61 research outputs found
Future non-linear stability for reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types II and VI
Using the methods developed for the Bianchi I case we have shown that a
boostrap argument is also suitable to treat the future non-linear stability for
reflection symmetric solutions of the Einstein-Vlasov system of Bianchi types
II and VI. These solutions are asymptotic to the Collins-Stewart solution
with dust and the Ellis-MacCallum solution respectively. We have thus
generalized the results obtained by Rendall and Uggla in the case of locally
rotationally symmetric Bianchi II spacetimes to the reflection symmetric case.
However we needed to assume small data. For Bianchi VI there is no
analogous previous result.Comment: 30 page
A global foliation of Einstein-Euler spacetimes with Gowdy-symmetry on T3
We investigate the initial value problem for the Einstein-Euler equations of
general relativity under the assumption of Gowdy symmetry on T3, and we
construct matter spacetimes with low regularity. These spacetimes admit, both,
impulsive gravitational waves in the metric (for instance, Dirac mass curvature
singularities propagating at light speed) and shock waves in the fluid (i.e.,
discontinuities propagating at about the sound speed). Given an initial data
set, we establish the existence of a future development and we provide a global
foliation in terms of a globally and geometrically defined time-function,
closely related to the area of the orbits of the symmetry group. The main
difficulty lies in the low regularity assumed on the initial data set which
requires a distributional formulation of the Einstein-Euler equations.Comment: 24 page
On Hawking's Local Rigidity Theorems for Charged Black Holes
We show the existence of a Hawking vector field in a full neighborhood of a
local, regular, bifurcate, non-expanding horizon embedded in a smooth
Einstein-Maxwell space-time without assuming the underlying space-time is
analytic. It extends one result of Friedrich, R\'{a}cz and Wald, which was
limited to the interior of the black hole region. Moreover, we also show, in
the presence of an additional Killing vector field which tangent to the
horizon and not vanishing on the bifurcate sphere, then space-time must be
locally axially symmetric without the analyticity assumption. This axial
symmetry plays a fundamental role in the classification theory of stationary
black holes.Comment: 20 page
The Bousso entropy bound in selfgravitating gas of massless particles
The Bousso entropy bound is investigated in a static spherically symmetric
spacetime filled with an ideal gas of massless bosons or fermions. Especially
lightsheets generated by spheres are considered. Statistical description of the
gas is given. Conditions under which the Bousso bound can be violated are
discussed and it is shown that a possible violating region cannot be
arbitrarily large and it is contained inside a sphere of unit Planck radius if
number of independent polarization states is small enough. It is also
shown that central temperature must exceed the Planck temperature to get a
violation of the Bousso bound for not too large.Comment: 14 pages, 4 figures, a paragraph added, version published in Gen.
Rel. Gra
Multipole radiation in a collisonless gas coupled to electromagnetism or scalar gravitation
We consider the relativistic Vlasov-Maxwell and Vlasov-Nordstr\"om systems
which describe large particle ensembles interacting by either electromagnetic
fields or a relativistic scalar gravity model. For both systems we derive a
radiation formula analogous to the Einstein quadrupole formula in general
relativity.Comment: 21 page
Existence of families of spacetimes with a Newtonian limit
J\"urgen Ehlers developed \emph{frame theory} to better understand the
relationship between general relativity and Newtonian gravity. Frame theory
contains a parameter , which can be thought of as , where
is the speed of light. By construction, frame theory is equivalent to general
relativity for , and reduces to Newtonian gravity for .
Moreover, by setting \ep=\sqrt{\lambda}, frame theory provides a framework to
study the Newtonian limit \ep \searrow 0 (i.e. ). A number of
ideas relating to frame theory that were introduced by J\"urgen have
subsequently found important applications to the rigorous study of both the
Newtonian limit and post-Newtonian expansions. In this article, we review frame
theory and discuss, in a non-technical fashion, some of the rigorous results on
the Newtonian limit and post-Newtonian expansions that have followed from
J\"urgen's work
The Newtonian Limit for Asymptotically Flat Solutions of the Vlasov-Einstein System
It is shown that there exist families of asymptotically flat solutions of the
Einstein equations coupled to the Vlasov equation describing a collisionless
gas which have a Newtonian limit. These are sufficiently general to confirm
that for this matter model as many families of this type exist as would be
expected on the basis of physical intuition. A central role in the proof is
played by energy estimates in unweighted Sobolev spaces for a wave equation
satisfied by the second fundamental form of a maximal foliation.Comment: 24 pages, plain TE
Regularity results for the spherically symmetric Einstein-Vlasov system
The spherically symmetric Einstein-Vlasov system is considered in
Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem
is the issue of global existence for initial data without size restrictions.
The main purpose of the present work is to propose a method of approach for
general initial data, which improves the regularity of the terms that need to
be estimated compared to previous methods. We prove that global existence holds
outside the centre in both these coordinate systems. In the Schwarzschild case
we improve the bound on the momentum support obtained in \cite{RRS} for compact
initial data. The improvement implies that we can admit non-compact data with
both ingoing and outgoing matter. This extends one of the results in
\cite{AR1}. In particular our method avoids the difficult task of treating the
pointwise matter terms. Furthermore, we show that singularities never form in
Schwarzschild time for ingoing matter as long as This removes an
additional assumption made in \cite{A1}. Our result in maximal-isotropic
coordinates is analogous to the result in \cite{R1}, but our method is
different and it improves the regularity of the terms that need to be estimated
for proving global existence in general.Comment: 25 pages. To appear in Ann. Henri Poincar\'
Post-Newtonian expansions for perfect fluids
We prove the existence of a large class of dynamical solutions to the
Einstein-Euler equations that have a first post-Newtonian expansion. The
results here are based on the elliptic-hyperbolic formulation of the
Einstein-Euler equations used in \cite{Oli06}, which contains a singular
parameter \ep = v_T/c, where is a characteristic velocity associated
with the fluid and is the speed of light. As in \cite{Oli06}, energy
estimates on weighted Sobolev spaces are used to analyze the behavior of
solutions to the Einstein-Euler equations in the limit \ep\searrow 0, and to
demonstrate the validity of the first post-Newtonian expansion as an
approximation
The formation of black holes in spherically symmetric gravitational collapse
We consider the spherically symmetric, asymptotically flat Einstein-Vlasov
system. We find explicit conditions on the initial data, with ADM mass M, such
that the resulting spacetime has the following properties: there is a family of
radially outgoing null geodesics where the area radius r along each geodesic is
bounded by 2M, the timelike lines are incomplete, and for r>2M
the metric converges asymptotically to the Schwarzschild metric with mass M.
The initial data that we construct guarantee the formation of a black hole in
the evolution. We also give examples of such initial data with the additional
property that the solutions exist for all and all Schwarzschild time,
i.e., we obtain global existence in Schwarzschild coordinates in situations
where the initial data are not small. Some of our results are also established
for the Einstein equations coupled to a general matter model characterized by
conditions on the matter quantities.Comment: 36 pages. A corollary on global existence in Schwarzschild
coordinates for data which are not small is added together with minor
modification
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