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

• URL: http://arxiv.org/abs/2510.01628v2
• Date: Tue, 28 Oct 2025 13:00:05 GMT
• Title: Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity
• Authors: Thomas Schürmann,
• Abstract summary: We prove a coordinate- and foliation-independent Heisenbergtype lower bound for quantum states strictly localized in geodesic balls of radius spacelike hypersurfaces of arbitrary spacetimes with matter and a constant.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\sigma_p r \ge \hbar/2$. Under the same assumption, a vector-field Barta-type argument improves this universal floor to the scale-invariant bound $\sigma_p r \ge \pi\hbar/2$, which provides a universal, foliation-independent floor. Any further sharpening of the constant requires eigenvalue-comparison results or other curvature-sensitive methods.

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