Black hole images as probes of thermodynamic evolution

TL;DR

This work investigates how horizon-scale images of black holes encode thermodynamic evolution in extended AdS gravity, using the RN--AdS black hole as a concrete testbed. By combining analytic arguments with numerical ray tracing, it shows that shadows and accretion-disk images reflect both phase transitions and ensemble information, with the photon-sphere radius $r_{\rm ps}$ governing the imaging signatures. Along isobaric paths the image size grows monotonically with the horizon radius $r_h$, whereas along isothermal paths the extended first law ${\rm d}M=T{\rm d}S+V{\rm d}P$ drives nonmonotonic behavior and Maxwell construction yields a critical reduced temperature $\tilde{T}_{0,c}$ separating two pre-transition patterns. Across the phase transition, image sizes jump upward in both ensembles, revealing temperature information in the isothermal case and establishing a paradigm for extracting thermodynamic imprints from optical appearances that can extend to other black-hole systems.

Abstract

We investigate how black hole images (shadows and accretion-disk images) encode thermodynamic evolution information across different ensembles, using the Reissner-Nordström-AdS black hole as an illustrative example. Through analytic treatment and numerical verification, we demonstrate that these images encode not only phase transition information but also ensemble information, including additional temperature information in the isothermal ensemble. Phase transition information appears as a sudden increase in image size, which we prove occurs in both isobaric and isothermal ensembles. The ensemble and temperature information originates from a fundamental difference between isobaric and isothermal evolution: image size varies monotonically with the horizon radius along isobars, whereas it exhibits nonmonotonic behavior along isotherms. This contrast serves as a diagnostic tool to distinguish isobaric from isothermal evolution. In the isothermal ensemble, the nonmonotonic behavior introduces an extremal radius whose relative ordering with the small- and large-black hole radii at the phase transition admits three logical possibilities. Our analysis reveals that only two of these possibilities are physically realized, separated by a critical reduced temperature. Furthermore, image evolution in the two resulting temperature intervals exhibits qualitatively differences, demonstrating that black hole images indeed encode temperature information. These results not only enrich the set of observational avenues for probing black hole thermodynamic properties, but also introduce a new paradigm. This paradigm studies phase transitions in conjunction with nonmonotonic evolution, providing a useful framework for exploring thermodynamic imprints in other black hole systems.

Figures (10)

• Figure 1: Effective potential $U(r)$ for the RN--AdS and asymptotically flat RN black holes.
• Figure 2: Evolution of the photon--sphere radius $r_{\rm ps}$ as a function of the horizon radius $r_h$ in the isobaric and isothermal ensembles.
• Figure 3: Evolution of the black hole shadow angle $\xi_{\rm c}$ and the accretion--disk image angle $\xi_{\rm n}$ as functions of the horizon radius $r_h$ in the isobaric ensemble, with $Q=0.5$ and $P=0.8\,P_{\rm c}$.
• Figure 4: Evolution of the black hole shadow angle $\xi_{\rm c}$ and the accretion--disk image angle $\xi_{\rm n}$ as functions of the horizon radius $r_h$ in the isobaric ensemble, with $Q=1$ and $P=0.8\,P_{\rm c}$.
• Figure 5: Evolution of the black hole shadow angle $\xi_{\rm c}$ and the accretion--disk image angle $\xi_{\rm n}$ as functions of the horizon radius $r_h$ in the isothermal ensemble, with $Q=0.5$ and $T_0=0.8\,T_{\rm c}$.