Revised Upper Limit to Energy Extraction from a Kerr Black Hole

We present a new upper limit on the energy that may be extracted from a Kerr black hole by means of particle collisions in the ergosphere (i.e., the “collisional Penrose process”). Earlier work on this subject has focused largely on particles with critical values of angular momentum falling into an extremal Kerr black hole from infinity and colliding just outside the horizon. While these collisions are able to reach arbitrarily high center-of-mass energies, it is very difficult for the reaction products to escape back to infinity, effectively limiting the peak efficiency of such a process to roughly 130%. When we allow one of the initial particles to have impact parameter $b>2M$, and thus not get captured by the horizon, it is able to collide along outgoing trajectories, greatly increasing the chance that the products can escape. For equal-mass particles annihilating to photons, we find a greatly increased peak energy of $E_{\text{out}}\approx 6\times E_{\text{in}}$. For Compton scattering, the efficiency can go even higher, with $E_{\text{out}}\approx 14\times E_{\text{in}}$, and for repeated scattering events, photons can both be produced and escape to infinity with Planck-scale energies.

Figure 1

Radial turning points in the effective potential $V_{\text{eff}}(r,b)$ for (top) massive and (bottom) massless particles, for a black hole with maximal spin $a=1$. Any particle in the yellow region can escape from the black hole, but in the blue regions, only particles with outgoing radial velocities can escape. The static limit is located at $r=2$ and the horizon is at $r=1$.

Figure 2

Peak efficiency for annihilation of equal-mass particles falling from rest at infinity, as a function of the radius at which the annihilation occurs. The impact parameter $b_1=2$ is fixed at the critical value, and $b_2$ varies. The black hole spin is maximal: $a=1$. (Compare to Fig. 2 of Ref. [9])

Figure 3

Peak efficiency for annihilation of equal-mass particles falling from rest at infinity, as a function of the radius at which the annihilation occurs. Unlike in Fig. 2, here we allow $p_r^{(1)}>0$, which greatly increases the fraction and energy of escaping photons. The impact parameter $b_1$ is fixed at 2 and $b_2$ varies.

Figure 4

Center-of-mass energy for the annihilation reactions shown in Figs. 2 and 3. The color scheme is as in Fig. 3, with solid lines corresponding to $p_r^{(1)}>1$ and dashed lines for $p_r^{(1)}<1$. The highest center-of-mass energies naturally occur for the prograde-retrograde collisions, which are closest to “head-on.”

Figure 5

Peak efficiency for Compton-like scattering events between a photon with $b_1=2$, $p_r^{(1)}>1$ and a massive particle falling from rest at infinity with rest mass $m_2$ and impact parameter $b_2=-2(1+\sqrt{2})$.

Figure 6

Spin dependency for efficiency of the scattering events plotted in Fig. 5 with ${ϵ}_1/m_2=10$.