Fermion scattering by a class of Bardeen black holes
Abstract: In this paper the scattering of fermions by a class of Bardeen black holes is investigated. After obtaining the scattering modes by solving the Dirac equation in this geometry, we use the partial wave method to derive an analytical expression for the phase shifts that enter into the definitions of partial amplitudes that define the scattering cross sections and the induced polarization. It is then showed that, like in the case of Schwarzschild and Reissner-Nordström, the phenomena of glory and spiral scattering are present.
pacs:04.70.-s, 03.65.Nk, 04.62.+v.
The existence of black holes it is well motivated theoretically in gravity theories models. Thus far, black holes were not directly detected or observed. However, there exists plenty of indirect evidences that indicate that such objects are real and do exist in nature. As showed by Penrose and Hawking hawking.book () in general relativity the occurrence of singularities is inevitable. This brings many issues to the table like the black hole information paradox SW1 (); SW2 (); SW3 () or the paradox discovered more recently of black hole’s ”firewalls” fire1 (); fire2 (); fire3 (); fire4 (). These paradoxes emerge because of the incompatibility between quantum theory and general relativity. It is widely believed that in a quantum theory of gravity the singularities contained in black holes will be removed. However, even in the early stages of investigations on singularities sakharov (); gliner () in general relativity, there were proposed models of black holes that could avoid the occurrence of a singularity. These black holes are said to be ”regular” in the sense of being singularity-free.
The first proposal of a regular black hole solution was made by Bardeen in ref. Bardeen1 (), and since then many other models of spherically symmetric regular black holes where presented in the literature borde (); rbh1 (); rbh2 (); rbh3 (); rbh4 (); rbh5 (); rbh6 (); rbh7 (); rbh8 (). In Ref. Ayon () the authors showed that the Bardeen black hole model can be physically interpreted as the gravitational filed produced by a nonlinear magnetic monopole. Later on this interpretation was extended to include also nonlinear electric charges so that one can now say that regular black holes models can have as a source a nonlinear electromagnetic field. More recently, in ref. wang () the authors extended the Bardeen solution to an entire class of Bardeen-like black holes that can be regular or not.
In this paper we will study the scattering of fermions (spin ) by a Bardeen regular black hole and by a Bardeen-class of black holes (as constructed in wang ()). We will use the partial wave method to obtain analytical expressions for the phase shifts that enter into the definition of partial amplitudes defining the scattering cross sections and the induced polarization. To our knowledge this is the first study to report analytical phase shifts for spin wave scattering by regular black holes. In previous works only the absorbtion of fermions by Bardeen black holes was investigated numerically in huang2 (), while in refs. huang1 (); Macedo2 (); Macedo2a (); Macedo2b () the case of massless scalar scattering by regular black holes was treated again numerically. Studies dedicated to fermion scattering by other types of spherically symmetric black holes can be found for example in refs. unruh (); Das (); Jin (); Doran (); Jung (); Gaina (); ChaoLin (); dolan (); Rogatko (); Liao (); Ghosh (); sporea1 (); sporea2 (); sporea3 ().
Both Bardeen regular black holes and the Bardeen-class type of black holes possess nonlinear magnetic (monopole) charges Ayon (); wang (). This implies the existence of an electromagnetic potential of the form with the total magnetic charge. In this work we will neglect the interaction between this potential and the charge of the fermion and focus instead only on studying the scattering resulted from the ”pure” gravitational interaction between the fermion and the black hole. However, even in this approximation the black hole magnetic monopole charge will still influence the scattering patterns through the presence of into the metric function and into the resulted scattering modes of the radial Dirac equation.
The rest of the paper is structured as follows. In Section II the Bardeen-class of black holes is presented very briefly. Section III starts with a very short review of the Dirac equation in spherically symmetric black hole geometries and continues with the search for scattering modes in the Bardeen spacetime. In the last part of this section the main result of the paper is presented, namely the form of the analytical phase shifts resulted from applying the partial wave method on the scattering modes derived earlier. The next Section IV is dedicated to a graphical analysis of the induced polarization and the scattering cross sections in which the presence of a backward ”glory” and ”spiral scattering” (orbiting) oscillations is showed to be present. The main conclusions and some final remarks are given in the last Section V of the paper.
