# Scalar field quasinormal modes on asymptotically locally flat rotating black holes in three dimensions

## Abstract

The pure quadratic term of New Massive Gravity in three dimensions admits asymptotically locally flat, rotating black holes. These black holes are characterized by their mass and angular momentum, as well as by a hair of gravitational origin. As in the Myers–Perry solution in dimensions greater than five, there is no upper bound on the angular momentum. We show that, remarkably, the equation for a massless scalar field on this background can be solved in an analytic manner and that the quasinormal frequencies can be found in a closed form. The spectrum is obtained requiring ingoing boundary conditions at the horizon and an asymptotic behavior at spatial infinity that provides a well-defined action principle for the scalar probe. As the angular momentum of the black hole approaches zero, the imaginary part of the quasinormal frequencies tends to minus infinity, migrating to the north pole of the Riemann sphere and providing infinitely damped modes of high frequency. We show that this is consistent with the fact that the static black hole within this family does not admit quasinormal modes for a massless scalar probe.

## 1 Introduction

Three dimensional gravity has been a useful arena to explore gravitational models with simpler properties than their counterpart in four dimensions. Since the Weyl tensor identically vanishes in three dimensions, Einstein equations imply that the spacetime is locally flat or (A)dS depending on whether the theory has a null, negative or positive cosmological term, respectively. This implies that General Relativity has no local degrees of freedom in vacuum, but nevertheless in the case with a negative cosmological constant, its spectrum contains black holes (the BTZ black hole) [1]. These black holes have been of fundamental importance on the tests of the holographic correspondence between physics in Anti de Sitter spacetime and that of a dual Conformal Field Theory living at the boundary [2, 3, 4, 5]. Just to mention two examples of this relation, the entropy of the BTZ black holes can be obtained by a microscopic counting of microstates in the dual theory [6] and their quasinormal ringing correlates precisely with relaxation time in the dual field theory at finite temperature [7, 8]. By the end of the last decade it was also realized that the lack of local degrees of freedom of General Relativity in three dimensions allows to construct a parity invariant, self-interacting theory for a massive particle of spin 2, i.e., a massive gravity theory. The Einstein-Hilbert Lagrangian is supplemented by quadratic terms in the curvature, and despite the fact that the field equations are of fourth order, their linearization around flat space correctly reproduces the Fierz–Pauli equation for a massive spin 2 excitation [9, 10]. The addition of a cosmological term allows to further explore the ideas of the holographic correspondence in the presence of a massive graviton in the bulk. As it generically occurs in theories containing quadratic powers in the curvature, such theory possesses two maximally symmetric (and therefore of constant curvature) solutions. For a precise relation of the couplings both vacua coincide and actually in this case one can find asymptotically locally (A)dS black holes that are characterized by the mass, angular momentum and an extra parameter that stands for a gravitational hair [12, 13]. As shown in [14] the entropy of such black holes can be reproduced by counting microstates in the boundary theory, providing a new test for such relation on a theory with local degrees of freedom in the bulk. Recently, there has been a revived interest in the study of asymptotic symmetries to null surfaces, where such surface could be null infinity or the event horizon of a black hole. It is expected that these studies might shed some light on the information paradox (see e.g. [15, 16, 17, 18, 19, 20] and references therein). New Massive Gravity provides for a simple setup to carry on such studies, since the spectrum of the purely quadratic theory (that is healthy and intrinsically of fourth-order [10]) contains asymptotically locally flat, rotating black holes in 2 + 1 dimensions [12]. Such black holes do not exist in General Relativity in vacuum. Within the realm of NMG, these black holes can be generalized to construct non-circular black objects, dubbed black flowers [25], a family of metrics whose simplest representative is the rotating black hole constructed in [12] in the massless limit of NMG.^{1} The aim of this paper is to show that the equation for the massless scalar, remarkably, can be solved analytically on these backgrounds, and that the quasinormal frequencies can be found in a closed manner. As usual, we require ingoing boundary condition at the horizon. We show that at infinity the natural boundary condition that makes the action principle for the scalar to be well-defined is that the field must vanish sufficiently fast. This is one of the few known, rotating black holes, that admit such integration.

This paper is organized as follows: Sect. 1 introduces the theory of new massive gravity and presents the asymptotically locally flat, rotating black hole solution. In Sect. 2 we consider a massless scalar field perturbation and solve it in an exact and analytic manner. We find the quasinormal modes in a closed form in the rotating case and show that as the angular momentum of the black hole decreases the imaginary part of the frequencies approach minus infinity. This is consistent with the fact that in the static case, the algebraic equations that determine the spectrum cannot be fulfilled. We therefore explore the behavior of the quasinormal frequencies in terms of the global charges of the background solution. Even though our interests is in the propagation of a scalar probe on the black hole background, let us mentions that our results could shed some light on the stability of the rotating black hole in NMG. Since the full non-linear field equations of NMG are of fourth order, their linearized version around a generic background will be in general of the same order. Nevertheless, there could be an affective quantity, constructed with second derivatives of the perturbation, which may have a second order dynamics, as it occurs in the Teukolsky equation, where the unknowns are linearized expressions for the components of the Weyl tensor in a null tetrad [28]. It is also worth mentioning that in some particular cases in GR it is known that the dynamics of some of the modes of the full gravitational perturbation coincide with the dynamics of a scalar probe with a given mass. This happens for example in the massless topological black hole [27]. A less striking connection between a scalar probe and a gravitational perturbation occurs with the Regge–Wheeler equation, which has the same functional form than the equation for a massless scalar probe in Schwarzschild background, but in this case there is a single numeric coefficient of difference in the effective Schroedinger potential (see [28]). Section 3 contains conclusions and further comments.

