Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 309 / 1 / b

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- Solution b

Solution

Affinely parametrized geodesics are the critical curves of the geodesic Lagrangian

L = \frac{1}{2} [- w \dot{t}^{2} + \frac{r ˙ ^{2}}{w} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \dot{ϕ}^{2}] .

For

w = 1 + r^{2}

and

w^{'} = 2 r

, the nonzero Christoffel symbols, up to symmetry in the lower indices, are

Γ^{t}_{t r} = \frac{w ^{'}}{2 w} = \frac{r}{w}, Γ^{r}_{tt} = \frac{w w ^{'}}{2} = r w, Γ^{r}_{rr} = - \frac{w ^{'}}{2 w} = - \frac{r}{w},

Γ^{r}_{θθ} = - r w, Γ^{r}_{ϕϕ} = - r w sin^{2} θ, Γ^{θ}_{r θ} = Γ^{ϕ}_{r ϕ} = \frac{1}{r},

Γ^{θ}_{ϕϕ} = - sin θ cos θ, Γ^{ϕ}_{θϕ} = cot θ .

They follow either from the Euler-Lagrange equations or directly from the Levi-Civita connection formula.

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