Repetitive Penrose Process in Accelerating Kerr Black Holes

This paper investigates the repetitive Penrose process in accelerating Kerr black holes and explores the influence of the acceleration factor on the repetitive Penrose process. After a brief review of accelerating Kerr black holes, we study the fundamental equations of the Penrose process in this spacetime, examine the stopping conditions required for the repetitive Penrose process, and obtain corresponding numerical results. The conclusions indicate that, apart from the third law of thermodynamics similar to previous cases, accelerating Kerr black holes exhibit stronger energy extraction capabilities compared to Kerr black holes during the repetitive Penrose process. Moreover, in prior studies, the energy utilization efficiency was difficult to exceed $50%$. However, in accelerating Kerr black holes, when the decay radius is relatively small, the energy utilization efficiency can exceed $50%$, indicating that the reduced extractable energy primarily transforms into extracted energy rather than irreducible mass. On the other hand, when the initial value of the acceleration factor is large, the extractable energy can decrease to nearly zero, which also differs from the case of Kerr black holes in previous studies.

💡 Research Summary

The paper investigates the repetitive Penrose process in accelerating Kerr black holes, focusing on how the acceleration parameter A influences the extraction of rotational energy. Starting from the C‑metric description of an accelerating Kerr spacetime, the authors write down the metric (Eq. 1) with the acceleration factor A, the spin a, and the mass M. Setting A = 0 recovers the ordinary Kerr solution. The event horizon r₊, the acceleration horizon r_A, and the ergosphere boundaries r_E (for the event horizon) and r_EA (for the acceleration horizon) are derived analytically (Eqs. 3‑5). For an extremal black hole (a = M) the ergosphere exists only for Â = AM below a critical value ≈ 0.30028; beyond this the ergosphere disappears and no energy extraction is possible.

The authors then formulate the Penrose process using four‑momentum conservation (Eq. 10) for three particles: the incident particle 0, the negative‑energy particle 1 that falls into the black hole, and the escaping particle 2. By imposing the “turning‑point” condition—zero radial momentum at the decay radius—their energies equal the effective potentials V₊ (Eq. 12). Under this optimal condition an analytical solution for the conserved quantities is obtained (Eqs. 13‑14). After each decay the black‑hole mass Mₙ and angular momentum Lₙ are updated (Eqs. 15‑17), which in turn modify the dimensionless spin â = a/M and the dimensionless acceleration Â = AM. The changes in the event horizon radius, the irreducible mass M_irr, and the extractable energy ΔE_extractable are expressed in Eqs. 18‑20. Two performance metrics are introduced: the energy return on investment ξₙ (Eq. 22) and the energy‑utilization efficiency Ξₙ (Eq. 23).

Because the iterative extraction cannot continue indefinitely, five stopping conditions are imposed (Eqs. 24‑25). They require a positive mass deficit, a negative energy for particle 1, a positive remaining extractable energy, non‑decrease of the irreducible mass (to respect the second law), and precise alignment of the decay radius with the peaks of the effective potentials. The minimal spin a_min that allows each particle to satisfy its condition is derived. For particle 0 the limit is set by the corotating marginally bound orbit (Eqs. 26‑27); for particle 2 by the corotating photon sphere (Eq. 29); for particle 1 by the requirement that the decay radius exceeds the event horizon, yielding a simple expression a_min,1 = √