Third law of repetitive electric Penrose process

TL;DR

The paper analyzes the repetitive electric Penrose process (EPP) in a Reissner-Nordström black hole to determine whether all extractable electromagnetic energy can be retrieved through classical iterations. By enforcing energy and charge conservation across successive decays and updating the black hole's mass and charge, the authors show that the irreducible mass $M_{\mathrm{irr}}$ grows according to $M_{\mathrm{irr},n}=\frac{M_n+\sqrt{M_n^2-Q_n^2}}{2}$, in line with Hawking's area theorem, which prevents full extraction of $E_{\mathrm{extractable}}=M-M_{\mathrm{irr}}$ and motivates a thermodynamic third-law analogy for EPP. They introduce two stopping conditions that bound the viable parameter space, demonstrate that the charge can be driven toward zero but not exactly zero by classical EPP, and define energy return on investment (EROI) and energy utilization efficiency (EUE). Numerical exploration reveals that EROI can exceed unity for suitable parameters while EUE remains below ~0.5, indicating substantial entropy growth and energy sequestration in $M_{\mathrm{irr}}$, with a concrete nine-step example illustrating the tradeoffs and outlining future work on Kerr-Newman evolution and repetitive superradiance.

Abstract

Recently, Ruffini et al. [Phys. Rev. Lett. 134 (2025) 8, 081403] pointed out that the repetitive Penrose process cannot drain the entire extractable energy of a Kerr black hole. In this Letter, we further point out that the charge of a Reissner-Nordström black hole cannot drop down to exactly zero via the repetitive electric Penrose process that is terminated after a finite number of iterative steps, indicating a thermodynamical third-law analog for the repetitive electric Penrose process.

Figures (4)

• Figure 1: The allowed region of the electric Penrose process derived from two iterative stopping conditions for various values of $\hat{E}_0$ and $\hat{q}_1$. The allowed region (shaded region) is to the left of the solid curve and to the right of the dashed curve.
• Figure 2: Schematic diagram of particle 0 decay. Parameters are the same as Tab. \ref{['tab:1']}. The red dashed line indicates the decay position at $\hat{r}=2.4$, the green dashed line indicates the incident energy $\hat{E}_0$, and the blue solid curves represent the effective potential at each iteration. It can be seen that the decay position is always to the right of the peak of the effective potential.
• Figure 3: Variation of EROI and EUE of the final state with the dimensionless decay radius, the charge-to-mass ratio of particle 1, and the reduced energy of particle 0. Here we keep the mass ratio $\eta=0.78345$ but choose a smaller mass of the incident particle $\mu_0=M_0/1000$ to eliminate the "oscillation" of the curves as much as possible. The basic choice of three variables is $\hat{r}_d=2.4$, $\hat{q}_1=-5$, and $\hat{E}_0=1.1$. Each time one of the three variables is changed while keeping the other two fixed. It is intuitive to see that $\xi_{n_\mathrm{end}}$ can easily be much greater than $1$, but $\Xi_{n_\mathrm{end}}$ is difficult to exceed $0.5$.
• Figure 4: The allowed region of the original Penrose process derived from the iterative stopping condition, which is the overlapping region to the upper right of the red dashed curve (previous constraint) and to the lower left of a solid curve (newly added constraint). The parameter of the blue curve $\hat{p}_{\phi1}=-19.434$ is taken from Ruffini:2024dwq, indicating that the condition $\hat{E}_1<0$ must be considered.