We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed by standard arguments of geodesic completion.

We consider momentum operators on intrinsically curved manifolds. Given that the momentum operators are Killing vector fields whose integral curves are geodesics, it is shown that the corresponding manifold is either flat, or otherwise of compact type with positive constant sectional curvature and dimension equal to 1, 3 or 7. Explicit representations of momentum operators and the associated Casimir element will be discussed for the 3-sphere. It will be verified that the structure constants of the underlying Lie algebra are proportional (...) to 2ħ/R, where R is the curvature radius of the sphere. This results in a countable energy and momentum spectrum of freely moving particles. It is shown that the maximum resolution of the possible momenta is given by the de-Broglie wave length λ=πR, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates and the associated commutator relations of position and momentum are established. (shrink)

In this paper, we make a proposal for addressing the cosmological constant problem. Our approach will be based on a reinterpretation of two non-standard de Sitter solutions given by the Einstein vacuum equations with Λ>0. As a first result, we derive an uncertainty principle for both variants of the de Sitter space (Theorem). Subsequently, a decomposition of the cosmological constant in a pair of time-dependent pieces is introduced (Corollary). The time-dependence of the corresponding energy and dark energy density is discussed (...) and especially matched at the Planck scale. Furthermore, we show that for every instant of cosmic time this approach can be revealed in terms of a Schwarzschild-Anti-de Sitter cosmology with Λ>0. The corresponding field equations are provided. (shrink)