Effective Potential and Topological Photon Spheres: A Novel Approach to Black Hole Parameter Classification

TL;DR

This paper introduces a topological approach to classify black hole parameters by analyzing photon spheres, providing a new method to distinguish black holes from naked singularities using effective potential structures.

Contribution

It presents a novel topological framework based on photon spheres to categorize gravitational models and determine black hole parameters, including complex scenarios with multiple horizons.

Findings

Presence of a stable photon sphere indicates a naked singularity or horizonless structure.

The method remains effective even in models with multiple event horizons.

Analysis of models with Perfect Fluid Dark Matter demonstrates the approach's versatility.

Abstract

In this paper, we base our analysis on the assumption that the existence of a photon sphere is an intrinsic feature of any ultra-compact gravitational structure with spherical symmetry. Utilizing the concept of a topological photon sphere, we categorize the behaviors of various gravitational models based on the structure of their photon spheres. This innovative approach enables us to define boundaries for black hole parameters, subsequently allowing us to classify the model as either a black hole or a naked singularity. Indeed, we will demonstrate that the presence of this interplay between the gravitational structure and the existence of a photon sphere is a unique advantage that can be utilized from both perspectives. Our observations indicate that a gravitational model typically exhibits the behavior of a horizonless structure (or a naked singularity) when a minimum effective potential (a stable photon sphere) appears within the studied spacetime region. Additionally, in this study, we tried to investigate the effect of this structure on the behavior of the photon sphere by choosing models that are affected by the Perfect Fluid Dark Matter (PFDM). Finally, by analyzing a model with multiple event horizons, we show that the proposed method remains applicable even in such scenarios.