Gravitationally induced uncertainty relations in curved backgrounds

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