627 research outputs found
The wave equation on the Schwarzschild metric II: Local decay for the spin 2 Regge Wheeler equation
Odd-type spin 2 perturbations of Einstein's equation can be reduced to the
scalar Regge-Wheeler equation. We show that the weighted norms of solutions are
in L^2 of time and space. This result uses commutator methods and applies
uniformly to all relevant spherical harmonics.Comment: AMS-LaTeX, 8 pages with 1 figure. There is an errata to this paper at
gr-qc/060807
Conformal scattering for a nonlinear wave equation on a curved background
The purpose of this paper is to establish a geometric scattering result for a
conformally invariant nonlinear wave equation on an asymptotically simple
spacetime. The scattering operator is obtained via trace operators at null
infinities. The proof is achieved in three steps. A priori linear estimates are
obtained via an adaptation of the Morawetz vector field in the Schwarzschild
spacetime and a method used by H\"ormander for the Goursat problem. A
well-posedness result for the characteristic Cauchy problem on a light cone at
infinity is then obtained. This requires a control of the nonlinearity uniform
in time which comes from an estimates of the Sobolev constant and a decay
assumption on the nonlinearity of the equation. Finally, the trace operators on
conformal infinities are built and used to define the conformal scattering
operator
Scattering of massive Dirac fields on the Schwarzschild black hole spacetime
With a generally covariant equation of Dirac fields outside a black hole, we
develop a scattering theory for massive Dirac fields. The existence of modified
wave operators at infinity is shown by implementing a time-dependent
logarithmic phase shift from the free dynamics to offset a long-range mass
term. The phase shift we obtain is a matrix operator due to the existence of
both positive and negative energy wave components.Comment: LaTex, 17 page
Recovering the mass and the charge of a Reissner-Nordstr\"om black hole by an inverse scattering experiment
In this paper, we study inverse scattering of massless Dirac fields that
propagate in the exterior region of a Reissner-Nordstr\"om black hole. Using a
stationary approach we determine precisely the leading terms of the high-energy
asymptotic expansion of the scattering matrix that, in turn, permit us to
recover uniquely the mass of the black hole and its charge up to a sign
The Dirac system on the Anti-de Sitter Universe
We investigate the global solutions of the Dirac equation on the
Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the
Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove
that there exists unitary dynamics, but its uniqueness crucially depends on the
ratio beween the mass of the field and the cosmological constant
: it appears a critical value, , which plays a role
similar to the Breitenlohner-Freedman bound for the scalar fields. When
there exists a unique unitary dynamics. In opposite, for
the light fermions satisfying , we construct several asymptotic
conditions at infinity, such that the problem becomes well-posed. In all the
cases, the spectrum of the hamiltonian is discrete. We also prove a result of
equipartition of the energy.Comment: 33 page
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
Local energy decay of massive Dirac fields in the 5D Myers-Perry metric
We consider massive Dirac fields evolving in the exterior region of a
5-dimensional Myers-Perry black hole and study their propagation properties.
Our main result states that the local energy of such fields decays in a weak
sense at late times. We obtain this result in two steps: first, using the
separability of the Dirac equation, we prove the absence of a pure point
spectrum for the corresponding Dirac operator; second, using a new form of the
equation adapted to the local rotations of the black hole, we show by a Mourre
theory argument that the spectrum is absolutely continuous. This leads directly
to our main result.Comment: 40 page
Universal Bound on Dynamical Relaxation Times and Black-Hole Quasinormal Ringing
From information theory and thermodynamic considerations a universal bound on
the relaxation time of a perturbed system is inferred, , where is the system's temperature. We prove that black holes
comply with the bound; in fact they actually {\it saturate} it. Thus, when
judged by their relaxation properties, black holes are the most extreme objects
in nature, having the maximum relaxation rate which is allowed by quantum
theory.Comment: 4 page
Asymptotic quasinormal modes of a coupled scalar field in the Gibbons-Maeda dilaton spacetime
Adopting the monodromy technique devised by Motl and Neitzke, we investigate
analytically the asymptotic quasinormal frequencies of a coupled scalar field
in the Gibbons-Maeda dilaton spacetime. We find that it is described by , which depends on the structure
parameters of the background spacetime and on the coupling between the scalar
and gravitational fields. As the parameters and tend to zero,
the real parts of the asymptotic quasinormal frequencies becomes ,
which is consistent with Hod's conjecture. When , the formula
becomes that of the Reissner-Nordstr\"{o}m spacetime.Comment: 6 pages, 1 figur
On the massive wave equation on slowly rotating Kerr-AdS spacetimes
The massive wave equation is
studied on a fixed Kerr-anti de Sitter background
. We first prove that in the Schwarzschild case
(a=0), remains uniformly bounded on the black hole exterior provided
that , i.e. the Breitenlohner-Freedman bound holds. Our proof
is based on vectorfield multipliers and commutators: The usual energy current
arising from the timelike Killing vector field (which fails to be
non-negative pointwise) is shown to be non-negative with the help of a Hardy
inequality after integration over a spacelike slice. In addition to , we
construct a vectorfield whose energy identity captures the redshift producing
good estimates close to the horizon. The argument is finally generalized to
slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing
vectorfield with for an
appropriate , which is also Killing and--in contrast to the
asymptotically flat case--everywhere causal on the black hole exterior. The
separability properties of the wave equation on Kerr-AdS are not used. As a
consequence, the theorem also applies to spacetimes sufficiently close to the
Kerr-AdS spacetime, as long as they admit a causal Killing field which is
null on the horizon.Comment: 1 figure; typos corrected, references added, introduction revised; to
appear in CM
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