  In this paper, we consider very rough solutions to Cauchy problem for the
Einstein vacuum equations in CMC spacial harmonic gauge, and obtain the local
well-posedness result in $H^s, s>2$. The novelty of our approach lies in that,
without resorting to the standard paradifferential regularization over the
rough, Einstein metric $\bg$, we manage to implement the commuting vector field
approach to prove Strichartz estimate for geometric wave equation $\Box_\bg
\phi=0$ directly.
