# Stability under scalar perturbations and quasinormal modes of 4D Einstein–Born–Infeld dilaton spacetime: exact spectrum

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## Abstract

We study the stability under scalar perturbations, and we compute the quasinormal modes of the Einstein–Born–Infeld dilaton spacetime in \(1+3\) dimensions. Solving the full radial equation in terms of hypergeometric functions, we provide an exact analytical expression for the spectrum. We find that the frequencies are purely imaginary, and we confirm our results by computing them numerically. Although the scalar field that perturbs the black hole is electrically neutral, an instability similar to that seen in charged scalar perturbations of the Reissner–Nordström black hole is observed.

## 1 Introduction

Black holes (BHs), a generic prediction of Einstein’s General Relativity (GR) [1], are way more than just mathematical objects. After Hawking’s seminal work [2, 3] in which it was shown that BHs emit radiation from their horizon, these objects have attracted a lot of attention over the last decades, as they comprise an excellent laboratory to study and understand several aspects of gravitational theories. In particular, the study of BHs has received considerable attention in the context of scale dependent theories (see e.g. [4, 5, 6, 7, 8]), where certain deviations from the classical solution appear.

On the other hand, how a system responds to small perturbations has always been an important issue in Physics. The work of [9] marked the birth of BH perturbations, and it was later extended by [10, 11, 12, 13, 14]. The state-of-the art in BH perturbations is summarized in Chandrasekhar’s monograph [15]. When BHs are perturbed, the geometry of spacetime undergoes dumbed oscillations. Quasinormal modes (QNMs), with a non-vanishing imaginary part, carry unique information about the few BH parameters, since they do not depend on the initial conditions. After the LIGO direct detections of gravitational waves [16, 17, 18], that offer us the strongest evidence so far that BHs exist and merge, QNMs of black holes are more relevant than ever. By observing the quasinormal spectrum, that is frequencies and damping rates, we can determine the mass, angular momentum and charge of the BH, and even falsify the theoretical paradigm of the no-hair conjecture [19]. QNMs of BHs have been extensively studied in the literature. For a review on the subject see [20], and for a more recent one [21].

Relativistic scattering of waves has been traditionally studied in asymptotically flat spacetimes without a cosmological constant, such as Schwarzschild [22], Kerr [23] and Reissner–Norstrom [24] BHs, see e.g. [25, 26, 27, 28, 29, 30, 31]. However, due to inflation [32], the current cosmic acceleration [33, 34] and the AdS/CFT correspondence [35, 36], asymptotically non-flat spacetimes with a positive or negative cosmological constant have also been studied over the years [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. In [48, 49], however, the authors have found black hole solutions in three and four dimensions that are neither asymptotically flat nor asymptotically (anti) de Sitter. In those works the model is described by the Einstein–Born–Infeld dilaton action. Originally the Born–Infeld non-linear electrodynamics was introduced in the 30’s in order to obtain a finite self-energy of point-like charges [50]. During the last decades this type of action reappears in the open sector of superstring theories [51, 52] as it describes the dynamics of D-branes [53, 54]. Furthermore, in the closed sector of all superstring theories at the massless level the graviton is accompanied by the dilaton that determines the string coupling constant. Since superstring theory is so far the only consistent theory of quantum gravity, it is more than natural to study the QNM of black hole solutions obtained in the framework of Einstein–Born–Infeld dilaton models.

Computing the QNM frequencies in an analytical way is possible in a few cases only [55, 56, 57, 58, 59, 60, 61], while in most of the cases some numerical scheme [62, 63, 64, 65] or semi-analytical methods are employed, such as the well-known from standard quantum mechanics WKB method used extensively in the literature [66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. In the present work we obtain an exact analytical expression for the quasinormal spectrum of a four-dimensional Einstein–Born–Infeld dilaton spacetime, which is well motivated since it contains ingredients found in superstring theory. For a neutral BH, such as the Schwarzschild one, the scalar, vector and tensor perturbations can be studied separately. If, however, the BH is electrically charged then electromagnetic and gravitational perturbations are coupled and must be studied simultaneously [69]. Therefore, in this work we take the first step to study the stability against scalar perturbations by perturbing the BH with a probe scalar field \(\varPhi \), not to be confused with the dilaton \(\phi \) (see the discussion below), hoping to be able to address the problem of the coupled electromagnetic-gravitational perturbation in a future work.

