# Partial wave analysis of the Dirac fermions scattered from Reissner–Nordström charged black holes

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## Abstract

The asymptotic form of Dirac spinors in the field of the Reissner–Nordström black hole is derived for the scattering states (with \(E>mc^2\)) obtaining the phase shifts of the partial wave analysis of Dirac fermions scattered from charged black holes. Elastic scattering and absorption are studied giving analytic formulas for the partial amplitudes and cross sections. A graphical study is performed for analyzing the differential cross section (forward/backward scattering) and the polarization degrees as functions of the scattering angle.

## Keywords

Black Hole Event Horizon Dirac Equation Differential Cross Section Absorption Cross Section## 1 Introduction

The problem of the quantum fermions scattered from Schwarzschild black holes was studied either in particular cases [1, 2] or by using combined analytical and numerical methods [3, 4, 5, 6, 7]. Recently we performed an analytic study of this process proposing a version of partial wave analysis that allowed us to write down closed formulas for the scattering amplitudes and cross sections [8]. The analytic approach improves our understanding of the quantum mechanisms that governs fermion scattering by black holes.

In the present paper we would like to extend this analytic study to the problem of the Dirac fermions scattered from Reissner–Nordström charged black holes since it seems that this problem was neglected so far. The studies performed in the existing literature have been concentrating mainly on scalar field [9, 10, 11, 12] and electromagnetic scattering [13, 14, 15, 16] on charged black holes. For these reasons we concentrate on studying the problem of fermion scattering on a Reissner–Nordström charged black hole by using analytical and graphical methods. This phenomenon could be interesting since along with the gravitational interaction we can study the effect of the interaction between the charges of the black hole and the fermion charge upon the scattering process. In addition the results related to the fermion absorption by the Schwarzschild black hole will be modified depending on the electric attraction/repulsion between the black hole charge and the fermion charge.

We deduce the asymptotic form of the Dirac spinors in the Reissner–Nordström geometry deriving the phase shifts and the partial amplitudes of elastic scattering as well as the absorption cross section. We present the principal analytic results without extended examples or comments that exceed the space of this short paper. We use the methods and notations of Ref. [8] and Planck’s natural units with \(G=c=\hbar =1\).

## 2 Asymptotic spinors in Reissner–Nordström geometry

The Dirac equation in curved spacetimes is defined in frames \(\{x;e\}\) formed by a local chart of coordinates \(x^{\mu }\), labeled by natural indices, \(\alpha ,\ldots ,\mu , \nu ,\ldots =0,1,2,3\), and an orthogonal local frame and coframe defined by the gauge fields (or tetrads), \(e_{\hat{\alpha }}\) and, respectively, \(\hat{e}^{\hat{\alpha }}\), labeled by the local indices \(\hat{\alpha },\ldots ,\hat{\mu },\ldots \) with the same range. In local-Minkowskian manifolds (*M*, *g*), having as a flat model the Minkowski spacetime \((M_0,\eta )\) of metric \(\eta =\mathrm{diag}(1,-1,-1,-1)\), the gauge fields satisfy the usual duality conditions, \(\hat{e}^{\hat{\mu }}_{\alpha }\, e_{\hat{\nu }}^{\alpha }=\delta ^{\hat{\mu }}_{\hat{\nu }},\,\, \hat{e}^{\hat{\mu }}_{\alpha }\, e_{\hat{\mu }}^{\beta }=\delta ^{\beta }_{\alpha }\) and the orthogonality relations, \(e_{\hat{\mu }}\cdot e_{\hat{\nu }}=\eta _{\hat{\mu }\hat{\nu }}\,,\, \hat{e}^{\hat{\mu }}\cdot \hat{e}^{\hat{\nu }}=\eta ^{\hat{\mu }\hat{\nu }}\). The gauge fields define the 1-forms \(\omega ^{\hat{\mu }}=\hat{e}^{\hat{\mu }}_{\nu }\mathrm{d}x^{\nu }\) giving the line element \(\mathrm{d}s^2=\eta _{\hat{\alpha }\hat{\beta }}\omega ^{\hat{\alpha }}\omega ^{\hat{\beta }}=g_{\mu \nu }\mathrm{d}x^{\mu }\mathrm{d}x^{\nu }\).

*m*, written with our previous notations [8] in the frame \(\{x;e\}\) defined by the Cartesian gauge [17, 18],

*M*and charge \(Q>0\) with the Reissner–Nordström line element

^{1}In the following we study the scattering solutions of the Dirac equation in the asymptotic domain where \(r\gg r_+\).

