Electromagnetic Waves: Transverse Dynamics and Free Maxwell Representation

01Propagation is a further question beyond local interaction

The magnetic configuration describes a common motion's transverse interaction change; potentials represent continuous response. Electromagnetic waves ask what degrees of freedom can propagate after the source's local action, carrying energy and polarization. ASD begins with transverse spatial dynamics and normalizes it into electromagnetic variables.

$t$ is the chosen process parameter and $c$ the homogeneous baseline's characteristic speed. Translation of a closed matter structure enters LVE through $\gamma(v)$. Free-wave propagation is not that matter translation; setting $v=c$ in the two-charge length factor is inappropriate. Emission, exterior propagation and material reception are modeled separately before forming a comparable signal.

02Restoring and kinetic energies form the transverse equation

In a homogeneous, long-wavelength, weak-perturbation continuum sector, choose length-valued state $\mathbf u_T(\mathbf x,t)$ with $\nabla\cdot\mathbf u_T=0$ under decaying or compatible boundaries. The constraint selects transverse freedom. Constant $\rho_s>0$ is effective inertia with units kg/m³; $K_T>0$ is transverse restoring coefficient with units Pa. The lowest quadratic Lagrangian density is:

The first term records kinetic energy of process change and the second the curl's restoring energy. $K_T$ is this restoring coefficient, not directed length factor $K(\phi)$. Positive coefficients give positive free energy. The quadratic form is a declared effective constitutive choice, without prescribing a microscopic lattice.

Let $S_T=\int\mathcal L_T^{(2)}\,d^3x\,dt$ with variations vanishing at time endpoints and spatial boundaries. Integrating the kinetic term in time and curl term in space yields:

With constant coefficients and compatible boundaries the bracket lies in the transverse sector, so transverse variation gives:

The second form uses $\nabla\times(\nabla\times\mathbf u_T)=\nabla(\nabla\cdot\mathbf u_T)-\nabla^2\mathbf u_T=-\nabla^2\mathbf u_T$. A free transverse wave has now been obtained with $c_T=\sqrt{K_T/\rho_s}$; variation alone has not established $c_T=c$.

03Normalize the spatial state into free electromagnetic fields

Fix the corresponding gauge and boundaries in a source-free radiation region, with transverse projection $\mathbf A_T=\mathbf A_\tau^\perp$. A constant $N_T$ relates the state, spatial curl and process change:

Subscript $\tau$ marks the electromagnetic torsion channel and $T$ the transverse component. $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$ is dimensionless and $N_T$ has units V·s/m². Magnetic field expresses spatial torsion, and radiation electric field the same state's process variation. The identification does not extend to all potentials in arbitrary gauges.

Insert these definitions into the free electromagnetic action and match temporal and spatial coefficients separately:

Thus $c_T=c$ depends on this coefficient matching and electromagnetic normalization. Observing light propagation is not itself a derivation of arbitrary constitutive parameters. Microscopic coefficients still require underlying dynamics; FCC is a subsequent candidate implementation.

The transverse constraint gives $\nabla\cdot\mathbf E_T=0$, curl divergence gives $\nabla\cdot\mathbf B=0$, and commuting space and time derivatives gives Faraday's relation. The remaining curl equation follows from the spatial wave equation:

The four relations are consequently:

The free Maxwell representation has been derived within the selected quadratic transverse model. Transverse constraint, boundaries, normalization and coefficient matching are explicit premises. Maxwell structure is an equivalent effective representation of these spatial dynamics, not an independent basic postulate replacing the construction. This has not simultaneously determined electrostatic sources, charge quantization or general high-speed topological cores.

04Vacuum constants and impedance in one physical channel

Section VI.4 connects the static Coulomb baseline, moving magnetic difference and free transverse normalization. The magnetic-difference coefficient $1/(\varepsilon_0c^2)$ is written as $\mu_0$ in SI magnetic quantities. In the free transverse channel the energy density is

This is manuscript (3.47d). Matching the two coefficients to $\varepsilon_0E_T^2/2+B^2/(2\mu_0)$ gives (3.47e): $\varepsilon_0=\rho_s/N_T^2$ and $1/\mu_0=K_T/N_T^2$. With the stated normalization, electric energy represents transverse kinetic energy and magnetic energy represents torsional restoring energy. The relation $\varepsilon_0\mu_0=\rho_s/K_T=c^{-2}$ connects propagation speed to both coefficients.

