Repetitive Penrose process in Kerr-de Sitter black holes

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.

💡 Research Summary

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The paper investigates the repetitive Penrose process in Kerr‑de Sitter (Kerr‑dS) black holes, focusing on how a positive cosmological constant Λ influences the efficiency of energy extraction. After a brief historical overview of the original Penrose mechanism and its many extensions, the authors introduce the Kerr‑dS metric in Boyer‑Lindquist coordinates, defining the radial functions Δ_r, Δ_θ, ρ², and Σ that depend on the mass M, spin a, and Λ. The locations of the inner (Cauchy) horizon r₋, the event horizon r₊, the cosmological horizon r_c, and the ergosphere boundaries r_E are obtained from Δ_r=0 and g_tt=0, respectively. They note that increasing Λ expands both the event horizon and the ergosphere while simultaneously reducing the maximum extractable rotational energy E_extractable, as illustrated in Figure 1.

The authors then formulate the basic Penrose decay: a particle 0 of mass μ₀ and energy E₀ falls into the ergosphere and splits into two particles μ₁ (negative‑energy) and μ₂ (positive‑energy). Conservation of energy, angular momentum, and radial momentum (Eq. 8) together with the normalization condition g_{μν}p^μp^ν=−μ² lead to effective potentials V_± for each particle (Eq. 12). For optimal extraction, the negative‑energy particle must have the most negative possible energy, which occurs when its energy equals the upper branch of the effective potential at a turning point (V_+¹). The authors prove that the radial momenta of all three particles must vanish at the decay radius r_x to satisfy the timelike geodesic condition and to maximize extracted energy.

In the repetitive scenario, after each extraction the black hole’s mass M_n and angular momentum L_n are updated according to Eq. 20, which changes the dimensionless spin â = a/M and the scaled cosmological parameter Λ̂ = ΛM² (Eq. 21). The irreducible mass M_irr and the remaining extractable energy E_extractable are recomputed after each iteration. The total extracted energy after n steps is E_extracted,n = M₀ – M_n. Two performance metrics are introduced: the energy return on investment (EROI) ξ_n = E_extracted,n/(n E₀) and the energy utilization efficiency (EUE) Ξ_n = E_extracted,n/(E_extractable,0 – E_extractable,n).

The iteration cannot continue indefinitely; three stopping conditions must be satisfied: (i) the mass deficit μ₀ – μ₁ – μ₂ > 0, (ii) the negative‑energy particle retains E₁ < 0, and (iii) the turning points of particles 0, 1, and 2 must lie on appropriate sides of their effective‑potential peaks (Eq. 26). The limiting case where a turning point coincides with a potential peak defines a minimal spin limit a_min for each particle. By solving the marginally bound orbit condition (Eq. 27) for particle 0 and the photon‑sphere condition (Eq. 28) for particle 2, the authors obtain a_min⁰, a_min¹, and a_min² as functions of the decay radius r_x and Λ̂. Figures 2‑4 show that larger Λ̂ raises all three minimal spins, while larger r_x lowers them, reflecting the interplay between the cosmological expansion and the local spacetime geometry.

Numerical experiments explore two regimes of the decay radius. In the “low‑radius” regime (r_x ≈ 1.2 M), the standard Kerr black hole (Λ=0) yields a higher EUE and a larger total extracted energy after the process terminates. In the “high‑radius” regime (r_x approaching the event horizon), Kerr‑dS black holes with sizable Λ̂ exhibit a stronger EROI, a higher single‑extraction energy capability, and ultimately a greater EUE and total extracted energy. This reversal is attributed to the fact that a larger Λ increases the effective local mass M_loc(r) = M + Λ r⁶/(r² + a²) (Eq. 5), which reduces the growth of the irreducible mass during each step and allows more rotational energy to be tapped before the stopping conditions are met.

A key conclusion is the persistence of a “third‑law‑like” behavior: even after many iterations, a non‑zero irreducible mass remains, and the extractable energy never reaches zero. This mirrors the third law of black‑hole thermodynamics, which forbids reaching an extremal state (zero surface gravity) through finite processes. The presence of Λ does not eliminate this limitation but modifies the quantitative bounds.

Overall, the paper demonstrates that a positive cosmological constant can enhance certain aspects of energy extraction—particularly the return on investment and single‑step yield—while also shifting the parameter space where the Kerr‑dS geometry outperforms the pure Kerr case in terms of overall efficiency. The findings have implications for theoretical models of advanced energy‑harvesting civilizations, for the thermodynamics of rotating black holes in de Sitter backgrounds, and for our understanding of how cosmological expansion influences local high‑energy processes.