Finite-resolution measurement induces topological curvature defects in spacetime

Abstract

We show that regularizing $(2+1)$-dimensional Minkowski spacetime with a finite-resolution Gaussian probe, analogous to Weyl-Heisenberg (Gabor) signal analysis and related quantization, induces a curved geometry with a topological defect. The regularized metric replaces $r^2$ by $r^2+{\sigma }^2$ in the angular part, where $\sigma$ is the resolution scale from the width of the Gaussian probe. The resulting Gaussian curvature integrates to $-2\pi$, independently of $\sigma$. This curvature defines an effective stress-energy source with universal total energy $E_{\text{eff}}=-1/(4G)$. The limit $\sigma \to 0$ leads to distributional Dirac-delta curvature and to appearance of topological defect at the origin. These results show that finite spatial resolution measurement does not merely smooth singularities but can shape spacetime geometry.