Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 2 / a / ii

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- Solution ii

Solution

The nonrelativistic fluid assumptions give

T_{00} = ρ + O (ϵ^{2} ρ), T_{0 i} = O (ϵ ρ), T_{ij} = O (ϵ^{2} ρ) .

Assume again a stationary field and asymptotic flatness. The harmonic-gauge Linearized Einstein equations become

\nabla^{2} \overset{ˉ}{h}_{00} = - 16 π ρ,

while the other trace-reversed components are smaller. Trace reversal then gives

h_{00} = \overset{ˉ}{h}_{00} /2

to leading order, so

\nabla^{2} h_{00} = - 8 π ρ .

Using

h_{00} = - 2Φ

yields

\nabla^{2} Φ = 4 π ρ

in units

G = 1

, which is the Poisson equation of Newtonian gravity.

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