Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 38B / a / Solution

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 ${\nabla }^{\nu }{T}_{\mu \nu }=0$.