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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 4

The <Levi-Civita connection> is
$$
\Gamma^\alpha{}_{\beta\gamma}
=\frac12g^{\alpha\delta}
(g_{\delta\beta,\gamma}+g_{\delta\gamma,\beta}-g_{\beta\gamma,\delta}).
$$
For $g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$, retain only first-order terms in $h$. Substitution into the curvature formula and contraction gives the <linearized Ricci tensor and scalar>
$$
2R_{\alpha\beta}
=h_\alpha{}^\gamma{}_{,\gamma\beta}
+h_\beta{}^\gamma{}_{,\gamma\alpha}
-h_\gamma{}^\gamma{}_{,\alpha\beta}
-h_{\alpha\beta,\gamma\rho}\eta^{\gamma\rho}.
$$
Hence
$$
\boxed{A=1,\qquad B=1,\qquad C=-1,\qquad D=-1.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 37B / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 37B / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 37B / a / Solution

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