Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 2 / a / Solution

The contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Combining the proposed equation with stress-energy conservation therefore imposes ${\nabla }_{b}R=0$. Taking its trace gives $R=\kappa T$, with $T=g^{ab}{T}_{ab}$, so for nonzero $\kappa$ the matter trace must also be constant. Ordinary matter need not satisfy this extra condition. For example, if asymptotic flatness forces the constant scalar curvature to vanish, the proposal would require $T=0$ everywhere.

The problem is an unjustified restriction on the matter trace, rather than the absence of every possible solution. The divergence-free Einstein tensor avoids it:The Einstein field equations $G_{ab}+\Lambda g_{ab}=\kappa T_{ab}$, with constant cosmological constant $\Lambda$, are consequently compatible with stress-energy conservation.