Eigenvalue Problem of Scalar Fields in BTZ Black Hole Spacetime

TL;DR

This paper investigates the eigenvalue problem of scalar fields in BTZ black hole spacetime, analyzing how black hole rotation and scalar mass influence eigenvalues and discussing implications for super-radiant instability.

Contribution

It provides a detailed analysis of scalar field eigenvalues in BTZ spacetime with boundary conditions, highlighting effects of rotation and mass, and explores super-radiant instability.

Findings

Eigenvalues depend on black hole rotation and scalar mass.

Boundary conditions significantly influence eigenvalue spectrum.

Relation between rotation and super-radiant instability is discussed.

Abstract

We studied the eigenvalue problem of scalar fields in the (2+1)-dimensional BTZ black hole spacetime. The Dirichlet boundary condition at infinity and the Dirichlet or the Neumann boundary condition at the horizon are imposed. Eigenvalues for normal modes are characterized by the principal quantum number $(0<n)$ and the azimuthal quantum number $(-\infty<m< \infty)$. Effects to eigenvalues of the black hole rotation and of the scalar field mass are studied explicitly. Relation of the black hole rotation to the super-radiant instability is discussed.