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

TL;DR

This paper proves that any horizon can be assigned an entropy proportional to its area by analyzing the partition function of spacetimes, revealing gravity's intrinsic holographic nature through a relation between entropy, horizon area, and action.

Contribution

It provides a general proof linking horizon entropy to area for all horizons and shows how this relation allows constructing the gravitational action, emphasizing gravity's holographic aspect.

Findings

Entropy proportional to horizon area for various spacetimes

Partition function form confirms entropy-area relation

Gravity's action can be derived from horizon properties

Abstract

I provide a general proof of the conjecture that one can attribute an entropy to the area of {\it any} horizon. This is done by constructing a canonical ensemble of a subclass of spacetimes with a fixed value for the temperature $T=\beta^{-1}$ and evaluating the {\it exact} partition function $Z(\beta)$. For spherically symmetric spacetimes with a horizon at $r=a$, the partition function has the generic form $Z\propto \exp[S-\beta E]$, where $S= (1/4) 4\pi a^2$ 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.