The role of the irreducible mass in repetitive Penrose energy extraction processes in a Kerr black hole

The concept of the irreducible mass ($M_{\rm irr}$) has led to the mass-energy ($M$) formula of a Kerr black hole (BH), in turn leading to its surface area $S=16\pi M_{\rm irr}^2$. This also allowed the coeval identification of the reversible and irreversible transformations, soon followed by the concepts of "extracted" and "extractable" energy. This new conceptual framework avoids inconsistencies recently evidenced in a repetitive Penrose process. We consider repetitive decays in the ergosphere of an initially extreme Kerr BH and show the processes are highly irreversible. For each decay, the particle that the BH captures causes an increase of the irreducible mass (so the BH horizon), much larger than the extracted energy. The energy extraction process stops {when the BH reaches a positive spin lower limit set by the process boundary conditions}. Thus, the reaching of a final non-rotating Schwarzschild BH state through this accretion process is impossible. We have assessed such processes for selected decay radii and incoming particle with rest mass $1\%$ of the BH initial mass $M_0$. For $r= 1.2 M$ and $1.9 M$, the sequence stops after $8$ and $34$ decays, respectively, at a spin $0.991$ and $0.857$, the energy extracted has been only $1.16\%$, and $0.42\%$, the extractable energy is reduced by $17\%$ and $56\%$, and the irreducible mass increases by $5\%$ and $22\%$, all values in units of $M_0$. These results show the highly nonlinear change of the BH parameters, dictated by the BH mass-energy formula, and that the BH rotational energy is mainly converted into irreducible mass. Thus, evaluating the irreducible mass increase in any energy extraction processes in the Kerr BH ergosphere is mandatory.