Minimum length metric and horizon area variation

Most of the efforts in unifying general relativity and quantum mechanics come out with two consequences: the presence of a minimum length scale and the non-locality of the spacetime at small scale. The qmetric, or minimum length metric, is a bitensor (it embodies non-locality) acting as a renormalized metric tensor with a minimum length built in: at large scale it approximates the classical metric tensor while the more we approach small scales the more the effects of the presence of a minimum length are relevant. After a review of the general description we construct the qmetric explicitly for Euclidean space and Minkowski spacetime, studying what happens to the area and volume elements of a geodesic congruence cross section. The relevant result is the presence of an irreducible minimum area for the cross section of a geodesic congruence emanating from a point: we can give a notion of a transverse area around any event of the spacetime upholding past results in literature. We exploit this result in the context of black hole horizon area variation, in the approximation such that the flat description can be used locally, showing that the qmetric proves that the presence of a minimum length brings with it a minimum step of area variation, i.e. a quantum of area.