Observing Black Hole Phase Transitions in Extended Phase Space and Holographic Thermodynamics Approaches from Optical Features

TL;DR

This paper investigates black hole phase transitions through optical features like photon trajectories and images, revealing that optical parameters can serve as order parameters and follow van der Waals-like critical behavior.

Contribution

It introduces optical parameters such as orbital half-period and Lyapunov exponents as novel order parameters for black hole phase transitions, with analytical validation of their critical scaling behavior.

Findings

Optical parameters exhibit multi-valued behavior near critical points.

Discontinuous changes in optical parameters indicate first-order phase transitions.

Critical exponents of optical parameters are 1/2, matching van der Waals fluid.

Abstract

The phase transitions of charged Anti-de Sitter (AdS) black holes are characterized by studying null geodesics in the vicinity of the critical curve of photon trajectories around black holes as well as their optical appearance as the black hole images. In the present work, the critical parameters including the orbital half-period $\tau$, the angular Lyapunov exponent $\lambda_L$, and the temporal Lyapunov exponent $\gamma_L$ are employed to characterize black hole phase transitions within both the extended phase space and holographic thermodynamics frameworks. Under certain conditions, we observe multi-valued function behaviors of these parameters as functions of bulk pressure and temperature in the respective approaches. We propose that $\tau$, $\lambda_L$, and $\gamma_L$ can serve as order parameters due to their discontinuous changes at first-order phase transitions. To validate this, we provide detailed analytical calculations demonstrating that these optical critical parameters follow scaling behavior near the critical phase transition point. Notably, the critical exponents for these parameters are found to be $1/2$, consistent with those of the van der Waals fluid. Our findings suggest that static and distant observers can study black hole thermodynamics by analyzing the images of regions around the black holes.