Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 37E / a

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- Solution a

Solution

For a slowly moving particle in the weak-field approximation,

d t / d τ = 1 + O (Φ, v^{2})

and terms containing two spatial velocities may be neglected. The spatial geodesic equation reduces to

\frac{d ^{2} x ^{i}}{d t ^{2}} = - Γ^{i}_{00} .

For

g_{00} = - (1 + 2Φ)

, the Levi-Civita connection is

Γ^{i}_{00} = \frac{1}{2} g^{ij} (- \partial_{j} g_{00}) = \partial_{i} Φ + O (Φ^{2}) .

Hence the Newtonian limit of general relativity is

\frac{d ^{2} x}{d t ^{2}} = - \nablaΦ .

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