Linear stability matrix and Lyapunov exponents for stationary spacetime
Abstract
Lyapunov exponents characterize the average rate at which nearby trajectories in phase space converge or diverge and therefore provide a useful criterion for dynamical stability. In this work, we develop a general framework for analyzing Lyapunov exponents in stationary spacetimes and extend previous studies beyond circular orbits. For a generic stationary spacetime described by gμν(r,θ), we show that the phase space relevant to linear perturbations is spanned by (r,pr,θ,pθ) and the corresponding linear stability matrix has rank four. When the spacetime admits a rank-2 Killing tensor, the linear stability matrix can be reduced to a block-diagonal form, allowing the Lyapunov exponents to be obtained analytically. By imposing the appropriate constraint conditions, the formalism further yields the linear stability matrices and Lyapunov exponents for circular and spherical orbits. The validity of the method is illustrated by several representative examples, including the (inverted) harmonic oscillator, planar Kepler motion, and particle motion in Schwarzschild and Kerr spacetimes. These results provide a unified framework for investigating the stability properties of particle orbits in both classical and relativistic dynamical systems.
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BibTeX
@misc{ye2026linear,
title = {{Linear stability matrix and Lyapunov exponents for stationary spacetime}},
author = {Xu Ye and Shao-Wen Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19418},
eprint = {2610.19418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19418},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Linear stability matrix and Lyapunov exponents for stationary spacetime
AU - Xu Ye
AU - Shao-Wen Wei
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19418
UR - https://arxiv.org/abs/2610.19418
AB - Lyapunov exponents characterize the average rate at which nearby trajectories in phase space converge or diverge and therefore provide a useful criterion for dynamical stability. In this work, we develop a general framework for analyzing Lyapunov exponents in stationary spacetimes and extend previous studies beyond circular orbits. For a generic stationary spacetime described by g_μν(r,θ), we show that the phase space relevant to linear perturbations is spanned by (r,p_r,θ,p_θ) and the corresponding linear stability matrix has rank four. When the spacetime admits a rank-2 Killing tensor, the linear stability matrix can be reduced to a block-diagonal form, allowing the Lyapunov exponents to be obtained analytically. By imposing the appropriate constraint conditions, the formalism further yields the linear stability matrices and Lyapunov exponents for circular and spherical orbits. The validity of the method is illustrated by several representative examples, including the (inverted) harmonic oscillator, planar Kepler motion, and particle motion in Schwarzschild and Kerr spacetimes. These results provide a unified framework for investigating the stability properties of particle orbits in both classical and relativistic dynamical systems.
ER -