Cosmic string and black hole limits of toroidal Vlasov bodies in general relativity

Abstract

We numerically investigate limits of a two-parameter family of stationary solutions to the Einstein-Vlasov system. The solutions are toroidal and have non-vanishing angular momentum. As one tunes to more relativistic solutions (measured for example by an increasing redshift) there exists a sequence of solutions which approaches the extreme Kerr black hole family. Solutions with angular momentum larger than the square of the mass are also investigated, and in the relativistic limit the near-field geometry of such solutions is observed to become conical in the sense that there is a deficit angle. Such solutions may provide self-consistent models for rotating circular cosmic strings.

Stationary solutions of the axially symmetric Einstein–Vlasov system: present status and open problems

“…In the axially symmetric case the situation is different. The numerical methods that have been used in previous works [2,3,46,47] rely on an iteration procedure which must converge in order to produce a solution. Such iterative methods have also been employed in the spherically symmetric setting, where it is observed that convergence is only achieved for a certain range of the parameters.…”

“…Here K is a positive constant, which we call the amplitude, and Φ is a given function. In principle the amplitude could be incorporated into Φ but since it plays an important role in the numerical fixed point methods used in [2,3,46,47] we separate it out. The quantities E and L z are given by…”

Kinetic gas disks surrounding Schwarzschild black holes

“…In [4] these configurations are modeled by a one-particle distribution function (DF) depending only on the energy E, azimuthal L z and total angular momentum L of the particles. Examples are given in which the DF is described by a generalized polytropic ansatz [5,6] depending only on E and L z . In this article, we provide additional examples where the DF is a function of E and the inclination angle i defined by cos i = L z /L.…”

Existence of Steady States of the Massless Einstein–Vlasov System Surrounding a Schwarzschild Black Hole

“…Several simplifications and generalizations have since then been obtained, and we refer to [22] for a simplified and general approach, to [6] for the existence of highly relativistic static solutions and to [11] for the existence of stationary solutions in the axisymmetric case. There are several other existence results and also results about the properties of the static solutions in the literature, and we refer to [8] for a review and to [2,3,9] for more recent results.…”