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