The characteristic initial value problem for plane symmetric spacetimes with weak regularity

Abstract

We investigate the existence and the global causal structure of plane symmetric spacetimes with weak regularity when the matter consists of an irrotational perfect fluid with pressure equal to its mass-energy density. Our theory encompasses the class of W1, 2 regular spacetimes whose metric coefficients have square-integrable first-order derivatives and whose curvature must be understood in the sense of distributions. We formulate the characteristic initial value problem with data posed on two null hypersurfaces intersecting along a two-plane. Relying on Newman–Penrose's formalism and expressing our weak regularity conditions in terms of the Newman–Penrose scalars, we arrive at a fully geometrical formulation in which, along each initial hypersurface, two scalar fields describing the incoming radiation must be prescribed in L1 and W−1, 2, respectively. To analyze the future boundary of such a spacetime and identify its global causal structure, we introduce a gauge that reduces the Einstein equations to a coupled system of wave equations and ordinary differential equations for well-chosen unknowns. We prove that, within the weak regularity class under consideration and for generic initial data, a true spacetime singularity forms in finite proper time. Our formulation is robust enough so that propagating discontinuities in the curvature or in the matter variables do not prevent us from constructing a spacetime whose curvature generically blows up on the future boundary. Earlier work on the problem studied here was restricted to sufficiently regular and vacuum spacetimes.