Entropic corrected Newton's law of gravitation and the Loop Quantum Black Hole gravitational atom

Abstract: One proposal by Verlinde \cite{Verlinde:2010hp} is that gravity is not a fundamental, but an entropic force. In this way, Verlinde has provide us with a way to derive the Newton's law of gravitation from the Bekenstein-Hawking entropy-area formula. On the other hand, since it has been demonstrated that this formula is susceptible to quantum gravity corrections, one may hope that these corrections could be inherited by the Newton's law. In this way, the entropic interpretation of Newton's law could be a prolific way in order to get verifiable or falsifiable quantum corrections to ordinary gravity in an observationally accessible regimes. Loop quantum gravity is a theory that provide a way to approach the quantum properties of spacetime. From this theory, emerges a quantum corrected semiclassical black hole solution called loop quantum black holes or self-dual black holes. Among the interesting features of loop quantum black holes is the fact that they give rise to a modified entropy-area relation where quantum gravity corrections are present. In this work, we obtain the quantum corrected Newton's law from the entropy-area relation given by loop quantum black holes. In order to relate our results with the recent experimental activity, we consider the quantum mechanical properties of a huge gravitational atom consisting in a light neutral elementary particle in the presence of a loop quantum black hole.