*E*are the common eigenspinors of the operators \(\{H_D,\, K,\, J_3\}\) corresponding to the eigenvalues \(\{ E,\kappa ,m_j\}\),

*l*and \(j=l\pm \frac{1}{2}\) as defined in Refs. [20, 21] (while in Ref. [7] \(\kappa \) is of opposite sign). We note that the antiparticle-like energy eigenspinors can be obtained directly using charge conjugation as in the flat case [22].

*x*, we can use the Taylor expansion of these equations with respect to \(\frac{1}{x}\), neglecting the terms of the order \(O(1/x^2)\). We obtain thus the asymptotic radial problem [8, 25], which can be rewritten as \(\mathcal{E}'\mathcal{F}=0\) where the new matrix operator takes the form

*x*by using the matrix

## 3 Partial wave analysis

*U*whose asymptotic form,

### 3.1 Phase shifts

The partial wave analysis exploits the asymptotic form of the exact analytic solutions which satisfy suitable boundary conditions that in our case might be fixed at the (exterior) event horizon (where \(x=0\)). Unfortunately, here we have only the asymptotic solutions (21) and (22) whose integration constants cannot be related to those of the solutions near the event horizon without resorting to numerical methods [1, 2, 3, 4, 5, 6, 7]. Therefore, we must choose suitable asymptotic conditions for determining the integration constants.

The arguments presented in Appendix C of Ref. [8] and briefly summarized here in Appendix A show that for \(Q=0\) we may obtain elastic collisions, with real phase shifts and a correct Newtonian limit for large angular momenta, only if we adopt the general asymptotic condition \(C_2^+=C_2^-=0\), eliminating thus the terms that diverge for \(x\rightarrow 0\). We believe that it is natural to keep this condition in the most general case of the Reissner–Nordström metric if we desire to have a smooth limit for \(Q\rightarrow 0\).

*l*are related as in Eq. (10), i.e. \(l=|\kappa |-\frac{1}{2}(1-\mathrm{sign}\,\kappa )\). The remaining point-dependent phase,

*k*obeying

*s*has a special role to play since it can take either real values or pure imaginary ones regardless of the fermion mass.

### 3.2 Elastic scattering

*s*the scattering is elastic since in this case the identity (31) guarantees that the phase shifts of Eq. (29) are real numbers such that \(|S_{\kappa }|=1\). Obviously, this happens only when \(\kappa \) (at given

*p*) satisfies the condition

*k*. Then the scalar amplitudes of Eq. (27),

### 3.3 Absorption

*l*for the values of

*p*(or

*E*) satisfying the condition \(k>l\). This means that for any fixed value of

*l*there is a threshold, \(E_l\), defined as the positive solution of the equation

## 4 Graphical discussion of the results

Let us now discuss some physical consequences of our results encapsulated by the formulas presented in the previous sections. For a better understanding of the analytical results we perform a graphical analysis of the differential cross section in terms of the scattering angle \(\theta \).

*m*th reduced ones,

In the following we focus our analysis on scattering from small or micro black holes (with \(M \sim 10^{15} - 10^{22}\) kg) since in this case the wave length of the fermion (\(\lambda =2\pi h/p\)) and the Schwarszchild radius (\(r_S=2M\)) have the same order of magnitude so that we can observe the presence of glory and orbiting scattering.

*r*.

The dependence of the absorption cross section in terms of energy (*E* / *m*) is given in Fig. 5, where the plots were obtained for \(l=1,2,3,4\). We observe that the modes with small angular momenta have the most important contribution to the absorption, because as we increase *l* the maxima observed in Fig. 5 becomes smaller. Also we observe that the maxima are shifted to the right as we increase the value of *l*. Our graphical result for the absorption cross section is similar to those obtained in [27], where the absorption of scalar particles on a dilaton black hole was studied.

## 5 Concluding remarks

This is the basic framework of the relativistic partial wave analysis of the Dirac fermions scattered by charged black holes in which we consider exclusively the contribution of the scattering modes. Our results are in accordance with the Newtonian limit since in the large-*l* limits and for very small momentum we can take \(s\sim |\kappa |\sim l\) and \(\lambda \sim q\) so that our phase shifts (29) become just the Newtonian ones [1, 2].

## Footnotes

- 1.
In this system \(\alpha \simeq \frac{1}{137}\) is the fine structure constant while the electron mass is \(m_e=\sqrt{\alpha _G}\simeq 4.178\, 10^{-23}\).

## Notes

### Acknowledgments

C.A. Sporea was supported by a grant of the Romanian National Authority for Scientific Research, Programme for research-Space Technology and Advanced Research-STAR, project nr. 72/29.11.2013 between Romanian Space Agency and West University of Timisoara.

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