Vector Potential: A Continuous Spatial Representation

01Why represent local interaction by continuous potentials?

The magnetic correction between two charges describes how motion changes transverse interaction in a specified configuration. Currents, time-varying fields and propagation require continuous variables for the surrounding state, beyond a force at two points. ASD describes response with shear, torsion and orientation, then organizes its effective representation with electromagnetic potentials.

For charge $q$ outside its core in a static homogeneous baseline, scalar potential and electric field are:

$r$ is radial distance in this baseline, and $\hat{\mathbf r}$ points from source to field point. $\Phi_\tau$ is measured in volts, its subscript marking the electromagnetic torsion channel. A radial field describes far-field geometry; the shear channel describes ASD's spatial interpretation. They are different notions of direction. Normalization between potential and spatial state comes from charge–field coupling. This potential is not interchangeable, without definition, with the gravity module's longitudinal density or potential variables.

02How potential changes organize electric and magnetic fields

Time-dependent fields involve more than the scalar potential's spatial variation. Introduce vector potential $\mathbf A_\tau$ and define:

$\mathbf A_\tau$ has units V·s/m, so $-\partial_t\mathbf A_\tau$ has the same electric-field units as $-\nabla\Phi_\tau$. The static case recovers the gradient; a changing field cannot omit the process term. $\mathbf B$ is defined by curl, representing a local circulating structure rather than an additional basic entity imposed on the preceding charge interaction.

Potentials are not unique. For a sufficiently smooth scalar $\chi$, take:

The magnetic change is $\nabla\times\nabla\chi=0$ and electric terms $\nabla\partial_t\chi$ and $-\partial_t\nabla\chi$ cancel. These potentials therefore represent the same $\mathbf E,\mathbf B$. Gauge freedom is representational redundancy; the value of an arbitrary-gauge $\mathbf A_\tau$ is not a unique physical displacement. With smooth fields and commuting derivatives, the definitions also give magnetic divergence and induction identities. Source dynamics needs additional coupling.

03Transverse projection and spatial state

With decaying or compatible boundaries, separate the transverse potential $\mathbf A_T=\mathbf A_\tau^\perp$, satisfying $\nabla\cdot\mathbf A_T=0$. Transverse projection removes the gradient representation and retains divergence-free degrees of freedom; its definition includes boundary conditions.

After fixing the corresponding gauge in a source-free radiation region, Chapter 3 takes $\mathbf A_T=N_T\mathbf u_T$. The transverse spatial state $\mathbf u_T$ has length units; constant $N_T$ has units V·s/m². Subscript $\tau$ marks the electromagnetic channel and $T$ its transverse component. This correspondence is restricted to the specified free transverse effective sector, rather than identifying the full scalar potential, arbitrary vector potential and sourced matter core with one displacement. Magnetic field and propagation continue with curl, energy and coefficient matching.

04How slow sources produce an effective vector potential

Let source velocity be $\mathbf u$, a velocity rather than the spatial state $\mathbf u_T$. At lowest order in a slow-source, near-zone quasistatic expansion, choose:

$\Phi_{\mathrm{static}}=q/(4\pi\varepsilon_0r)$ is the static reference potential. Velocity supplies orientation and the $1/r$ kernel spatial decay. If $\mathbf u$ is spatially constant, curl contributes $\nabla\Phi_{\mathrm{static}}\times\mathbf u/c^2$; spatially varying velocity adds $\Phi_{\mathrm{static}}\nabla\times\mathbf u/c^2$. A moving source also contributes the vector potential's time derivative to the electric field.

For a localized, steady divergence-free macroscopic current $\mathbf J$, decaying boundaries select the linear static effective solution:

$\mathbf r$ is the field point, $\mathbf r'$ the source integration position, $R=|\mathbf r-\mathbf r'|$, and $d^3r'$ the source volume element. Superposition and the $1/R$ kernel follow from the chosen linear field equation, not a unique arbitrary-source potential inferred from LVE. Quasistatic and full retarded potentials occupy different approximation levels and cannot be interchanged for high-speed sources.

05Reconstruct the free matter action from ASD dynamics

Matter response requires its own variational basis. An action integrates a Lagrangian over the process, with equations of motion obtained from stationarity under trajectory variations. Chapter 2 provides mechanical momentum $\mathbf p=m_0\gamma\mathbf v$, free energy including rest energy $E_f=m_0c^2\gamma$, and $dE_f=\mathbf v\cdot d\mathbf p$. Rest mass is $m_0$, and $\mathbf v=d\mathbf x/dt$ uses the same variables and process parameter.

The Legendre pairing reconstructs the free Lagrangian from energy and momentum:

Here $\mathbf p\cdot\mathbf v-E_f=m_0\gamma(v^2-c^2)=-m_0c^2/\gamma$ and $dL_0=\mathbf p\cdot d\mathbf v+\mathbf v\cdot d\mathbf p-dE_f=\mathbf p\cdot d\mathbf v$. Thus the velocity derivative restores mechanical momentum. Manuscript equation (3.20c) further gives:

$\delta_{ij}$ is the identity matrix. Transverse and longitudinal velocity-response eigenvalues are $m_0\gamma$ and $m_0\gamma^3$, respectively, corresponding to Chapter 2's reconstruction impedances. They are positive for $m_0>0,v<c$, making the local energy–momentum correspondence invertible. Rest energy fixes the additive constant. The form comes from ASD momentum and work–energy, rather than replacing that construction with a standard free-particle premise.

06From free motion to effective field coupling

The free term alone does not couple charge to potentials. For macroscopic effective currents, slow matter sources and quasistatic or low-frequency response, Chapter 3 separately chooses minimal coupling:

$-q\Phi_\tau$ couples charge to scalar potential, and $q\mathbf v\cdot\mathbf A_\tau$ couples moving charge to vector potential. The free term is reconstructed within ASD; interaction terms are effective constitutive choices. They are not uniquely fixed by the special magnetic correction and do not yet form a complete general high-speed topological source model.

Variation of this declared action gives mechanical response and canonical momentum. The magnetic-field calculation performs the derivatives explicitly so that the potential-to-force connection is traceable.