69 research outputs found
Comments on the paper 'Static solutions of the Vlasov-Einstein system' by G. Wolansky
In this note we address the attempted proof of the existence of static
solutions to the Einstein-Vlasov system as given in \cite{Wol}. We focus on a
specific and central part of the proof which concerns a variational problem
with an obstacle. We show that two important claims in \cite{Wol} are incorrect
and we question the validity of a third claim. We also discuss the variational
problem and its difficulties with the aim to stimulate further investigations
of this intriguing problem: to answer the question whether or not static
solutions of the Einstein-Vlasov system can be found as local minimizers of an
energy-Casimir functional.Comment: 9 page
On the rotation curves for axially symmetric disk solutions of the Vlasov-Poisson system
A large class of flat axially symmetric solutions to the Vlasov-Poisson
system is constructed with the property that the corresponding rotation curves
are approximately flat, slightly decreasing or slightly increasing. The
rotation curves are compared with measurements from real galaxies and
satisfactory agreement is obtained. These facts raise the question whether the
observed rotation curves for disk galaxies may be explained without introducing
dark matter. Furthermore, it is shown that for the ansatz we consider stars on
circular orbits do not exist in the neighborhood of the boundary of the steady
state.Comment: 27 pages, 17 figures. Final versio
On the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system
The present status on the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system is reviewed. Under the assumptions that a spherically symmetric static object has isotropic pressure and non-increasing energy density outwards, Buchdahl showed 1959 the bound M/R<4/9, where M is the ADM mass and R the outer radius. Most static solutions of the Einstein-Vlasov system do not satisfy these assumptions. The bound M/R<4/9 nevertheless holds and it is sharp. An analogous bound in the charged case is also given. The important question of stability of spherically symmetric static solutions is presently open but numerical results are available and these are reviewed. A natural question is to go beyond spherical symmetry and consider axially symmetric solutions, and a recent result on the existence of axially symmetric stationary solutions is also discussed
Existence of steady states of the massless Einstein-Vlasov system surrounding a Schwarzschild black hole
We show that there exist steady states of the massless Einstein-Vlasov system
which surround a Schwarzschild black hole. The steady states are (thick) shells
with finite mass and compact support. Furthermore we prove that an arbitrary
number of shells, necessarily well separated, can surround the black hole. To
our knowledge this is the first result of static self-gravitating solutions to
any massless Einstein-matter system which surround a black hole. We also
include a numerical investigation about the properties of the shells.Comment: 30 pages, 13 figure
Static solutions to the Einstein-Vlasov system with non-vanishing cosmological constant
We construct spherically symmetric, static solutions to the Einstein-Vlasov
system with non-vanishing cosmological constant . The results are
divided as follows. For small we show existence of globally regular
solutions which coincide with the Schwarzschild-deSitter solution in the
exterior of the matter sources. For we show via an energy estimate
the existence of globally regular solutions which coincide with the
Schwarzschild-Anti-deSitter solution in the exterior vacuum region. We also
construct solutions with a Schwarzschild singularity at the center regardless
of the sign of . For all solutions considered, the energy density and
the pressure components have bounded support. Finally, we point out a
straightforward method to obtain a large class of globally non-vacuum
spacetimes with topologies and which arise from our solutions using the periodicity of the
Schwarzschild-deSitter solution. A subclass of these solutions contains black
holes of different masses.Comment: 31 pages, 7 figure
Models for Self-Gravitating Photon Shells and Geons
We prove existence of spherically symmetric, static, self-gravitating photon
shells as solutions to the massless Einstein-Vlasov system. The solutions are
highly relativistic in the sense that the ratio is close to ,
where is the Hawking mass and is the area radius. In 1955 Wheeler
constructed, by numerical means, so called idealized spherically symmetric
geons, i.e. solutions of the Einstein-Maxwell equations for which the energy
momentum tensor is spherically symmetric on a time average. The structure of
these solutions is such that the electromagnetic field is confined to a thin
shell for which the ratio is close to , i.e., the solutions are
highly relativistic photon shells. The solutions presented in this work provide
an alternative model for photon shells or idealized spherically symmetric
geons
Bounds on M/R for Charged Objects with positive Cosmological constant
We consider charged spherically symmetric static solutions of the
Einstein-Maxwell equations with a positive cosmological constant . If
denotes the area radius, and the gravitational mass and charge of
a sphere with area radius respectively, we find that for any solution which
satisfies the condition where and
are the radial and tangential pressures respectively,
is the energy density, and for which
the inequality holds. We also investigate
the issue of sharpness, and we show that the inequality is sharp in a few cases
but generally this question is open.Comment: 12 pages. Revised version to appear in Class. Quant. Gra
On the steady states of the spherically symmetric Einstein-Vlasov system
Using both numerical and analytical tools we study various features of
static, spherically symmetric solutions of the Einstein-Vlasov system. In
particular, we investigate the possible shapes of their mass-energy density and
find that they can be multi-peaked, we give numerical evidence and a partial
proof for the conjecture that the Buchdahl inequality , the quasi-local mass, holds for all such steady states--both
isotropic {\em and} anisotropic--, and we give numerical evidence and a partial
proof for the conjecture that for any given microscopic equation of state--both
isotropic {\em and} anisotropic--the resulting one-parameter family of static
solutions generates a spiral in the radius-mass diagram.Comment: 34 pages, 18 figures, LaTe
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