 Solution

ID: past-exam-of-the-relativity-course-of-the-university-of-cambridge/2017/1/a/solution

Use the Christoffel symbol formula with its first index raised:

Γ^{i}_{jk} = \frac{1}{2} g^{i ℓ} (\partial_{j} g_{ℓ k} + \partial_{k} g_{ℓ j} - \partial_{ℓ} g_{jk}) .

g_{rr} = e^{2 χ}

and

g_{θθ} = r^{2} e^{2 χ}

, with inverse components

g^{rr} = e^{- 2 χ}

and

g^{θθ} = r^{- 2} e^{- 2 χ}

. Since

\partial_{r} χ = χ^{'}

and all angular derivatives vanish,

Γ^{r}_{rr} = χ^{'}, Γ^{r}_{θθ} = - r - r^{2} χ^{'}, Γ^{θ}_{θ r} = Γ^{θ}_{r θ} = \frac{1}{r} + χ^{'} .

For example,

Γ^{r}_{θθ} = - \frac{1}{2} e^{- 2 χ} \partial_{r} (r^{2} e^{2 χ})

. The other components vanish. The formula supplied in the paper gives the first-kind coefficients

Γ_{ijk}

; multiplication by the inverse metric tensor is needed for the requested second-kind coefficients. These polar coordinates are understood away from

r = 0

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