The speed fixes their product but leaves a coupling scale. Expressing it through vacuum wave impedance, (3.47f)–(3.47g) give

The impedance $Z_0$, measured in ohms, satisfies $|\mathbf E_T|/|\mathbf H|=Z_0$ for a uniform plane transverse wave, with vacuum $\mathbf H=\mathbf B/\mu_0$. It is a field-amplitude ratio, not dissipative resistance. The fine-structure constant $\alpha$ is dimensionless. Current SI fixes the defining numerical values of $e,h,c$; $\mu_0$ is no longer exactly stipulated as $4\pi\times10^{-7}\,\mathrm{H/m}$ and is linked to experimentally determined $\alpha$, as specified in CGPM Resolution 1 (2018). ASD’s microscopic account of the coupling scale still requires $N_T$ and consistent normalization of static sourcing and propagation.

05Reading wavevector, polarization and energy flux from the equations

In the free homogeneous region, omit the transverse subscript so that $\mathbf E$ denotes $\mathbf E_T$. Taking curl of Faraday's relation and using $\nabla\cdot\mathbf E=0$ with the other curl equation gives $(\nabla^2-c^{-2}\partial_t^2)\mathbf E=0$; the magnetic field obeys the same form. Substituting $\exp[i(\mathbf k\cdot\mathbf x-\omega t)]$ yields:

$\mathbf k$ is wavevector, $k=|\mathbf k|$, $\omega$ angular frequency and $\hat{\mathbf k}$ the direction of propagation. Both fields are transverse and $B=E/c$. The two transverse polarizations combine into linear, circular or elliptical polarization. Phase and group speeds are $c$ in this homogeneous, isotropic, nondispersive sector.

Propagation also carries energy. Dot the Ampère–Maxwell equation with $\mathbf E$, Faraday's equation with $\mathbf B/\mu_0$, and use $\nabla\cdot(\mathbf E\times\mathbf B)=\mathbf B\cdot\nabla\times\mathbf E-\mathbf E\cdot\nabla\times\mathbf B$ to obtain:

$u_{\mathrm{em}}$ is free-field energy density in J/m³, $\mathbf S$ energy flux in W/m², and $\mathbf J$ effective current density. With $\mathbf J=0$, energy is locally conserved. Electric and magnetic contributions are equal in a plane wave, with $|\mathbf S|=cu_{\mathrm{em}}$. With sources, $-\mathbf J\cdot\mathbf E$ has the opposite sign to matter's gain, making total exchange consistent.

Manuscript equations (3.49)–(3.50) also organize momentum exchange with $\mathbf g_{\mathrm{em}}=\mathbf S/c^2$ and electromagnetic stress. Matter and field jointly carry momentum, including boundary stress and radiation; free energy flux alone is not an apparatus force calculation.

06Additional choices for sourced fields

Free propagation establishes a causal channel, but source entry must be specified. With constant $\varepsilon_0$, charge conservation and Gauss constraint imply:

$\rho_e$ is signed charge density. The divergence relation shows how displacement current maintains consistency, while permitting additional divergence-free terms; conservation alone does not uniquely determine dynamics. Chapter 3 separately chooses a lowest-order local linear effective action:

The first two terms describe free fields; $-\rho_e\Phi_\tau+\mathbf J\cdot\mathbf A_\tau$ is source coupling. The source part applies to macroscopic effective currents, slow matter sources and quasistatic or low-frequency retarded response. Slow source speed, near zone and long wavelength impose distinct conditions. A source approximation does not make free-wave propagation a slow-speed process.

For scalar variation $\delta\mathbf E=-\nabla\delta\Phi_\tau$, integration by parts gives coefficient $\varepsilon_0\nabla\cdot\mathbf E-\rho_e$. For vector variation, $\delta\mathbf E=-\partial_t\delta\mathbf A_\tau$ and $\delta\mathbf B=\nabla\times\delta\mathbf A_\tau$ give $\varepsilon_0\partial_t\mathbf E-\mu_0^{-1}\nabla\times\mathbf B+\mathbf J$. Independent stationarity yields:

Together with the potential identities, these form the sourced effective Maxwell system. Its variation closes once the action is chosen; source coupling remains constitutive and does not replace an independent high-speed closed topological source equation.

Under gauge change the source action acquires a boundary term and the integral of $-\chi(\partial_t\rho_e+\nabla\cdot\mathbf J)$. The latter vanishes for conserved sources. Conversely, the field-equation divergence restores the same continuity relation, showing consistency of representation, coupling and conservation within the model.

07Retarded potentials connect emission and propagation

For localized conserved sources choose Lorenz gauge $\nabla\cdot\mathbf A_\tau+c^{-2}\partial_t\Phi_\tau=0$. The field equations become the potential wave equations (3.52). With no incident free field and appropriate sources and boundaries, the retarded solution is:

$\mathbf r$ is the field point, $\mathbf r'$ the source integration position, $R=|\mathbf r-\mathbf r'|$, and the source is evaluated at $t-R/c$. The kernel carries local source changes to the field point at $c$; incident waves add homogeneous solutions. Changing gauge preserves the causal physical electric response formed jointly by scalar and transverse vector potentials.

