DRUMS Theory · Black Hole Collimation in DRUMS

Foundational Field Representation

In DRUMS, the universe is modeled as a coherent superfluid membrane described by a complex order parameter:

\[ \Psi(\mathbf{x},t) = \rho(\mathbf{x},t) \, e^{i\theta(\mathbf{x},t)} \]

The velocity field emerges from the phase gradient:

\[ \mathbf{v} = \frac{\hbar}{m} \nabla \theta \]

Governing Hydrodynamic Equations

The system obeys the continuity equation:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \]

and a modified Euler equation including quantum pressure:

\[ m \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right) = -\nabla (V + Q) \]

\[ Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} \]

Black Hole as a Phase Singularity

A black hole corresponds to a topological defect where the phase winds non-trivially:

\[ \oint \nabla \theta \cdot d\mathbf{l} = 2\pi n \]

This produces quantized circulation:

\[ \Gamma = \oint \mathbf{v} \cdot d\mathbf{l} = n \frac{h}{m} \]

Formation of Axial Symmetry

Near the singularity, density collapses and gradients steepen. The resulting pressure anisotropy emerges from the quantum pressure term:

\[ \nabla Q \propto \nabla \left( \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} \right) \]

Because the phase defect enforces cylindrical symmetry, gradients are minimized along the axis and maximized radially.

Collimation Mechanism

The key to collimation is anisotropic stress:

\[ \sigma_{ij} = \rho v_i v_j + P \delta_{ij} + \sigma_{ij}^{(Q)} \]

Where the quantum stress tensor introduces directional dependence:

\[ \sigma_{ij}^{(Q)} \sim \partial_i \partial_j \ln \rho \]

This produces:

Radial confinement (large transverse gradients)
Axial escape channels (minimal longitudinal gradients)

Jet Acceleration

Acceleration occurs along the axis due to pressure release:

\[ F_z = -\frac{\partial}{\partial z} (P + Q) \]

Since radial pressure is higher than axial pressure:

\[ \frac{\partial}{\partial r} (P + Q) \gg \frac{\partial}{\partial z} (P + Q) \]

Flow is forced into narrow axial jets.

Stability and Self-Collimation

The system naturally stabilizes via vorticity conservation:

\[ \frac{d \mathbf{\omega}}{dt} = (\mathbf{\omega} \cdot \nabla) \mathbf{v} \]

Vorticity aligns with the axis, reinforcing collimation.

“Black hole collimation in DRUMS is not an added mechanism but a direct consequence of the governing equations.”

Final Interpretation

Black hole collimation in DRUMS is a direct consequence of:

Phase singularities enforcing topology
Quantum pressure generating anisotropic stress
Density gradients shaping flow geometry
Vorticity alignment stabilizing jets

Highly collimated jets emerge inevitably from the governing equations, without additional free parameters or ad hoc assumptions.

Conclusion: Jets as a Geometric Necessity

In the DRUMS framework, relativistic jets are not a separate phenomenon requiring a dedicated acceleration mechanism. They are the natural outflow geometry imposed by the superfluid's quantum pressure and phase topology around a black hole. The same principles that give rise to the black hole's existence — phase winding and density collapse — also dictate how matter escapes from its vicinity.

This unified description explains why jets are collimated over vast distances, why they remain stable, and why they emerge from black holes of vastly different scales — from stellar-mass systems to supermassive galactic nuclei. In each case, the underlying superfluid dynamics are the same.

The collimation mechanism is not a puzzle to be solved but a signature to be read. Every observed jet is a direct probe of the quantum pressure tensor and the topological structure of the superfluid phase field around the black hole. The fact that jets are ubiquitous and geometrically similar across scales is strong evidence that the superfluid description captures the essential physics.