Frolov Black Hole Surrounded by Quintessence -- I: Thermodynamics, Geodesics and Shadows

TL;DR

The paper analyzes a Frolov black hole embedded in a quintessence field, characterized by a length scale $\alpha_0$ and charge $q$, with a quintessence coupling $c$. Using the metric $f(r)=1-\frac{(2Mr-q^2)r^2}{r^4+(2Mr+q^2)\alpha_0^2}-\frac{c}{r^{3w+1}}$ and fixing $M=1$, it derives thermodynamic quantities, examines null and timelike geodesics via an effective potential $V_{eff}$, and computes shadows from the photon-sphere condition. Key findings include a three-horizon structure, local but not global thermodynamic stability (positive Helmholtz free energy across horizons), strong sensitivity of null geodesics to $c$ (outward-shifting photon orbits and enhanced repulsion), and shadow radii that depend mainly on $\alpha_0$ and $q$ with modest influence from $c$. Confronting the shadow with EHT observations of Sgr A$^*$ yields constraints on $c$ (roughly $0\le c\lesssim 0.012$ at $1\sigma$, $0\le c\lesssim 0.025$ at $2\sigma$) and bounds on $\alpha_0$ and $q$, illustrating the strong-field constrainability of such non-singular BHs in quintessence backgrounds.

Abstract

The Frolov black hole (BH) is a charged extension of the Hayward BH, having regularity at the central point $r = 0$ and an asymptotically Schwarzschild form for large values of $r$. Such a BH is parameterized by a length scale parameter, \( α_0 \). In this paper, we analyze the thermodynamic properties, null and timelike geodesics, and shadows of a Frolov BH immersed in a quintessence field. Our results indicate that the smaller BH is locally thermodynamically stable yet globally unstable at all horizon radii. Neither the quintessence parameter nor the other model parameters like the charge $q$ and length scale parameter $α_0$ change this global instability. We extend the study of the null and timelike geodesics to the vicinity of the BH by analyzing how the geodesic motion depends on the model parameters. Finally, we analyze the shadow of the BH system and find that the shadow radii are sensitively dependent on model parameters. In contrast, the influence of the quintessence parameter itself on the size of the shadow is found to be rather weak.

Figures (17)

• Figure 1: The lapse function is plotted for different combinations of the parameters $c$, $\alpha_0$, and charge $q$.
• Figure 2: The Hawking temperature is plotted for different values of the model parameters $\alpha_0$, $q$ and the quintessence parameter $c$.
• Figure 3: The ADM mass is plotted for different combinations of the parameters $c$, $\alpha_0$, and charge $q$.
• Figure 4: The specific heat is plotted for different combinations of $\alpha_0$, $q$ and $c$.
• Figure 5: The free energy is plotted for different values of $\alpha_0$, $q$ and $c$.