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The Dirac and Klein-Gordon equations and Greybody Radiation for the Regular Hayward Black Hole

We investigate the Dirac and Klein-Gordon equations, as well as greybody radiation, for the Hayward black hole (BH) spacetime. We first consider the Dirac equation using a null tetrad in the Newman- Penrose (NP) formalism. The equations are then separated into angular and radial parts. A pair of one-dimensional Schr\"odinger like wave equations with effective potentials is obtained from the radial part. In order to examine the behavior of potentials, they are plotted with respect to radial distances. Additionally, the Klein-Gordon equation is considered in the Hayward BH spacetime. At the end, we compute greybody factors for bosons and fermions and our results are shown graphically and discussed.


I. INTRODUCTION
Regular black holes (BHs) were discovered first by Bardeen and came to be known as . These types of BH do have an event horizon but no spacetime singularities. The singularity theorem is bypassed in such a way that these BH solutions only satisfy the weak energy condition and not the strong one. Although a regular BH is not the solution to the Einstein vacuum field equations, it can be obtained either by using nonlinear electrodynamics; or by reinventing gravity. The charged version of Bardeen BH was derived by the coupling of Einstein and Maxwell field equations by Ayon-Beato and Garcia [4]. They introduced a nonlinear electric field as a source of charge for the solution of Maxwell field equations. Bronnikov [5][6][7] later demonstrated analytically that charged regular BHs are not correct solutions of the field equations because the electromagnetic Lagrangian in these solutions is invariably different in various portions of space.
Hayward [8], on the other hand, proposed a regular (non singular) BH solution that is comparable to Bardeen BH and has no charge term. This BH solution consists of a compact spacetime region of trapped surfaces with circularly joined inner and outer bounds as a single smooth trapping horizon [8]. The Hayward BH has well-defined asymptotic limits, such as in the limit r → ∞, it is Schwarzschild BH and de-Sitter BH r → 0. The regular Hayward BH has been the subject of a lot of research [9][10][11][12][13][14]. The rotating and modified BH of Hayward is discussed in [15][16], along with its implications. Recently, the quasinormal frequencies for a class of regular black hole solutions generalizing Bardeen and Hayward spacetimes were studied by considering the sixth order WKB approximation [17]. In addition, Hayward spacetime has also been studied by solving Maxwell's equations for Hayward geometry and considering a Hayward spacetime immersed in a magnetic field [18].
The organization of the paper is as follows. Sect. 2 discusses the Hayward BH, writes the Dirac equation in Hayward spacetime, separates the equation into angular and radial parts, and examines the radial equations to end up with a pair of wave equations with effective potentials. In sect. 3, we study the Klein-Gordon equation in Hayward BH spacetime, and we calculate the greybody factors of Hayward BH spacetime for fermions as well as bosons.
The paper ends with the conclusion in Sect. 4.

II. DIRAC EQUATION IN HAYWARD BH SPACETIME
The spherically symmetric static Hayward non singular BH is given by the following line element [8]: where the lapse function f (r) has the following form , 0, 0 , where a bar over a quantity denotes complex conjugation and the dual co-tetrad are given by The non-zero spin coefficients can then be computed as Now, we shall employ the Chandrasekar-Dirac equations (CDEs) [33] (D + − ρ) where F 1 , F 2 , G 1 and G 2 represent the components of the wave function "Dirac spinors" and µ 0 = √ 2µ p is the mass of the particle. The directional derivatives in CDEs are defined by Substituting the spin coefficients and the directional derivatives into CDEs leads to where the operators are We assume a separable solution in the form of is the frequency of the incoming wave and m is the azimuthal quantum number of the wave: Substituting Eqs. (10) into Eq.s (8), the CDEs transform into In order to separate the equations, we define a separation constant. This is carried out by using the angular equations. Actually, it is already known from the literature that the separation constant can be expressed in terms of the spin-weighted spheroidal harmonics.
The radial parts of CDEs become where λ is the separation constant. We further assume that R 1 (r) = P 1 (r) r , R 2 (r) = P 2 (r) r , then Eqs. (12,13) transform into, Using the new functions P 1 (r) = T 1 (r) , P 2 (r) = f 2 T 2 (r), and defining the tortoise coordinate r * as d dr * = f d dr , Eqs. (14,15) become To write Eqs. (16,17) in a more compact form, we combine the solutions in the following way: ψ + = T 1 + T 2 , ψ − = T 2 − T 1 . Hence, we end up with a pair of one-dimensional Schrödinger like equations with effective potentials, The massless Dirac particle (fermions) can be obtained by setting µ 0 = 0 in (20) namely is similar to that shown in Figures 2 and 3.

