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

Abstract In quantum theory on curved backgrounds, Heisenberg’s uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann–Lüders projection onto a geodesic ball B Σ ( r ) of radius r on a Cauchy slice ( Σ , h ) , with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B Σ ( r ) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace–Beltrami operator on the ball, σ p ⩾ ℏ λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus | ψ | from phase-gradient fluctuations and clarifies how the spectral geometry of ( Σ , h ) controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ⩾ π ℏ / 2 , depending only on the proper radius r .