Mode stabilities and instabilities for scalar fields on Kerr exterior spacetimes

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+.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2015

Author dc:creator

• Shlapentokh-Rothman, Yakov

Advisor dc:contributor.advisor

• Igor Rodnianski.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/99323

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/99323