Time-reversed Shannon entropy as a chaos indicator for non-integrable systems

TL;DR

The paper addresses the challenge of distinguishing chaotic from regular motion in non-integrable general relativistic systems. It introduces time-reversed Shannon entropy (TRSE), which quantifies forward-vs-time-reversed evolution via the entropy difference $\Delta H$ and demonstrates its effectiveness in Kerr and Schwarzschild-Melvin spacetimes. TRSE correctly discriminates chaotic dynamics and aligns with the complementary MIPP indicator across parameter scans, outperforming traditional Shannon entropy in some regimes. This work provides a unified, efficient framework for chaos diagnosis in curved spacetimes with potential applications to modified gravity, gravitational-wave emission, and explorations of quantum chaos.

Abstract

We propose a novel chaos indicator -- time-reversed Shannon entropy (TRSE) -- that leverages the interplay between time-reversal symmetry breaking and information entropy in curved spacetimes. By quantifying statistical discrepancies between forward and backward temporal evolution of particle orbits, TRSE robustly distinguishes chaotic from regular dynamics in non-integrable systems. In contrast, integrable systems exhibit stable, symmetric probability distributions preserved by conserved quantities such as the Carter constant. We validate the method through high-precision numerical simulations in both Kerr and Schwarzschild-Melvin black hole geometries, evolving trajectories forward and backward in time. Furthermore, we refine our previously introduced particle-pair mutual information (MIPP) and perform comprehensive parameter-space scans, revealing a strong quantitative agreement between MIPP and TRSE. The two indicators emerge as complementary probes of chaos: TRSE captures symmetry breaking in orbital evolution, while MIPP measures statistical correlations. Together, they establish a unified framework for diagnosing chaos in general relativistic systems, paving a new path to understand the fundamental nature of chaos in non-integrable systems.

Figures (5)

• Figure 1: Orbital trajectories and probability distributions in Schwarzschild spacetime. (a) and (b) The system parameters for Orbit 1 are $E=0.983$, $L_{z}=0.1$, $r=29.49$, and $\theta=1^{\circ}$. The system parameters for Orbit 2 are $E=0.983$, $L_{z}=4$, $r=29.49$, and $\theta=90^{\circ}$. The system parameters for Orbit 3 are $E=0.983$, $L_{z}=4$, $r=6$, and $\theta=90^{\circ}$. (c) and (d) The system parameters for Orbit 4 are $E=0.98$, $L_{z}=4.6$, B=0.01, $r=16$, and $\theta=90^{\circ}$.
• Figure 2: MIPP and Shannon entropy scanplots for the magnetic field parameter B in Schwarzschild spacetime. The range of the magnetic field parameter B is from 0.0002 to 0.01, and a total of 50 orbits were calculated. The system parameters are $E=0.99$, $L_{z}=4$, $r=16$, and $\theta=90^{\circ}$.
• Figure 3: Orbital trajectories and probability distributions in Kerr spacetime. (a) and (b) The system parameters for Orbit 1 are $E=0.946$, $a=0.9$, $L_{z}=0$, $r=7.535$, and $\theta=90^{\circ}$. The system parameters for Orbit 2 are $E=0.946$, $a=0.9$, $L_{z}=3$, $r=7.535$, and $\theta=90^{\circ}$. The system parameters for Orbit 3 are $E=0.95$, $a=0.9$, $L_{z}=3$, $r=7.535$, and $\theta=90^{\circ}$. (c) and (d) The system parameters for Orbit 4 are $E=0.998$, $a=0.9$, $L_{z}=3$, B=0.01, $r=7.535$, and $\theta=90^{\circ}$.
• Figure 4: TRSE, MIPP and FLI scanplots for the key parameter($B, E, L_{z}$) in Kerr spacetime. (a)-(c) The range of the magnetic field parameter B is from 0.0002 to 0.01, and a total of 50 orbits were calculated. The system parameters are $E=0.989$, $L_{z}=3$, $a=0.998$, $r=8$, and $\theta=90^{\circ}$. (d)-(f) The range of the parameter $E$ is from 0.9902 to 0.998, and a total of 40 orbits were calculated. The system parameters are $L_{z}=4$, $B=0.001$, $a=0.01$, $r=10$, and $\theta=90^{\circ}$. (g)-(i) The range of the parameter $L_{z}$ is from 2.05 to 4, and a total of 40 orbits were calculated. The system parameters are $E=0.99$, $B=0.001$, $a=0.01$, $r=11$, and $\theta=90^{\circ}$.
• Figure 5: MIPP, TRSE, and Shannon entropy scanplots for the parameter $E$ and orbital probability distributions in Schwarzschild-Melvin spacetime. The range of the parameter $E$ is from 0.561 to 0.61, and a total of 50 orbits were calculated. The system parameters are $B=0.1$, $L_{z}=4$, $r=10.656$, and $\theta=90^{\circ}$.