Definition and stability of Lorentzian manifolds with distributional curvature

Abstract

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.2000 Mathematics Subject Classification. Primary : 35L65. Secondary : 76L05, 76N.

On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

“…Another remarkable example is provided by Ori's solution [32], where the effect of the pointwise blow-up of the Kretschmann scalar at the Cauchy horizon, corresponding to infinite tidal forces there, does not necessarily lead to the "destruction" of an observer crossing the horizon; more precisely, a double integral of the Kretschmann scalar remains finite. Other solutions to the Einstein equations with lower regularity than C 2 have been studied in the literature (see for instance [5,27,28] and references therein). In fact, we will see in this series of papers that there even exist classical solutions of the Einstein equations which are not necessarily C 2 .…”

“…This paper is the first part of a trilogy devoted to the study of Problem 1.1. We study the relation between the spherically symmetric Einstein-Maxwellscalar field equations with a cosmological constant and the first order PDE system (18)− (27), for the quantities (6)−(12). We establish its well posedness under the minimal regularity conditions leading to classical solutions.…”

“…We start by showing that the Einstein equations imply the first order system of PDE (18)− (27). More precisely, this system consists of ten equations for the seven unknowns r, ν, λ, ̟, θ, ζ and κ (with ν = ∂ u r, λ = ∂ v r, θ = r∂ v φ and ζ = r∂ u φ): two equations for the first partial derivatives of r, ¶ Interestingly, the first spherical harmonic mode of the field decays exponentially along the event horizon; thus, strictly speaking, the decay is exponential in spherical symmetry.…”

“…Notice that this system is overdetermined, as there are two equations for r and two equations for ̟, besides (27). We choose to regard equations (18), (23) and (27) as constraints on the initial data; we will then show that they are preserved by the evolution.…”

“…We choose to regard equations (18), (23) and (27) as constraints on the initial data; we will then show that they are preserved by the evolution. Therefore, we solve equations (19)−(22) and (24)−(26) only, with initial data satisfying appropriate restrictions.…”