Repetitive Penrose process in Kerr-de Sitter black holes

TL;DR

This work extends the study of the repetitive Penrose process to Kerr–de Sitter spacetimes, examining how a positive cosmological constant $\Lambda$ affects energy extraction. By deriving the Kerr-dS metric properties, ergosphere structure, and the maximum extractable energy $E_{extractable}$, the authors establish the framework for nonlinear, iterative energy extraction and define metrics such as $EROI$ and $EUE$. Through an analytical treatment of the equatorial plane and a controlled iterative procedure, they show that a third-law–like bound persists: not all extractable rotational energy can be retrieved, with irreducible mass absorbing much of the energy deficit. Comparisons with Kerr reveal that Kerr-dS generally yields higher $EROI$ and single-extraction capability, and larger $\Lambda$ amplifies these effects, though the iteration stopping condition can limit the total extracted energy and reduce $EUE$ at certain radii. These results illuminate the role of a cosmological constant in energy extraction processes near rotating black holes and suggest that Kerr-dS spacetimes offer enhanced energy-transfer potential under appropriate conditions.

Abstract

Recently, references [1,2] found that the repetitive Penrose process cannot extract all the extractable rotational energy of a Kerr black hole, and reference [3] found that the repetitive electric Penrose process cannot extract all the electrical energy of a Reissner-Nordström (RN) black hole. This suggests that a law analogous to the third law of thermodynamics exists for the repetitive Penrose process. In this paper, we intend to study the repetitive Penrose process in the Kerr-de Sitter (Kerr-dS) black hole. We will explore influences of the cosmological parameter on the repetitive Penrose process. The results show that, in addition to a similar third law of thermodynamics, the Kerr-dS black hole yields a higher energy return on investment (EROI) and single-extraction energy capability compared to the Kerr black hole. Specifically, the larger the cosmological parameter, the stronger the EROI and the single-extraction energy capability. Furthermore, we also find that at a lower decay radius, the Kerr black hole exhibits a higher energy utilization efficiency (EUE) and more extracted energy after the repetitive Penrose process is completed. However, at a higher decay radius, the situation is reversed, i.e., the Kerr-dS black hole exhibits a higher EUE and more extracted energy, which is due to the existence of stopping condition of the iteration.

Figures (7)

• Figure 1: Variation of $E_{extractable}/M$, $a/M$, $r_{+}/M$, and $r_{E}/M$ for an extremal black hole with respect to $\Lambda M^2$.
• Figure 2: Variation of $\hat{a}_{\min,0}$ with respect to $\hat{r}_x$ for different values of $\Lambda M^2$.
• Figure 3: Variation of $\hat{a}_{\min,2}$ with respect to $\hat{r}_x$ for different values of $\Lambda M^2$.
• Figure 4: Variation of $\hat{a}_{\min,1}$ with respect to $\hat{r}_x$ for different values of $\Lambda M^2$.
• Figure 5: Variation of $\hat{a}_{\min}$ for the three particles with respect to $\hat{r}_x$ for different values of $\Lambda M^2$.