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Thermodynamic Topology and Photon Spheres Analysis of Black Holes in Brane-World: Insights from Barrow Entropy

Usman Zafar, Abdul Jawad, Kazuharu Bamba, Mohammad Ali S. Afshar, Mohammad Reza Alipour, Saeed Noori Gashti, Jafar Sadeghi

Abstract

We explore the thermodynamics and geothermodynamics of black holes with Barrow entropy in a brane-world scenario, where the horizon geometry of the black hole is regarded as a fractal structure. Our analysis reveals the behavior of heat capacity, identifying both bound and divergence points. For the Bekenstein-Hawking entropy, the divergence point exhibits smooth behavior, indicating no phase transition. In contrast, we observe divergence with Barrow entropy as the deformation parameter increases, confirming the presence of a zero point in heat capacity through various thermodynamic geometry formalisms. Additionally, we delve into thermodynamic topology, detailing the classification of black holes in the brane-world context and comparing their characteristics determined from the Bekenstein-Hawking and the Barrow entropy. Notably, fixing the deformation and cosmological parameters results in a topological charge $-1$ predominately by the dark matter parameter, which remains unaffected despite variations in other parameters. In the dS model, the cosmological horizon prevents stable photon spheres, making topological charges of $0$ and $+1$ unattainable. Incremental increases in the cosmological parameter reduce the dark matter parameter-dominated region.

Thermodynamic Topology and Photon Spheres Analysis of Black Holes in Brane-World: Insights from Barrow Entropy

Abstract

We explore the thermodynamics and geothermodynamics of black holes with Barrow entropy in a brane-world scenario, where the horizon geometry of the black hole is regarded as a fractal structure. Our analysis reveals the behavior of heat capacity, identifying both bound and divergence points. For the Bekenstein-Hawking entropy, the divergence point exhibits smooth behavior, indicating no phase transition. In contrast, we observe divergence with Barrow entropy as the deformation parameter increases, confirming the presence of a zero point in heat capacity through various thermodynamic geometry formalisms. Additionally, we delve into thermodynamic topology, detailing the classification of black holes in the brane-world context and comparing their characteristics determined from the Bekenstein-Hawking and the Barrow entropy. Notably, fixing the deformation and cosmological parameters results in a topological charge predominately by the dark matter parameter, which remains unaffected despite variations in other parameters. In the dS model, the cosmological horizon prevents stable photon spheres, making topological charges of and unattainable. Incremental increases in the cosmological parameter reduce the dark matter parameter-dominated region.
Paper Structure (4 sections, 19 equations, 3 figures, 1 table)

This paper contains 4 sections, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Brane-World Black Hole: A Brief Discussion
  3. Thermal Stability of Brane-World BH through the Barrow entropy
  4. Thermodynamic Geometry of Brane-World Black Hole Through the Barrow entropy

Figures (3)

  • Figure 1: Heat capacity $C$ and temperature $T$ as the function of the Barrow entropy. In Fig. \ref{['CTGRAPH']}(\ref{['C, T versus S']}), we have plotted $T$ (red curve) and heat capacity $C$ (black curve) by inserting $\alpha=~0.3,~\beta=~0.7$ and deformation parameter $\delta=~1$. In Fig. \ref{['CTGRAPH']}(\ref{['Cgraph']}), heat capacity is presented by inserting $\delta=0$ (red curve), $\delta=0.5$ (black curve), $\delta=1$ (blue curve).
  • Figure 2: Small root (left), large root (middle), and divergence (right) in terms of dark matter parameter $\beta$. In Figs. \ref{['BDGRAPHB']}(\ref{['BP3']}), \ref{['BDGRAPHB']}(\ref{['BP4']}) and \ref{['BDGRAPHB']}(\ref{['DP2']}), the trajectories correspond to the different values of deformation parameters such as $\delta=0$ (red curve), $\delta=0.3$ (black curve), $\delta=0.7$ (blue curve) and $\delta=1$ (grey curve).
  • Figure :