A proof of the instability of AdS for the Einstein-massless Vlasov system

In recent years, the conjecture on the instability of Anti-de Sitter spacetime, put forward by Dafermos-Holzegel (Dynamic instability of solitons in 4 + 1 dimesnional gravity with negative cosmological constant, 2006. https://www.dpmms.cam.ac.uk/similar to md384/ADSinstability.pdf) and Dafermos (The Black Hole Stability problem, Talk at the Newton Institute, Cambridge, 2006. http://www-old.newton.ac.uk/webseminars/pg+ws/2006/gmx/1010/dafermos/) in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work (Phys Rev Lett 107(3):031102, 2011) of Bizon-Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). In order to circumvent problems associated with the trivial break down of the Einstein-null dust system occuring at the center r=0, Moschidis (2017) studied the evolution of the system in the exterior of an inner mirror placed at r=r0, r0>0. However, in view of additional considerations on the nature of the instability, it was necessary for Moschidis (2017) to allow the mirror radius r0 to shrink to 0 with the size of the initial perturbation; well-posedness in the resulting complicated setup (involving low-regularity estimates of uniform modulus with respect to r0) was obtained in Moschidis (The Einstein-null dust system in spherical symmetry with an inner mirror: structure of the maximal development and Cauchy stability, 2017. arXiv:1704.08685). In this paper, we establish the instability of AdS for the Einstein-massless Vlasov system in spherical symmetry; this will be the first proof of the AdS instability conjecture for an Einstein-matter system which is well-posed for regular initial data in the standard sense, without the addition of an inner mirror. The necessary well-posedness results for this system are obtained in our companion paper (Moschidis in The characteristic initial-boundary value problem for the Einstein-massless Vlasov system in spherical symmetry, 2018. arXiv:1812.04274). Our proof utilises an instability mechanism based on beam interactions which is superficially similar to the one appearing in Moschidis (A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017. arXiv:1704.08681). However, new difficulties associated with the Einstein-massless Vlasov system (such as the need for control on the paths of non-radial geodesics in a large curvature regime) will force us to develop a different strategy of proof involving a novel configuration of beam interactions. One of the main novelties of our construction is the introduction of a multi-scale hierarchy of domains in phase space, on which the initial support of the Vlasov field f is localised. The propagation of this hierarchical structure of the support of f along the evolution will be crucial both for controlling the geodesic flow under minimal regularity assumptions and for guaranteeing the existence of the solution until the time of trapped surface formation.