A sharp counterexample to local existence of low regularity solutions to   Einstein's equations in wave coordinates
  We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in Sobolev spaces H^s with regularity s>2. We give an example of data in Sobolev spaces with regularity s=2 for which there is no local solution in this space. 
Boris Ettinger and Hans Lindblad
math.AP math-ph math.MP