Vacuum Energy in Ultralocal metrics for TT tensors with Gaussian Wave Functionals

Abstract

We calculate, in a class of Gauge invariant functionals, by variational methods, the difference of vacuum energy between two different backgrounds: Schwarzschild and Flat Space. We perform this evaluation in an Hamiltonian formulation of Quantum Gravity by standard ''$3+1$'' decomposition. After the decomposition the scalar curvature is expanded to second order with respect to the Schwarzschild metric. We evaluate this energy difference in momentum space, in the lowest possible state (regardless of any negative mode). We find a singular behaviour in the UV-limit, due to the presence of the horizon when $r=2m.$ When $r>2m$ this singular behaviour disappears, which is in agreement with various other models presented in the literature.