3,689 research outputs found
Decay of the Maxwell field on the Schwarzschild manifold
We study solutions of the decoupled Maxwell equations in the exterior region
of a Schwarzschild black hole. In stationary regions, where the Schwarzschild
coordinate ranges over , we obtain a decay rate of
for all components of the Maxwell field. We use vector field methods
and do not require a spherical harmonic decomposition.
In outgoing regions, where the Regge-Wheeler tortoise coordinate is large,
, we obtain decay for the null components with rates of
, , and . Along the event horizon and in ingoing regions, where ,
and when , all components (normalized with respect to an ingoing null
basis) decay at a rate of C \uout^{-1} with \uout=t+r_* in the exterior
region.Comment: 37 pages, 5 figure
On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system
In a previous work \cite{An1} matter models such that the energy density
and the radial- and tangential pressures and
satisfy 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, satisfies
Moreover, given a sequence of solutions such that then the
limit supremum of was shown to be bounded by
In this paper we show that the hypothesis
that 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 is bounded, but that the limit is
since for Vlasov matter.
Thus, static shells of Vlasov matter can have arbitrary close to
which is interesting in view of \cite{AR2}, where numerical evidence is
presented that 8/9 is an upper bound of of any static solution of the
spherically symmetric Einstein-Vlasov system.Comment: 20 pages, Late
The dynamical stability of the static real scalar field solutions to the Einstein-Klein-Gordon equations revisited
We re-examine the dynamical stability of the nakedly singular, static,
spherical ly symmetric solutions of the Einstein-Klein Gordon system. We
correct an earlier proof of the instability of these solutions, and demonstrate
that there are solutions to the massive Klein-Gordon system that are
perturbatively stable.Comment: 13 pages, uses Elsevier style files. To appear in Phys. Lett.
Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes
We initiate the study of the spherically symmetric Einstein-Klein-Gordon
system in the presence of a negative cosmological constant, a model appearing
frequently in the context of high-energy physics. Due to the lack of global
hyperbolicity of the solutions, the natural formulation of dynamics is that of
an initial boundary value problem, with boundary conditions imposed at null
infinity. We prove a local well-posedness statement for this system, with the
time of existence of the solutions depending only on an invariant H^2-type norm
measuring the size of the Klein-Gordon field on the initial data. The proof
requires the introduction of a renormalized system of equations and relies
crucially on r-weighted estimates for the wave equation on asymptotically AdS
spacetimes. The results provide the basis for our companion paper establishing
the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this
system.Comment: 50 pages, v2: minor changes, to appear in Annales Henri Poincar\'
The spherically symmetric collapse of a massless scalar field
We report on a numerical study of the spherically symmetric collapse of a
self-gravitating massless scalar field. Earlier results of Choptuik(1992, 1994)
are confirmed. The field either disperses to infinity or collapses to a black
hole, depending on the strength of the initial data. For evolutions where the
strength is close to but below the strength required to form a black hole, we
argue that there will be a region close to the axis where the scalar curvature
and field energy density can reach arbitrarily large levels, and which is
visible to distant observersComment: 23 pages, 16 figures, uuencoded gzipped postscript This version omits
2 pages of figures. This file, the two pages of figures and the complete
paper are available at ftp://ftp.damtp.cam.ac.uk/pub/gr/rsh100
Phase-Transition Theory of Instabilities. II. Fourth-Harmonic Bifurcations and Lambda-Transitions
We use a free-energy minimization approach to describe the secular and
dynamical instabilities as well as the bifurcations along equilibrium sequences
of rotating, self-gravitating fluid systems. Our approach is fully nonlinear
and stems from the Ginzburg-Landau theory of phase transitions. In this paper,
we examine fourth-harmonic axisymmetric disturbances in Maclaurin spheroids and
fourth-harmonic nonaxisymmetric disturbances in Jacobi ellipsoids. These two
cases are very similar in the framework of phase transitions. Irrespective of
whether a nonlinear first-order phase transition occurs between the critical
point and the higher turning point or an apparent second-order phase transition
occurs beyond the higher turning point, the result is fission (i.e.
``spontaneous breaking'' of the topology) of the original object on a secular
time scale: the Maclaurin spheroid becomes a uniformly rotating axisymmetric
torus and the Jacobi ellipsoid becomes a binary. The presence of viscosity is
crucial since angular momentum needs to be redistributed for uniform rotation
to be maintained. The phase transitions of the dynamical systems are briefly
discussed in relation to previous numerical simulations of the formation and
evolution of protostellar systems.Comment: 34 pages, postscript, compressed,uuencoded. 7 figures available in
postscript, compressed form by anonymous ftp from asta.pa.uky.edu (cd
/shlosman/paper2 mget *.ps.Z). To appear in Ap
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
Self-Similar Scalar Field Collapse: Naked Singularities and Critical Behaviour
Homothetic scalar field collapse is considered in this article. By making a
suitable choice of variables the equations are reduced to an autonomous system.
Then using a combination of numerical and analytic techniques it is shown that
there are two classes of solutions. The first consists of solutions with a
non-singular origin in which the scalar field collapses and disperses again.
There is a singularity at one point of these solutions, however it is not
visible to observers at finite radius. The second class of solutions includes
both black holes and naked singularities with a critical evolution (which is
neither) interpolating between these two extremes. The properties of these
solutions are discussed in detail. The paper also contains some speculation
about the significance of self-similarity in recent numerical studies.Comment: 27 pages including 5 encapsulated postcript figures in separate
compressed file, report NCL94-TP1
Final fate of spherically symmetric gravitational collapse of a dust cloud in Einstein-Gauss-Bonnet gravity
We give a model of the higher-dimensional spherically symmetric gravitational
collapse of a dust cloud in Einstein-Gauss-Bonnet gravity. A simple formulation
of the basic equations is given for the spacetime with a perfect fluid and a cosmological constant. This is a
generalization of the Misner-Sharp formalism of the four-dimensional
spherically symmetric spacetime with a perfect fluid in general relativity. The
whole picture and the final fate of the gravitational collapse of a dust cloud
differ greatly between the cases with and . There are two
families of solutions, which we call plus-branch and the minus-branch
solutions. Bounce inevitably occurs in the plus-branch solution for ,
and consequently singularities cannot be formed. Since there is no trapped
surface in the plus-branch solution, the singularity formed in the case of
must be naked. In the minus-branch solution, naked singularities are
massless for , while massive naked singularities are possible for
. In the homogeneous collapse represented by the flat
Friedmann-Robertson-Walker solution, the singularity formed is spacelike for , while it is ingoing-null for . In the inhomogeneous collapse with
smooth initial data, the strong cosmic censorship hypothesis holds for and for depending on the parameters in the initial data, while a
naked singularity is always formed for . These naked
singularities can be globally naked when the initial surface radius of the dust
cloud is fine-tuned, and then the weak cosmic censorship hypothesis is
violated.Comment: 23 pages, 1 figure, final version to appear in Physical Review
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