Entropy density of spacetime from the zero point length

TL;DR

This paper links the surface term in Einstein's gravitational action to the entropy density of null surfaces, incorporating zero-point length effects via a nonlocal qmetric to support emergent gravity ideas.

Contribution

It derives the gravitational heat density from the surface term in the action using a nonlocal qmetric, connecting quantum discreteness to classical gravitational thermodynamics.

Findings

Surface term reproduces entropy density in emergent gravity

Zero-point length effects incorporated through qmetric

Links quantum discreteness to classical gravitational thermodynamics

Abstract

It is possible to obtain the gravitational field equations in a large class of theories from a thermodynamic variational principle which uses the gravitational heat density $\mathcal{S}_g$ associated with null surfaces. This heat density is related to the discreteness of spacetime at Planck scale, $L_P^2 = (G\hbar / c^3)$, which assigns $A_{\perp}/L_P^2$ degrees of freedom to any area $A_{\perp}$. On the other hand, it is also known that the surface term $K\sqrt{h}$ in the gravitational action principle correctly reproduces the heat density of the null surfaces. We provide a link between these ideas by obtaining $\mathcal{S}_g$, used in emergent gravity paradigm, from the surface term in the action in Einstein's gravity. This is done using the notion of a nonlocal qmetric -- introduced recently [arXiv:1307.5618, arXiv:1405.4967] -- which allows us to study the effects of zero-point-length of spacetime at the transition scale between quantum and classical gravity. Computing $K\sqrt{h}$ for the qmetric in the appropriate limit directly reproduces the entropy density $\mathcal{S}_g$ used in the emergent gravity paradigm.