Uncertainty Principle on 3-Dimensional Manifolds of Constant Curvature

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature K. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann–Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and K. For hyperbolic spaces, we also obtain a global lower bound \(\sigma _p\ge |K|^\frac{1}{2}\hbar \), which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by \(r_s\ge 2\,l_P\), where \(l_P\) is the Planck length.