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-L\"uders 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 λ₁ of the Laplace-Beltrami operator on the ball, σ_p ≥ ℏ√λ₁, 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 ≥ πℏ/2, depending only on the proper radius r.