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 Incoming links: Expanding flat slicing of de Sitter spacetime

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 37E / a / Solution Created 2026-09-24 Updated 2026-10-03

For the Expanding flat slicing of de Sitter spacetime, take the geodesic Lagrangian

L = \frac{1}{2} g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = \frac{1}{2} [- \dot{t}^{2} + e^{2 H t} (\overset{x}{˙}^{2} + \overset{y}{˙}^{2} + \overset{z}{˙}^{2})] .

The Euler-Lagrange equation for

t

is

\ddot{t} + H e^{2 H t} (\overset{x}{˙}^{2} + \overset{y}{˙}^{2} + \overset{z}{˙}^{2}) = 0,

while each spatial coordinate

x^{i}

obeys

\overset{x}{¨}^{i} + 2 H \dot{t} \overset{x}{˙}^{i} = 0.

Comparing these equations with the geodesic equation gives the complete list of nonzero Christoffel symbols:

Γ^{t}_{ij} = H e^{2 H t} δ_{ij}, Γ^{i}_{t j} = Γ^{i}_{j t} = H δ^{i}_{j}, i, j \in {x, y, z} .

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