Ultrahigh-Energy Debris from the Collisional Penrose Process

Soon after the discovery of the Kerr metric, Penrose realized that superradiance can be exploited to extract energy from black holes. The original idea (involving the breakup of a single particle) yields only modest energy gains. A variant of the Penrose process consists of particle collisions in the ergoregion. The collisional Penrose process has been explored recently in the context of dark matter searches, with the conclusion that the ratio $\eta$ between the energy of postcollision particles detected at infinity and the energy of the colliding particles should be modest ($\eta \lesssim 1.5$). Schnittman [Phys. Rev. Lett. 113, 261102 (2014)] has shown that these studies underestimated the maximum efficiency by about 1 order of magnitude (i.e., $\eta \lesssim 15$). In this work we show that particle collisions in the vicinity of rapidly rotating black holes can produce high-energy ejecta and result in high efficiencies under much more generic conditions. The astrophysical likelihood of these events deserves further scrutiny, but our study hints at the tantalizing possibility that the collisional Penrose process may power gamma rays and ultrahigh-energy cosmic rays.

Figure 1

Effective radial potential $V^{+}$ [cf. Eq. (2)] for an extremal black hole and a particle with energy $E_1=1$ and angular momentum $L_1=b_1{E}_1$. The horizon is located at $r=1$, and the ergoregion corresponds to $r<r_{\text{ergo}}=2$. The dash-dotted (blue online) curve shows the potential felt by particle 1 for the process discussed in Ref. [16]; particle 1 cannot access the dashed (blue online) region with $r<r_t$. The dashed (red online) curve corresponds to the process considered in this Letter. The solid (black online) curve depicts the potential for the critical value of $b_1$ separating these two processes.

Figure 2

Maximum efficiency ${\eta }_{\max }$ for the collision of equal-energy particles ($R=1$) as a function of the radius at which the reaction occurs. We consider $p_1^r>0$, $p_2^r<0$, $b_2=-2(1+\sqrt{2})$, and an extremal black hole ($a=1$). The curves for $b_1>2$ terminate at the turning point of particle 1.

Figure 3

Maximum efficiency ${\eta }_{\max }$ for the same process considered in Fig. 2 as a function of the radius at which the reaction occurs, for $a=1$. When $b_2<-2(1+\sqrt{2})$, particle 2 must also be produced close to the black hole.

Figure 4

Maximum efficiency as a function of the black hole spin $a$ for the same process as in Fig. 2. The maximum efficiency is attained at the photon sphere $r_c$, below which infalling particle 3 can not escape. For $a\simeq 1$, the maximum efficiency scales as $1/\sqrt{1-a^2}$.

Figure 5

Maximum gain for a Compton-like scattering between a massless particle with $p_1^r>0$, $b_1=1.99$ and a massive particle with $p_2^r<0$, $b_2=-2(1+\sqrt{2})$. We plot the maximum gain for different values of $R\equiv {\epsilon }_1/{\epsilon }_2$.