Entanglement entropy in quantum gravity and the Plateau problem

In a quantum gravity theory the entropy of entanglement S between the fundamental degrees of freedom spatially divided by a surface is discussed. Classical gravity is considered as an emergent phenomenon and arguments are presented that (1) S is a macroscopical quantity which can be determined without knowing a real microscopical content of the fundamental theory; (2) S is given by the Bekenstein-Hawking formula in terms of the area of a codimension 2 hypesurface B; (3) in static space-times B can be defined as a minimal hypersurface of a least volume separating the system in a constant-time slice. It is shown that properties of S are in agreement with basic properties of the von Neumann entropy. Explicit variational formulae for S in different physical examples are considered.