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

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- Solution c

Solution

The

x

Euler-Lagrange equation has the first integral

e^{2 H t} \overset{x}{˙} = v_{0},

where the constant follows from the initial data. Combining this with proper-time normalization gives the Timelike geodesic in the expanding flat slicing of de Sitter spacetime:

- \dot{t}^{2} + e^{2 H t} \overset{x}{˙}^{2} = - 1 ⟹ \dot{t} = 1 + v_{0}^{2} e^{- 2 H t} .

Consequently

\frac{d τ}{d t} = \frac{1}{1 + v _{0}^{2} e ^{- 2 H t}} .

Integrating from

t = 0

, where

τ = 0

, gives

τ (t) = \frac{1}{2 H} lo g [\frac{1 + v _{0}^{2} e ^{- 2 H t} + 1}{1 + v _{0}^{2} e ^{- 2 H t} - 1} \frac{1 + v _{0}^{2} - 1}{1 + v _{0}^{2} + 1}] .

t \to - \infty

, the first ratio tends to

1

, so

τ ⟶ τ_{-} = - \frac{1}{2 H} lo g \frac{1 + v _{0}^{2} + 1}{1 + v _{0}^{2} - 1} > - \infty.

Thus

t

tends to

- \infty

at the finite proper time

τ_{-}

.

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