Longitudinal–Transverse Coupling

01Task

OP-EM-005 · OPEN

Continue the established local variational model toward normalized sources, complete fields and stable finite-energy configurations.

02Established baseline

This task continues the nonlinear longitudinal–transverse candidate in final Chapter 3, Section VIII; it does not start by inventing a coupling. Under compatible boundaries, split displacement into curl-free $\mathbf u_L$ and divergence-free $\mathbf u_T$. Define the dimensionless $\eta=-\nabla\cdot\mathbf u_L$ and $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$. Positive $\eta$ means local compression; negative $\eta$ means expansion. Adopt the rotational-scalar coupling

The coefficient $g_{LT}$ has energy-density units. This term makes transverse torsion intensity a local input to longitudinal volumetric strain. The coupling is a chosen nonlinear constitutive relation; its variations are results that can be checked within that choice.

For variations vanishing at temporal endpoints and spatial boundaries, use $\delta\eta=-\nabla\cdot\delta\mathbf u_L$:

For the transverse variation, $\delta\boldsymbol\Omega_T=\nabla\times\delta\mathbf u_T$. Curl integration by parts preserves its sign under these boundary conditions, so

Vary together with the free fields and place the terms on the right-hand sides to obtain reciprocal generalized force densities:

Their units are N/m³. They are a longitudinal gradient source and a divergence-free transverse source, not electric current density $\mathbf J$. The coupling is cubic about a uniform zero background and leaves the lowest-order free-wave spectrum unchanged; finite backgrounds or amplitudes allow mutual driving.

03Local energy bound and induced expansion

A cubic term alone need not bound local energy below. Add a positive longitudinal quadratic term and a transverse quartic term, with constant $K_L>0$ and $\lambda_T$, both having pressure units. Completing the square gives

A sufficient condition is $\lambda_T\geq2g_{LT}^2/K_L$. Minimizing with respect to $\eta$ at fixed transverse intensity gives

Thus positive $g_{LT}$ makes transverse torsion induce expansion and density deficit. Local strain $\eta$ is not the exterior long-range gravitational scale potential. Connecting the local response to a long-range source requires longitudinal constitutive relations, complete gradient energy and a boundary solution.

In a closed conservative system, including the coupling potential preserves total energy. A loss in one channel can be a gain in another. Exchange is not irreversible dissipation, and a local lower bound is not an already obtained complete stable particle solution.

04Work to complete

Starting from this local candidate and both variations, determine coefficients and normalization in one longitudinal–transverse action and specify complete gradients, sources and boundaries. Solve a finite-energy configuration, perturbation spectrum about a nonzero background, channel exchange or dynamical-stability problem. The next result should distinguish assumed parameters, derived coefficients and computable response functions.

05Deliverables and decision criteria

Provide the complete equations of one action, boundary conditions, dimensional and conservation checks, a reproducible configuration or spectrum calculation, and an explicit map from local density response to a long-range source. If only the local square completion is tested, state its lower-bound scope without claiming electron or proton masses or stability in every perturbation direction.

06Review boundary and source

Based on final Chapter 3, equations (3.74)–(3.77) and the fourth scope item in Section IX. Variables, reciprocal sources, a local lower bound and induced expansion have been obtained. Microscopic coefficients and complete stable localized structures remain research tasks. The existing problem identity OP-EM-005 is retained.

07Formal continuation in Space Dynamics

The coupling page, OP-SD-001 and OP-SD-006 continue this task. They should calculate sources, fields and stable structures for the same candidate without reclassifying the completed local variations as absence of dynamics.