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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 1

Write the <Einstein field equations> in $D$ dimensions as
$$
R_{\mu\nu}-\frac12g_{\mu\nu}R=\kappa T_{\mu\nu}.
$$
Taking the trace gives
$$
\left(1-\frac D2\right)R=\kappa T,
\qquad
R=-\frac{2\kappa}{D-2}T.
$$
Therefore the <Trace-reversed Einstein field equations in D dimensions> are
$$
R_{\mu\nu}
=\kappa\left(T_{\mu\nu}-\frac1{D-2}g_{\mu\nu}T\right).
$$

For the values from part (a), let $Y=\nabla_\rho\phi\nabla^\rho\phi$. Then
$$
T_{\mu\nu}
=\frac12\nabla_\mu\phi\nabla_\nu\phi
-\frac14g_{\mu\nu}(Y+m^2\phi^2),
$$
and
$$
T=\frac{2-D}{4}Y-\frac D4m^2\phi^2.
$$
Substitution cancels the $Yg_{\mu\nu}$ terms and yields
$$
\boxed{
R_{\mu\nu}
=\frac\kappa2\left[
\nabla_\mu\phi\nabla_\nu\phi
+\frac{m^2}{D-2}g_{\mu\nu}\phi^2
\right]
}.
$$
For the convention $\kappa=8\pi G$, the prefactor is $4\pi G$. This is the <Ricci curvature sourced by a Klein-Gordon scalar field>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 38B / b / Solution

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