The emergent gravity's entropy Ansatz from an augmented variational principle

In the last decade, physical and geometrical investigations about the relationship between horizon thermodynamics and gravitational dynamics suggest that gravity could be an emergent phenomenon. Among the others, Padmanabhan's theory of "emergent gravity" focus on the concept of spacetime as an effective macroscopic description of a more fundamental microscopic theory ("atoms of spacetime") at the Planck scales, thus aiming to reconcile the large scale description of gravity and the small one of quantum physics. However, some mathematical aspects of this approach are still not clear as, for example, the derivation of Einstein equations from a suitable entropy functional (and not from an action as in standard General Relativity). Thus in this work, a direct and non-trivial link between Padmanabhan's entropy used in emergent gravity and standard General Relativity action is established. To do that, Augmented Variational Principles and Kerr-Schild metrics will be used. We shall also discuss how this link accounts for the details of the variation of Padmanabhan's action based on gravitational entropy. It will also clarify the role of the background metric used in Padmanabhan's functional and its non-dynamical role.