# Chaos in self-gravitating many-body systems: Lyapunov time dependence of $N$ and the influence of general relativity

**Authors:** Simon F. Portegies Zwart, Tjarda C. N. Boekholt, Emiel Por, Adrian S. Hamers, Steve L. W. McMillan  
**arXiv:** [2109.11012](https://arxiv.org/abs/2109.11012)  
**Date:** 2021-09-22  
**Summary page:** https://www.summarizepaper.com/en/paper/2109.11012v2/

## Key points

- The study explores chaotic behavior of self-gravitating many-body systems and investigates the Lyapunov time dependence on the number of bodies (N) and the influence of general relativity.
- Small perturbations introduced at the start or due to numerical errors grow exponentially with time in Newtonian gravity.
- For relatively homogeneous systems, the rate of growth per crossing time increases with N up to N ~ 30, but for larger systems, the growth rate has a weaker scaling with N. However, for concentrated systems, the rate of exponential growth continues to scale with N.
- In relativistic self-gravitating systems, the rate of growth is almost independent of N but only noticeable when the system's mean velocity approaches the speed of light within three orders of magnitude.
- Chaotic behavior in systems with more than a dozen bodies is qualitatively different when only solving pairwise interactions in Einstein-Infeld-Hoffmann equations compared to taking into account interaction terms (or cross terms).
- The research was made possible by open-source packages and libraries such as Python, MPI, AMUSE, Sapporo GPU library, MPFR library. All N-body codes used in this study are available in AMUSE repository at amusecode.org.
- The calculations using Brutus took about 107 CPU seconds while other two sets were comparable totaling about a year.
- This project was supported by funds from European Research Council (ERC) under Horizon 2020 research and innovation program under grant agreement No 638435 (GalNUC), National Science Foundation under grant AST-1814772. It was performed using resources provided by Academic Leiden Interdisciplinary Cluster Environment (ALICE) and LGM-II (NWO grant # 621.016.701).
- The authors acknowledge discussions with Clifford Will, Ina Sellentin, and Arend Moerman.

## Summary

This study explores the chaotic behavior of self-gravitating many-body systems and investigates the Lyapunov time dependence on the number of bodies ($N$) and the influence of general relativity. The researchers confirm earlier findings that small perturbations introduced at the start or due to numerical errors grow exponentially with time in Newtonian gravity. They observe that for relatively homogeneous systems, this rate of growth per crossing time increases with $N$ up to $N \sim 30$, but for larger systems, the growth rate has a weaker scaling with $N$. However, for concentrated systems, the rate of exponential growth continues to scale with $N$. In relativistic self-gravitating systems, they find that the rate of growth is almost independent of $N$, but this effect is only noticeable when the system's mean velocity approaches the speed of light within three orders of magnitude. The study also highlights that chaotic behavior in systems with more than a dozen bodies is qualitatively different when only solving pairwise interactions in Einstein-Infeld-Hoffmann equations compared to taking into account interaction terms (or cross terms). This finding motivates further studies on the microscopic effect of general relativity on orbital chaos and higher-order cross-terms in Taylor-series expansion. The research was made possible by open-source packages and libraries such as Python, MPI, AMUSE, Sapporo GPU library, MPFR library. All N-body codes used in this study are available in AMUSE repository at amusecode.org. The calculations using Brutus took about 107 CPU seconds while other two sets were comparable totaling about a year. This project was supported by funds from European Research Council (ERC) under Horizon 2020 research and innovation program under grant agreement No 638435 (GalNUC), National Science Foundation under grant AST-1814772. It was performed using resources provided by Academic Leiden Interdisciplinary Cluster Environment (ALICE) and LGM-II (NWO grant # 621.016.701). The authors acknowledge discussions with Clifford Will, Ina Sellentin, and Arend Moerman.