Abstract
We prove the global asymptotic stability of the Minkowski space for the massless Einstein–Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in v is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at the top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein–Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.
Highlights
There are several well-established strategies to address this problem, such as the original approach of [12] or the one by Lindblad and Rodnianski [30] based on the formulation of the Einstein equations in wave coordinates
The precise statement is given in Subsection 2.3, and can be summarized as follows: Theorem 1.1. (Main theorem, rough version) Consider smooth and asymptotically flat initial data ( 0, g, k, f), where 0 ≈ R3, to the massless Einstein–Vlasov system which are sufficiently close to the ones of Minkowski spacetime (R3, δ, 0, 0)
The Cauchy Problem in Wave Coordinates and Initial Data. It is well-known that the Einstein equations can be formulated as a Cauchy problem and in the case of the Einstein–Vlasov system, the well-posedness is guaranteed by a theorem of Choquet-Bruhat [11]
Summary
The nonlinear stability of the Minkowski space, first established in the fundamental work of Christodoulou and Klainerman [12], is one of the most important results in mathematical relativity. There are several well-established strategies to address this problem, such as the original approach of [12] or the one by Lindblad and Rodnianski [30] based on the formulation of the Einstein equations in wave coordinates. These pioneering works were generalized in different ways to more general sets of initial perturbations as well as to various Einstein-matter models [5,17,22,23,24,27,31,42,45]. Other stability results for the massive EVS were established in the cosmological setting [1,14,15,36]
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