Black Holes Thermodynamics in a new kind of Noncommutative Geometry

Motivated by the energy dependent metric in gravity's rainbow, we will propose a new kind of energy dependent noncommutative geometry. It will be demonstrated that like gravity's rainbow, this new noncommutative geometry is described by an energy dependent metric. We will analyse the effect of this noncommutative deformation on the Schwarzschild black holes and Kerr black holes. We will perform our analysis by relating the commutative and this new energy dependent noncommutative metrics using an energy dependent Moyal star product. We will also analyze the thermodynamics of these new noncommutative black hole solutions. We will explicitly derive expression for the corrected entropy and temperature for these black hole solutions. It will be demonstrated that for these deformed solutions black remnants cannot form. This is because these correction increase rather than reduce the temperature of the black holes.