On the single versus the repetitive Penrose process in a Kerr black hole

Extracting the rotational energy from a Kerr black hole (BH) is one of the crucial topics in relativistic astrophysics. Here, we give special attention to the Penrose ballistic process based on the fission of a massive particle $\mu_0$ into two particles $\mu_1$ and $\mu_2$, occurring in the ergosphere of a Kerr BH. Bardeen et al. indicated that for the process to occur, some additional "hydrodynamical forces or superstrong radiation reactions" were needed. Wald and Chandrasekhar further expanded this idea. This animosity convinced T. Piran and collaborators to move from a simple three-body system characterizing the original Penrose process to a many-body system. This many-body approach was further largely expanded by others, some questionable in their validity. Here, we return to the simplest original Penrose process and show that the solution of the equations of motion, imposing the turning point condition on their trajectories, leads to the rotational energy extraction from the BH expected by Penrose. The efficiency of energy extraction by a single process is quantified for three different single decay processes occurring respectively at $r=1.2 M$, $r=1.5 M$, and $r=1.9 M$. An interesting repetitive model has been proposed by Misner, Thorne \& Wheeler (hereafter MTW73). Indeed, it would appear that a repetitive sequence of $246$ decays of the above injection process at $r=1.2 M$ and the corresponding ones at $r=1.5 M$ and $r=1.9 M$ could extract $100\%$ of the rotational energy of the BH, so violating energy conservation. The accompanying paper, accounting for the existence of the BH irreducible mass, introduces a non-linear approach that avoids violating energy conservation and leads to a new energy extraction process.