Our work is organized as follows: After this introduction, we present the model and the BH solution in the next section. In Sect. 3 we discuss scalar perturbations where we present the effective potential of the Schrödinger-like equation, while in the fourth section we solve the radial equation in terms of hypergeometric functions. In Sect. 5 we obtain an exact expression for the quasinormal modes, and in Sect. 6 we compare our solution with numerical results obtained with a recently developed method [62]. Finally, we conclude our work in the last section. We use natural units such that \(c = \hbar = 1\) and metric signature \((-, +, +, +)\).

## 2 The model and the BH background

*R*is the Ricci scalar,

*g*is the determinant of the metric tensor \(g_{\mu \nu }\), \(F_{\mu \nu }\) is the electromagnetic field strength, \(\phi \) is the dilaton with a self-interaction potential \(V(\phi )=2 \varLambda e^{-2 \kappa \phi }\), \(\gamma \) is the Born–Infeld parameter, and \(\kappa \) is the dilaton coupling constant. Assuming static spherically symmetric solutions, the line element of the metric is found to be [49]

*b*,

*c*are constants of integration. In the following we set for convenience and without loss of generality \(b=1\) and \(c=0\). Since the model is string inspired, in the following we shall consider the string coupling case \(\kappa =1\). Then the line element takes the form

*L*is given by

*H*is given by [49]

*Q*of the black hole is given by [49]

*L*does not depend on the mass of the black hole.

## 3 Scalar perturbations

*r*. Next, we recast the equation for the radial part into a Schrödinger-like equation of the form

*x*being the so-called tortoise coordinate, and

*d*is a constant of integration which will be taken as unity. Therefore, we obtain the expression for the effective potential

*r*(or at large

*x*, since when \(r \gg r_H\), \(r \simeq e^{x/L}\)) the potential tends to the constant \(V_0\), and therefore defining \(\varOmega \equiv \sqrt{\omega ^2-V_0}\) the solution for \(\psi \) is given by

*x*takes the form of ingoing and outgoing plane waves provided that \(\omega ^2 > V_0\), while the QNMs are determined by requiring that \(D_+=0\).

## 4 Solution of the full radial equation in terms of hypergeometric functions

*z*reads

*F*satisfies the following differential equation

*a*,

*b*,

*c*satisfy the condition \(c-a-b=1-2 \beta \). Therefore, the solution for the radial part is given by

*D*is an arbitrary coefficient, and the hypergeometric function can be expanded in a Taylor series as follows

## 5 Exact quasinormal spectrum

*R*(

*r*) for \(r \gg r_H\) can be written down as follows

*Q*). In particular, for low charge (large \(\gamma \)) the imaginary part is positive, whereas as the charge grows at a certain point the imaginary part becomes negative. Interestingly enough, a behaviour similar to that found in [80, 81] is observed, although the scalar field that perturbs the BH in the present work is not electrically charged. Contrary to these works, however, where it was found that all modes with \(l >0\) were stable, our results show that for any value of the angular momentum there is a certain value of the overtone number after which the modes become unstable.

Scalar QNMs of Einstein–Born–Infeld dilaton black hole for various values of \(\gamma \) and \(\varLambda \). \(l,\,n\) are the angular momentum and overtone number, respectively, and \(L^{-1}=2(1-\varLambda -2H)\), see text. The values without the parenthesis are the exact QNMs, while the ones in the parenthesis are the numerical values

