Impact of Painleve–Gullstrand coordinates on the thermodynamics of Kiselev black holes

We investigate the implications provided by the modified Painleve–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, while also including the deformation parameter, which signals deviation from the standard Painleve–Gullstrand metric. We address two specific issues related to the problems of radiation and dust, as 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 perform a thermodynamical analysis of our results by determining the Hawking temperature and entropy for the cases of radiation and dust. The results are obtained in closed form and show their sensitivity to quintessence. We set up a framework wherein we compare the effect of the deformation parameter on the velocity of an infalling particle. We analyze and comment on the dependence of the quintessence parameter on the mass and charge of the Kiselev black hole. We show explicitly the entropy behaviour for the radiation and dust cases in the plots made with respect to the variation in black hole mass M and black hole charge Q for the quintessence parameter \alpha.