L = \frac{1}{2} (- λ^{2} \dot{t}^{2} + μ^{2} \overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \dot{ϕ}^{2}),

where here the overdot on a coordinate means differentiation with respect to the affine parameter. Applying the Euler-Lagrange equation to each coordinate and comparing with the geodesic equation

\overset{x}{¨}^{α} + Γ_{β γ}^{α} \overset{x}{˙}^{β} \overset{x}{˙}^{γ} = 0

determines the Christoffel symbols. To display derivatives of the metric coefficients unambiguously, write

λ_{t} = \partial_{t} λ

λ_{r} = \partial_{r} λ

, and similarly for

μ

. The nonzero symbols, together with those obtained by symmetry in the two lower indices, are

Γ_{tt}^{t} = \frac{λ _{t}}{λ}, Γ_{tt}^{r} = \frac{λ λ _{r}}{μ ^{2}}, Γ_{θθ}^{r} = - \frac{r}{μ ^{2}}, Γ_{r θ}^{θ} = Γ_{θ r}^{θ} = \frac{1}{r}, Γ_{r ϕ}^{ϕ} = Γ_{ϕ r}^{ϕ} = \frac{1}{r}, Γ_{t r}^{t} = Γ_{r t}^{t} = \frac{λ _{r}}{λ}, Γ_{t r}^{r} = Γ_{r t}^{r} = \frac{μ _{t}}{μ}, Γ_{ϕϕ}^{r} = - \frac{r sin ^{2} θ}{μ ^{2}}, Γ_{ϕϕ}^{θ} = - sin θ cos θ, Γ_{θϕ}^{ϕ} = Γ_{ϕθ}^{ϕ} = cot θ . Γ_{rr}^{t} = \frac{μ μ _{t}}{λ ^{2}}, Γ_{rr}^{r} = \frac{μ _{r}}{μ},

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