Probing Black Hole Thermodynamics and Microstructure via the Shadow of Sagittarius A*

Abstract

We explore the connection between black hole shadows, thermodynamic phase structure, and microstructure of charged and rotating black holes within General Relativity and Geometrothermodynamics. Focusing on Reissner-Nordström and Kerr solutions, we establish a criterion to select the most suitable Geometrothermodynamic metric for a system, revealing that the first metric from enthalpy and the second from mass correctly reproduce heat capacity singularities. We show that the shadow radius encodes the same phase information as entropy and introduce Shadow-Microstructure diagrams to extract insights into stability and microscopic interaction types directly from observational bounds. Applying this framework to Sagittarius A*, we constrain the macroscopic parameters and the allowed microscopic thermodynamic phases. Our findings indicate that shadow measurements offer a novel probe of thermodynamic and microscopic aspects of black holes, enabling tests of alternative theories of gravity and thermodynamic frameworks.

Figures (13)

• Figure 1: (a) Black hole existence lies above the blue parabola $S = \pi Q^2$; the orange parabola $S_m = 3\pi Q^2$ is the critical curve of $C_Q$. Local thermodynamic stability at fixed charge occurs between the two curves. (b) Temperature, (c) heat capacity at constant $Q$, and (d) mass, all expressed in terms of $x \equiv S/\pi Q^2$.
• Figure 2: (a) Helmholtz free energy of the RN black hole; (b) parametrized entropy branches in terms of the reduced variable $x \equiv S/\pi Q^2$. The limits $\chi \to 0$ and $\chi \to \pi$ correspond to $x \to 1$ ($T=0$) and $x \to 3$ ($T = T_{\max}$), respectively.
• Figure 3: (a) Black hole existence lies above the blue curve $S = 2\pi |J|$; the red curve $S = S_{m}$ is the critical curve of $C_J$. Local thermodynamic stability at fixed $J$ occurs between the two curves. (a) Temperature, (b) heat capacity at constant $J$, and (c) mass, all expressed in terms of $x \equiv S/2\pi|J|$.
• Figure 4: (a) Helmholtz free energy of the Kerr black hole; (b) entropy branches in terms of $x \equiv S/2\pi|J|$.
• Figure 5: Reduced GTD scalars and heat capacity of the RN black hole. (a) $\mathcal{R}^{I}$ computed from the potential $H(S,U)$, (b) $\mathcal{R}^{II}$ and (c) $\mathcal{R}^{III}$ computed from the potential $M(S,Q)$. Gray regions indicate non-physical configurations where $T < 0$. All plots have been rescaled for clarity.