## 2 Massless limit of NMG and its rotating black hole

*b*,

*a*and \(\mu \) which determine the global charges associated to the asymptotic Killing vectors \(\partial _{t}\) and \(\partial _{\phi }\), i.e. the mass and the angular momentum, respectively, which are given by [25]

## 3 Massless scalar on the rotating black hole

*x*, such that

*a*.

*F*stands for a hypergeometric function \(_{2}F_{1}\), and

*p*and

*q*take integer values \(0,1,2,3,.\ldots \). Conditions (32) lead to the determination of the spectrum of quasinormal frequencies for ingoing boundary condition at the horizon and on-shell vanishing boundary term of the variation of the action functional at infinity (30). We can introduce the reduced quantities

*p*and

*q*, determine the overtones. As mentioned above, note that \({\mathcal {C}}_1({\hat{J}},n)={\mathcal {C}}_2(-{\hat{J}},-n)\) with

*p*interchanged by

*q*. We observe that the constraint \({\mathcal {C}}_1=0\)(\({\mathcal {C}}_2=0\)) can be solved only for positive(negative) values of the black hole angular momentum \({\hat{J}}\), but due to the symmetry mentioned above, as expected, both constraints will lie on the same locus.

*x*, defined such that \(\sigma _0=1\).

Some comments on the boundary conditions are in order. For asymptotically flat black holes in four dimensions, as for example in the Kerr family, the scalar field has two possible behaviors at infinity which are plane waves that describe the out-going and the in-coming modes, and therefore the leading contribution goes as \(e^{-i\omega \left( t\pm r\right) }\) with the plus(minus) sign standing for the in-coming(out-going) mode. The background of the rotating asymptotically, locally flat, black hole of New Massive Gravity allows different behaviors at infinity, which are given in (27). Even in terms of the proper radial distance at infinity \(\rho =\frac{2}{\sqrt{b}}\sqrt{r}\), one would have \(x\sim \rho ^{2}\), and the behavior (27) would still have a power law fashion and would not be compatible with the interpretation of an out-going or in-coming plane wave. The situation at hand is actually more similar to the analysis of quasinormal modes in asymptotically AdS black holes where the leading asymptotic behaviors are \(r^{-\Delta _{\pm }}\) and \(\Delta _{\pm }\) are given in terms of the mass of the field and the dimension of the spacetime, defining the conformal weights of the operator dual to the scalar on the boundary CFT. Therefore, with the asymptotic behaviors given by Eq. (27) a natural physical condition that allows to compute the quasinormal frequencies is to require that the field at infinity decays fast enough as to lead to a well defined variational principle (30) and this is in fact the strategy we have followed.

Below we present plots of the QNM spectrum of the scalar probe in terms of the physical variables, as well as closed analytic expressions for the asymptotic behavior of the frequencies.

### 3.1 Static case and asymptotic frequencies

*p*changed by

*q*). This is a peculiar feature of the static solution: there are no massless scalar field quasinormal modes for the asymptotically flat, static black hole with gravitational hair in NMG. The constraints (34) and (35) can be mapped to the Riemann sphere, where it can be seen that at the leading order when \(\omega _{i}\rightarrow -\infty \) (for constant \(\omega _{r}\)) the constraint is indeed solved since (37) reads

^{2}

### 3.2 The quasinormal spectrum

### 3.3 Asymptotic frequencies

*j*on Schwarzschild black hole. In that case one has [36]

*J*, since this cannot be reached within the Kerr family. (It can be reached in Myers–Perry in \(d>5\) [37]). Quasinormal frequencies \({\hat{\omega }}\) in the limit \({\hat{J}}\rightarrow \infty \) becomes purely imaginary (damped) and depend only on the overtone integer

*p*approaches a straight line with slope \(2\sqrt{2}\).

## 4 Conclusions

In this paper we have shown that a massless scalar probe on the asymptotically locally flat, rotating black hole of pure NMG can be analytically solved leading to an exact expression for the quasinormal mode frequencies. We have imposed ingoing boundary conditions at the horizon and a decay at infinity which is fast enough leading to a well defined variational principle for the scalar probe. For a given black hole, parametrized by the mass, angular momentum and gravitational hair, the quasinormal spectrum is defined by the angular momentum of the field as well as an integer counting overtones. We showed that under these boundary conditions the imaginary part of the frequencies is always negative, not leading to any superradiance, as expected for a massless scalar. Even more, when the angular momentum of the black hole approaches zero, the imaginary part of the frequencies tends to minus infinity leading to infinitely damped modes. Remarkably, this is one of the few examples of rotating black holes in which the quasinormal frequencies for a scalar probe can be obtained in an analytic, closed form.^{3}

Since the rotating black hole, even in the case with cosmological constant, has vanishing Cotton tensor [12, 13], it is a conformally flat spacetime, therefore the solution for a conformal scalar could also be obtained in an analytic manner by applying a conformal transformation to the solution of the free, massless scalar of 3D Minkowski or AdS space (since the latter is also conformally flat). Once the mapping is done on the general solution, one would have to impose the boundary conditions at the horizon and infinity taking care of the potential divergences induced by the conformal mapping that leads to the black hole.

The stable propagation of scalar probes on the rotating background may lead to the existence of rotating black holes with both scalar and gravitation hair. In NMG static black holes with self-interacting scalar hair already exist, with Lifshitz and AdS asymptotics [42, 43], and it would be interesting to construct rotating black holes with these hairs.

## Footnotes

## Notes

### Acknowledgements

The authors want to thank Gaston Giribet and Nicolas Grandi for enlightening comments. This work was partially supported by FONDECYT Grants 1170279 and 1181047. O.F. is supported by the Project DINREG 19/2018 of the Dirección de Investigación of the Universidad Católica de la Santísima Concepción.

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