Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length (2002.04029v1)

Abstract: Chaos is present in most stellar dynamical systems and manifests itself through the exponential growth of small perturbations. Exponential divergence drives time irreversibility and increases the entropy in the system. A numerical consequence is that integrations of the N-body problem unavoidably magnify truncation and rounding errors to macroscopic scales. Hitherto, a quantitative relation between chaos in stellar dynamical systems and the level of irreversibility remained undetermined. In this work we study chaotic three-body systems in free fall initially using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic. We demonstrate that the fraction of irreversible solutions decreases as a power law with numerical accuracy. This can be derived from the distribution of amplification factors of small initial perturbations. Applying this result to systems consisting of three massive black holes with zero total angular momentum, we conclude that up to five percent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution, thus rendering them fundamentally unpredictable.

• The paper demonstrates a power-law relationship between chaos and irreversibility scale in gravitational three-body systems, finding some require numerical accuracy smaller than the Planck length for reversibility.
• Using the Brutus code, researchers showed that even with high precision, a significant fraction of chaotic three-body systems remain irreversible due to large amplification factors of initial perturbations.
• The findings highlight the inherent unpredictability of chaotic systems, emphasizing the need for improved precision in astrophysical simulations and suggesting focus on statistical reliability rather than deterministic outcomes.

Gargantuan Chaotic Gravitational Three-Body Systems and Their Irreversibility to the Planck Length

The dynamics of three-body systems in astrophysics are paramount for understanding complex gravitational interactions, particularly those involving massive entities such as black holes. This paper extensively explores the chaotic nature of these systems and the implications of chaos on the reversibility of their solutions, pushing the frontiers of numerical accuracy to unprecedented scales.

Key Findings and Methodology

The paper employs the high-precision N-body code, Brutus, to examine three-body gravitational systems initially in a free-fall state. A principal focus is the profound sensitivity of these systems to minute perturbations in initial conditions, characterized by exponential divergence and a consequent increase in entropy. The research succeeds in establishing a quantitative relation between chaos and the scale of irreversibility across varying levels of numerical precision.

The researchers demonstrate that a power-law relationship governs the fraction of irreversible solutions relative to numerical precision—some chaotic three-body systems necessitate an accuracy smaller than the Planck length for reversibility. This finding suggests a fundamental unpredictability within them. Specifically, they note that up to five percent of triple black hole systems with zero total angular momentum may require such precision to achieve reversibility.

The paper also introduces a methodical approach to studying the homology map of the Agekyan-Anosova, allowing for the characterization of a diverse array of initial conditions. The reversibility tests conducted reveal that even with increased computational precision beyond machine capabilities, a significant fraction remains irreversible due to large amplification factors of initial perturbations.

Theoretical Implications

The chaotic nature of three-body systems highlights the broader concept of time irreversibility in stellar dynamics, intimately connected to the arrow of time through entropy production. The exponential growth of small perturbations elucidates the challenge of predicting the evolution of such systems beyond a few Lyapunov timescales. The propagation of numerical errors in simulations mirrors physical perturbations, raising questions about the validity of long-term predictions in chaotic systems—even when employing sophisticated numerical methods.

Practical Implications and Future Prospects

From a practical standpoint, these results emphasize the need for improved precision in astronomical simulations, especially for systems involving supermassive black holes, which play a crucial role in galaxy formation and evolution theories. The inaccessibility of absolute precision in physical and numerical experiments suggests researchers will benefit from focusing on statistical reliability rather than deterministic outcomes.

Further development of high-precision numerical methods and enhanced computational power will be critical. The paper's insights point toward a deeper examination of the sensitivity of chaotic systems to ensure that the trajectories calculated are representative of realistic configurations, despite the inherent unpredictability at macroscopic scales.

Conclusion

The research explores the intrinsic complexity of three-body chaotic systems by combining sophisticated numerical techniques with a theoretical understanding of chaos and irreversibility. The findings have significant implications for the predictability of chaotic systems in astrophysics, highlighting both the capabilities and limitations of current numerical simulations. As computational resources expand and methods continue to evolve, future work in this domain will likely refine these models, potentially unlocking deeper insights into the chaotic universe we inhabit.