Deciphering black hole phase transitions through photon spheres

TL;DR

This work examines how quantum trace anomaly corrections to black hole thermodynamics in AdS spacetime influence the photon-sphere structure. It derives an analytic coexistence curve for the small/large black hole phase transition in extended phase space and shows the reduced equation of state becomes charge-independent, with a special merging point at $\alpha_c=Q^2/8$ where new critical behavior and swallowtail features arise. The authors demonstrate that the photon sphere and shadow parameters oscillate with the phase transition, and define the reduced photon-sphere radius as an order parameter, finding a critical exponent $\beta=1$ due to the anomaly. These results suggest that gravitational observables such as shadows can encode thermodynamic phase structure, offering potential observational tests and motivating extensions to rotating or de Sitter cases.

Abstract

Black hole thermodynamics is a crucial and foundational aspect of black hole physics, yet its observational verification remains exceptionally challenging. The photon sphere of a black hole, a manifestation of strong gravitational effects, is intrinsically linked to its shadow, which has been directly captured through observations made by the Event Horizon Telescope. Investigating black hole thermodynamics from a gravitational perspective presents an intriguing avenue for research. This paper obtains an analytical formula for the coexistence curve and investigates the relationship between the thermodynamic phase transition and the photon sphere of a black hole with quantum anomaly. It proposes that the photon sphere encodes information about the black hole phase transition, arguing that the change in the photon sphere radius can serve as an order parameter characterizing the black hole's phase transition.

Figures (8)

• Figure 1: The Gibbs free energy $G$ vs temperature $T$ and pressure $P$. The Gibbs free energy exhibits swallowtail behavior below and above the critical point. The swallowtail behavior disappears at critical temperature and critical pressure. Here we have set $\alpha_{\text{c}}=Q^2/8$ and $Q=1$.
• Figure 2: Maxwell's equal area law in the $P-V$ plane. The curve oscillates for a pair of conjugate thermodynamic variables as a first-order phase transition takes place and the areas in the two shadowed regions are the same. Here we have chosen $T=1.3T_{\text{c}}$.
• Figure 3: The coexistence curve of phase transition for the black hole with quantum anomaly. In terms of the reduced parameters, the coexistence curve is independent of charge $Q$. The red dot denotes the critical points. In contrast to conventional black hole thermodynamic systems, first-order phase transitions occur both below and above the critical point.
• Figure 4: The effective potential for a photon outside the horizon of the four-dimensional black hole with quantum anomaly. The parameters are set to $M=1$, $\alpha_{\text{c}}=Q/8$, $Q=1$, and $P=0.5P_{\text{c}}$. The angular momentum $L/E$ of the photon varies from $1.5$ to $4.0$ from bottom to top. The red thick line corresponds to the effective potential with critical angular momentum $L_{\text{c}}/E=b_{\text{ps}}\approx 2.6$.
• Figure 5: Phase transition temperature as a function of the reduced photon sphere radius. (a). Phase transition temperature as a function of the reduced photon sphere radius at thermodynamic pressure $P=0.5P_{\text{c}}$. (b). Phase transition temperature as a function of the reduced photon sphere radius at different pressures.