Fermion absorption cross section of a Schwarzschild black hole

We study the absorption of massive spin-half particles by a small Schwarzschild black hole by numerically solving the single-particle Dirac equation in Painlevé-Gullstrand coordinates. We calculate the absorption cross section $\sigma$ over a range of gravitational couplings $Mm/m_P{}^2$ and incident particle energies $E$. At high couplings, where the Schwarzschild radius $R_S$ is much greater than the particle wavelength $\lambda$, we find that $\sigma (E)$ approaches the classical result for a point particle. At intermediate couplings, where $R_S\sim \lambda$, we find oscillations around the classical limit whose precise form depends on the particle mass. At low couplings, where $R_S\ll \lambda$, we demonstrate that the minimum-possible cross section approaches $\pi R_S{}^2/2$. For high incident particle energies, the cross section converges on the geometric-optics value of $27\pi R_S{}^2/4$. At low particle energies we find agreement with an approximation derived by Unruh.

Figure 1

Classical and quantum absorption cross section. The plot compares the absorption cross section for the Dirac wave [solid curve] with the classical prediction for a point particle [dotted curve], for a gravitational coupling of $\alpha =Mm/m_P{}^2=0.2$. The cross section is plotted in units of $(GM/c^2{)}^2$ (proportional to the event horizon area), and the energy in units of the rest mass energy $m{c}^2$.

Figure 2

Partial wave cross sections at $\alpha =0.2$. This plot shows the contribution from each partial wave [labeled by angular momentum $\kappa =\pm (j+1/2)$] to the total cross section ${\sigma }_{\mathrm{tot}}$, for a gravitational coupling of $\alpha =Mm/m_P{}^2=0.2$. The cross section is plotted in units of $(GM/c^2{)}^2$ (proportional to the event horizon area), and the incident particle energy in units of the rest mass energy $m{c}^2$.

Figure 3

Quantum absorption cross section for a range of couplings. The plot shows the cross section at a range of gravitational couplings, $0.05\le Mm/m_P{}^2\le 0.2$. The horizontal line is the photon limit.

Figure 4

Quantum absorption cross sections in the ${\lambda }_C\gg R_S$ limit. The plot shows the absorption cross section as a function of energy, for small couplings, $\alpha =Mm/m_P{}^2\ll 1$. In the low-energy region plotted here, the wavelength of the Dirac particle is large compared to the black hole event horizon. Absorption is dominated by the lowest $j=1/2$ angular momentum states ($\kappa =1,-1$ states), and the low-energy approximation of Unruh [7] is valid (see text). As $\alpha \to 0$, the minimum of the cross section approaches $2\pi$ (dotted line). In the high-energy limit, all cross sections return to the photon limit, $27\pi$.