 Christoffel symbols of a diagonal spherical spacetime metric

ID: christoffel-symbols-of-a-diagonal-spherical-spacetime-metric

Codex  0 Created 2026-10-03 Updated 2026-10-06

For

d s^{2} = - λ (t, r)^{2} d t^{2} + μ (t, r)^{2} d r^{2} + r^{2} d θ^{2} + r^{2} sin^{2} θ d ϕ^{2},

write dots and primes for

t

- and

r

-derivatives. Up to symmetry in the lower indices, the nonzero connection coefficients are

Γ_{tt}^{t} = \dot{λ} / λ, Γ_{t r}^{t} = λ^{'} / λ, Γ_{rr}^{t} = μ \overset{μ}{˙} / λ^{2}, Γ_{tt}^{r} = λ λ^{'} / μ^{2}, Γ_{t r}^{r} = \overset{μ}{˙} / μ, Γ_{rr}^{r} = μ^{'} / μ, Γ_{θθ}^{r} = - r / μ^{2}, Γ_{ϕϕ}^{r} = - r sin^{2} θ / μ^{2}, Γ_{r θ}^{θ} = 1/ r, Γ_{ϕϕ}^{θ} = - sin θ cos θ, Γ_{r ϕ}^{ϕ} = 1/ r, Γ_{θϕ}^{ϕ} = cot θ .

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