Entropy density of spacetime from the zero point length

It is possible to obtain gravitational field equations in a large class of theories from a thermodynamic variational principle which uses the gravitational heat density Sg associated with null surfaces. This heat density is related to the structure of spacetime at Planck scale, LP2=(Għ/c3), which assigns A⊥/LP2 degrees of freedom to any area A⊥. On the other hand, it is also known that the surface term Kh in the gravitational action correctly reproduces the heat density of the null surfaces. We provide a link between these ideas by obtaining Sg, used in emergent gravity paradigm, from the surface term in the Einstein–Hilbert action. 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 Kh for the qmetric in the appropriate limit directly reproduces the entropy density Sg used in the emergent gravity paradigm.