186 research outputs found
Resonance expansions for tensor-valued waves on asymptotically Kerr-de Sitter spaces
In recent joint work with Vasy, we analyze the low energy behavior of
differential form-valued waves on black hole spacetimes. In order to deduce
asymptotics and decay from this, one in addition needs high energy estimates
for the wave operator acting on sections of the form bundle. The present paper
provides these on perturbations of Schwarzschild-de Sitter spaces in all
spacetime dimensions . In fact, we prove exponential decay, up to a
finite-dimensional space of resonances, of waves valued in any finite rank
subbundle of the tensor bundle, which in particular includes differential forms
and symmetric tensors. As the main technical tool for working on vector bundles
that do not have a natural positive definite inner product, we introduce
pseudodifferential inner products, which are inner products depending on the
position in phase space.Comment: 29 pages, 1 figure. v2 is the published version, with improved
expositio
Global well-posedness of quasilinear wave equations on asymptotically de Sitter spaces
We establish the small data solvability of suitable quasilinear wave and
Klein-Gordon equations in high regularity spaces on a geometric class of
spacetimes including asymptotically de Sitter spaces. We obtain our results by
proving the global invertibility of linear operators with coefficients in high
regularity -based function spaces and using iterative arguments for the
non-linear problems. The linear analysis is accomplished in two parts: Firstly,
a regularity theory is developed by means of a calculus for pseudodifferential
operators with non-smooth coefficients, similar to the one developed by Beals
and Reed, on manifolds with boundary. Secondly, the asymptotic behavior of
solutions to linear equations is studied using standard b-analysis, introduced
in this context by Vasy; in particular, resonances play an important role.Comment: 94 pages, 7 figures. v2: many minor corrections; added details in
section 6.3.2. v3: published version, with an expanded introduction and
further details in section
The geometry of embedded pseudo-Riemannian surfaces in terms of Poisson brackets
Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature
of a surface embedded in a Riemannian manifold in terms of Poisson brackets of
the embedding coordinates. We generalize these expressions to the
pseudo-Riemannian setting and derive explicit formulas for the case of surfaces
embedded in with indefinite metric.Comment: 6 page
Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime
Adapting and extending the techniques developed in recent work with Vasy for
the study of the Cauchy horizon of cosmological spacetimes, we obtain
boundedness, regularity and decay of linear scalar waves on subextremal
Reissner-Nordstr\"om and (slowly rotating) Kerr spacetimes, without any
symmetry assumptions; in particular, we provide simple microlocal and
scattering theoretic proofs of analogous results by Franzen. We show polynomial
decay of linear waves relative to a Sobolev space of order slightly above
. This complements the generic blow-up result of Luk
and Oh.Comment: 31 pages, 9 figures. v2 is the published version, with minor
improvement
Non-trapping estimates near normally hyperbolic trapping
In this paper we prove semiclassical resolvent estimates for operators with
normally hyperbolic trapping which are lossless relative to non-trapping
estimates but take place in weaker function spaces. In particular, we obtain
non-trapping estimates in standard spaces for the resolvent sandwiched
between operators which localize away from the trapped set in a rather
weak sense, namely whose principal symbols vanish on .Comment: 20 pages, 3 figures. The main results of this paper were in the
appendix of arXiv:1306.4705v1; with v2 the appendix has been removed and it
is posted independently, in an expanded form, as this paper. v2 is the
published version, with minor improvements of the exposition and corrected
reference
Reconstruction of Lorentzian manifolds from boundary light observation sets
On a time-oriented Lorentzian manifold with non-empty boundary
satisfying a convexity assumption, we show that the topological,
differentiable, and conformal structure of suitable subsets of
sources is uniquely determined by measurements of the intersection of future
light cones from points in with a fixed open subset of the boundary of ;
here, light rays are reflected at according to Snell's law. Our
proof is constructive, and allows for interior conjugate points as well as
multiply reflected and self-intersecting light cones.Comment: 34 pages, 11 figures. v2 is the published version, with typos fixe
Analysis of linear waves near the Cauchy horizon of cosmological black holes
We show that linear scalar waves are bounded and continuous up to the Cauchy
horizon of Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter spacetimes, and in
fact decay exponentially fast to a constant along the Cauchy horizon. We obtain
our results by modifying the spacetime beyond the Cauchy horizon in a suitable
manner, which puts the wave equation into a framework in which a number of
standard as well as more recent microlocal regularity and scattering theory
results apply. In particular, the conormal regularity of waves at the Cauchy
horizon - which yields the boundedness statement - is a consequence of radial
point estimates, which are microlocal manifestations of the blue-shift and
red-shift effects.Comment: 56 pages, 14 figures. v2 is the published version, with fewer typos
and updated reference
Wave decay for star-shaped obstacles in : papers of Morawetz and Ralston revisited
The purpose of this expository note is to revisit Morawetz's method for
obtaining a lower bound on the rate of exponential decay of waves for the
Dirichlet problem outside star-shaped obstacles, and to discuss the uniqueness
of the sphere as the extremizer of the sharp lower bound proved by Ralston.Comment: 16 pages, 3 figure
Resonances for obstacles in hyperbolic space
We consider scattering by star-shaped obstacles in hyperbolic space and show
that resonances satisfy a universal bound which is optimal in dimension . In odd dimensions we also show
that for a universal constant
, where is the radius of a ball containing the obstacle; this gives
an improvement for small obstacles. In dimensions and higher the proofs
follow the classical vector field approach of Morawetz, while in dimension
we obtain our bound by working with spaces coming from general relativity. We
also show that in odd dimensions resonances of small obstacles are close, in a
suitable sense, to Euclidean resonances.Comment: 37 pages, 10 figures. v2: added dedication to C. S. Morawetz, fixed
typos. v3: published version, added section 6.
Asymptotics for the wave equation on differential forms on Kerr-de Sitter space
We study asymptotics for solutions of Maxwell's equations, in fact of the
Hodge-de Rham equation without restriction on the form degree,
on a geometric class of stationary spacetimes with a warped product type
structure (without any symmetry assumptions), which in particular include
Schwarzschild-de Sitter spaces of all spacetime dimensions . We prove
that solutions decay exponentially to or to stationary states in every form
degree, and give an interpretation of the stationary states in terms of
cohomological information of the spacetime. We also study the wave equation on
differential forms and in particular prove analogous results on
Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our
analysis and deduce asymptotics and decay for solutions of Maxwell's equations,
the Hodge-de Rham equation and the wave equation on differential forms on
Kerr-de Sitter spacetimes with small angular momentum.Comment: 47 pages. v2 is the published version, with improved expositio
- …