Painlevé-Gullstrand coordinates for Kiselev black holes

TL;DR

Explores a modified Painlevé-Gullstrand coordinate system in ADM gravity to study $Kiselev$ black holes embedded in quintessence. By introducing a lapse $\mathcal{N}$ and a deformation parameter $\lambda$, it derives a static metric with shift $v(r)$ and analyzes radiation ($\omega = \frac{1}{3}$) and dust ($\omega = 0$) cases, deriving horizon structures and shift constraints. Closed-form Hawking temperature and entropy expressions reveal explicit $\alpha$-dependent corrections; increasing $\alpha$ widens horizon gaps and leads to remnant masses where $T_H \to 0$, with entropy behaving monotonically in $M$ and showing $\alpha$-induced modifications. The results demonstrate how quintessence influences black hole thermodynamics in a regular coordinate framework, offering insights for horizon physics and quantum effects in non-flat backgrounds.

Abstract

We investigate the implications provided by the modified Painlevé-Gullstrand coordinates in the context of quintessence for the Kiselev black hole. In this regard, we set up a fully static line element in terms of lapse and shift functions, apart from including the deformation parameter signaling deviation from the standard Painlevé-Gullstrand metric. We address two specific issues pertaining to the problems of radiation and dust furnished by the corresponding barotropic index parameter and study the related consequences by performing a range of analyses to explore the influence imposed by quintessence. We also discuss the thermodynamical consequences by evaluating the expressions of the Hawking temperature and the entropy function in closed forms.

Figures (20)

• Figure 1: Variation of $f(r)$ with $r$ for different values of $\alpha$ for the radiation case. The blue dashed line represents the outer horizon, the red one represents the inner horizon, and the green dotted line corresponds to where $r_0$ is minimum. We have taken $M=1$ and $Q=0.75$ to exhibit the sub-extremal condition.
• Figure 2: Diagrams illustrating variation of $v(r)$ with $r$ for the radiation case for different values of $\alpha$. We have taken $M=1$ and $Q=0.75$ to exhibit the sub-extremal condition.. The panels correspond to (a) $r_0 = 0.5625$ and (b) $r_0 = 0.0625$. The red dashed line is indicative of the inner horizon $r^{rad}_-$ and the blue dashed line is indicative of the outer horizon $r^{rad}_+$. The green dashed line shows the point $r_0$.
• Figure 3: $f(r)$ vs $r$ for different values of $\alpha$ for the dust case. The blue dashed line represents the outer horizon, the red one represents the inner horizon, and the green dotted line denotes the minimum point $r_0$. We have taken $M=1$ and $Q=0.75$ to show the sub-extremal case.
• Figure 4: Diagrams illustrating variation of $v(r)$ with $r$ for the dust case for different values of $\alpha$. We have taken $M=1$ and $Q=0.75$. The panels correspond to (a) $r_0 = 0.5625$ and (b) $r_0 = 0.45$. The red dashed line is indicative of the inner horizon $r^{dust}_{-}$ and the blue dashed line is indicative of the outer horizon $r^{dust}_{+}$. The green dashed line shows $r_0$.
• Figure 5: Hawking temperature $T_H$ plotted versus mass $M$ and $\alpha$ for the radiation case. Here we have taken $Q=0.75$.