Global hyperbolicity for spacetimes with continuous metrics
  We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points. 
Clemens S\"amann
math.DG gr-qc math-ph math.MP