Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 3 / b / ii

Table of contents
- Solution ii

Solution

Stationarity changes

□

to the spatial Laplace operator. The Green function of the Poisson equation therefore gives

\overset{ˉ}{h}_{00} (r) = 4 \int_{R^{3}} \frac{ρ ( r ^{'} )}{∣ r - r ^{'} ∣} d^{3} r^{'},

\overset{ˉ}{h}_{0 i} (r) = 4Ω \int_{R^{3}} \frac{ρ ( r ^{'} )}{∣ r - r ^{'} ∣} (y^{'}, - x^{'}, 0)_{i} d^{3} r^{'}, \overset{ˉ}{h}_{ij} = 0 .

Here the bar denotes the trace-reversed metric perturbation; this is the variable denoted by

h

in the displayed field equation of the question.

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