Instability and Information Production Around Kerr Black Holes: Effects on Entropy and the Shadow

TL;DR

This work analyzes the information produced by unstable geodesics around a Kerr black hole using Lyapunov exponents and KS entropy. It extends shadow theory by including non equatorial photon orbits to refine mass and spin inferences from subrings, and it connects the unstable massive particle dynamics to black hole thermodynamics by interpreting the KS entropy as a physical entropy bounded by the Bekenstein limit. For massive particles, a Visser style volume regularization yields an entropy S that satisfies S≤2πER, supporting a link between information theory and black hole thermodynamics. The results imply observable consequences for the black hole shadow and hint at deeper connections between gravitational dynamics, information, and thermodynamics, with future work needed to define KS type entropy for null orbits and to extend the framework to other gravity theories and near-horizon observables.

Abstract

Massless or massive particles in unstable orbits around a Kerr black hole exhibit exponentially unstable motion when perturbed. They either plunge into the black hole or escape to infinity after making some oscillations around the equatorial plane. This exponentially unstable motion causes information production. In the case of the photons that escape to infinity, it was recently suggested that this information can be used to resolve the subring structure of the shadow image and obtain more precise data about the black hole mass and spin. Here, we extend this method to obtain more precise results by including th non-equatorial contributions to the Lyapunov exponents. For massive particles plunging into the Kerr black hole, we show that the associated Kolmogorov-Sinai entropy derived from the Lyapunov exponents can be interpreted in the context of black hole thermodynamics and obeys Bekenstein's bound on the entropy of a physical system. Thus, the perturbed unstable orbits, either ending inside the black hole or at the observer's screen, have physical consequences.

Figures (8)

• Figure 1: The Lyapunov exponent for the prograde binding orbit where the black hole parameter is $M=1$ with $E_{\text{circ},+}=m=1$ is plotted as a function of the rotation parameter.
• Figure 2: The Lyapunov exponent for the retrograde binding orbit where the black hole parameter is $M=1$ with $E_{\text{circ},+}=m=1$ is plotted as a function of the rotation parameter.
• Figure 3: The angular momentum is plotted as a function of radius for the case where $M=1$, $m=E=1$, and $\alpha=a=0.5$.
• Figure 4: The Carter's constant is plotted as a function of radius for the case where $M=1$, $m=E=1$, and $\alpha=a=0.5$.
• Figure 5: The angular dependence of the Lyapunov exponent for the case where $M=1$, $E=m=1$, $x=r=3$ and $\alpha=a=0.5$ is plotted.