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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}

For the <Levi-Civita connection>, the all-lowered <Riemann curvature tensor> has the algebraic symmetries
$$
\boxed{R_{\lambda\mu\nu\kappa}=-R_{\mu\lambda\nu\kappa}
=-R_{\lambda\mu\kappa\nu},\qquad
R_{\lambda\mu\nu\kappa}=R_{\nu\kappa\lambda\mu}.}
$$
The <first Bianchi identity> is the cyclic identity
$$
\boxed{R_{\lambda\mu\nu\kappa}+R_{\lambda\nu\kappa\mu}+R_{\lambda\kappa\mu\nu}=0.}
$$
These statements concern the <Riemann curvature tensor> at each point and are unchanged by reversing its overall <curvature sign convention>.

Solution

Past exam of the general relativity course of the University of Oxford / 2022 / 1 / a / Solution

Past exam of the general relativity course of the University of Oxford / 2022 / 1 / a / Solution

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