Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity

Abstract: We prove a coordinate- and foliation-independent Heisenberg-type lower bound for quantum states strictly localized in geodesic balls of radius $r$ on spacelike hypersurfaces of arbitrary spacetimes (with matter and a cosmological constant). The estimate depends only on the induced Riemannian geometry of the slice; it is independent of the lapse, shift, and extrinsic curvature, and controls the canonical momentum variance/uncertainty $\sigma_p$ by the first Dirichlet eigenvalue of the Laplace-Beltrami operator (Theorem). On weakly mean-convex balls we obtain the universal product inequality $\sigma_p\,r \ge \hbar/2$, whose constant is optimal and never attained (Corollary). This result abstracts and extends the framework recently introduced for black-hole slices in [10].