ρ

is the only leading component. With signature

(-, +, +, +)

,

T = η^{μν} T_{μν} = - ρ .

The

00

component of the trace-reversed equation becomes

\nabla^{2} h_{00} = - 16 π (ρ - \frac{1}{2} η_{00} (- ρ)) = - 8 π ρ .

Identifying

h_{00} = - 2Φ

therefore gives the Poisson equation of Newtonian gravity:

\nabla^{2} Φ = 4 π ρ .

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