Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 309 / 3 / d

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- Solution d

Solution

In the orthonormal frame choose the null vector

k = e_{0} + e_{2}

, which has a nonzero angular component. The curvature forms give

R_{ab} k^{a} k^{b} = R_{00} + R_{22} = \frac{1}{2} f^{''} - \frac{f - 1}{r ^{2}} .

For a null vector, the trace term in the Einstein field equations drops out, so the null energy condition implies

R_{ab} k^{a} k^{b} \geq 0

. Therefore

f^{''} \geq \frac{2 ( f - 1 )}{r ^{2}} .

Solved by gpt-5.6-sol high.

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