Canonical Uncertainty Relations for Madelung Variables in Curved Spacetime: Relating Dark Matter Cores and Unruh Radiation

Abstract

We establish fundamental uncertainty relations for the hydrodynamic variables arising from the Madelung representation of quantum fields in curved spacetime. Through canonical quantization of the density \(n\) and phase \(\theta\) variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function \(N\) and spatial metric \(\gamma_{ij}\). These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for Scalar Field Dark Matter (SFDM) models and stochastic quantum gravity. We present two groundbreaking applications. First, By coupling the density-velocity uncertainty to the Newtonian condition of hydrostatic equilibrium, we derive the exact scaling law for the minimum core radius of a self-gravitating scalar field halo. We naturally find the scale \(r_c \propto m^{-1/2}\) which is a testable prediction. Second, by evaluating the phase--momentum commutator in Rindler coordinates and relating the canonical momentum \(\Pi_\theta\) to the energy density via the Hamiltonian, we obtain the acceleration scale \(k_B T = C\hbar a/c\). The proportionality constant \(C=1/(2\pi)\) is fixed by the topological quantization of the Madelung phase circulation around the Euclidean horizon. This dual achievement demonstrates that quantum uncertainty on curved backgrounds is the unifying principle behind both galactic structure and black hole thermodynamics.