Ii A class of Bardeen-like black holes
In Ref. wang () a class of spherically symmetric and asymptotically flat Bardenn-like black holes depending on two-parameters was constructed, having the following line element
These black holes have a singularity if the parameter , referred from now one as the ”Schwarzschild mass”, takes a nonzero value. Otherwise, if regular black holes are obtained and the Bardeen black hole corresponds to the particular choice and . The term can be interpreted as the mass of the nonlinear magnetic monopole. Moreover, the sum
constitutes the ADM mass of the black hole resulted from the asymptotic form of the metric function . The parameter entering eq. (1) is related to the magnetic monopole charge by the relation .
where has dimension of length squared and is a dimensionless constant. In the week field limit one gets a vector field that is sightly stronger when compared with a Maxwell field.
Iii Dirac fermions and scattering cross sections
iii.1 Dirac equation. Preliminaries
The Dirac equation
can be brought to the following explicit form
where and the covariant derivative is defined by , with the generators of the group; the point-independent Dirac matrices obeying and is a spin-connection
with the usual Christoffel symbols. The tetrad fields and are point dependent defining (non-holonomic) local frames and co-frames and the following relations hold
the Dirac equation (5) can be reduced to only a radial equation. The angular part of the Dirac equation is the same as in the Dirac theory from flat spacetime and it’s solutions are the usual 4-component angular spinors Thaller (); Landau (). This is due to the fact that in the Cartesian gauge (8) the Dirac equation is manifestly covariant under rotations cota2 (). Using this gauge it was possible to find complete analytical solutions to the Dirac equation on de Sitter/anti-de Sitter space time cota3 (); cota3a (); crucean1 (); crucean2 () and approximative analytical solutions in black hole geometries cota (); sporea4 (); sporea1 (); sporea2 (); sporea3 () that were later on used to study different aspects of the scattering problem on those spacetimes sporea1 (); sporea2 (); sporea3 (); crucean3 (); crucean4 (); crucean5 (); crucean6 ().
The remaining unsolved radial part of the Dirac equation is found by assuming the following type of particle-like solutions with a given energy
iii.2 Scattering modes
Inserting the line element (1) into eq. (10) produces a system of two differential equations that has no known analytical solutions due to the complex form of the line element (1). However, because we are interested here only in finding the scattering modes, one can approximate eq. (10) in the asymptotic region of the Bardeen black hole class specified by relation (1) and find approximative analytical solutions. On these new solutions a partial wave method will be used in order to calculate the (elastic) scattering cross section and the induced polarisation resulted after the interaction of a fermion beam with the black hole.
Let us start by introducing the new variable
with the radius of the black hole horizon. In terms of the new introduced variable the function becomes
and where the notation was introduced. The radial Dirac equation (10) is in fact a system of two differential equations for the radial wave functions that, after multiplying each equation by and expressing all terms as a function of the new variable given by eq. (11), is equivalent to eq. (13)
The system of differential equations obtained in (13) can not be solved analytically as it is. However, if we are interested in obtaining an analytical solution this can be done if we restrict to a domain far away from the black hole event horizon. One obtains a more simple system of differential equations that have analytical solutions. By making a Taylor expansion with respect to and discarding the terms of the order and higher, while keeping all the other remanning terms, the system of equations, valid in the asymptotic region of the black hole for the two radial wave function, reduces to
We observe that if then the ratio and recovering the Schwarzschild case described in ref. cota ().
