On the maximal efficiency of the collisional Penrose process

TL;DR

This paper analytically investigates the collisional Penrose process near Kerr black holes, revealing that despite diverging CM energies, the maximum energy and efficiency of escaping particles are modest, with a peak efficiency of about 13.92.

Contribution

It provides new analytic formulas for the CM energy and escaping particle energy, refining previous results and quantifying the maximal efficiency in realistic scenarios.

Findings

Maximal efficiency is approximately 13.92 for certain scattering scenarios.

Escaping particle energy is only about ten times the incident particles' rest mass energy.

CM energy can diverge, but the energy of escaping particles remains modest.

Abstract

The center of mass (CM) energy in a collisional Penrose process - a collision taking place within the ergosphere of a Kerr black hole - can diverge under suitable extreme conditions (maximal Kerr, near horizon collision and suitable impact parameters). We present an analytic expression for the CM energy, refining expressions given in the literature. Even though the CM energy diverges, we show that the maximal energy attained by a particle that escapes the black hole's gravitational pull and reaches infinity is modest. We obtain an analytic expression for the energy of an escaping particle resulting from a collisional Penrose process, and apply it to derive the maximal energy and the maximal efficiency for several physical scenarios: pair annihilation, Compton scattering, and the elastic scattering of two massive particles. In all physically reasonable cases (in which the incident particles initially fall from infinity towards the black hole) the maximal energy (and the corresponding efficiency) are only one order of magnitude larger than the rest mass energy of the incident particles. The maximal efficiency found is ~ 13.92 and it is obtained for the scattering of an outgoing massless particle by a massive particle.