Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner-Nordström black holes
arXiv:2303.15338 · doi:10.1007/s00205-024-02060-1
Abstract
We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner-Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis' results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.
References in corpus (11)
- Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
- Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II
- Gravitational instability of an extreme Kerr black hole
- Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime
- The geometry of the tangent bundle and the relativistic kinetic theory of gases
- Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole
- Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds
- The Stability of the Minkowski space for the Einstein-Vlasov system
- Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes
- A proof of the instability of AdS for the Einstein--null dust system with an inner mirror
- The trapping effect on degenerate horizons