 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-course-of-the-university-of-oxford/2013/1/a/solution

(-, +, +, +)

x^{0} = c t

, and retain terms linear in

Φ/ c^{2}

Ψ/ c^{2}

g^{00} = - (1 - 2Φ/ c^{2})

and

g^{ij} = (1 + 2Ψ/ c^{2}) δ^{ij}

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

the differentiated metric perturbation is already first order, so the prefactor can be replaced by the flat inverse metric. For example

Γ^{0}_{ij} = \frac{1}{2} \partial_{0} g_{ij} = - δ_{ij} \partial_{0} Ψ/ c^{2}

and

Γ^{k}_{00} = - \frac{1}{2} \partial_{k} g_{00} = \partial_{k} Φ/ c^{2}

. Evaluating the remaining components gives

Γ^{0}_{00} Γ^{k}_{00} = \frac{\partial _{0} Φ}{c ^{2}}, = \frac{\partial _{k} Φ}{c ^{2}}, Γ^{0}_{0 i} Γ^{k}_{0 i} = \frac{\partial _{i} Φ}{c ^{2}}, = - \frac{\partial _{0} Ψ}{c ^{2}} δ_{i}^{k}, Γ^{0}_{ij} Γ^{k}_{ij} = - \frac{\partial _{0} Ψ}{c ^{2}} δ_{ij}, = - \frac{\partial _{i} Ψ δ _{j}^{k} + \partial _{j} Ψ δ _{i}^{k} - \partial ^{k} Ψ δ _{ij}}{c ^{2}} .

The lower-index symmetry follows from the torsion-free connection. Products of these first-order Christoffel symbols are beyond the retained weak-field approximation.

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