Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 36D / a / Solution

{c}

For
$$
ds^2=-\lambda(t,r)^2dt^2+\mu(t,r)^2dr^2
+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2,
$$
write dots and primes for $t$- and $r$-derivatives. Up to symmetry in the lower indices, the nonzero connection coefficients are
$$
\begin{gathered}
\Gamma^t_{tt}=\dot\lambda/\lambda,
\quad \Gamma^t_{tr}=\lambda'/\lambda,
\quad \Gamma^t_{rr}=\mu\dot\mu/\lambda^2,\\
\Gamma^r_{tt}=\lambda\lambda'/\mu^2,
\quad \Gamma^r_{tr}=\dot\mu/\mu,
\quad \Gamma^r_{rr}=\mu'/\mu,\\
\Gamma^r_{\theta\theta}=-r/\mu^2,
\quad \Gamma^r_{\phi\phi}=-r\sin^2\theta/\mu^2,\\
\Gamma^\theta_{r\theta}=1/r,
\quad \Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta,
\quad \Gamma^\phi_{r\phi}=1/r,
\quad \Gamma^\phi_{\theta\phi}=\cot\theta.
\end{gathered}
$$


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 Incoming links: Christoffel symbols of a diagonal spherical spacetime metric