Gravitationally induced uncertainty relations in curved backgrounds
Luciano Petruzziello, Fabian Wagner
Abstract
This paper aims at investigating the influence of space-time curvature on the uncertainty relation. In particular, relying on previous findings, we assume the quantum wave function to be confined to a geodesic ball on a given spacelike hypersurface whose radius is a measure of the position uncertainty. On the other hand, we concurrently work out a viable physical definition of the momentum operator and its standard deviation in the nonrelativistic limit of the $3+1$ formalism. Finally, we evaluate the uncertainty relation, which to second order depends on the Ricci scalar of the effective 3-metric and the corresponding covariant derivative of the shift vector. For the sake of illustration, we apply our general result to a number of examples arising in the context of both general relativity and extended theories of gravity.