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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 4

Use the <geodesic Lagrangian>
$$
\mathcal L=\frac12\left(
-\lambda^2\dot t^{\,2}
+\mu^2\dot r^{\,2}
+r^2\dot\theta^{\,2}
+r^2\sin^2\theta\,\dot\phi^{\,2}
\right),
$$
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>
$$
\ddot x^\alpha+\Gamma^\alpha_{\beta\gamma}
\dot x^\beta\dot x^\gamma=0
$$
determines the <Christoffel symbols>. To display derivatives of the metric coefficients unambiguously, write $\lambda_t=\partial_t\lambda$, $\lambda_r=\partial_r\lambda$, and similarly for $\mu$. The nonzero symbols, together with those obtained by symmetry in the two lower indices, are
$$
\boxed{
\begin{aligned}
&\Gamma^t_{tt}=\frac{\lambda_t}{\lambda},
&&\Gamma^t_{tr}=\Gamma^t_{rt}=\frac{\lambda_r}{\lambda},
&&\Gamma^t_{rr}=\frac{\mu\mu_t}{\lambda^2},\\
&\Gamma^r_{tt}=\frac{\lambda\lambda_r}{\mu^2},
&&\Gamma^r_{tr}=\Gamma^r_{rt}=\frac{\mu_t}{\mu},
&&\Gamma^r_{rr}=\frac{\mu_r}{\mu},\\
&\Gamma^r_{\theta\theta}=-\frac r{\mu^2},
&&\Gamma^r_{\phi\phi}=-\frac{r\sin^2\theta}{\mu^2},\\
&\Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=\frac1r,
&&\Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta,\\
&\Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r}=\frac1r,
&&\Gamma^\phi_{\theta\phi}=\Gamma^\phi_{\phi\theta}=\cot\theta.
\end{aligned}}
$$
This is the <Christoffel symbols of a diagonal spherical spacetime metric>.

Solution

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

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

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

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