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 $B_{\Sigma }(r)$ 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, ${\sigma }_p\ge \hslash \sqrt{{\lambda }_1}$, 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, ${\sigma }_{p}r\ge \pi \hslash /2$, depending only on the proper radius $r$.

23 pages