Analytical proof of the isospectrality of quasinormal modes for Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter
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Analytical proof of the isospectrality of quasinormal modes for Schwarzschild-de Sitter and Schwarzschild-Anti de Sitter spacetimes Flora Moulin1 · Aurélien Barrau1 Received: 2 April 2020 / Accepted: 28 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective. Keywords Black holes · Quasinormal modes · General relativity · Gravitational waves · Isospectrality
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conditions for isospectrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Newman–Penrose formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Preliminaries on isospectrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Derivation of the radial equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Proof of isospectrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Phantom gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Aurélien Barrau [email protected] Laboratoire de Physique Subatomique et de Cosmologie, CNRS/IN2P3, Université Grenoble-Alpes, 53, avenue des Martyrs, 38026 Grenoble Cedex, France 0123456789().: V,-vol
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F. Moulin, A. Barrau
1 Introduction The direct measurement of gravitational waves emitted by the coalescence of black holes (BHs) is now possible. Since the seminal detection by LIGO [1], several other events were recorded and a catalogue is already available [2]. The recent improvement in sensitivity has even led to a dramatic increase in the detection rate. The recorded gravitational waves carry fundamental informations about the structure of spacetime, BHs being vacuum solutions of the Einstein field equations. Three phases can be distinguished during a coalescence: the inspiral, the merger and the ringdown. The later can be partially treated perturbatively as a superposition of damped oscillations with different complex frequencies, called quasinomal modes (QNMs). An intuitive introduction can be found in [3] and a review in [4]. The ringdown does not lead to pure “normal” modes because the system looses energy through the emission of gravitational waves. The equations for the metric perturbations are somehow un
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