Absorption of planar massless scalar waves by Kerr black holes

We consider planar massless scalar waves impinging upon a Kerr black hole, for general angles of incidence. We compute the absorption cross section via the partial wave approach, and present a gallery of results. In the low-frequency regime, we show that the cross section approaches the horizon area; in the high-frequency regime, we show that the cross section approaches the geodesic capture cross section. In the aligned case, we extend the complex angular momentum method to obtain a “sinc” approximation, which relates the regular high-frequency oscillations in the cross section to the properties of the polar null orbit. In the nonaligned case, we show, via a semianalytic approximation, that the reduction in symmetry generates a richer, less regular absorption cross section. We separate the absorption cross section into corotating and counterrotating contributions, showing that the absorption is larger for counterrotating waves, as expected.

Figure 1

Illustrating a planar wave impinging upon a Kerr black hole. The left plot shows a segment of planar wave impinging upon a black hole at angle of incidence $\gamma$ (where $\gamma$ is angle between the black-hole rotation axis and the direction of incidence). The right plot shows the locus of absorption, corresponding to that part of the wavefront which is absorbed in the geometric-optics limit. The locus is described by $b_c(\chi )$, where $\chi$ is the angle between a point on the surface and the projection of the BH rotation axis, and $b_c$ is the critical impact parameter.

Figure 2

Schematic illustration of a slice of a Kerr black hole. Here, $M=1$, $a/M=0.8$ and the event horizon (black) and ergoregion (red) are shown as solid lines. The photon orbit zone, marked in beige, is spanned by the family of constant-radius null geodesics, shown as dotted lines. Special cases include the polar orbit, which runs from pole to pole (and precesses around the black hole), shown as a dashed line at $r/M\approx 2.67$; and the co- and counterrotating equatorial orbits marked by blue dots at $r/M\approx 1.81$ and 3.82, respectively.

Figure 3

Geodesic capture cross section $\sigma$ as a function of angle of incidence $\gamma$, where $\gamma =0$ corresponds to incidence along the black hole’s axis of rotation (and $\sigma$ is symmetric under $\gamma \to \pi -\gamma$). The geodesic capture cross section is the high-frequency asymptote for the planar wave absorption cross section.

Figure 4

The sinc approximation for on-axis incidence ($\gamma =0$). The solid lines show the numerically determined cross section (cf. Fig. 6), and the dashed lines show the sinc approximation, Eq. (25), for a range of $a/M$ and $\omega M$.

Figure 5

Transmission factors ${\Gamma }_{\omega lm}$ for angular multipoles $l=5$ and $-l\le m\le l$. The solid lines show numerical solutions of Eq. (13), and the dashed lines show the semianalytic approximation, Eq. (26).

Figure 6

On-axis ($\gamma =0$) absorption cross section for $a/M=0.00$, 0.60 and 0.99. The horizontal lines represent the high-frequency limits. We see that the general pattern of the on-axis absorption cross section, even for rapidly rotating BHs, is similar to the case of spherical ($a=0$) BHs.

Figure 7

Off-axis absorption cross section for $a/M=0.30$, 0.60, 0.90 and 0.99. For comparison, we also exhibit the on-axis case ($\gamma =0$), and its high-frequency limit (horizontal lines). We see that the oscillation pattern for the off-axis cases differs considerably from the regular one exhibited in the on-axis case.

Figure 8

Co- (${\sigma }^{+}$) and counterrotating (${\sigma }^{-}$) absorption cross section for $\gamma =30$, 60, and 90 deg. The curves are plotted for different BH rotation parameters, namely $a/M=0.30$, 0.60, 0.90, and 0.99. The counterrotating absorption cross sections are larger than the correspondent corotating ones, and their separation becomes larger as $a$ increases.

Figure 9

Partial absorption cross sections for $a/M=0.90$ and $\gamma =60\mathrm{deg}$. We plot the main contributions for fixed values of $l$ and $m$, both for corotating (curves with lower peaks) and counterrotating (higher peaks) cases.

Figure 10

Zoom at the mode with $l=m=1$ for $\gamma =30$, 60, and 90 deg, with $a/M=0.99$. Superradiance is more evident for this mode, although being very small even in this case.