Electromagnetism: From Coulomb Interaction to Transverse Waves

01Begin with the interaction between two charges

Static charges attract or repel, while moving charges also exhibit magnetic interactions. ASD asks a common question: if motion changes the surrounding spatial interaction structure, must the distance entering the charge interaction be specified again? When the moving interaction is compared with Coulomb interaction calculated at the static reference separation, how does the difference become a magnetic contribution?

Chapter 3 draws on two upstream results. LVE specifies length relations attached to a motion state and direction; dynamics establishes matter momentum, energy and their changes. The chapter first chooses a configuration in which two charge interactions can be compared clearly, then introduces continuous spatial states for local response and outward propagation. Its basic magnetic derivation starts with Coulomb interaction, motion and true interaction distance, rather than transforming electromagnetic fields between frames.

“True distance” denotes the physical interaction distance assigned by the ASD constitutive model. Its relation to reference coordinates and standards requires a measurement mapping; the name does not make it an unconditional instrument reading. The process parameter is $t$, and the characteristic propagation speed in the homogeneous baseline is $c$. Matter velocities refer to a local homogeneous spatial background. An apparatus calculation must also specify the laboratory's relation to that background.

02From a static baseline to a motion-dependent correction

Let the charges be $q_1,q_2$, with exterior reference separation $r$. Neglect finite size, external gradients and boundary corrections. The static baseline magnitude at the same charges and reference separation is:

$\varepsilon_0$ is the vacuum permittivity used in this electromagnetic normalization; the charge signs determine attraction or repulsion. Now give both charges the same constant parallel velocity $\mathbf v$, with their separation perpendicular to it. Write $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The directed LVE factor $K(\phi)=\gamma(1-\beta\cos\phi)$ at $\phi=\pi/2$ gives:

Geometry alone does not specify a force. At this step Chapter 3 makes an explicit constitutive choice: Coulomb inverse-square interaction occurs at the true distance $r_T$. The denominator becomes $\gamma^2r^2$, giving the moving interaction magnitude:

Let $\hat{\mathbf r}$ point from $q_1$ to $q_2$. The vectors $\mathbf F_0,\mathbf F_T$ retain the charge signs and represent the baseline and true physical forces. Their difference provides the chapter's entry into magnetic interaction:

For like charges the correction points inward and reduces repulsion; for unlike charges it points outward and reduces the attraction magnitude. The total interaction remains $\mathbf F_T=\mathbf F_0+\Delta\mathbf F_m$. Adding the correction again after calculating the interaction at $r_T$ would count the same scale effect twice. No low-speed truncation is used in this chain for the stated configuration and $0\leq v<c$. It does not specify general unequal-velocity, antiparallel, oblique or accelerating sources. Coulomb interaction and magnetic force develop the inputs, signed correction and virtual-work pairing.

03Why continuous spatial fields are also needed

A two-charge law answers one local configuration question. Currents, magnetic needles, induction and changes arriving at distant points require a state that varies with position and process parameter. ASD follows the motion-dependent transverse interaction by introducing shear and torsion: shear describes relative transverse changes between neighboring spatial regions; torsion describes changes of local orientation. Continuous potentials organize this response into calculable field variables.

With effective scalar potential $\Phi_\tau$ and vector potential $\mathbf A_\tau$, the subscript $\tau$ marking the electromagnetic torsion channel, define:

$\mathbf E$ includes the scalar gradient and the vector potential's process change; $\mathbf B$ is its curl. Here curl measures a local circulating variation. These definitions organize the representation, but do not alone determine how sources create potentials, matter responds, or fields propagate. Gauge freedom allows different potentials to represent the same $\mathbf E,\mathbf B$, so an arbitrary-gauge vector potential cannot be identified pointwise with a unique displacement.

Vector potential explains gauge freedom, transverse projection and the slow-source near-zone approximation, then reconstructs the free matter action from Chapter 2's momentum–energy relations. A separately chosen minimal coupling supplies linear interactions of scalar potential with charge and vector potential with current; variation then gives charge response. Magnetic field develops this calculation, currents, moments and induction. Reproducing the leading magnetic correction is a consistency check of the effective representation; a comparison of total forces must retain electric corrections of the same order.

The angular field around a moving source is a further problem. Moving-charge field compares a radial-intensity candidate with a potential-gradient candidate. Giving $R_T=K(\phi)R$ alone does not fix field components, area elements, retardation or a stationary probe's signal. The co-moving two-charge law is not automatically the complete high-speed prediction for another source–probe configuration.

04How transverse degrees of freedom become propagating modes

Local interaction is not yet a free wave. In a homogeneous, long-wavelength, weak-perturbation sector, Chapter 3 chooses a length-valued transverse state $\mathbf u_T$ with $\nabla\cdot\mathbf u_T=0$. This divergence-free condition selects the transverse channel rather than longitudinal volume or density variation. Positive inertia coefficient $\rho_s$ and transverse restoring coefficient $K_T$ enter a quadratic action:

Variation with constant coefficients and compatible boundaries gives a transverse wave equation with $c_T^2=K_T/\rho_s$. A constant normalization $\mathbf A_T=N_T\mathbf u_T$ represents the spatial curl as magnetic field and process change as transverse electric field. Only after matching $\rho_s=\varepsilon_0N_T^2$ and $K_T=N_T^2/\mu_0$ does one obtain $c_T=c$ and the free Maxwell representation.

This free-propagation derivation has been completed within that effective model. The quadratic action, transverse constraint, normalization and coefficient matching are its premises. LVE alone has not determined microscopic coefficients or solved general high-speed topological charge sources. Electromagnetic waves follows the variation, four free-field relations, polarization and energy flux, then explains effective source coupling and retarded solutions.

A free transverse wave propagating at $c$ and a translating closed matter structure are different physical processes. The wave's speed must not be inserted as matter velocity in $\gamma(v)$. Emission, exterior propagation and reception are modeled separately before forming a complete measurement process.

05Connecting the results to experiments and further research

The ASD construction proceeds from the static Coulomb baseline to a specified motion configuration, LVE true distance, magnetic correction, continuous fields and transverse propagation. Each transition has identifiable geometrical, constitutive or approximation inputs; the preceding step does not uniquely determine every later result by itself.

Experiments ultimately compare readings from the same apparatus. True physical force and reference-coordinate generalized force pair the same virtual work with different variables. Numerical agreement or disagreement cannot bypass displacement, momentum, process parameters and detector coupling to establish equivalence or discrimination. The experimental matrix connects current and propagation cases; the high-γ field case still requires a source model and electro-optic response, without an invented fit here.

Space Dynamics connects transverse degrees of freedom with the longitudinal structure associated with gravity. Chapter 3 provides candidate longitudinal–transverse variational sources and a local energy boundedness condition. Stable matter solutions and long-range response still require full gradient energy, boundaries and nonlinear equations. Mathematical correspondence, candidate microscopic interpretation and experimental confirmation retain their separate meanings.