Chapter 17
Coulomb Flow and Charge Potential
Field and Energy Systems
Author: Mark Anthony Higgins
Revised Date: 23 February 2026
Chapter 17
What this flow captures
CoulombFlow_ChargePotential reformulates classical electrostatics as a problem of boundary coherence and curvature tension, rather than forces acting through empty space.
In HST, a charge is not a material point but a localized phase imbalance — a region where the curvature of the coherence field diverges slightly from equilibrium.
The electric field is the gradient of that imbalance; the potential is the stored curvature tension between boundaries.
Thus, Coulomb’s law becomes a specific case of coherence redistribution:
\[ \partial_{\mu} J^{\mu}_{\mathrm{coh}} = 0 \;\Longleftrightarrow\; \nabla\!\cdot\!\mathbf{E} = \frac{\rho}{\varepsilon_{0}}\,. \]
The inverse-square field reflects how curvature coherence spreads over spherical boundaries to maintain constant flux.
Core intuition (HST view)
In the conventional view, a charge emits an electric field described by
\[ \mathbf{E} = \frac{1}{4\pi \varepsilon_{0}}\,\frac{q}{r^{2}}\,\hat{r}, \qquad \Phi = \frac{1}{4\pi \varepsilon_{0}}\,\frac{q}{r}\,. \]
In HST, this same relationship describes how boundary phase curvature relaxes through space: the field lines are coherence trajectories and the potential is their curvature energy density per unit phase.
When two charged boundaries interact, coherence flows from higher tension to lower — what we perceive as an electric force.
Mathematical structure
The electrostatic potential obeys Poisson’s equation:
\[ \nabla^{2}\Phi = -\frac{\rho}{\varepsilon_{0}}\,. \]
The field energy density is
\[ \rho_{E} = \tfrac{1}{2}\,\varepsilon_{0}\,|\mathbf{E}|^{2}\,. \]
Total stored energy:
\[ U = \tfrac{1}{2} \int \rho\,\Phi\, dV = \tfrac{1}{2}\,\varepsilon_{0} \int |\mathbf{E}|^{2}\, dV\,. \]
In HST, this stored energy is the curvature tension of the field:
\[ \Sigma \propto |\mathbf{E}|^{2}\,. \]
representing the amount of geometric distortion sustained by coherence at the boundary.
Electric field lines are curvature contours; equipotential surfaces are regions of constant coherence tension.
Interpreting charge through boundary coherence
| Classical term | HST interpretation |
|---|---|
|
Charge
\( q \)
|
Localised curvature asymmetry — net phase offset
|
|
Electric field \( \mathbf{E} \)
|
Gradient of coherence realignment
|
|
Potential \( \Phi \)
|
Scalar curvature tension energy per unit charge
|
|
Field energy
\( q\,\mathbf{E} \)
|
Stored curvature tension density
|
|
Force
𝑞
𝐸
|
Restoring flow of coherence seeking boundary equilibrium
|
|
Superposition
|
Linear coherence overlay (additivity of phase gradients)
|
The field equations describe not action at a distance but the redistribution of curvature in a connected coherence fabric.
Boundary flux and conservation
The field flux through a closed surface,
\[ \oint_{S} \mathbf{E} \cdot d\mathbf{A} = \frac{q_{\mathrm{enc}}}{\varepsilon_{0}}\,. \]
is identical to the continuity of coherence current:
\[ \int_{V} \partial_{\mu} J^{\mu}_{\mathrm{coh}}\, dV = 0\,. \]
This ensures that curvature flow is divergence-free everywhere except where phase discontinuity (charge) exists.
When two charges approach, their curvature boundaries deform; the resulting overlap defines the potential energy of interaction.
Superposition and linearity
At low curvature tension, the field obeys perfect linearity:
\[ \Phi_{\mathrm{total}} = \sum_{i} \Phi_{i}, \qquad \mathbf{E}_{\mathrm{total}} = \sum_{i} \mathbf{E}_{i}\,. \]
Each charge contributes additively — coherence flows independently through the global field.
When field intensity rises, curvature feedback appears (nonlinear terms like leading to self-interaction or dielectric breakdown — the geometric expression of nonlinearity.
Energy and coherence flow
The Poynting vector
\[ \mathbf{S} = \frac{1}{\mu_{0}}\,(\mathbf{E} \times \mathbf{B})\,. \]
in dynamic fields represents the directional flux of coherence energy.
In static Coulomb systems, that flux is zero on average — coherence tension is stored, not propagated.
A radiating charge is simply a boundary where coherence oscillates and releases stored curvature energy as electromagnetic waves.
Philosophical coherence
In the language of HST:
Charge is phase; potential is tension; field is geometry seeking equilibrium.
Significance
The Coulomb flow demonstrates that even the simplest field — electrostatics — obeys the same coherence law as gravity, fluid dynamics, or quantum phase.
Every “force” in physics is the movement of coherence tension toward equilibrium across a boundary.
When expressed this way, electromagnetism becomes not a separate interaction but a manifestation of curvature coherence in U(1) symmetry space — the harmonic of phase conservation predicted by Noether’s theorem.
At its heart, the Coulomb field is geometry remembering its imbalance, and potential is that memory’s stored tension.
In essence
Charge is not a thing but a curvature note in the field’s coherence song.
Potential is the pitch of that note — the strain of geometry awaiting release.
All fields, even the electric, are coherence seeking to heal their own asymmetry.
