## “Exploring Photon Rings in Axisymmetric Black Holes: A Penrose Limit Perspective”

arXiv:2403.10605v1 Announce Type: new

Abstract: We study the physics of photon rings in a wide range of axisymmetric black holes admitting a separable Hamilton-Jacobi equation for the geodesics. Utilizing the Killing-Yano tensor, we derive the Penrose limit of the black holes, which describes the physics near the photon ring. The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation, as well as the instabilities and Lyapunov exponents of the null geodesics. Consequently, the Lyapunov exponents and frequencies of the photon geodesics, along with the quasinormal modes, can be all extracted from a Hamiltonian in the Penrose limit plane wave metric. Additionally, we explore potential bounds on the Lyapunov exponent, the orbital and precession frequencies, in connection with the corresponding inverted harmonic oscillators and we discuss the possibility of photon rings serving as holographic horizons in a holographic duality framework for astrophysical black holes. Our formalism is applicable to spacetimes encompassing various types of black holes, including stationary ones like Kerr, Kerr-Newman, as well as static black holes such as Schwarzschild, Reissner-Nordstr”om, among others.

**Future Roadmap: Challenges and Opportunities on the Horizon**

## Introduction

In this study, we delve into the fascinating realm of photon rings in a diverse range of axisymmetric black holes. Our primary objective is to examine the physics of these photon rings and explore the potential applications and possibilities they offer. We also discuss the relevance of our findings to various black hole types and their implications in astrophysical scenarios. Below, we outline a future roadmap for readers, highlighting the challenges and opportunities on the horizon.

## Understanding the Physics of Photon Rings

To comprehend the physics behind photon rings, we start by investigating black holes that allow for a separable Hamilton-Jacobi equation for the geodesics. Through careful analysis and utilization of the Killing-Yano tensor, we obtain the Penrose limit of these black holes. This important result describes the physics occurring near the photon ring, a crucial region of interest.

### Linking the Plane Wave Geometry and Wave Equation

The obtained plane wave geometry is directly linked to the frequency matrix of the massless wave equation. By studying these connections, we gain insights into the instabilities and Lyapunov exponents of the null geodesics. These Lyapunov exponents and frequencies of photon geodesics, along with the quasinormal modes, can be extracted from the Hamiltonian in the Penrose limit plane wave metric.

## Potential Bounds and Inverted Harmonic Oscillators

We further explore the potential bounds on the Lyapunov exponent, the orbital and precession frequencies. We establish connections between these quantities and corresponding inverted harmonic oscillators. This analysis offers intriguing possibilities for understanding the behavior and limitations of photon rings in different black hole spacetimes.

## Holographic Duality Framework for Astrophysical Black Holes

Our investigation also delves into the concept of holographic horizons and their applicability to astrophysical black holes. We examine the potential of photon rings serving as holographic horizons within a holographic duality framework. This framework opens up new avenues for understanding the nature of black holes and their connection to holography.

### Applicability to Various Black Hole Types

Our formalism is applicable to a wide range of black hole types. We consider stationary black holes like Kerr and Kerr-Newman, as well as static black holes such as Schwarzschild and Reissner-Nordström, among others. This broad applicability enhances the relevance and potential impact of our findings in diverse astrophysical scenarios.

## Conclusion

By delving into the physics of photon rings in a range of axisymmetric black holes, we have uncovered valuable insights and potential applications. Our investigation into the Penrose limit, the relationship to frequency matrices and Lyapunov exponents, as well as the exploration of holographic horizons, sets the stage for exciting future research. Despite potential challenges in terms of computational complexity and theoretical formulation, the opportunities for advancing our understanding of black holes and their dynamics are vast.