  We investigate the local regularity of pointed spacetimes, that is,
time-oriented Lorentzian manifolds in which a point and a future-oriented, unit
timelike vector (an observer) are selected. Our main result covers the class of
Einstein vacuum spacetimes. Under curvature and injectivity bounds only, we
establish the existence of a local coordinate chart defined in a ball with
definite size in which the metric coefficients have optimal regularity. The
proof is based on quantitative estimates, on one hand, for a constant mean
curvature (CMC) foliation by spacelike hypersurfaces defined locally near the
observer and, on the other hand, for the metric in local coordinates that are
spatially harmonic in each CMC slice. The results and techniques in this paper
should be useful in the context of general relativity for investigating the
long-time behavior of solutions to the Einstein equations.
