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

An orthonormal frame in spacetime is a local basis $\{e_a\}$ satisfying $g(e_a,e_b)={\eta }_{ab}$. The metric determines it only up to a point-dependent Lorentz transformation, with orientation and time orientation optionally restricting the component of the Lorentz group. The orthonormal coframe in spacetime is defined byand the metric is

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The connection 1-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. Metric compatibility gives ${\omega }_{ab}=-{\omega }_{ba}$, while vanishing torsion gives Cartan's first structure equationThe curvature 2-forms arewhich displays their relation to the Riemann curvature tensor.

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ChooseCartan's first structure equation gives the independent nonzero connection 1-formswith the remaining forms obtained by metric antisymmetry. Cartan's second structure equation then givesThese signs follow the curvature convention stated in Question 1.

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In the orthonormal frame choose the null vector $k=e_0+e_2$, which has a nonzero angular component. The curvature forms giveFor a null vector, the trace term in the Einstein field equations drops out, so the null energy condition implies $R_{ab}{k}^a{k}^b\ge 0$. Therefore

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