Relativistic Extended Uncertainty Principle from Spacetime Curvature

TL;DR

This paper derives a relativistic extension of the uncertainty principle influenced by spacetime curvature, generalizing previous nonrelativistic results and applicable to both massive and massless particles.

Contribution

It introduces a relativistic uncertainty relation based on spacetime curvature using the ADM formalism, extending prior nonrelativistic work to a general relativistic context.

Findings

Uncertainty relation corrections depend on Ricci scalar and metric gradients.

Results are applicable to both massive and massless particles.

Provides a noncovariant but consistent relativistic framework.

Abstract

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.