On the non existence of thermal equilibrium between Hawking radiation and thermal dust in the context of Rastall gravity

In this paper, we investigate the possibility of thermal equilibrium between Hawking radiation emitted by a RN-AdS black hole and dust like matter absorbing thermal energy surrounding it, called thermal dust, in the context of Rastall gravity with cosmological constant term. The energy density of thermal dust increases linearly with its temperature. Therefore, thermal dust represent the dust like matter that absorb thermal energy and gets heated up linearly with temperature. In the context of Rastall gravity, we derive the metric of RN-AdS black hole surrounded by thermal dust and it possesses the temperature term of the thermal dust. Hence, using this metric we can check if there exists thermal equilibrium between Hawking radiation emitted by a RN-AdS black hole and surrounding thermal dust by simply demanding the temperature of thermal dust equal to Hawking temperature. Then, we show that the entropy term become unphysical for positive Hawking temperature, when we demand this thermal equilibrium condition. Hence, we conclude that there cannot exist thermal equilibrium between Hawking radiation emitted by a RN-AdS black hole and its surrounding thermal dust background in the context of Rastall gravity.