Generalised Hyperbolicity in Space-times with Conical Singularities

A desirable property of any space-time used to model a physically plausible scenario is that the evolution of the Einstein’s equations is well posed i.e. the initial value problem has a unique solution. Space-times whose metrics are at least C, which guarantees the existence of unique geodesics, fall within the context of the Cosmic Censorship Hypothesis of Penrose (1979). The hypothesis states that that the space-time will be globally hyperbolic, i.e. strong causality is satisfied and J(p) ∩ J(q) is compact ∀p, q ∈M , and hence the evolution of Einstein’s equations is well defined. There are however a number of space-times with weak singularities which model physically plausible scenarios such as thin cosmic strings (Vickers, 1987), impulsive gravitational waves (Penrose, 1972) and dust caustic space-times (Clarke and O’Donnell, 1992). Typically such a space-time has a locally bounded metric whose differentiability level is lower than C, but whose curvature is well defined as a distribution, often with its support on a proper submanifold. Although cosmic censorship may be violated for such space-times, it does not rule out the possibility that the evolution of some fields is well posed. A concept of hyperbolicity for such space-times was proposed by Clarke (1998). This was based on the extent to which singularities disrupted the local evolution of the initial value problem for the scalar wave equation. φ = f