A. Klein-Gordon equation
The massless Klein-Gordon equation of scalar field in spherical coordinates is written as where g is the determinant of the spacetime metric (1), so that √ −g = r 2 sin θ. For the metric (1) the Klein-Gordon equation becomes To separate the variables in (24), we assume the solutions to the wave equation in the form where ω denotes the frequency of the wave and m is the azimuthal quantum number.
Therefore, the radial part of Klein-Gordon equation (24) becomes where λ is the eigenvalue coming from the physical solution of the angular equation of Θ (θ), namely the standard spherical harmonics [34]. By changing the variable in a new form as R (r) = U (r) r , the radial wave equation (26) recasts into a one dimensional Schrödinger like equation as follows in which r * is the tortoise coordinate: dr * dr = 1 f , and V ef f is the effective potential given by Based on the above equation, the effective potential (28) is dependent on Hayward parameters. Recall that Fig. 1 shows that Hayward spacetimes can have two horizons, which correspond to the BH spacetimes, one horizon, which corresponds to the extremal BH spacetimes, and no horizons, which correspond to no-horizon spacetimes, depending on the mass M and Hayward parameter l. The effective potential vanishes at the two horizons, as the lapse function also vanishes there. Moreover, the lapse function in metric (1) governing the event horizon location, (r approaches the event horizon as r * approaches −∞). Hence, the effective potential will be restricted here only to the regions located outside the event horizon. Figure 5 illustrates how the Hayward parameter affects the effective potential. It indicates that with Hayward parameter, the potentials become more stable than with no Hayward parameter (l = 0), implying that their primary function is to formalize potentials and reduce their peaks. As a result, we can deduce that when the potential peaks decrease, the greybody factor will increase.

B. Greybody Factor of bosons
The greybody factor is used to calculate the deviation of thermal spectra from the black body or Hawking radiation [35,36]. In essence, the greybody factor is a physical quantity related to the quantum nature of a BH. A high greybody factor value indicates a high probability that Hawking radiation can reach spatial infinity [37,38]. In this section, we assume that Hawking radiation is a massless scalar field that obeys the Klein-Gordon equation for bosons and the Dirac equation for fermions. We will use the general method of semi-analytical bounds for the greybody factors [39,40] to compute the greybody factor of the Hayward BH spacetime as follows, in which r * is the tortoise coordinate and where h(r * ) is a positive function satisfying h (−∞) = h (−∞) = ω. For more details, one can see [39,40]. We select h = ω. Thus, Eq. 30 becomes The greybody factor for bosons is computed using the potential from (28). However, we use the classical term-by-term integration technique for obtaining asymptotic integral expansions [41,42], so (31) becomes We plot σ (ω) with various values of the Hayward parameter in Figure 5 to see how this bosonic greybody factor behaves. It can be seen that the Hayward parameter suppresses the greybody factor.
The behavior of spin-(+1/2) and spin-(-1/2) under the effect of the Hayward parameter is depicted in Figures 6 and 7. The figures indicate that, like with bosons, when the Hayward parameter l grows, the greybody factor drops.

IV. CONCLUSION
In this work, we investigated the Dirac and Klein-Gordon equations, as well as greybody radiation in Hayward spacetime, which describes a non-singular BH. In section II, we used null tetrads in the NP formalism to investigate the Dirac equation in a Hayward spacetime In section III, we studied the Klein-Gordon equation in Hayward spacetime and reduced the radial part to one-dimensional Schrödinger equations with effective potentials. Furthermore, we looked into the effect of the Hayward parameter on the effective potentials (Fig.   4). In our study, we found that the potential become more stable when the Hayward parameter was reduced. We also calculated an analytical solution for the greybody factor, which is one of the fundamental pieces of information that can be obtained from BHs for both Finally, the interactions between BHs and fundamental fields help us understand BHs better. We hope to investigate the Dirac and Klein-Gordon equations, as well as the Hawking radiation of the rotating modified Hayward BH [43] in the near future.