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Gravitational Axial Perturbations and Quasinormal Modes of Loop Quantum Black Holes

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Gravitational Axial Perturbations and Quasinormal Modes of Loop Quantum Black Holes Eur. Phys. J. C (2019) 79:157 https://doi.org/10.1140/epjc/s10052-019-6565-2 Regular Article - Theoretical Physics Gravitational axial perturbations and quasinormal modes of loop quantum black holes M. B. Cruz1,a,C.A.S.Silva2,b,F.A.Brito1,3,c 1 Departamento de Física, Universidade Federal da Paraíba, João Pessoa, Paraíba 58051-970, Brazil 2 Instituto Federal de Educação Ciência e Tecnologia da Paraíba (IFPB), Campus Campina Grande-Rua Tranquilino Coelho Lemos 671, Jardim Dinamérica I, João Pessoa, Brazil 3 Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, Campina Grande 58429-900, Paraíba, Brazil Received: 3 November 2018 / Accepted: 3 January 2019 / Published online: 21 February 2019 © The Author(s) 2019 Abstract Loop quantum gravity (LQG) is a theory that pro- ular, in the context of this theory, a black hole metric known as poses a way to model the behavior of the spacetime in sit- loop quantum black hole (LQBH), or self-dual black hole, has uations where its atomic characteristic arises. Among these been proposed [10,11]. This solution corresponds to a quan- situations, the spacetime behavior near the Big Bang or black tum corrected Schwarzschild solution and possess the inter- hole’s singularity. The detection of gravitational waves, on esting property of self-duality. From this property, the black the other hand, has opened the way to new perspectives in the hole singularity is removed and replaced by another asymp- investigation of the spacetime structure. In this work, by the totically flat region, which is an expected effect in a quantum use of a WKB method introduced by Schutz and Will (Astro- gravity regime. Moreover, LQBHs have been pointed as a phys J 291:L33, 1985), and after improved by Iyer and Will possible candidate for dark matter [11,12] and as the building (Phys Rev D 35:3621, 1987), we study the gravitational wave blocks of a holographic version of loop quantum cosmology spectrum emitted by loop quantum black holes, which corre- [13]. spond to a quantized version of the Schwarzschild spacetime In order to move black holes from a simple mathemati- by LQG techniques. From the results obtained, loop quantum cal solution of the gravitational equations to objects whose black holes have been shown stable under axial gravitational existence in nature is possible, a key point consists in to inves- perturbations. tigate black hole’s stability under perturbations. It is due to the fact that an isolated black hole would never be found in nature. In fact, complex distributions of matter such as 1 Introduction accretion disks, galactic nuclei, strong magnetic fields, other stars, etc are always present around black holes, which in turn One of most exciting predictions of general relativity is the actively interact with their surroundings. Even if all macro- existence of black holes, objects from which no physical bod- scopic objects and fields in space have been removed, a black ies or signals can get loose of their drag due to its strong grav- hole will interact with the vacuum around it, creating pairs itational field. Going far beyond astrophysics, black holes of particles and evaporating due to Hawking phenomena. appear as objects that may help us to clarify one of the most Besides, in the first moments after its formation, a black hole intriguing bone of contention in the current days, the quan- is in a perturbed state due to gravitational collapse of matter. tum nature of gravity. It is because, in the presence of a black In this way, a real black hole will be always in a perturbed hole’s strong gravitational field, quantum features of space- state. time must be manifested [3–9]. A black hole’s response to a perturbation occurs by emit- Loop quantum gravity, on the other hand, is a theory that ting gravitational waves whose evolution corresponds, firstly, has given ascent to models that provide a portrait of the quan- to a relatively short period of initial outburst of radiation fol- tum features of spacetime unveiled by a black hole. In partic- lowed by a phase where the black hole get going to vibrat- ing into exponentially decaying oscillations, “quasinormal modes”, whose frequencies and decay times depend only a e-mail: [email protected] on the intrinsic characteristics of the black hole itself, being b e-mail: [email protected] indifferent to the details of the collapse. Finally, at a very large c e-mail: [email protected] 123 157 Page 2 of 10 Eur. Phys. J. C (2019) 79 :157 time, the quasinormal modes are slapped down by power-law perturbations must be included. Investigation on the quasi- or exponential late-time