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-L\"uders projection onto a geodesic ball $B_\Sigma(r)$ of radius $r$ on a Cauchy slice $(\Sigma,h)$, 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 $\sigma_p$ and show that strict confinement to $B_\Sigma(r)$ enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue $\lambda_1$ of the Laplace-Beltrami operator on the ball, $\sigma_p \ge \hbar\sqrt{\lambda_1}$, and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus $|\psi|$ from phase-gradient fluctuations and clarifies how the spectral geometry of $(\Sigma,h)$ controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, $\sigma_p r \ge \pi\hbar/2$, depending only on the proper radius $r$.