1,720,965 research outputs found
Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime
No full textWe give a quantitative refinement and simple proofs of mode stability type statements for the wave equation on Kerr backgrounds in the full sub-extremal range (|a| < M). As an application, we are able to quantitatively control the energy flux along the horizon and null infinity and establish integrated local energy decay for solutions to the wave equation in any bounded-frequency regime. Keywords: Black Hole, Wave Equation, Half Plane, Quasinormal Mode, Mode Stabilit
Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes
Full text linkFor any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation.National Science Foundation (U.S.) (Grant DMS-0943787
The wave equation on the exterior of a Schwarzschild black hole
No full textThis thesis investigates a wave equation on the Minkowski
and Schwarzschild spacetimes. We rely on the vector field method and
the rp-weighted inequalities discovered by Dafermos and Rodnianski to
show the decay of energy and solutions in the cases studied
Twisted Self-Similarity and the Einstein Vacuum Equations
Full text linkIn the previous works [I. Rodnianski and Y. Shlapentokh-Rothman, Naked
Singularities for the Einstein Vacuum Equations: The Exterior Solution,
arXiv:1912.08478 and Y. Shlapentokh-Rothman, Naked Singularities for the
Einstein Vacuum Equations: The Interior Solution, arXiv:2204.09891] we have
introduced a new type of self-similarity for the Einstein vacuum equations
characterized by the fact that the homothetic vector field may be spacelike on
the past light cone of the singularity. In this work we give a systematic
treatment of this new self-similarity. In particular, we provide geometric
characterizations of spacetimes admitting the new symmetry and show the
existence and uniqueness of formal expansions around the past null cone of the
singularity which may be considered analogues of the well-known
Fefferman--Graham expansions. In combination with previous results, our
analysis will show that the twisted self-similar solutions are sufficiently
general to describe all possible asymptotic behaviors for spacetimes in the
small data regime which are self-similar and whose homothetic vector field is
everywhere spacelike on an initial spacelike hypersurface. We present an
application of this later fact to the understanding of the global structure of
Fefferman--Graham spacetimes and the naked singularity exteriors of [I.
Rodnianski and Y. Shlapentokh-Rothman, Naked Singularities for the Einstein
Vacuum Equations: The Exterior Solution, arXiv:1912.08478]. Lastly, we observe
that by an amalgamation of techniques from previous works, one may associate
true solutions to the Einstein vacuum equations to each of our formal
expansions in a suitable region of spacetime.Comment: 45 pages, 5 figures, final versio
Mode stabilities and instabilities for scalar fields on Kerr exterior spacetimes
No full textThesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.Cataloged from PDF version of thesis.Includes bibliographical references (pages 131-136).In this thesis we study wave and Klein-Gordon equations on Kerr exterior spacetimes. For the wave equation, we give a quantitative refinement and simple proofs of mode stability type statements on Kerr backgrounds in the full sub-extremal range ([absolute value of]a < M). As an application, we are able to quantitatively control the energy flux along the horizon for solutions to the wave equation in any bounded-frequency regime. This estimate plays a crucial role in the author's recent proof, joint with Mihalis Dafermos and Igor Rodnianski, of boundedness and decay for the solutions to the wave equation on the full range of sub-extremal Kerr spacetimes. For the Klein-Gordon equation, we show that given any Kerr exterior spacetime with non-zero angular momentum, we may find an open family of non-zero Klein-Gordon masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein-Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to [absolute value of]am/2Mr+.by Yakov Shlapentokh-Rothman.Ph. D
Full text linkThe weak cosmic censorship conjecture posits that, generically, all singularities in General Relativity arising from regular asymptotically flat initial data should have a complete future null infinity. While this conjecture remains wide open, it has inspired many mathematical works concerning topics such as trapped surface formation and the construction of naked singularities. In this article we will review some of these works and attempt to emphasize their interconnectedness
Naked Singularities for the Einstein Vacuum Equations: The Exterior Solution
Full text linkIn this work we initiate the mathematical study of naked singularities for
the Einstein vacuum equations in dimensions by constructing solutions
which correspond to the exterior region of a naked singularity. A key element
is our introduction of a new type of self-similarity for the Einstein vacuum
equations. Connected to this is a new geometric twisting phenomenon which plays
the leading role in singularity formation.
Prior to this work, the only known examples of naked singularities were the
solutions constructed by Christodoulou for the spherically symmetric
Einstein-scalar-field system, as well as other solutions explored numerically
for either the spherically symmetric Einstein equations coupled to suitable
matter models or for the Einstein equations in higher dimensions.Comment: 116 pages, final versio
Polynomial time decay for solutions of the Klein--Gordon equation on a subextremal Reissner--Nordstr\"{o}m black hole
Full text linkWe consider the massive scalar field equation
on any subextremal Reissner--Nordstr\"{o}m exterior metric . We prove
that solutions with localized initial data decay pointwise-in-time at the
polynomial rate in any spatially compact region
(including the event horizon), for some small .
Moreover, assuming the validity of the Exponent Pair Conjecture on exponential
sums in Number Theory, our result implies that decay upper bounds hold at the
rate , for any arbitrarily small .
In our previous work, we proved that each fixed angular mode decays at the
exact rate , thus the upper bound
is sharp, up to a loss. Without the restriction to a fixed
angular mode, the solution turns out to have an unbounded Fourier transform due
to discrete frequencies associated to quasimodes, and caused by the occurrence
of stable timelike trapping. Our analysis nonetheless shows that
inverse-polynomial asymptotics in still hold after summing over all angular
modes.Comment: 76 pages, 2 figure
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