Scattering by regular black holes: Planar massless scalar waves impinging upon a Bardeen black hole

Singularities are common features of general relativity black holes. However, within general relativity, one can construct black holes that present no singularities. These regular black hole solutions can be achieved by, for instance, relaxing one of the energy conditions on the stress-energy tensor sourcing the black hole. Some regular black hole solutions were found in the context of nonlinear electrodynamics, the Bardeen black hole being the first one proposed. In this paper, we consider a planar massless scalar wave scattered by a Bardeen black hole. We compare the scattering cross section computed using a partial-wave description with the classical geodesic scattering of a stream of null geodesics, as well as with the semiclassical glory approximation. We obtain that, for some values of the corresponding black hole charge, the scattering cross section of a Bardeen black hole has a similar interference pattern to a Reissner-Nordström black hole.

Figure 1

Geodesics approaching a Bardeen BH from infinity with an impact parameter of $b=5.2M$, for different values of the BH charge. The Schwarzschild case ($Q=0$) is also exhibited (solid line).

Figure 2

Left panel: Classical scattering cross section of Bardeen BHs, with $Q=0.5,0.8$ and 1, and for the Schwarzschild BH ($Q=0$). Right panel: Classical scattering cross section of Bardeen BHs, normalized by the Schwarzschild case.

Figure 3

Glory scattering parameters for Bardeen BHs with varying charge, considering only the dominant contribution ($\mathrm{\Theta }=\pi$). We note that, with the exception of $|db/d\theta {|}_{\theta =\pi }$, all the important parameters related to the glory scattering decrease monotonically with the increase of $Q$.

Figure 4

Scattering cross sections for Bardeen BHs considering different BH charges. We also plot the Schwarzschild case, for comparison. We see that the value of the charge affects the fringe widths, while the (avarage) amount of scattered flux remains basically the same.

Figure 5

Comparison of partial-wave, semiclassical glory and classical geodesic approaches for the differential scattering cross section, for $M\omega =3$ (in the first two cases), and different values of the Bardeen BH charge [$Q=0.5$ (left) and $Q=1$ (right)]. The semiclassical glory approximation reproduces very well the results for backscattered waves ($\theta \sim \pi$), while the classical approach works well for small scattering angles.

Figure 6

Glory intensity in the backward direction ($\theta =\pi$) as a function of the Bardeen BH charge. We see that the amplitude computed through the partial-wave method oscillates around the one computed through the glory approximation.

Figure 7

Comparison between Bardeen and RN BH scalar scattering. Top-left panel: The case of extremal Bardeen and RN BHs for $M\omega =3.0$. Top-right panel: Bardeen extreme BH scattering compared with the scattering from a RN BH with $Q=0.753$, for $M\omega =0.5$. Bottom panels: The same as the top-right panel, but with $M\omega =1.5$ (left) and $M\omega =3.0$ (right).