Role of Horizons in Semiclassical Gravity: Entropy and the Area Spectrum

In any space-time, it is possible to have a family of observers who have access to only part of the space-time manifold, because of the existence of a horizon. We 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}. In the coordinate frame in which these observers are at rest, the horizon manifests itself as a (coordinate) singularity in the metric tensor. Regularization of this singularity removes the inaccessible region, and leads to the following consequences: (a) The non-trivial topological structure for the effective manifold allows one to obtain the standard results of quantum field theory in curved space-time. (b) In case of gravity, this principle requires that the effect of the unobserved degrees of freedom should reduce to a boundary contribution $A_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}}$ to the gravitational action. When the boundary is a horizon, $A_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}}$ reduces to a single, well-defined term proportional to the area of the horizon. Using the form of this boundary term, it is possible to obtain the full gravitational action in the semiclassical limit. © This boundary term must have a quantized spectrum with uniform spacing, $\Delta A_{boundary}=2\pi\hbar$, in the semiclassical limit. This, in turn, yields the following results for semiclassical gravity: (i) The area of any one-way membrane is quantized. (ii) The information hidden by a one-way membrane amounts 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.