Relativistic extended uncertainty principle from spacetime curvature

The investigations presented in this study are directed at relativistic modifications of the uncertainty relation derived from curvature background spacetime. These findings generalize previous work that is recovered nonrelativistic limit. Applying $3+1$ splitting accordance with ADM formalism, we find physical momentum operator and compute its standard deviation for wave functions confined to a geodesic ball on spacelike hypersurface. Its radius can then be understood as measure position uncertainty. Under assumption small uncertainties comparison length scales, obtain corresponding corrections flat space. Those depend Ricci scalar effective spatial metric, particle moving on, and, if there nonvanishing time-space components spacetime gradients shift vector lapse function. Interestingly, result applicable not only massive but also massless particles. Over all, covariant, yet consistently general approach. We further speculate possible covariant extension.