30 research outputs found
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
A scattering theory construction of dynamical vacuum black holes
We construct a large class of dynamical vacuum black hole spacetimes whose
exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr
metric. The construction proceeds by solving a backwards scattering problem for
the Einstein vacuum equations with characteristic data prescribed on the event
horizon and (in the limit) at null infinity. The class admits the full
"functional" degrees of freedom for the vacuum equations, and thus our
solutions will in general possess no geometric or algebraic symmetries. It is
essential, however, for the construction that the scattering data (and the
resulting solution spacetime) converge to stationarity exponentially fast, in
advanced and retarded time, their rate of decay intimately related to the
surface gravity of the event horizon. This can be traced back to the celebrated
redshift effect, which in the context of backwards evolution is seen as a
blueshift.Comment: 88 pages, 14 figures, v2: minor changes, references adde
Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes
We construct quasimodes for the Klein-Gordon equation on the black hole
exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are
associated with time-periodic approximate solutions of the Klein Gordon
equation and provide natural candidates to probe the decay of solutions on
these backgrounds. They are constructed as the solutions of a semi-classical
non-linear eigenvalue problem arising after separation of variables, with the
(inverse of the) angular momentum playing the role of the semi-classical
parameter. Our construction results in exponentially small errors in the
semi-classical parameter. This implies that general solutions to the Klein
Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The
latter result completes previous work by the authors, where a logarithmic decay
rate was established as an upper bound
Shock Formation in Small-Data Solutions to Quasilinear Wave Equations: An Overview
In his 2007 monograph, D. Christodoulou proved a remarkable result giving a
detailed description of shock formation, for small -initial conditions
( sufficiently large), in solutions to the relativistic Euler equations in
three space dimensions. His work provided a significant advancement over a
large body of prior work concerning the long-time behavior of solutions to
higher-dimensional quasilinear wave equations, initiated by F. John in the mid
1970's and continued by S. Klainerman, T. Sideris, L. H\"ormander, H. Lindblad,
S. Alinhac, and others. Our goal in this paper is to give an overview of his
result, outline its main new ideas, and place it in the context of the above
mentioned earlier work. We also introduce the recent work of J. Speck, which
extends Christodoulou's result to show that for two important classes of
quasilinear wave equations in three space dimensions, small-data shock
formation occurs precisely when the quadratic nonlinear terms fail the classic
null condition
Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes
We prove that there are no non-stationary (with respect to the Hawking
vectorfield) real mode solutions to the Teukolsky equations on all
-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further
prove that stationary solutions do not exist if the black hole parameters
satisfy the Hawking-Reall bound and
. We conclude with the
statement of mode stability which preludes boundedness and decay estimates for
general solutions which will be proven in a separate paper. Our boundary
conditions are the standard ones which follow from fixing the conformal class
of the metric at infinity and lead to a coupling of the two Teukolsky
equations. The proof relies on combining the Teukolsky-Starobinsky identities
with the coupled boundary conditions. In the stationary case the proof exploits
elliptic estimates which fail if the Hawking-Reall bound is violated. This is
consistent with the superradiant instabilities expected in that regime.Comment: 30 pages, 4 figures. A Mathematica notebook containing all the
computations is attached as ancillary file. v2 includes Teukolsky-Starobinsky
conservation laws and a comparison with the asymptotically flat cas