65 research outputs found
On "time-periodic" black-hole solutions to certain spherically symmetric Einstein-matter systems
This paper explores ``black hole'' solutions of various Einstein-wave matter
systems admitting an isometry of their domain of outer communications taking
every point to its future. In the first two parts, it is shown that such
solutions, assuming in addition that they are spherically symmetric and the
matter has a certain structure, must be Schwarzschild or Reissner-Nordstrom.
Non-trivial examples of matter for which the result applies are a wave map and
a massive charged scalar field interacting with an electromagnetic field. The
results thus generalize work of Bekenstein [1] and Heusler [12] from the static
to the periodic case. In the third part, which is independent of the first two,
it is shown that Dirac fields preserved by an isometry of a spherically
symmetric domain of outer communications of the type described above must
vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations
or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster,
Smoller, and Yau [9], [7], [8], and also [6].Comment: 17 pages, 5 figure
The black hole stability problem for linear scalar perturbations
We review our recent work on linear stability for scalar perturbations of
Kerr spacetimes, that is to say, boundedness and decay properties for solutions
of the scalar wave equation \Box_g{\psi} = 0 on Kerr exterior backgrounds. We
begin with the very slowly rotating case |a| \ll M, where first boundedness and
then decay has been shown in rapid developments over the last two years,
following earlier progress in the Schwarzschild case a = 0. We then turn to the
general subextremal range |a| < M, where we give here for the first time the
essential elements of a proof of definitive decay bounds for solutions {\psi}.
These developments give hope that the problem of the non-linear stability of
the Kerr family of black holes might soon be addressed. This paper accompanies
a talk by one of the authors (I.R.) at the 12th Marcel Grossmann Meeting,
Paris, June 2009.Comment: 48 pages, 5 figures, to appear in Proceedings of the 12 Marcel
Grossmann Meetin
Small-amplitude nonlinear waves on a black hole background
Let G(x) be a C^0 function such that |G(x)|\le K|x|^{p} for |x|\le c, for
constants K,c>0. We consider spherically symmetric solutions of
\Box_g\phi=G(\phi) where g is a Schwarzschild or more generally a
Reissner-Nordstrom metric, and such that \phi and \nabla \phi are compactly
supported on a complete Cauchy surface. It is proven that for p> 4, such
solutions do not blow up in the domain of outer communications, provided the
initial data are small. Moreover, |\phi|\le C(\max\{v,1\})^{-1}, where v
denotes an Eddington-Finkelstein advanced time coordinate.Comment: 24 pages, 8 figure
On the nonlinear stability of higher-dimensional triaxial Bianchi IX black holes
In this paper, we prove that the 5-dimensional Schwarzschild-Tangherlini
solution of the Einstein vacuum equations is orbitally stable (in the fully
non-linear theory) with respect to vacuum perturbations of initial data
preserving triaxial Bianchi IX symmetry. More generally, we prove that
5-dimensional vacuum spacetimes developing from suitable asymptotically flat
triaxial Bianchi IX symmetric data and containing a trapped or marginally
trapped homogeneous 3-surface possess a complete null infinity whose past is
bounded to the future by a regular event horizon, whose cross-sectional volume
in turn satisfies a Penrose inequality, relating it to the final Bondi mass. In
particular, the results of this paper give the first examples of vacuum black
holes which are not stationary exact solutions.Comment: 15 pages, 5 figures, v2: minor change
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