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

TL;DR

The paper introduces a new chaos indicator called time-reversed Shannon entropy (TRSE) that detects chaos in non-integrable systems by analyzing symmetry breaking in orbital evolution within curved spacetimes.

Contribution

It presents TRSE as a novel, robust chaos indicator and demonstrates its effectiveness alongside particle-pair mutual information in general relativistic systems.

Findings

TRSE effectively distinguishes chaotic from regular dynamics in black hole spacetimes.

TRSE and MIPP show strong quantitative agreement as complementary chaos probes.

Numerical simulations validate TRSE's ability to detect symmetry breaking in orbital evolution.

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.