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

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 $A=B=1$.