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Exotic Compact Objects Interacting with Fundamental Fields Engineering Exotic compact objects interacting with fundamental fields Nuno André Moreira Santos Thesis to obtain the Master of Science Degree in Engineering Physics Supervisors: Prof. Dr. Carlos Alberto Ruivo Herdeiro Prof. Dr. Vítor Manuel dos Santos Cardoso Examination Committee Chairperson: Prof. Dr. José Pizarro de Sande e Lemos Supervisor: Prof. Dr. Carlos Alberto Ruivo Herdeiro Member of the Committee: Dr. Miguel Rodrigues Zilhão Nogueira October 2018 Resumo A astronomia de ondas gravitacionais apresenta-se como uma nova forma de testar os fundamentos da física − e, em particular, a gravidade. Os detetores de ondas gravitacionais por interferometria laser permitirão compreender melhor ou até esclarecer questões de longa data que continuam por responder, como seja a existência de buracos negros. Pese embora o número cumulativo de argumentos teóricos e evidências observacionais que tem vindo a fortalecer a hipótese da sua existência, não há ainda qualquer prova conclusiva. Os dados atualmente disponíveis não descartam a possibilidade de outros objetos exóticos, que não buracos negros, se formarem em resultado do colapso gravitacional de uma estrela suficientemente massiva. De facto, acredita-se que a assinatura do objeto exótico remanescente da coalescência de um sistema binário de objetos compactos pode estar encriptada na amplitude da onda gravitacional emitida durante a fase de oscilações amortecidas, o que tornaria possível a distinção entre buracos negros e outros objetos exóticos. Esta dissertação explora aspetos clássicos da fenomenologia de perturbações escalares e eletromagnéticas de duas famílias de objetos exóticos cuja geometria, apesar de semelhante à de um buraco negro de Kerr, é definida por uma superfície refletora, e não por um horizonte de eventos. Emgeral, tais objetos registam instabilidades quando caracterizados por condições de fronteira totalmente refletoras. No entanto, mostra-se que podem ser estáveis se se considerar condições de fronteira parcialmente ou sobre-refletoras. Os resultados sugerem que, pelo menos no que respeita a esta instabilidade, estes objetos exóticos podem ser viáveis do ponto de vista astrofísico. Palavras-chave: instabilidade de ergo-região, objetos compactos, buracos negros, relatividade geral Abstract Gravitational-wave astronomy offers a novel testing ground for fundamental physics, namely by unfolding new prospects of success in probing the nature of gravity. Current and near-future gravitational-wave interferometers are expected to provide deeper insights into long-standing open questions in gravitation such as the existence of black holes. Although a cumulative number of both theoretical and observational arguments has been strengthening the black-hole hypothesis, some sort of proof is still lacking. Up-to-date gravitational-wave data does not preclude other exotic compact objects rather than black holes from being the ultimate endpoint of compact binary mergers. The late-time gravitational-wave ringdown signal from compact binary coalescences has been argued to encode the signature of the compact object left behind the merger, which hints at the possibility of distinguishing black holes from other exotic compact objects. The present thesis addresses classical phenomenological aspects of scalar and electromagnetic field perturbations of two families of Kerr-like exotic compact objects featuring a surface with reflective properties instead of an event horizon. While these horizonless alternatives are prone to ergoregion instabilities when their surface is perfectly-reflecting, it is shown that stability can be achieved when considering partially- or over-reflecting boundary conditions. The results suggest that, at least inwhat regards this instability, Kerr-like exotic compact objects may be astrophysically viable. Keywords: ergoregion instability, exotic compact objects, black holes, general relativity Table of contents List of figures ix List of tables xiii Acronyms xv 1 Introduction 1 1.1 Black holes in general relativity . .2 1.2 Black holes as an endpoint of stellar evolution . .4 1.3 Observing astrophysical black holes . .5 1.4 Exotic compact objects . .6 1.5 Thesis scope and outline . .7 2 The Kerr metric 9 2.1 The Kerr metric in the Boyer-Lindquist form . .9 2.2 Symmetries . 11 2.2.1 Discrete symmetries . 11 2.2.2 Continuous symmetries . 11 2.3 Curvature singularity and maximal analytical extension . 12 2.4 Zero angular momentum observer (ZAMO) and frame dragging . 15 2.5 Ergoregion . 15 2.6 Penrose process . 17 2.7 Superradiance . 18 3 Quasinormal modes 21 3.1 Black-hole perturbation theory . 21 3.2 Quasinormal modes . 24 3.3 Methods for computing quasinormal modes . 26 3.3.1 Direct-integration shooting method . 26 4 Scalar perturbations of exotic compact objects 29 4.1 Klein-Gordon equation on Kerr spacetime . 29 viii Table of contents 4.2 Quasinormal modes . 31 4.2.1 Schwarzschild-like exotic compact objects . 