Yafet Sanchez Sanchez
A desirable property of any spacetime is that the evolution of any physical field is locally well-defined. For smooth spacetimes this is guaranteed by standard local well-posedness results. Moreover, there are physically reasonable spacetimes for which the initial value problem is well-posed but the spacetime has low differentiability.
In this talk we will show that in certain spacetimes with hypersurface and stringlike singularities one still has local well-posedness of the wave equation in the Sobolev space $H^1$. This function space is chosen as it allows us to define the energy-momentum tensor of the field distributionally. It is also the one needed for solutions of the linearised Einstein equations. The methods we employ are therefore also relevant to finding low regularity solutions of the Einstein equations which, as shown by Dafermos, is an important issue when considering Strong Cosmic Censorship.
Motivated by this work we propose a definition of a strong gravitational singularity as an obstruction to the evolution of a test field rather than the usual definition as an obstruction to the evolution of a test particle along a causal geodesic. This definition has the advantage that it is directly related to the physical effect of the singularity on the field (the energy-momentum tensor fails to be Integrable) and also that it may be applied to situations where the regularity of the metric falls below $C^{1,1}$ where one no longer has existence and uniqueness of geodesics.
This is joint work with James Vickers.
