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 θ variables and their conjugate momenta, we derive exact uncertainty principles that depend on spacetime geometry through the lapse function N and spatial metric γ_ij. These relations reveal how gravitational fields modulate quantum fluctuations and provide first-principles constraints for scalar field dark matter models and stochastic quantum gravity.