Asymptotic Generalized Extended Uncertainty Principle

TL;DR

This paper develops a formalism to derive an asymptotic extended uncertainty principle in curved spaces, incorporating curvature effects and applying it to various models, including black hole scenarios.

Contribution

It introduces a new method to derive the asymptotic extended uncertainty principle for arbitrary curvature models, combining it with the GUP through deformed commutators.

Findings

Derived curvature-induced corrections to the EUP involving Ricci scalar and Cartan invariants.

Verified the formalism for homogeneous space models.

Applied the formalism to inhomogeneities, rotation, and black hole 'hair' effects.

Abstract

We present a formalism which allows for the perturbative derivation of the Extended Uncertainty Principle (EUP) for arbitrary spatial curvature models and observers. Entering the realm of small position uncertainties, we derive a general asymptotic EUP. The leading 2nd order curvature induced correction is proportional to the Ricci scalar, while the 4th order correction features the 0th order Cartan invariant Psi^2 (a scalar quadratic in curvature tensors) and the curved space Laplacian of the Ricci scalar all of which are evaluated at the expectation value of the position operator, i.e. the expected position when performing a measurement. This result is first verified for previously derived homogeneous space models and then applied to other non-trivial curvature related effects such as inhomogeneities, rotation and an anisotropic stress fluid leading to black hole "hair". Our main achievement combines the method we introduce with the Generalized Uncertainty Principle (GUP) by virtue of deformed commutators to formulate a generic form of what we call the Asymptotic Generalized Extended Uncertainty Principle (AGEUP).