Perturbations of the Gravitational Energy in the TEGR: Quasinormal Modes of the Schwarzschild Black Hole

José Wadih Maluf,Karlúcio H C Castello-Branco,Fernando Lessa Carneiro,Sérgio Ulhoa

doi:10.3390/universe7040100

Abstract

We calculate the gravitational energy spectrum of the perturbations of a Schwarzschild black hole described by quasinormal modes, in the framework of the teleparallel equivalent of general relativity (TEGR). We obtain a general formula for the gravitational energy enclosed by a large surface of constant radius r, in the region m<<r<<∞, where m is the mass of the black hole. Considering the usual asymptotic expression for the perturbed metric components, we arrive at finite values for the energy spectrum. The perturbed energy depends on the two integers n and l that describe the quasinormal modes. In this sense, the energy perturbations are discretized. We also obtain a simple expression for the decrease of the flux of gravitational radiation of the perturbations.

Highlights

The response of a black hole or neutron star to external, nonradial perturbations is described by quasinormal modes
In the context of the field equations of the teleparallel equivalent of general relativity (TEGR), one finds a suitable and consistent framework for the definitions of the gravitational energy–momentum and 4-angular momentum [5,6], which satisfy the algebra of the Poincaré group
In the course of the calculations, we find that the perturbed gravitational energy depends on the square of the field quantities h0 and h1 that appear in (12)

Summary

Introduction

The response of a black hole or neutron star to external, nonradial perturbations is described by quasinormal modes. A comprehensive review of the physics related to this interesting phenomena is found in [1,2] These modes are damped oscillations of the space-time geometry that may be used to characterize the intrinsic properties of the physical system. The equations for the metric perturbations admit solutions provided the frequencies are discrete and under the imposition of special boundary conditions The solutions of these equations are the quasinormal modes. In the context of the field equations of the TEGR, one finds a suitable and consistent framework for the definitions of the gravitational energy–momentum and 4-angular momentum [5,6], which satisfy the algebra of the Poincaré group. A similar analysis regarding the spectrum of the energy perturbations of the Schwarzschild space-time has not been presented far.

Review of the Gravitational Energy–Momentum Definition in the TEGR

Axial Perturbations of the Schwarzschild Black Hole

The Gravitational Energy of the Axial Perturbations

Polar Perturbations of the Schwarzschild Black Hole

The Gravitational Energy of the Polar Perturbations

Final Remarks