Scattering of massless scalar waves by Reissner-Nordström black holes

We present a study of scattering of massless planar scalar waves by a charged nonrotating black hole. Partial wave methods are applied to compute scattering and absorption cross sections, for a range of incident wavelengths. We compare our numerical results with semiclassical approximations from a geodesic analysis, and find excellent agreement. The glory in the backward direction is studied, and its properties are shown to be related to the properties of the photon orbit. The effects of the black hole charge upon scattering and absorption are examined in detail. As the charge of the black hole is increased, we find that the absorption cross section decreases, and the angular width of the interference fringes of the scattering cross section at large angles increases. In particular, the glory spot in the backward direction becomes wider. We interpret these effects under the light of our geodesic analysis.

Figure 1

Geodesics in the Reissner-Nordström spacetime for different values of the ratio $q=|Q|/M$. Here, the impact parameter has been chosen to be $b=5.2M$. We see that the black hole has a stronger influence in the particle trajectory for smaller values of $q$.

Figure 2

The effective potential for the extreme Reissner-Nordström black hole as a function of the tortoise coordinate (16), plotted for different choices of $l$.

Figure 3

The effective potential given by Eq. (14) with $l=0$ is plotted for $q=0$ (solid line), $q=0.5$ (dashed line), and $q=1$ (dotted line). The effective potential goes to zero at the event horizon and at infinity. As we can see, the maximum of the potential increases as the black hole charge increases.

Figure 4

The partial absorption cross section with $l=0$ plotted for Reissner-Nordström black holes, with $q=0$ (solid line), $q=0.5$ (dashed line), and $q=1$ (dotted line). We see that ${\sigma }_{\mathrm{abs}}^{(0)}\to A$ as $\omega M\to 0$.

Figure 5

Reissner-Nordström absorption cross section for $q=0$, $q=0.5$, and $q=1$. We see that the total absorption cross section decreases as the black hole charge increases.

Figure 6

Comparison between the scattering cross section of Reissner-Nordström black holes and the glory approximation at $\omega M=3$, for $q=0$, 0.5, and 1. As we can see, for the three cases, the numerical results are in excellent agreement with the glory approximation for angles close to 180°.

Figure 7

Scattering cross section for Reissner-Nordstrom black holes with charges $q=0$ (solid line), $q=0.5$ (dashed line), and $q=1$ (dotted line), for $\omega M=1.0$. The width of the glory peak gets wider for bigger values of the black hole charge.

Figure 8

Scattering cross section for Reissner-Nordstrom black holes with charges $q=0$, $q=0.5$, and $q=1$, for $\omega M=3.0$. Here, as in Fig. 7, the width of the glory peak increases as the black hole charge increases.

Figure 9

Glory parameters $b_g$ and $|db/d\theta {|}_{\theta =\pi }$ shown as a function of $q$. The plots compare the approximation Eq. (a13) (dotted line) with accurate results from numerical integration (solid line). The approximation for $|db/d\theta {|}_{\theta =\pi }$ is clearly less good than the approximation of $b_g$, and the accuracy diminishes further as $q\to 1$.

Figure 10

Intensity of the glory peak $\mathcal{A}$ as a function of $q$. The prediction of the logarithmic approximation is compared with the exact solution of the orbital equation. We also show the intensity of the glory peak computed numerically via the partial wave method for $\omega M=1.0$, 3.0, and 5.0.