Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 38B (General Relativity)

Contract the Riemann tensor to obtainand define the Einstein tensorContracting the differential Bianchi identity twice, using the curvature symmetries and metric compatibility, gives the contracted Bianchi identityIt follows immediately thatThe Einstein equations with cosmological constant areSince $\nabla g=0$, the left side is divergence-free, so consistency requires and implies the local conservation law ${\nabla }^{\nu }{T}_{\mu \nu }=0$.

Write $H=\dot{a}/a$. The nonzero Christoffel symbols for the flat FLRW metric areThe mixed Einstein tensor isFor a mixed tensor,For $\mu =t$, direct substitution givesFor each spatial $\mu$, homogeneity makes the partial derivatives vanish and the remaining diagonal connection terms cancel pairwise. This verifies the flat FLRW Einstein-tensor divergence directly.

In vacuum,The $tt$ equation isChoosing the positive root gives the expanding de Sitter scale factor in flat slicingand the spatial equation is then automatically satisfied.