The Global Nonlinear Stability of Minkowski Space for the Massless Einstein–Vlasov System

Abstract

Minkowski space is shown to be globally stable as a solution to the EinsteinVlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value problem for the massless Einstein-Vlasov system in this region. This relies on weighted estimates for the solution which, for the Vlasov part, are obtained by introducing the Sasaki metric on the mass shell and estimating Jacobi fields with respect to this metric by geometric quantities on the spacetime. The stability of Minkowski space result for the vacuum Einstein equations is then appealed to for the remaining regions.

Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

“…The spherically symmetric cases in dimension (3 + 1) have been treated in [25,26] for the massive case and in [4] for the massless case with compactly supported initial data. A proof of stability for the massless case without spherical symmetry but with compact support in both x and v has recently been obtained in [31]. As in [4], the compact support assumptions and the fact that the particles are massless are important as they allow to reduce the proof to that of the vacuum case outside from a strip going to null infinity.…”

“…As in [4], the compact support assumptions and the fact that the particles are massless are important as they allow to reduce the proof to that of the vacuum case outside from a strip going to null infinity. Interestingly, the argument in [31] is quite geometric, relying for instance on the double null foliation, in the spirit of [13], as well as several structures associated with the tangent bundle of the tangent bundle of the base manifold. The case of massive particles introduces however several extra difficulties, some of them already present in the case of the Einstein-massive scalar field system, for which a small data global existence result was only obtained very recently [14,32].…”

Vector field methods for kinetic equations with applications to classical and relativistic systems

“…As discussed earlier, the difficulty originates in large v and, at large v, v 0 ∼ |v| holds, meaning that it becomes increasingly hard to distinguish massive from massless particles. However, the vector field v i ∂ v i , which appears 6 We do not also assume any compact support in x, but we do start from some hyperboloid. To go from an initial t = const slice to a future hyperboloid typically requires strong initial decay in x, see the discussion in [3, Appendix A].…”

“…The spherically symmetric cases in dimension (3 + 1) have been treated in [17] for the massive case and in [5] for the massless case with compactly supported initial data. A proof of stability for the massless case without spherical symmetry but with compact support in both x and v has been given in [6]. As in [5], the compact support assumptions and the fact that the particles are massless are important as they allow to reduce the proof to that of the vacuum case outside from a strip going to null infinity.…”

A vector field method for massless relativistic transport equations and applications

“…The asymptotic behavior of the solutions is derived using vector field methods for both the particle density and the potential. We refer to [19], [10], [3] and [5] for similar results on other massless Vlasov systems.…”