On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system

Abstract

In a previous work \cite{An1} matter models such that the energy density $\rho \ge 0,$ and the radial- and tangential pressures $p\ge 0$ and $q,$ satisfy $p+q\le \Omega \rho ,\Omega \ge 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1],R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega +1{)}^2-1)/(2\Omega +1{)}^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega +1{)}^2-1)/(2\Omega +1{)}^2=8/9,$ since $\Omega =1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.

20 pages, Latex

References in corpus (1)

• On the steady states of the spherically symmetric Einstein-Vlasov system

Cited by in corpus (29)

• Sharp bounds on the critical stability radius for relativistic charged spheres
• Sharp bounds on $2m/r$ of general spherically symmetric static objects
• The Einstein-Vlasov System/Kinetic Theory
• A numerical investigation of the steady states of the spherically symmetric Einstein-Vlasov-Maxwell system
• On the steady states of the spherically symmetric Einstein-Vlasov system
• Sharp bounds on 2m/r for static spherical objects
• Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case - A simple proof for finite extension
• Analytical families of 2-component anisotropic polytropes and their relativistic extensions
• Bounds on M/R for Charged Objects with positive Cosmological constant
• Models for Self-Gravitating Photon Shells and Geons
• Bounds on M/R for static objects with a positive cosmological constant
• The formation of black holes in spherically symmetric gravitational collapse
• Extreme $\ell$-boson stars
• Stability and instability of self-gravitating relativistic matter distributions
• On Axisymmetric and Stationary Solutions of the Self-Gravitating Vlasov System
• Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system
• Black hole formation from a complete regular past for collisionless matter
• Bounds on the mass-to-radius ratio for non-compact field configurations
• Stability for the spherically symmetric Einstein-Vlasov system - a coercivity estimate
• Buchdahl type inequalities in $d$-dimensions
• Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas -- A review
• Does space-time torsion determine the minimum mass of gravitating particles?
• Newtonian and General Relativistic Models of Spherical Shells - II
• Static spherically symmetric Einstein-Vlasov shells made up of particles with a discrete set of values of their angular momentum
• The QCD mass gap and quark deconfinement scales as mass bounds in strong gravity
• Existence of steady states of the massless Einstein-Vlasov system surrounding a Schwarzschild black hole
• Gravitational collapse and the formation of black holes for the spherically symmetric Einstein-Vlasov system
• Static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant
• Uniqueness of static, isotropic low-pressure solutions of the Einstein-Vlasov system