Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 3

The metric directly giveswith no mixed products. Henceis an orthonormal frame.

The dual orthonormal coframe isbecause $e^{\mu }(e_{\nu })={\delta }^{\mu }{}_{\nu }$.

One hasCartan's torsion-free first structure equation and metric compatibility give the only nonzero connection 1-forms:

Cartan's second structure equation givesand, for $i\ne j$,With all indices lowered and the curvature convention in the question, the independent nonzero orthonormal Riemann curvature tensor components aretogether with those obtained from the Riemann symmetries.

PutContracting the curvature components givesin the orthonormal frame, with no sum on $i$. Vacuum requires $S_1=S_2$ and $p_i(S_1-1)=0$ for every $i$. Since the $p_i$ are not all zero, some $p_i\ne 0$, forcing $S_1=1$ and then $S_2=1$. Conversely these conditions make every Ricci component vanish. Thus the Kasner metric is vacuum exactly whenso $A=B=1$.