| \(l=0\) | \(l=1\) | \(l=2\) |
---|---|---|---|

\(\gamma =0.112, \, \varLambda =0.01\) (\(L=3.00993\)) | |||

0 | 0 | \(-\,\)2i | \(-\,\)6i |

(0.000) | (\(-\,\)2.000i) | (\(-\,\)6.000i) | |

1 | 0.221489i | \(-\,\)0.445178i | \(-\,\)1.77851i |

(0.221489i) | (\(-\,\)0.445178i) | (\(-\,\)1.77851i) | |

2 | 0.39868i | \(-\,\)0.00131968i | \(-\,\)0.80132i |

(0.39868i) | (\(-\,\)0.00131968i) | (\(-\,\)0.80132i) | |

3 | 0.569543i | 0.283829i | \(-\,\)0.2876i |

(0.569543i) | (0.283829i) | (\(-\,\)0.2876i) | |

\(\gamma =0.5, \, \varLambda =0.1\) ( | |||

0 | 0 | \(-\,\)2i | \(-\,\)6i |

(0.000) | (\(-\,\)2.000i) | (\(-\,\)6.000i) | |

1 | 0.375955i | \(-\,\)0.290712i | \(-\,\)1.62405i |

(0.375955i) | (\(-\,\)0.290712i) | (\(-\,\)1.62405i) | |

2 | 0.676718i | 0.276718i | \(-\,\)0.523282i |

(0.676718i) | (0.276718i) | (\(-\,\)0.523282i) | |

3 | 0.966741i | 0.681026i | 0.109598i |

(0.966741i) | (0.681026i) | (0.109598i) | |

\(\gamma =2, \, \varLambda =0.001\) ( | |||

0 | 0 | \(-\,\)2i | \(-\,\)6i |

(0.000) | (\(-\,\)2.000i) | (\(-\,\)6.000i) | |

1 | 0.623828i | \(-\,\)0.0428385i | \(-\,\)1.37617i |

(0.623828i) | (\(-\,\)0.0428385i) | (\(-\,\)1.37617i) | |

2 | 1.12289i | 0.722891i | \(-\,\)0.0771093i |

(1.12289i) | (0.722891i) | (\(-\,\)0.0771093i) | |

3 | 1.60413i | 1.31842i | 0.746987i |

(1.60413i) | (1.31842i) | (0.746987i) |

## 6 Numerical results

We review a non grid-based interpolation scheme, proposed by Lin et al. [62]. This method makes use of data points in a small region of a query point to estimate its derivatives by employing Taylor expansion. The data points can be scattered, therefore they do not sit on a grid. A key step of the method is to discretize the unknown eigenfunction in order to transform a differential equation and its boundary conditions into a homogeneous matrix equation. Based on the information about *N* scattered data points, Taylor series are carried out for the unknown eigenfunction up to *N*-th order for each discretized point. The resulting homogeneous system of linear algebraic equations is solved for the eigenvalue. A huge advantage of this method is that the discretization of the wave function and its derivatives are made to be independent of any specific metric through coordinate transformation.

This method has been tested thoroughly for its accuracy and efficiency to various differential equation and eigenvalue problems in [62, 63], and [64]. The QNM results have been compared with WKB approximation [66, 67] (up to the sixth order), Horowit z-Hubeny method [82], and continued fraction method [83] achieving very good precision.

We have applied the present method to compute the scalar QNMs of a four-dimensional Einstein–Born–Infeld dilaton black hole, and our numerical results are summarized in Table 1.

We observe that the numerical results agree perfectly with our main result shown in eq. (43). We immediately see that for \(n=0\) the modes depend only on the angular momentum *l*, irrespectively of the choice of \(\gamma \) and \(\varLambda \), which is in agreement with \(\omega _{n=0}=-l (l+1) i\) obtained in the previous section. For \(l=0\), the aforementioned fundamental mode is exactly zero, while for \(l\ge 1\), as *l* increases, the modes decay faster while more stable overtones begin to appear. This is to be expected, since larger angular momentum increases the height of the potential stabilizing the system. Finally, we observe that as *L* decreases the stable overtones decay slower while the unstable ones grow faster.

## 7 Conclusions

To summarize, in this article we have studied the stability under scalar perturbations of (1+3)-dimensional Einstein–Born–Infeld dilaton spacetimes, and we have provided an exact analytical expression for the frequencies, which are found to be purely imaginary. We have confirmed our results computing the frequencies numerically using a recently developed non grid-based numerical scheme. In addition, an instability similar to that seen in charged scalar perturbations of the Reissner–Nordström black hole is observed, although in our work the scalar field that perturbs the BH is not electrically charged.

## Notes

### Acknowledgements

K. D. acknowledges financial support provided under the European Union’s H2020 ERC Consolidator Grant “Matter and strong field gravity: New frontiers in Einstein’s theory” grant agreement no. MaGRaTh-646597 under the supervision of Dr. Vitor Cardoso. G. P. thanks the Fundação para a Ciência e Tecnologia (FCT), Portugal, for the financial support to the Multidisciplinary Center for Astrophysics (CENTRA), Instituto Superior Técnico, Universidade de Lisboa, through the Grant No. UID/FIS/00099/2013. A.R. was supported by the CONICYT-PCHA/ Doctorado Nacional/2015-21151658.

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