Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy

Technical Summary: Repetitive Penrose Process in Rastall Rotating Black Holes Immersed in Quintessence Dark Energy

Problem Statement
While the Penrose process is a fundamental theoretical mechanism for extracting rotational energy from black holes via particle decay within the ergoregion, the original formulation faces astrophysical limitations due to the requirement of extremely high relative velocities between decay fragments. To address this, the "repetitive Penrose process" was proposed, where energy extraction is iterated following each decay event. However, the behavior of this iterative process within the context of modified gravity theories remains largely unexplored. Specifically, there is a lack of detailed analysis regarding how Rastall gravity—a theory where the conservation of energy-momentum is generalized to $\nabla_\mu T^{\mu\nu} = \alpha \nabla^\nu R$—and the presence of a quintessence dark energy field influence the efficiency, dynamics, and termination conditions of rotational energy extraction. This paper investigates the repetitive Penrose process in the spacetime of a Rastall rotating black hole surrounded by a quintessence field to determine how the Rastall structure parameter ($\hat{N}_s$) and the Rastall coupling parameter ($\alpha$) affect the extraction of rotational energy.

Methodology
The authors formulate the problem within the framework of a rotating Rastall black hole metric in Boyer-Lindquist coordinates, incorporating a surrounding quintessence field with an equation of state parameter $\omega_s = -2/3$. The study employs geometric units ($c=G=1$) and focuses on particle motion in the equatorial plane.

1. Theoretical Framework: The authors derive the conservation equations for energy, angular momentum, and radial momentum governing the splitting of a particle (Particle 0) into two fragments (Particle 1 and Particle 2) within the ergoregion. They establish iterative evolution equations for the black hole's mass ($M$), spin ($a$), irreducible mass ($M_{irr}$), and the dimensionless Rastall structure parameter ($\hat{N}_s = N_s / M^{2-\zeta}$) after each extraction step.
2. Termination Conditions: A set of physical constraints is formulated to determine when the iterative process must stop. These include:
   • Mass deficit constraints ($1 - \bar{\mu}_1 - \bar{\mu}_2 > 0$).
   • The requirement that the infalling fragment (Particle 1) possesses negative energy ($\hat{E}_1 < 0$).
   • The non-decrease of the irreducible mass ($\Delta M_{irr} \geq 0$) to satisfy the generalized second law of thermodynamics.
   • The condition that the decay radius must lie outside the event horizon and satisfy turning-point conditions relative to the effective potential peaks.
3. Numerical Analysis: The authors perform numerical simulations to track the evolution of the system. They analyze the minimum spin lower limits required to terminate the process for different decay radii ($\hat{r}_p$) and parameter values. They calculate key performance metrics:
   • Energy Return on Investment ($\xi$): The ratio of harvested energy to the total energy of incident particles.
   • Energy Utilization Efficiency ($\Xi$): The ratio of harvested energy to the total reduction in the black hole's extractable energy.
   • Extractable Energy: The remaining rotational energy available for extraction.

Key Contributions

• Extension of Repetitive Penrose Process: The work extends the repetitive Penrose process from the standard Kerr spacetime to the more general Rastall rotating black hole background immersed in quintessence.
• Identification of Critical Stopping Criterion: Through the analysis of minimum spin lower limits for the three particles involved in the decay, the authors demonstrate that the termination of the iterative process is consistently governed by Particle 0 (the incident particle). Particle 0 possesses the highest minimum spin threshold among all decay products, defining the critical stopping point for the extraction sequence.
• Parameter Sensitivity Analysis: The study systematically quantifies the influence of the Rastall structure parameter ($\hat{N}_s$) and the coupling parameter ($\alpha$) on the energetics of the process, distinguishing their specific roles in modifying the horizon structure and ergosphere dynamics.

Results

• Termination Dynamics: The iterative process terminates when the black hole's spin drops below the critical threshold determined by Particle 0. The irreducible mass increases monotonically, preventing the extraction of 100% of the rotational energy.
• Influence of $\hat{N}_s$:
  • Increasing $\hat{N}_s$ generally enhances the energy utilization efficiency ($\Xi$), particularly at lower decay radii.
  • Smaller values of $\hat{N}_s$ yield a larger maximum energy return on investment ($\xi$).
  • Increasing $\hat{N}_s$ shifts the location of the maximum extracted energy toward higher decay radii and accelerates the depletion of the remaining extractable energy reservoir.
  • The total extractable energy decreases as $\hat{N}_s$ increases.
• Influence of $\alpha$:
  • The Rastall coupling parameter $\alpha$ also influences the efficiency and the location of peak extracted energy, generally enhancing efficiency at higher decay radii and reducing the total extractable energy.
  • However, the paper notes that the effects of $\alpha$ on these energetics are very small compared to the significant impact of $\hat{N}_s$.
• Decay Radius Dependence: The efficiency of the process is highly favored at lower decay radii. As the decay radius increases, the energy return on investment and utilization efficiency generally decline, and the number of possible iterations decreases.
• Comparison with Kerr: In certain parameter regimes (e.g., specific combinations of $\hat{N}_s$ and $\alpha$), the remaining extractable energy after termination can be significantly different from standard Kerr black hole results, sometimes leaving a larger residual energy reservoir.

Significance
The paper claims that these findings highlight the pivotal role of Rastall gravity and quintessence dark energy in shaping the efficiency and dynamics of black hole rotational energy extraction. By demonstrating that the Rastall structure parameter significantly alters the energy utilization efficiency and the evolution of the extractable energy reservoir, the study suggests that deviations from General Relativity could leave distinguishable signatures in black hole energetics. The work provides a systematic analysis of how non-minimal coupling between matter and geometry influences the long-term evolution of energy extraction, offering a deeper understanding of the ultimate limits of rotational energy extraction in modified gravity frameworks. The authors emphasize that their results clarify the interplay between modified gravity and black hole energetics, specifically regarding the iterative extraction of rotational energy and the evolution of the irreducible mass.