The near-zone quasistatic potential is the lowest-order representation of this effective source response. A uniformly moving source in an infinite homogeneous background can carry an accompanying field without sustained flux at infinity. Acceleration, structural changes or boundaries can excite outward radiation. Complete high-speed radiation cannot be determined from the angular scale of $K(\phi)$ alone, and a received signal still passes through a material detector.

08How acceleration releases energy from an accompanying structure

For a slow point source in the far zone, retaining the acceleration-produced $1/R$ term of the retarded potentials gives final (3.57):

The unit vector $\mathbf n$ points from source to observer, $t_R$ is the retarded instant, and $\mathbf a(t_R)$ is source acceleration. The double cross product selects its component transverse to the observation direction. Substitution into energy flux and multiplication by $R^2d\Omega$ give (3.58):

Here $\theta_a$ is the angle to acceleration, not velocity. The total uses $\int\sin^2\theta_a\,d\Omega=8\pi/3$. The stated slow-source effective model therefore already yields an angular radiation pattern and total energy loss; it does not provide unlimited radiation energy without accounting for source reaction.

Separately denote the angle to velocity by $\phi$. For $\gamma\gg1,\phi\ll1$, (3.59) gives $K(\phi)\simeq(1+\gamma^2\phi^2)/(2\gamma)$, with comparable terms at $\phi_c\sim\gamma^{-1}$. This is a forward geometric angular scale, not a complete high-speed radiation power or an ASD-specific coherence residual. Source amplitude, polarization, acceleration and receiving standards must all enter the calculation.

09Comparing measurement states versus rewriting one field solution

Final (3.60) writes phase as $\Psi=\mathbf k\cdot\mathbf r-\omega t$, with $k=2\pi/\lambda$ and $\omega=2\pi f$. Here $\Psi$ denotes wave phase, not gravitational potential. Comparing specified measurement states, $\lambda'=K\lambda$ together with $f\lambda=c$ in both states gives $f'=f/K$ and $T'=KT$, where $T=1/f$. Rewriting one field solution using different spatial coordinates can instead retain the same process parameter $t$; that operation uses its own chain rule without adding the measurement-state frequency transformation a second time.

Equation (3.62) supplies a cancellation that must accompany experimental proposals:

Scaling one path segment and its wavelength by the same constant leaves the phase count unchanged. An actual comparison uses the same emission phase, reception event and frequency standard. A deformed wavefront picture cannot replace the residual calculated from their joint response.

10What the fixed three-axis mapping already establishes

Choose a $+x$ propagation branch in a uniform local region, with matter velocity also along $x$. Final (3.63)–(3.68) adopt constant $K_x=\gamma(1-\beta)$, $K_y=K_z=\gamma$ and $x_{T,i}=K_ix_i$, while retaining the same $t$. Express field components in the true orthogonal basis, then pull their arguments back to reference coordinates. The chain rule gives

Constant coefficients let the operators commute, preserving consistent divergence, curl identities and wave equations. Along the chosen positive branch $x=ct/K_x$, the reference coordinate speed is $c/K_x$, while the true speed is $dx_T/dt=K_x(c/K_x)=c$. The speed $c$ entered through the previous wave equation and coefficient matching; the last equality checks consistency of its coordinate representation.

11Why an ellipsoid cannot replace complete directed geometry

Defining $r_T=K(\phi)r$ and requiring $r_T=ct$ at the same $t$ gives final (3.69), $r(\phi,t)=ct/K(\phi)$. A fixed three-axis mapping instead writes the true sphere as $K_x^2x^2+K_y^2y^2+K_z^2z^2=c^2t^2$, with directional factor

It satisfies $K_{\rm axes}(-\hat{\mathbf n})=K_{\rm axes}(\hat{\mathbf n})$, whereas the complete directed $K$ generally does not. The fixed-branch calculation is established, but its symmetric ellipsoid is not a global realization of arbitrary-direction LVE. Equation (3.72) also gives $[K(\phi)+K(\phi+\pi)]/2=\gamma$. This even average is direction independent and does not by itself yield a nonspherical ellipsoid or round-trip interference difference.

If an orthogonal metric $g_{ij}=K_i^2\delta_{ij}$ is separately specified with position-dependent scales, (3.73) gives the scalar operator $\Delta_g f=J^{-1}\sum_i\partial_i[(J/K_i^2)\partial_i f]$, where $J=K_xK_yK_z$. Coefficient gradients must remain; vector fields additionally require changing bases, and time-dependent scales affect time derivatives. Complete directed geometry and an apparatus emission–reception chain should build on these established local results.