Entropic force approach to noncommutative Schwarzschild black holes signals a failure of current physical ideas

Recently, a new perspective of gravitational-thermodynamic duality as an entropic force arising from alterations in the information associated with the positions of material bodies is found. In this paper, we generalize some features of this model in the presence of noncommutative Schwarzschild black hole by applying the method of coordinate coherent states describing smeared structures. We employ two different distributions: a) Gaussian and b) Lorentzian. Both mass distributions provide the similar quantitative aspects for the entropic force. Our study show the entropic force on the smallest fundamental unit of the holographic screen with radius r 0 is zero. As a result, black-hole remnants are unconditionally inert and gravitational interactions do not exist therein. So, a distinction between gravitational and inertial mass in the size of black-hole remnant is observed, i.e. the failure of the principle of equivalence. In addition, if one considers the screen radius to be less than the radius of the smallest holographic surface at the Planckian regime, then one encounters some unusual dynamical features leading to gravitational repulsive force and negative energy. On the other hand, the significant distinction between the two distributions is perceived to occur around r 0 , and that is worth of mentioning: at this regime either our analysis is not the proper one, or nonextensive statistics should be applied.