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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 1

For the <Expanding flat slicing of de Sitter spacetime>, take the <geodesic Lagrangian>
$$
L=\frac12g_{\mu\nu}\dot x^\mu\dot x^\nu
=\frac12\left[-\dot t^2+e^{2Ht}(\dot x^2+\dot y^2+\dot z^2)\right].
$$
The <Euler-Lagrange equation> for $t$ is
$$
\ddot t+He^{2Ht}(\dot x^2+\dot y^2+\dot z^2)=0,
$$
while each spatial coordinate $x^i$ obeys
$$
\ddot x^i+2H\dot t\dot x^i=0.
$$
Comparing these equations with the <geodesic equation> gives the complete list of nonzero <Christoffel symbols>:
$$
\boxed{\Gamma^t{}_{ij}=He^{2Ht}\delta_{ij},
\qquad
\Gamma^i{}_{tj}=\Gamma^i{}_{jt}=H\delta^i{}_j,
\qquad i,j\in\{x,y,z\}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 37E / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 37E / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 37E / a / Solution

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