From Substrate Dynamics to Emergent Maxwell Equations

In Reactive Substrate Theory (RST), matter and fields emerge from the interaction between a deeper Substrate and a coherence field. Here we sketch how, in a slowly varying Substrate background, the phase dynamics of the coherence field Ψ give rise to an effective Maxwell-like gauge structure.

1. The Coupled System

Substrate (geometry / medium):

∂²S − c²∇²S + βS³ = σ · F_R(C[Ψ])

Matter / coherence field:

∂²Ψ − v²∇²Ψ + μΨ + λ|Ψ|²Ψ = κ S Ψ

Here, S(x,t) is the Substrate tension field, and Ψ(x,t) is a coherence (soliton) field coupled to the Substrate. The goal is to show that, in an appropriate limit, the phase of Ψ obeys equations structurally similar to Maxwell’s equations.

2. Polar Decomposition of Ψ

Write Ψ in amplitude–phase form:

Ψ(x,t) = √ρ(x,t) · e^iθ(x,t)

ρ(x,t): coherence / density of the soliton field
θ(x,t): phase field (where the EM-like structure will live)

Substituting this into the Ψ equation and separating real and imaginary parts yields two coupled equations.

3. Continuity-Type Equation (Imaginary Part)

From the imaginary part, one obtains a continuity-like equation:

∂ρ/∂t + ∇ · (ρ v² ∇θ) = 0

This has the form of a charge conservation law:

∂ρ/∂t + ∇ · J = 0

with an effective current:

J = ρ v² ∇θ

Thus, coherence conservation in Ψ naturally plays the role of charge conservation.

4. Phase Dynamics (Real Part)

The real part of the Ψ equation gives a Hamilton–Jacobi–like equation for the phase:

∂θ/∂t + (v²/2)(∇θ)² + μ + λρ − κS − Q = 0

where the “quantum pressure” term is

Q = −(v² / 2) (∇²√ρ / √ρ)

In the weak-gradient, smooth-density regime, Q is small and can be neglected, leaving a classical-like phase dynamics with explicit coupling to the Substrate S.

5. Effective Electromagnetic Potentials

Define effective gauge potentials in terms of the phase:

A = ∇θ        (vector potential)
Φ = −∂θ/∂t    (scalar potential)

Then the phase equation can be rewritten schematically as:

Φ = (v²/2)|A|² + μ + λρ − κS

The Substrate field S appears as a source term in the scalar potential Φ, analogous to how background geometry or matter content can influence electromagnetic potentials in curved spacetime.

6. Effective Fields: E and B

Define emergent fields:

E = −∇Φ − ∂A/∂t
B = ∇ × A

Since A = ∇θ, we have:

B = ∇ × ∇θ = 0

in smooth regions. However, when θ contains topological defects (vortices, solitons, phase slips), the curl of ∇θ can be nonzero, and B becomes nontrivial. This mirrors how magnetism emerges from phase structure in superfluids and superconductors.

7. Two Maxwell Equations as Identities

From the definitions of E and B alone, we obtain:

Gauss-like law for magnetism: ∇ · B = 0 (unless topological defects act as monopole analogs)
Faraday’s law: ∇ × E = −∂B/∂t

These are identities resulting from the definitions of A, Φ, E, and B in terms of θ, not independent postulates.

8. Charge and Current from Coherence

Recall the continuity equation:

∂ρ/∂t + ∇ · (ρ v² A) = 0

Define:

ρ_q = ρ          (effective charge density)
J = ρ v² A      (effective current)

Then:

∂ρ_q/∂t + ∇ · J = 0

Charge conservation thus emerges directly from coherence conservation in the Ψ field.

9. Substrate S as the Source of Gauss’s Law

Now consider the Substrate equation:

∂²S − c²∇²S + βS³ = σ · F_R(C[Ψ])

Linearize for weak S (small βS³):

∂²S − c²∇²S ≈ σ · F_R(ρ, θ)

Assume, to leading order:

F_R ≈ f₀ ρ

Then:

∂²S − c²∇²S ≈ f₀ σ ρ

So S obeys a wave equation sourced by the effective charge density ρ. Since Φ depends on S, variations in S drive variations in Φ, and hence in E.

Taking the divergence of E:

∇ · E = −∇²Φ − ∂/∂t (∇ · A)

With a Lorenz-like gauge condition:

∇ · A + (1/c²) ∂Φ/∂t = 0

one finds:

∇ · E ≈ κ ∇²S ≈ κ f₀ σ ρ

This is a Gauss-like law:

∇ · E = ρ_q / ε_eff

with an effective permittivity ε_eff set by the coupling constants.

10. Ampère–Maxwell Law

Taking the curl of B:

∇ × B = ∇ × (∇ × A)

Using vector identities and the gauge condition, one obtains:

∇ × B = (1/c²) ∂E/∂t + J_eff

where J_eff is proportional to ρ v² A, i.e. the same current that appeared in the continuity equation. This is the Ampère–Maxwell law in emergent form.

11. Physical Interpretation in RST

In this construction:

Electromagnetism emerges as the hydrodynamics of the phase θ of a coherent soliton field Ψ.
Charge is coherence density ρ.
Current is phase flow J = ρ v² A.
Fields E and B are derived from phase gradients and their dynamics.
Substrate S acts as a source for the scalar potential Φ and thus for E.

Gravity, in contrast, corresponds to the static, low-frequency deformation of the Substrate itself. Both electromagnetism and gravity arise from the same coupled system, but in different limits.

12. Regime of Validity and Predictions

This Maxwell-like structure holds under:

slowly varying Substrate S
smooth coherence density ρ
strong phase coherence in Ψ

In extreme environments (plasmas, magnetars, early-universe conditions, high-field laboratories), these assumptions break down. Quantum pressure terms reappear, nonlinearities in F_R become important, and S and Ψ fully entangle. RST therefore predicts controlled deviations from Maxwell’s equations in such regimes.

13. One-Line Summary

In the weak-gradient, slowly varying Substrate limit, the phase dynamics of the coherence field Ψ admit an effective gauge structure. Defining A = ∇θ and Φ = −∂θ/∂t, the resulting field variables satisfy a Maxwell-like system, with charge and current emerging as conserved coherence density and phase flow, while Substrate perturbations act as sources for the scalar potential.

Conspiranon