For finding the scattering modes, that corresponds to the case , it proves useful to introduce new radial wave functions , that consist in the following combination of the old wave functions
Using the above relation and after some computations one arrives at the following equations satisfied by the functions and , namely
where . Equations (16) can now be solved using Maple or Mathematica and the analytical solutions can be written as a combination of Whittaker functions
where and the following parameters were introduced
iii.2.1 Pure Bardeen black hole case
As already mentioned, by choosing and in eq. (1) the original Bardeen black hole solution is recovered. This solution is a black hole with two distinct horizons only if , it has degenerate horisons if and for there are no horizons present. Because the function has now a more simpler form,
one can find and write an analytical expression for the location of the black hole outer horizon, denoted from now one by . Moreover, one can easily show that , where the function depends only on the ratio . The existance of horizons imposes the constraint borde ().
and also now and . We will see in section IV that this is enough to produce a noticeable difference in the scattering patterns.
iii.3 Analytical phase shifts and scattering cross sections
that depend only on the scattering angle :
The resulted final form for the point-independent phase shifts is given by the following expression
where we used for the same sign convention as in Landau () such that and .
The series (22) are poorly convergent as a direct consequence of the singularity present at that requires an infinite number of Legendre polynomials to describe it. To make the series more convergent one can define the th reduced series
with and are taken from (22). We found that using only two iterations for function and one iteration for it is sufficient to make the series convergent enough without distorting too much the analytical results.
Iv Results and discussion
In this section we present and discuss the main features of fermion scattering by Bardeen regular black holes and also by a Bardeen-class of black holes. The analysis will focus on scattering by small or micro black holes (with Kg) because in this case the glory and orbiting scattering phenomena are showed to be significant.
In labeling the figures we will make use of the following parameters: the speed of incident fermions; that can be seen as a dimensionless measure of the gravitational coupling because (restoring the units) it forms the dimensionless quantity: , with the Schwazschild radius and the associated quantum particle wavelength; and the ratios and that appear when writing the black hole horizon radius as . Moreover, sice in the asymptotic zone , one can easily show that such that the condition is always satisfied. In the following analysis we will take in all the plots, because the same conclusions are obtained also for the cases with .
In Fig. 1 the differential scattering cross section as a function of the scattering angle is presented for fixed values of the ratios (left panels corresponding to pure Bardeen case); and ; a fixed value of the speed () of incoming fermions, while the parameter (or for Bardenn-class) takes different values. As it can be seen in Fig. 1 the scattering pattern takes a simple form for small values of . However, as the value of is increased more complex scattering patterns start to appear, including the presence of a maxima in the backward direction () known also as ”glory” scattering Ford1 (); Ford2 () and the presence of oscillations in the scattering intensity that give rise to orbiting or ”spiral scattering” Matzner1 (); Matzner2 () (that may occur when the particle’s ”classical” orbit passes the scattering center multiple times). As the speed of the incoming fermion is increased the peak in the -direction starts to move to the left and the scattering intensity maxima that was occurring at for non-relativistic fermions is transforming into a minima if the fermions are massless (). As it can be best seen in the bottom panels of Fig. 1 the magnitude of the spiral scattering oscillations and their angular frequency are increasing with the mass of the black hole.
Comparing the scattering intensity in Fig. 2 of Bardeen regular black hole (blue and red curves) with that of Schwarzschild black hole (the black dotted curve) we observe that the glory peak is higher for the scattering by a Schwarzschild black hole, while as the ratio increases the maxima in the backward direction becomes lower and at the same time the frequency of oscillations in the spiral scattering are slightly decreasing as well. The curve with corresponds to the Bardeen degenerate case, when only one horizon is present. From an observational astrophysical point of view if it will ever be possible to observe a scattering pattern produced by a beam of fermions when encountering in their path a black hole, then one can say if that black hole is a Schwarzschild black hole or a Bardeen one by analyzing the scattered signal and if the mass of the black hole is known, then the value of can also be found.
The scattering cross section for the Bardeen-class black holes (1) is plotted in Fig. 3. The left panels show the scattering patterns for Bardeen-class black holes that have the ”Schwarzschild mass” bigger than the mass of the magnetic monopoles, while in the right panels the opposite is true. From our analysis it resulted that when compared with the Schwarzschild black hole, the scattering intensity in the backward direction () is higher for the scattering by a Bardeen-class black hole. From Fig. 3 one can also observe that in all the plots as the value of is increasing more oscillations start to appear into the scattering intensity. Furthermore, if we assume fixed then it results that the -mass of the black hole is increasing (which is equivalent with increasing the mass , eq. 2, if remains constant) and as a consequence the angular frequency of the oscillations present in the scattering intensity are increasing as well. This feature was sowed to be true also for fermion scattering by Schwarzschild and Reissner-Nordström black holes dolan (); sporea1 (); sporea2 (); sporea3 (). Thus one can wonder if the frequency of the oscillations of the spiral scattering are increasing with the black hole mass for any type of (spherically symmetric) black holes.
