Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / a

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- Solution a

Solution

For coordinates

x^{α}

, the Christoffel symbols of the Levi-Civita connection are

Γ_{β γ}^{α} = \frac{1}{2} g^{α δ} (\partial_{β} g_{δ γ} + \partial_{γ} g_{δ β} - \partial_{δ} g_{β γ}),

where

g^{α δ}

is the inverse metric.

For the two-dimensional hyperbolic metric in polar coordinates,

g_{rr} = a^{2}, g_{ϕϕ} = a^{2} sinh^{2} r, g^{rr} = a^{- 2}, g^{ϕϕ} = \frac{1}{a ^{2} s i n h ^{2} r} .

Only

g_{ϕϕ}

has a nonzero derivative, namely

\partial_{r} g_{ϕϕ} = 2 a^{2} sinh r cosh r .

It follows that the nonzero connection coefficients are

Γ_{ϕϕ}^{r} = - sinh r cosh r, Γ_{r ϕ}^{ϕ} = Γ_{ϕ r}^{ϕ} = coth r .

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