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).
In the particular case , corresponding to the familiar Schwarzschild black hole, the function has the same expression as in (18) but with the variable and parameters computed for .
2.2. The Dirac Equation
The spinor of mass minimally coupled to gravity is described by the Dirac equationwith
In contrast to the Klein-Gordon case, the situation is more complicated in the case of the Dirac equation (22) and this complication is basically due to the square root , which appears in the expressions of and . Thus, with the term expressing the Ricci spin-connection given bythe Dirac equation becomesAs in the previous Klein-Gordon case, one can use the separation of the variableswith the function defined asand one obtains the explicit expression of the differential equation satisfied by :where
Using the Weyl representation for the matrices,withwhere denote the usual Pauli matrices and (28) becomesand one may use again the standard procedure based on the separation of the variables. Thus, with the bi-spinor written in terms of two components spinors aswhereone obtains the following system of coupled radial equations for the components and , that is,and take into account the following essential relations:
Thus, the angular parts , with , are satisfying the decoupled equationswith the solutions given by the spin-weighted spherical harmonics [29], for .
As for the radial equations, we employ the auxiliary function method and consider and asso that system (35) leads to the following simpler equations for the unknown functions :The differential equation for , that is,can not be analytically solved. Numerically, using Mathematica [30], with the initial conditionsthe absolute value of the radial part of given in (26), namely,is represented in Figure 2, for .
This is describing the fermionic ground state in the outer region, with just one maximum (as it should) and exponentially vanishing at infinity. We have not analyzed the corresponding modes located within the black hole, since, in the limit , the area of the sphere is zero so that this surface is singular. Once increases, the singular surface moves towards the event horizon , and one has to solve the problems related to the physically meaningful boundary conditions.
If one imposes and performs a series expansion of the last term multiplying in (40), to first order in , this last term can be approximated to and the solution is given by the Heun confluent functions [8, 9] aswith the parameters
Near the exterior event horizon, ; that is, ; the Heun confluent functions in (43) have a polynomial form if their parameters are satisfying the condition [8, 9]By replacing the expressions of (44) in (45), one gets the relationfor the Heun function multiplied by , and the relationfor the one multiplied by . These relations are pointing out a quantized part of the imaginary part of , which corresponds to resonant frequencies [28].
3. The Massless Case
The Dirac equation has been worked out for several physically important metrics, mainly using the NP formalism [25] and some of the solutions, especially in the massless case, have been expressed in terms of Heun confluent functions [13].
In view of the analysis developed in the previous section, the massless and chargeless fermions are described by the radial equations coming from system (35); namely,withThus, system (48) turns into the simpler formwhich leads to the following differential equation for :and similarly for . The solution of this equation is expressed in terms of Heun confluent functions aswhere the variable isand the corresponding parameters are
The solutions to Heun confluent equations are computed as power series expansions around the regular singular point ; that is, . The series converges for (the second regular singularity) and the analytic continuation is obtained by expanding the solution around the regular singularity and overlapping the series.
For large values, one may use the formula [8, 28]where is an arbitrary constant and is the phase shift. With the parameters given in (54), the component from (49) gets the asymptotic formwhich, for large values, behaves like
The necessary condition for a polynomial form of the Heun confluent functions in (45) leads to the following quantized imaginary quasi-spectrum:
In order to study the radiation emitted by the GHS black hole, one has to take the radial solution near the exterior event horizon, . For , the Heun functions can be approximated to and the (radial) components of defined in (26) and (33), with given in (49), can be written as andpointing out the in and out modesBy definition, the component should asymptotically have the formso that the relative scattering probability at the exterior event horizon surface is given byInspecting the above relations, it yields the well-known results: ,and the mean number of emitted particleswhere is the Hawking temperature.
4. Conclusions
In the present paper, we have used the free-of-coordinates formalism to write down both the Klein-Gordon and the Dirac equations, in their expression, for the GHS metric in (1).
Unlike the case for bosons, it turns out that, for the charged massive fermions interacting with the GHS dilaton black hole, the radial equation (40) does not have an analytic solution. However, to first order in , the corresponding equations are satisfied by the Heun confluent functions (43).
In the massless case, the Dirac equation can be analytically solved and the derived solution, given by (52), is valid for the whole space, which includes not only the near-horizon region but also the far-away-from-the-black-hole region. Once the relation in (45) among the Heun function’s parameters is imposed, the confluent Heun functions can be cast into a polynomial form and the energy spectrum is given by the imaginary quantized expression (58).
Finally, by identifying the out modes near the event horizon, we identified the Hawking black body radiation and the expected Hawking temperature is correctly recovered.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, Project no. PN-III-P4-ID-PCE-2016-0131, within PNCDI III.