Semiclassical quantization of gravity I: Entropy of horizons and the area spectrum

TL;DR

This paper develops a semiclassical approach to gravity, showing that horizon areas are quantized with uniform spacing, and deriving the entropy-area relation and thermodynamic interpretation of Einstein's equations.

Contribution

It introduces a principle that boundary terms in the gravitational action are quantized and links this to horizon entropy and area quantization, providing a new perspective on gravity's quantum aspects.

Findings

Horizon area spectrum is quantized with uniform spacing of 2πħ.

Horizon entropy is proportional to one-fourth of the horizon area.

Gravity's action has a thermodynamic interpretation in static space-times.

Abstract

The principle of equivalence provides a description of gravity in terms of the metric tensor and determines how gravity affects the light cone structure of the space-time. This, in turn, leads to the existence of observers (in any space-time) who do not have access to regions of space-time bounded by horizons. To take into account this generic possibility, it is necessary to demand that \emph{physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access}. This principle is powerful enough to obtain the following results: (a) The action principle of gravity must be of such a structure that, in the semiclassical limit, the action of the unobserved degrees of freedom reduces to a boundary contribution $A_{\rm boundary}$ obtained by integrating a four divergence. (b) When the boundary is a horizon, $A_{\rm boundary}$ essentially reduces to a single, well-defined, term. (c) This boundary term must have a quantized spectrum with uniform spacing, $\Delta A_{boundary}=2\pi\hbar$, in the semiclassical limit. Using this principle in conjunction with the usual action principle in gravity, we show that: (i) The area of any one-way membrane is quantized. (ii) The information hidden by a one-way membrane leads to an entropy which is always one-fourth of the area of the membrane, in the leading order. (iii) In static space-times, the action for gravity can be given a purely thermodynamic interpretation and the Einstein equations have a formal similarity to laws of thermodynamics.