The Holography of Gravity encoded in a relation between Entropy, Horizon area and Action for gravity

I provide a general proof of the conjecture that one can attribute an entropy to the area of any horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature T = β and evaluating the exact partition function Z(β). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z ∝ exp[S−βE], where S = (1/4)4πa and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the black-hole spacetimes and provides a simple and consistent interpretation of entropy and energy for De Sitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. Further, I show that the relationship between entropy and area allows one to construct the action for the gravitational field on the bulk and thus the full theory. In this sense, gravity is intrinsically holographic. Among the class of Lorentzian spacetime metrics which allow a positive definite continuation to the Euclidean time coordinate τ = it, there exists a subclass of spacetime metrics which require τ to be treated as periodic with some period β. It is natural to interpret such a feature as describing a finite temperature field theory with temperature T = β. (For a review, see e.g., [1].) In the case of black hole spacetimes, one can also associate an entropy with the horizon and construct a consistent parallel with thermodynamics. While there is some indication that we can associate an entropy with the area of any horizon (Rindler, De Sitter ......), any such association will also require defining the energy for such a spacetime in order to provide consistent thermodynamic relationships. This, however, is not an easy task in general relativity and hence progress has been somewhat limited in this issue. However, if the association of thermodynamical variables with horizons is of truly fundamental significance, then it is indeed necessary that our conclusions are applicable to any horizon and it must be possible to attribute an entropy to any horizon. In fact, if the entropy of spacetimes arise because some information is hidden by the horizon, then all horizons (even the observer dependent ones like Rindler or De Sitter horizons) must have an entropy. Further, one must be able to obtain such a result in an elegant manner, from standard statistical mechanical procedures. That is, results must “ fall out” of a proper argument allowing us to: (i) associate entropy with any horizon and (ii) identify the energy content of the spacetime. I will now show that this is indeed possible. What