2025/09/23 by Thomas Schürmann, Schürmann, Thomas · 1 citation
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Noncommutative and Quantum Gravity Theories #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.2509.19099
We establish a coordinate-invariant Heisenberg-type lower bound for quantum states strictly localized in geodesic balls of radius rg on horizon-regular spacelike slices of static, spherically symmetric, asymptotically flat (AF) black-holes. Via a variance-eigenvalue equivalence the momentum uncertainty reduces to the first Dirichlet eigenvalue of the Laplace-Beltrami operator, yielding a slice-uniform Hardy baseline σp rg ≥ ℏ/2 under mild convexity assumptions on the balls; the bound is never attained and admits a positive gap both on compact interior regions and uniformly far out. For the Schwarzschild Painlevé-Gullstrand (PG) slice, whose induced 3-geometry is Euclidean, one recovers the exact Euclidean scale σp rg ≥ πℏ, which is optimal among all admissible slices. The entire construction extends across the black-hole horizon, and it transfers to the static axisymmetric Weyl class, where the Hardy floor, strict gap, and AF π-scale persist (a global PG-like optimum need not exist).