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 (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 \(Π_θ\) to the energy density via the Hamiltonian, we obtain the acceleration scale \(k_B T = C\hbar a/c\). The proportionality constant \(C=1/(2π)\) 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.
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 (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 (Π_θ) to the energy density via the Hamiltonian, we obtain the acceleration scale (k_B T = C\hbar a/c). The proportionality constant (C=1/(2π)) 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.
The Heisenberg Uncertainty Principle is traditionally formulated within the mathematical idealization of perfectly flat Euclidean space. However, physical reality dictates thatspace is inherently curved by gravitational fields, cosmological dynamics, and perpetualquantum vacuum fluctuations. This paper proposes a novel theoretical framework that generalizes three fundamental uncertainty principles—Position-Momentum, Energy-Time, andAngle-Angular Momentum—by directly incorporating spatial curvature on a cosmologicalscale. We introduce a continuous spatial modulating function, f(x), governed by the netspatial curvature R(x), representing the dynamic interplay between attractive and repulsive geometric forces. To rigorously preserve the quantum mechanical axiom that physicalobservables correspond to Hermitian operators, we employ Weyl symmetrization to construct modified differential operators for momentum and angular momentum. Through exactmathematical derivations utilizing wave packet phase-space dynamics, a modified Ehrenfesttheorem, and the resolution of topological boundary discontinuities via mapping to a Universal Covering Space with strict Dirichlet boundary conditions, we establish absolute lowerbounds for these uncertainties. Ultimately, we demonstrate that the minimum bound ofquantum uncertainty is not a rigid universal constant, but is strictly positive and dynamically dictated by local geometric curvature, providing a robust bridge between quantummechanics and spacetime geometry.Keywords: Generalized Uncertainty Principle; Spatial Curvature; Weyl Symmetrization; Hermitian Operators; Modified Ehrenfest Theorem; Universal Covering Space; Cosmological Scale.PACSNumbers: 03.65.Ta(Foundationsof quantummechanics); 04.60.-m (Quantum gravity);03.65.Ca (Formalism).MSC2020 Codes: 81P05 (General and philosophical foundations of quantum mechanics);81S05 (Canonical quantization, commutation relations and statistics)
Inverted Relativistic Phase-Fluid Mechanics on Curved Spacetime introduces a covariant matter-phase model in which the phase field defines the local four-momentum, written in readable form as: p_mu = hbar * nabla_mu phi or equivalently: four-momentum = Planck constant × phase gradient The density n weights the conserved phase current and determines how strongly the phase flow contributes to the stress-energy tensor and therefore to spacetime curvature. In the minimal model, n has no independent gradient term. It acts as an algebraic density variable enforcing a relativistic Hamilton-Jacobi or mass-shell relation. This gives an inverted relativistic phase fluid: the phase determines motion, while the density weights its physical and gravitational effect. The model is then extended through the Madelung decomposition: psi = sqrt(n) * exp(i phi) This adds the amplitude-gradient term and produces the relativistic quantum-pressure contribution: Q_rel = - hbar^2 * Box(sqrt(n)) / sqrt(n) where Box is the covariant d’Alembert operator on curved spacetime. In this way, the minimal phase-fluid picture becomes connected to the full Klein-Gordon/Madelung quantum-fluid structure on curved spacetime. The paper provides a covariant bridge between: proper-time matter phase, de Broglie momentum-phase relations, relativistic fluid mechanics, Madelung quantum hydrodynamics, quantum pressure, stress-energy generation, and gravitational curvature. The model does not replace general relativity and does not claim to be a completed theory of quantum gravity. Instead, it formulates a compact theoretical framework in which matter phase, density, quantum amplitude structure and spacetime curvature are coupled without introducing a preferred reference frame. A compact summary of the construction is: The phase carries the motion, the density carries the weighting, the quantum pressure carries the wave nature, and the metric carries gravity. This preprint is intended as a theory note and starting point for further analytical and numerical work, including fixed-background simulations, weak gravitational backreaction, and full Einstein-Klein-Gordon/Madelung evolution.
The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, PLμ and PRμ, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling (q/2)σμνFμν while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation -- representing the first law of thermodynamics for Dirac fields -- we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.
Inverted Relativistic Phase-Fluid Mechanics on Curved Spacetime introduces a covariant matter-phase model in which the phase field defines the local four-momentum, written in readable form as: p_mu = hbar * nabla_mu phi or equivalently: four-momentum = Planck constant × phase gradient The density n weights the conserved phase current and determines how strongly the phase flow contributes to the stress-energy tensor and therefore to spacetime curvature. In the minimal model, n has no independent gradient term. It acts as an algebraic density variable enforcing a relativistic Hamilton-Jacobi or mass-shell relation. This gives an inverted relativistic phase fluid: the phase determines motion, while the density weights its physical and gravitational effect. The model is then extended through the Madelung decomposition: psi = sqrt(n) * exp(i phi) This adds the amplitude-gradient term and produces the relativistic quantum-pressure contribution: Q_rel = - hbar^2 * Box(sqrt(n)) / sqrt(n) where Box is the covariant d’Alembert operator on curved spacetime. In this way, the minimal phase-fluid picture becomes connected to the full Klein-Gordon/Madelung quantum-fluid structure on curved spacetime. The paper provides a covariant bridge between: proper-time matter phase, de Broglie momentum-phase relations, relativistic fluid mechanics, Madelung quantum hydrodynamics, quantum pressure, stress-energy generation, and gravitational curvature. The model does not replace general relativity and does not claim to be a completed theory of quantum gravity. Instead, it formulates a compact theoretical framework in which matter phase, density, quantum amplitude structure and spacetime curvature are coupled without introducing a preferred reference frame. A compact summary of the construction is: The phase carries the motion, the density carries the weighting, the quantum pressure carries the wave nature, and the metric carries gravity. This preprint is intended as a theory note and starting point for further analytical and numerical work, including fixed-background simulations, weak gravitational backreaction, and full Einstein-Klein-Gordon/Madelung evolution.
We develop a comprehensive thermodynamic description for a zero-temperature boson gas in curved spacetime, integrating energy conservation with information-theoretic principles. Using the hydrodynamic Madelung representation within the ADM formalism, we establish two fundamental relationships: an energy balance equation representing the first law of thermodynamics from a spacetime perspective, and an information-theoretic constraint connecting Fisher entropy to the dynamical evolution of the boson density. This dual formulation clearly separates energy transport from information conservation while revealing how quantum information is preserved in curved backgrounds. The introduction of a stochastic velocity provides a bridge between quantum potential effects and underlying spacetime fluctuations. We demonstrate the consistency of our framework through detailed analyses of quantum systems in both Minkowski and Schwarzschild spacetimes. This work provides a unified foundation for studying relativistic bosonic systems, with direct relevance to boson stars and scalar field dark matter models.
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature (1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as Γ=L2/ℓ2 for states that saturate the uncertainty relations, where L and ℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit L→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both ℓ→0 and L→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.