31 4.2.2 Kerr-like exotic compact objects . 32 4.2.3 Superspinars . 39 4.3 Superradiant scattering . 41 4.4 Summary . 45 5 Electromagnetic perturbations of exotic compact objects 47 5.1 The Newman-Penrose formalism . 47 5.2 Maxwell’s equations . 48 5.2.1 Maxwell’s equations on Kerr spacetime . 49 5.3 Electric and magnetic fields in the ZAMO frame . 53 5.4 Perfectly-reflecting boundary conditions . 54 5.5 Detweiler transformation . 55 5.6 Quasinormal modes . 57 5.6.1 Schwarzschild-like exotic compact objects . 57 5.6.2 Kerr-like exotic compact objects . 58 5.6.3 Superspinars . 62 5.7 Summary . 64 6 Conclusion and Future Work 65 References 67 Appendix A Teukolsky-Starobinsky identities 71 A.1 Definitions and operator identities . 71 A.2 Teukolsky-Starobinsky identities for spin-1 fields . 72 List of figures 2.1 Maximal analytical extension of Kerr solution for a2 > M 2.................. 13 2.2 Carter-Penrose of the maximal analytical extension of Kerr spacetime along the axis of symmetry (θ = 0) for a2 < M 2 and a2 = M 2.......................... 14 2.3 Proper volume of the ergoregion of Kerr spacetime as a function of |a/M|.......... 16 4.1 Real and imaginary parts of the fundamental |l| = 1, 2 scalar quasinormal mode frequencies of a Schwarzschild-like exotic compact object with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M, where rH is the would-be event horizon of the corresponding Schwarzschild black hole, as a function of δ/M, for both Dirichlet and Neumann boundary conditions. 32 4.2 Real and imaginary parts of the fundamental l = m = 1 scalar quasinormal mode frequencies of a Kerr-like exotic compact object with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, δ ≪ M, where rH is the would-be event horizon of the corresponding Kerr black hole, as a function of δ/M, for both Dirichlet and Neumann boundary conditions. 34 4.3 Critical value of the rotation parameter above which the fundamental l = m = 1 scalar quasinormal mode frequency of a perfectly-reflecting (|R|2 = 1) Kerr-like exotic compact object is unstable, for both Dirichlet and Neumann boundary conditions. 36 4.4 Detailed view of the imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequencies of a Kerr-like exotic compact object with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M, where rH is the would-be event horizon of the corresponding Kerr black hole, as a function of the rotation parameter a/M in the range [0.8,1], for both Dirichlet and Neumann boundary conditions. 36 4.5 Timescale of the scalar ergoregion instability of rapidly-rotating Kerr-like exotic compact 2 objects with a perfectly-reflecting (|R| = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M, where rH is the would-be event horizon of the corresponding Kerr black hole, as a function of δ/M, for l = m = 1 and both Dirichlet and Neumann boundary conditions. 37 x List of figures 4.6 Imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequencies of a Kerr-like exotic compact objects with a partially-reflecting|R| ( 2 < 1) surface at r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr black hole and δ/M = 10−5, as a function of a/M, for quasi-Dirichlet and quasi-Neumann boundary conditions. 38 4.7 Fit of the maximum value of the imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequency of a Kerr-like exotic compact object with reflectivity R in the range [−0.9980,−1] (quasi-Dirichlet boundary conditions) to the polynomial (4.22), for different values of δ/M...................................... 38 4.8 Real and imaginary parts of the fundamental l = m = 1 scalar quasinormal frequencies of 2 a superspinar with a perfectly-reflecting (|R| = 1) surface at r = r0 > 0, as a function of a/M, for both Dirichlet and Neumann boundary conditions. 40 4.9 Critical value of the rotation parameter below which the fundamental l = m = 1 scalar quasinormal mode frequency of a perfectly-reflecting (|R|2 = 1) superspinar is unstable, for both Dirichlet and Neumann boundary conditions. 41 4.10 Timescale of the scalar ergoregion instability of superspinars with a perfectly-reflecting 2 (|R| = 1) surface at r = r0 > 0, as a function of r0, for l = m = 1.............. 41 4.11 Imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequencies of a superspinar featuring a surface with reflectivity R at r = r0 => 0, as a function of a/M.. 41 4.12 Amplification factors for superradiant l = m = 1 scalar field perturbations scattered off Kerr-like exotic compact objects with a/M = 0.9 and featuring a surface with reflectivity R at r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr black hole and δ/M = 10−5.................................... 42 4.13 Numerical and analytical values for the amplification factors of superradiant l = m = 1 scalar field perturbations scattered off Kerr-like exotic compact objects with a/M = 0.9 and featuring a surface with reflectivity R at r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr black hole and δ/M = 10−5............
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