tails. normal mode spectrum in the context of LQBHs may reveal The issue of black hole stability under perturbations was some advantages front others scenarios under the experimen- firstly addressed by Regge and Wheeler [14], and by Zerilli tal point of view since the quantum corrections present in [15], which based on the black hole perturbation theory, have this scenario depend on the dimensionless Barbero–Immirzi demonstrated the stability of the Schwarzschild metric. The parameter [32], which as it has been pointed in [33] does methods used are familiar from quantum mechanics: pertur- not suffers with the problem of mass suppression, as occurs bations caused by an external (e.g., gravitational or electro- with the parameters of other quantum gravity theories like magnetic) field are taken into account as waves scattering off superstring theory or noncommutative theory. In these the- the respective potential. It is due to the fact that the formalism ories, the quantum corrections appears as proportional to m provided by Regge–Wheller and Zerilli removes the angular (lqg/M) , where lqg is a quantum gravity motivated dimen- dependence in the perturbation variables by the use of a ten- sionful parameter of the theory, M is the black hole mass, sorial generalization of the spherical harmonics which makes and m is some positive number. In this way, for large black possible to translate the solution of the perturbed Einstein’s holes, the corrections coming from quantum gravity effects equations in the form of a Schrodinger-like wave equation are mass suppressed. treatment. Posteriorly, the Regge–Wheeler/Zerilli formalism The article is organized as follows: in Sect. 2,werevisethe was extended to the charged [16–18] and rotating [19,20] LQBH’s theory; in Sect. 3, we calculate the Regge–Wheeler black hole scenarios. A full description of the black hole equation for LQBHs; in Sect. 4, we calculate the quasinormal perturbation framework can be found in the text by Chan- modes of the LQBHs. Section 5 is devoted to conclusions and drasekhar [21]. perspectives. Throughout this paper, we have used h¯ = c = The most valuable phase in the evolution of black hole G = 1. perturbations is given by its quasinormal modes which can give us information not only about the black hole stability, but also, as emphasized by Berti [22], ‘how much stable it is’. 2 Loop quantum black holes In other words, quasinormal modes tell us which timescale a black hole radiate away its matter contend after formation. Some efforts in order to find out black hole solutions in LQG By the way, in this context the prefix ‘quasi’ means that the have been done by several authors [10,34–43]. In this sec- black hole consists in an open system that loses energy due tion, we shall analyze a particular solution, called self-dual, to the emission of gravitational waves. The issue of black that appeared at the primary time from a simplified model of hole quasinormal modes is interesting not only by the inves- LQG consisting in symmetry reduced models correspond- tigation of black hole stability, but also because gravitational ing to homogeneous spacetimes (see [10] and the references waves have been pointed as a possible experimental way to therein). make contact with black holes and, in this way, with the In this way, the LQBH framework which we shall work quantum spacetime characteristics revealed by them. Indeed, here is delineated by a quantum gravitationally corrected recent results from LIGO, which has detected a gravitational Schwarzschild metric, written as wave signal from black holes [23–26], and the last gravita- ds2 =−G(r)dt2 + F−1(r)dr2 + H(r)dΩ2 tional wave detection from neutron stars [27] have open the (1) floodgates for a large range of possibilities in gravitational with physics in the same way that the discovery of infrared radia- Ω2 = θ 2 + 2 θ φ2, tion by William Hershel the ‘1800s’ opened the possibility of d d sin d (2) the investigation and application of the electromagnetic spec- where, in the Eq. (1), the metric functions are given by trum beyond to the visible. It can yet be reinforced by oth- (r − r+)(r − r−)(r + r∗)2 ers results from others gravitational antennas such as LISA, G(r) = , 4 + 2 VIRGO, and others. In a timely way, the quasinormal funda- r a0 mental mode constitutes the predominating contribution to (r − r+)(r − r−)r 4 such signals in the long period of damping proper oscillations F(r) = , (3) (r + r∗)2(r 4 + a2) [28]. Moreover, the quasinormal modes’ ringdown phase of 0 the signal is useful to distinguish between black holes and and other possible sources [29]. a2 H(r) = r 2 + 0 , (4) In this work, we shall address the gravitational wave pro- r 2 duction by LQBHs. Quasinormal frequencies for such black where holes have been addressed in the scalar perturbations case in 2 [30,31].
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