In Fig. 4 we have plotted the differential scattering cross section as a function of the scattering angle for a Schwarzschild black hole (blue solid lines), a regular Bardeen black hole (black dotted lines) and a Bardeen-class black hole (red dash-dotted lines).
An incident unpolarized beam of massive fermions becomes partially polarized after it gets scattered by the black hole. The polarization degree induced can be computed using the following formula Landau ()
with a unit vector specifying the direction of the scattered wave.
In Fig.5 the dependence of the polarization on the scattering angle is plotted for given values of the parameters . As can be seen it has a pronounced oscillatory behaviour. From the top panels in Fig.5 one can observe that if the parameter is increasing then the frequency of the oscillations present in the polarization are increasing as well. Now if we assume that the energy of the fermion is fixed, then this means that the oscillations present in the polarization are more pronounced for black holes with higher masses. Compared with the Schwarzschild polarization (see bottom panels in Fig. 5) the Bardeen and Bardeen-class black hole polarizations have a slightly less oscillatory behaviour. The oscillations present in the polarization can be seen as a consequence of the oscillations present in the glory and spiral scattering.
The alignment of the scattered fermions with the forward on-axis direction can be visualized using polar plots representations of the polarization degree as showed in Fig.6. One can observe the Mott polarization in the direction orthogonal to the scattering plane, phenomena also reported before in the literature for Schwarzschild dolan (); sporea1 () and Reissner-Nordström sporea2 () black holes.
In this work we have studied the scattering of fermions by a class of Bardeen black holes that include also the original Bardeen regular black hole solution. A partial wave method was used on a set of scattering modes obtained by solving the Dirac equation in the asymptotic region of these black hole geometries. In this way we were able to obtain for the first time analytical phase shifts as in our previous studies sporea1 (); sporea2 (); sporea3 () on fermion scattering by Schwarzschild and Reissner-Nordström black holes. In Section V it was showed that the phenomena of glory (scattering in the backward direction) and spiral scattering (oscillations in the scattering intensity) are present. We also saw that an incident unpolarized beam could become partially polarized after the interaction with the black hole.
In figs. 1-6 besides the parameters (that can be associated with a measure of the gravitational coupling) and (speed of the fermion) we also used the ratios , and to label the figures. The departure of the scattering pattern from the Schwarzschild case becomes significant as and increases. For the original Bardeen regular black hole, is the maximum value allowed and it corresponds to the degenerate case, when the two horizons coincide. If the magnetic charge , that also implies and , then one recovers the scattering by a Schwarzschild black hole discussed in our previous paper sporea1 ().
As was showed the glory and spiral scattering start to become significant for values of the parameter or bigger (in geometrical units with ) due to the fact that for this values the associated wavelength of the incident fermions is of the same order of magnitude as the black hole horizon radius and thus diffraction patterns start to occur. Another feature, that was showed to be present also for fermion scattering by Schwarzschild dolan (); sporea1 () and Reissner-Nordström black holes sporea2 (), is that as the total mass of the black hole is increasing the oscillations present in the scattering intensity become more frequent meaning that spiral scattering is significantly enhanced with the black hole mass.
Magnetic monopoles, although long theoreticized, have not yet been detected or observed experimentally. Moreover, in grand unified theories (GUT), magnetic monopoles are inevitable. However, it is assumed that in the present day era of the universe there are almost no magnetic monopole relics that survived. Now if one assumes that such relics can be contained in a black hole, then by analyzing the signal produced by the scattering of fermions (or other particles) by a black hole containing magnetic monopoles (such as the Bardeen-class discussed here) then it would be possible to obtain at least some indirect evidences for the existence of magnetic monopoles.
Acknowledgements.This work was supported by a grant of Ministery of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-PD-2016-0842, within PNCDI III.
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