I gave recently some lectures (Berlin, Rutgers, College Park, Miami) about the construction of local canonical foliations of observers in Einstein spacetimes of general relativity when the curvature is solely assumed to be bounded and no assumption on its derivatives is made. In this joint work with B.-L. Chen, under geometric bounds on the curvature and injectivity radius near the observer, I proved that there exist a CMC (constant mean curvature) foliation as well as CMC–harmonic coordinates. These objects are defined in geodesic balls with definite size depending only on the assumed bounds, and the components of the Lorentzian metric has optimal regularity in these coordinates. The proof combines geometric estimates (Jacobi field, comparison theorems) and quantitative estimates for nonlinear elliptic equations with low regularity.

The lectures are based on the following two papers:

B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713.

B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, J. Geom. Phys. 59 (2009), 913–-941.