Estimate Of The Black Hole Mass With A Single Quasinormal Mode Open Access

Liu, Yuke (Spring 2022)

Permanent URL: https://etd.library.emory.edu/concern/etds/wm117q27b?locale=en
Published

Abstract

When astrophysical black holes are under perturbation, they emit gravitational waves which are characteristic of the black hole. The frequencies of the waves are called quasinormal modes. There is a large amount of research and physics literature showing that it is possible to infer black hole parameters (e.g. mass, angular momentum etc) from the quasinormal modes. The problem is similar to the famous Kac’s problem “Can you hear the shape of a drum?” The main difference for the black hole problem, which makes it more challenging and interesting is that in practice only a few quasi-normal modes can be acquired. Thus we are not supposed to use all quasinormal modes to determine the black hole parameter as usually done in the inverse spectral problems.

In this work, we study a non-rotating black hole called the de Sitter-Schwartzchilde black hole, which is characterized only by its mass. We develop mathematical methods to obtain a lower bound of the mass from a single quasinormal mode. In particular, we study Zerilli’s equation which describes the black hole perturbation. We estimate the lower bound of resonance width by adapting a method due to Harrell for one-dimensional Schrodinger equations. The lower bound yields the desired estimate of the mass. For a toy model scattering problem, we show by numerical examples the feasibility of the method.

1 Introduction 1

1.1 InverseSpectralProblem......................... 1

1.2 QuasinormalModesOfABlackHole .................. 4

1.3 TheOutline................................ 8

2 Scattering Theory 11

2.1 TheFreeScattering............................ 11

2.2 ThePotentialScattering ......................... 13

2.3 RecoverThePotentialFromAResonance . . . . . . . . . . . . . . . 17

3 Estimates of Black Hole Mass 21

3.1 TheMainResult ............................. 21

3.2 TheProof................................. 24