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General relativity course of the University of Oxford

Past exam of the general relativity course of the University of Oxford

2022

{c}

Each lower index contributes a minus connection term to the <covariant derivative> of the <metric tensor>:
$$
\nabla_\lambda g_{\mu\nu}
=\partial_\lambda g_{\mu\nu}
-\Gamma^\rho_{\mu\lambda}g_{\rho\nu}
-\Gamma^\rho_{\nu\lambda}g_{\mu\rho}.
$$
Substitute the <Christoffel symbols> of the <Levi-Civita connection> and use the <inverse metric> identity $g_{\rho\nu}g^{\rho\sigma}=\delta^\sigma_\nu$:
$$
\begin{aligned}
g_{\rho\nu}\Gamma^\rho_{\mu\lambda}
&=\tfrac12(\partial_\mu g_{\nu\lambda}
+\partial_\lambda g_{\nu\mu}-\partial_\nu g_{\mu\lambda}),\\
g_{\mu\rho}\Gamma^\rho_{\nu\lambda}
&=\tfrac12(\partial_\nu g_{\mu\lambda}
+\partial_\lambda g_{\mu\nu}-\partial_\mu g_{\nu\lambda}).
\end{aligned}
$$
Adding cancels the first and third terms and leaves exactly $\partial_\lambda g_{\mu\nu}$. Hence
$$
\boxed{\nabla_\lambda g_{\mu\nu}=0.}
$$
This is <metric compatibility>: the <Levi-Civita connection> preserves the <inner product> under <parallel transport>.

Solution

Past exam of the general relativity course of the University of Oxford / 2022 / 2 / c / Solution

Past exam of the general relativity course of the University of Oxford / 2022 / 2 / c / Solution

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Past exam of the general relativity course of the University of Oxford / 2022 / 2 / c / Solution

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