Advances in High Energy Physics

Volume 2019, Article ID 5769564, 8 pages

https://doi.org/10.1155/2019/5769564

## Heun-Type Solutions of the Klein-Gordon and Dirac Equations in the Garfinkle-Horowitz-Strominger Dilaton Black Hole Background

^{1}Faculty of Physics, “Alexandru Ioan Cuza” University, 11 Blvd. Carol I, Iasi 700506, Romania^{2}Research Department, Faculty of Physics, “Alexandru Ioan Cuza” University, 11 Blvd. Carol I, Iasi 700506, Romania

Correspondence should be addressed to Cristian Stelea; or.ciau@aelets.naitsirc

Received 20 December 2018; Revised 28 January 2019; Accepted 11 February 2019; Published 26 February 2019

Guest Editor: Subhajit Saha

Copyright © 2019 Marina-Aura Dariescu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study the Klein-Gordon and the Dirac equations in the background of the Garfinkle-Horowitz-Strominger black hole in the Einstein frame. Using a gauge covariant approach, as an alternative to the Newman-Penrose formalism for the Dirac equation, it turns out that these solutions can be expressed in terms of Heun confluent functions and we discuss some of their properties.

#### 1. Introduction

In recent years, black holes with electric or magnetic charge, in presence of a scalar field called* dilaton*, have been studied mainly in string theories. These charged black holes are solutions of the low-energy four-dimensional effective theories obtained by dimensional compactification of the heterotic string theories. Generically, the effective action of these theories describes a massless dilaton coupled to an abelian vector field [1]. Due to the dimensional compactification process, the dilaton is also nonminimally coupled to the Ricci scalar, with the effective solution being described in the so-called string frame. However, to facilitate comparison with the standard black holes in general relativity, it is convenient to go to the so-called Einstein frame by performing conformal rescaling of the metric (for a review, see [2]).

A remarkable black hole solution of the effective four-dimensional compactified theory was found by Gibbons and Maeda [3, 4] and independently rediscovered in a simpler form, few years later, by Garfinkle, Horowitz, and Strominger (GHS) [5] (for a review of its properties, see [6]). Even though, in terms of the string metric, the electric and magnetic black holes have very different properties, in the Einstein frame the metric does not change when we go from an electrically charged to a magnetically charged black hole (this is basically due to the electromagnetic duality present in the Einstein frame; in the string frame, the electromagnetic field strength is also modified by the dilaton field [7]).

Using the GHS metric in the Einstein frame, the present work is devoted to a study of the Klein-Gordon and Dirac equations, which describe charged particles evolving in the Garfinkle-Horowitz-Strominger (GHS) dilaton black hole spacetime. Within a gauge covariant approach, it turns out that the solutions can be expressed in terms of Heun confluent functions [8, 9]. Special attention is given to the resonant frequencies, which arise here by imposing a polynomial form of the Heun functions. In general, the so-called quasi-normal modes have discrete spectra of complex characteristic frequencies, with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping. By comparing these modes with the gravitational waves observed in the universe, one should be able to identify the presence of a GHS black hole [10, 11] (see also in [12] the effect of the dilaton field imprint on the gravitational waves emitted in the collision of two GHS black holes).

When the parameter related to the dilaton field goes to zero, one obtains the Klein-Gordon and Dirac equations for the usual Schwarzschild metric, which have been intensively worked out both in their original form and in different types of extensions. For instance, recently, for the Schwarzschild metric in the presence of an electromagnetic field, the Klein-Gordon and Dirac equations for massless particles have been put into a Heun-type form [13, 14]. One should note that Heun functions are often encountered when studying the propagation of various test fields in the background of various black holes or relativistic stars [15–20] and also in cosmology in the context of extended effective field theories of inflation [21].

The method used in the present paper, while based on Cartan’s formalism, is an alternative to the Newman-Penrose (NP) formalism [22], which is usually employed for solving Dirac equation describing fermions in the vicinity of different types of black holes [23–27].

The structure of this paper is as follows: in the next section, we present the solutions of the Klein-Gordon and Dirac equations in the background of the GHS dilatonic black hole. In Section 3, we discuss the solutions of the massless Dirac equations in this background and show how to recover the expression of the Hawking temperature. The final section is dedicated to conclusions.

#### 2. Klein-Gordon and Dirac Equations on the GHS Dilaton Black Hole Metric

In Einstein frame, the static and spherically symmetric GHS dilaton black hole metric is given by [5]wherewith and being the mass and the charge of this black hole, which has an event horizon at and two singularities located at and . Obviously, if the electric charge of the GHS black hole is zero, the metric in (1) reduces to the Schwarzschild one.

The parameter is related to the dilaton field as (Note that we set the asymptotic value of the dilaton field .)where the* minus* and* plus* signs are for, respectively, the magnetically electrically charged black holes.

Within the gauge covariant formulation, we introduce the pseudoorthonormal frame ; that is,whose corresponding dual base isso that the metric in (1) becomes the usual Minkowsky metric , with .

Using the first Cartan’s equation,with and , we obtain the following connection one-forms , where ; namely,

In the pseudoorthonormal bases (with ), the fourth component of the one-form potential isand it corresponds to an electric field:

##### 2.1. The Klein-Gordon Equation

For the complex scalar field of mass , minimally coupled to gravity, the Klein-Gordon equation has the general -gauge covariant formthat is,wherewith .

The two terms in the left-hand side of relation (11) are, respectively, given byandand the Klein-Gordon equation in (11) can be cast into the following explicit form:Using the separation of the variables with the ansatzwhere are the spherical harmonics, it turns out that the unknown function is the solution of the differential equationThis equation can be solved exactly, with its solutions being expressed in terms of the confluent Heun functions [8, 9] aswith the variableand parameters

The parameters and are purely imaginary, and the radial part of the density probability is given by the square modulus of the Heun functions in (18). The two independent solutions have the generic behavior represented in Figure 1, for . The main features of the probability curve are quite nice; that is, it satisfies all the text-book requirements imposed to physically meaningful wave functions. In this respect, on the horizon and it gets a series of local decreasing maxima, finally vanishing rapidly, at the spatial infinity. If the number of these maxima was finite, the state would be bounded. Otherwise, it could asymptotically radiate. More details on the physical phenomena related to these properties are thoroughly discussed in [28], where the authors are computing the complex values of the energy spectrum coming from the polynomial condition imposed on the Heun functions. In [19, 28], the authors were working in coordinate bases, withso that , where is given in (8).