Holographic Gravity and the Surface term in the Einstein-Hilbert Action

Abstract

Certain peculiar features of Einstein-Hilbert (EH) action provide clues towards a holographic approach to gravity which is independent of the detailed microstructure of spacetime. These features of the EH action include: (a) the existence of second derivatives of dynamical variables; (b) a non trivial relation between the surface term and the bulk term; (c) the fact that surface term is non analytic in the coupling constant, when gravity is treated as a spin-2 perturbation around flat spacetime and (d) the form of the variation of the surface term under infinitesimal coordinate transformations. The surface term can be derived directly from very general considerations and using (d) one can obtain Einstein's equations {\it just from the surface term of the action}. Further one can relate the bulk term to the surface term and derive the full EH action based on purely thermodynamic considerations. The features (a), (b) and (c) above emerge in a natural fashion in this approach. It is shown that action $A_{grav}$ splits into two terms $-S+βE$ in a natural manner \textit{in any stationary spacetime with horizon}, where $E$ is essentially an integral over ADM energy density and $S$ arises from the integral of the surface gravity over the horizon. This analysis shows that the true degrees of freedom of gravity reside in the surface term of the action, making gravity intrinsically holographic. It also provides a close connection between gravity and gauge theories, and highlights the subtle role of the singular coordinate transformations.