Some perspective of thermodynamical and optical properties of black holes in Maxwell-dilaton-dRGT-like massive gravity

TL;DR

This work analyzes charged black holes in Maxwell-dilaton-dRGT-like massive gravity, combining a dilaton UV correction with a dRGT-like massive gravity IR correction. It derives the full thermodynamics, including a modified entropy and an AMD mass, and demonstrates four regimes of thermal stability with van der Waals–like phase transitions, confirmed via geometrothermodynamics where the HPEM metric accurately tracks heat-capacity divergences and zeros. In the optical sector, the authors compute the photon-sphere and shadow radii, showing how the dilaton coupling $\alpha$, charge $q$, graviton mass $m_{g}$, and massive-gravity parameter $\eta_{1}$ jointly shape the black hole silhouette, and they confront EHT observations of Sgr A* to constrain $\alpha$ for various parameter choices. The emission spectrum is analyzed to reveal how these parameters modulate the high-frequency peak, linking near-horizon geometry to observable radiative signatures. Overall, the paper provides a coherent framework connecting modified gravity, black-hole thermodynamics, optical signatures, and observational constraints, with implications for testing extensions of general relativity.

Abstract

Motivated by integrating the dilaton field (as a UV correction) with dRGT-like massive gravity (as an IR correction) into Einstein gravity, we investigate the thermodynamic and optical properties of black holes within this gravitational framework. We begin by reviewing the black hole solutions in Maxwell-dilaton-dRGT-like massive gravity, followed by an analysis of how various parameters influence on the asymptotical behavior of the spacetime and the event horizon of these black holes. In the subsequent section, we examine the conserved and thermodynamic quantities associated with these black holes, paying particular attention to the effects of parameters like $β$, $α$, and the massive parameters ($η_{1}$ and $η_{2}$) on their local stability by simultaneously evaluating the heat capacity and temperature. We also adopt an alternative method to study phase transitions using geometrothermodynamics. Furthermore, we explore how the parameters of Maxwell-dilaton-dRGT-like massive gravity impacts the optical characteristics and radiative behavior of black holes. In particular, we analyze the effects of the dilaton coupling constant ($α$), charge ($q$), the massive gravity parameter ($η_1$), and the graviton mass ($m_g$) on the radius of the photon sphere and the resulting black hole shadow. Moreover, the theoretical shadow radius is compared to the observational data from $Sgr A^*$. Additionally, we investigate the energy emission rate of these black holes, revealing that these parameters substantially influence the emission peak.

Figures (13)

• Figure 1: The mertic function $f(r)$ versus $r$ for different values of parameters.
• Figure 2: The heat capacity ($C_{Q}$) and the Hawking temperature ($T$) versus entropy ($S$) for different values of $\alpha$ by considering $Q=b=\beta =-\eta _{2}=0.1$, $\eta _{1}=0.5$, $\Lambda =-0.1$, $c_{0}=1$, and $m_{g}=0.2$. Bold lines represent the Hawking temperature ($T$), while thin lines indicate the heat capacity ($C_{Q}$)
• Figure 3: The heat capacity ($C_{Q}$) and the Hawking temperature ($T$) versus entropy ($S$) for different values of $\beta$ by considering $Q=b=-\eta _{2}=0.1$, $\alpha =0.3$, $\eta _{1}=0.5$ , $\Lambda =-0.1$, $c_{0}=1$, and $m_{g}=0.2$. Bold lines represent the Hawking temperature ($T$), while thin lines indicate the heat capacity ($C_{Q}$)
• Figure 4: The heat capacity ($C_{Q}$) and the Hawking temperature ($T$) versus entropy ($S$) for different values of $\eta _{1}$ by considering $Q=b=\beta =-\eta _{2}=0.1$, $\alpha =0.3$, $\Lambda =-0.1$, $c_{0}=1$, and $m_{g}=0.2$. Bold lines represent the Hawking temperature ($T$), while thin lines indicate the heat capacity ($C_{Q}$)
• Figure 5: The heat capacity ($C_{Q}$) and the Hawking temperature ($T$) versus entropy ($S$) for different values of $\eta_{2}$ by considering $Q=b=\beta=0.1$, $\eta_{1}=1$, $\alpha =0.3$, $\Lambda=-0.1$, $c_{0}=1$, and $m_{g}=0.2$. Bold lines represent the Hawking temperature ($T$), while thin lines indicate the heat capacity ($C_{Q}$)