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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

In recent years, black holes with electric or magnetic charge, in presence of a scalar field called

A remarkable black hole solution of the effective four-dimensional compactified theory was found by Gibbons and Maeda [

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

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 [

The method used in the present paper, while based on Cartan’s formalism, is an alternative to the Newman-Penrose (NP) formalism [

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

In Einstein frame, the static and spherically symmetric GHS dilaton black hole metric is given by [

The parameter

Within the

Using the first Cartan’s equation,

In the pseudoorthonormal bases (with

For the complex scalar field of mass

The two terms in the left-hand side of relation (

The parameters

The square modulus of the function

In the particular case

The spinor of mass

In contrast to the Klein-Gordon case, the situation is more complicated in the case of the Dirac equation (

Using the Weyl representation for the

Thus, the angular parts

As for the radial equations, we employ the auxiliary function method and consider

Function (

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

If one imposes

Near the exterior event horizon,

The Dirac equation has been worked out for several physically important metrics, mainly using the NP formalism [

In view of the analysis developed in the previous section, the massless and chargeless fermions are described by the radial equations coming from system (

The solutions to Heun confluent equations are computed as power series expansions around the regular singular point

For large

The necessary condition for a polynomial form of the Heun confluent functions in (

In order to study the radiation emitted by the GHS black hole, one has to take the radial solution near the exterior event horizon,

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

Unlike the case for bosons, it turns out that, for the charged massive fermions interacting with the GHS dilaton black hole, the radial equation (

In the massless case, the Dirac equation can be analytically solved and the derived solution, given by (

Finally, by identifying the

No data were used to support this study.

The authors declare that they have no conflicts of interest.

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.