Preliminary Paper

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We treat the ether as a magnetoelectric medium with field-responsive parameters μ(x,t) and ε(x,t). The constitutive relations and (slightly generalized) Maxwell system are

D = ε(x,t) E, 

B = μ(x,t) H,

 ∇×E = − {∂B/∂t} − Jm,

∇×H = {∂D/∂t} + Je,

∇⋅D = ρe,

∇⋅B = ρm,

where ρm and Jm​ are infinitesimal monopole charge/current densities (curl reservoir). In charge-free radiation zones we use ρe ⁣≈ ⁣0, Je ⁣≈ ⁣0, with ρm, Jm retained only as regulators that allow finite curl storage.

The local light speed and impedance are

c(x,t) = 1/sqrt{με}, Z(x,t) = sqrt{μ/ε}.

It follows that

ln⁡ c = −1/2 ln ⁡(με),

ln⁡ Z = 1/2 ln⁡ μ − 1/2ln⁡ε.  

The instantaneous energy densities and Poynting vector are

uE = 1/2 {ε ∣E∣^2},

uB = 1/2 {∣B∣^2}/μ,                                         

S = E×H.

With variable μ, ε, the power balance becomes

∂/∂t(uE + uB) + ∇⋅S = − 1/2 ∣E∣^2 {∂ε/∂t} (electric storage),  − 1/2 ∣B∣^2 {∂/∂t} ⁣(1/μ) (magnetic storage) − E⋅Je − H⋅Jm.

The right-hand terms quantify exchange with the ether (curl reservoir). In radiation zones (Je ⁣≈ ⁣0) the two explicit storage terms govern the kinetic ↔ potential energy choreography that underpins inertia and “gravitational” delay.

Energy partition:

E  total = E kinetic + E  potential,

ΔEpotential ∝ ∫ ⁣(1/2∣E∣^2 dε+1/2∣B∣^2 d(1/μ)). 

The instantaneous energy densities and Poynting vector are

uE = 1/2 ε ∣E∣^2,

uB =  1/2 ∣B∣^2/μ,

S = E × H.

With variable μ,ε, the power balance becomes

∂/∂t(uE + uB) + ∇⋅S = − 1/2 ∣E∣^2 {∂ε/∂t} (electric storage) − 1/2 ∣B∣^2 {∂/∂t ⁣(1/μ)} (magnetic storage) − E⋅Je − H⋅Jm.

The right-hand terms quantify exchange with the ether (curl reservoir). In radiation zones (Je ⁣≈ ⁣0) the two explicit storage terms govern the kinetic ↔ potential energy choreography that underpins inertia and “gravitational” delay.

Energy partition:

E total = E  kinetic + E potential,

ΔE  potential ∝ ∫ ⁣(1/2∣E∣^2 dε + 1/2∣B∣^2  d(1/μ)). 

Define the refractive index

n(x) = c0/c(x) = c0 sqrt με, 

where c0​ is the measured vacuum speed near equilibrium. Fermat’s principle yields the ray equations

d/ds ⁣(n k^) = ∇n,

dx/ds = k^,

with k^ the unit tangent and s arclength. In terms of c,

dk^/ds = − ∇ ln ⁡c + (k^⋅∇ ln ⁡c)k^.

Thus rays deflect toward decreasing c (increasing n), i.e., toward higher με—our “impedance gravity.” No spacetime curvature is required; bending is geometric optics in a graded medium.

An effective potential convenient for comparisons:

ΦZ(x) ≡ c0^2 ln ⁡n(x) = − c0^2 ln⁡ ⁣{c(x)/c0}.

Then the ray acceleration obeys

d^2x/ds^2 ≈  − ∇ ⁣(ΦZ/c0^2),

and for massive charges (see §8.6) an analogous force law appears with appropriate coupling.

Define the impedance curvature tensor and scalar:

Kij ≡ ∂i∂j ln⁡ Z,

K ≡ ∇^2 ln⁡ Z.

We postulate the macroscopic field equation

∇^2 ln ⁡Z  ∝  E ≡ ⟨ uE + uB⟩,

i.e., impedance curvature tracks coarse-grained energy density. This is the μ–ε analog of ∇^2 Φ ∝ ρ in Newtonian gravity and mirrors GR’s Gμν ∝ Tμνat the level of averages (cf. §7.1).

Because Z = sqrt {μ/ε}​ and c = 1/sqrt {με} ​, gradients in ln ⁡Z and ln⁡ c separate magnetic–electric asymmetry from speed:

∇ ln ⁡Z = 1/2 ∇ ln⁡ μ − 1/2 ∇ ln⁡ ε,

∇ ln ⁡c = −1/2 ∇ ln ⁡μ −1/2 ∇ ln⁡ ε.

Curl amplification increases ∣∇ ln Z∣| (asymmetry) while generally lowering c (delay).

Let a charge q with velocity v move through a rotating impedance field. Define an impedance rotation (vorticity) vector

ΩZ ≡ 1/2 ∇ × (∇ ln ⁡Z × v^),

evaluated along the trajectory. The leading transverse (Coriolis-like) force is

F⊥ ≈ qeff  v × ΩZ, 

qeff = αZ q,

where αZ​ is a dimensionless coupling that captures how strongly the ether’s  impedance vorticity interacts with the charge’s EM structure. Opposite charges spiral oppositely (bubble-chamber helices), consistent with your observation.

A complementary longitudinal drag/boost from graded c is

F∥ ≈ − γZ m v (v^⋅∇ ln ⁡c),

with γZ > 0. When a leading high-energy particle reduces c behind it (imprinting μ, ε  wake), followers can be accelerated (−∇ ln ⁡c points forward), realizing the ether accelerator mechanism.

Introduce the product field

Π(x, t) ≡ μ(x, t) ε(x, t )= 1/c^2.

Gravitational strength is governed by gradients of Π\PiΠ. The saturation hypothesis states that there exists Πmax⁡​ such that

∂Π/∂ρ → 0 as energy density ρ → ρ⋆,

implying

∇Π → 0 ⇒ ∇ ln⁡ c → 0,

so gravitational forcing weakens at extreme density—your “high mass, low gravity” result. The medium stiffens; further energy adds volume or internal modes, not deeper gradients.

The energy-equilibrium sheet (equal electric and magnetic participation) satisfies

uE = uB ⟺ μ = ε/c^2(local).

On this sheet the impedance ratio and speed co-determine a balanced propagation state; departures from the sheet quantify asymmetry and potential curl storage (mass tendency). Let

Δasym ≡ ln⁡ ⁣(μc^2/ε) = 2 ln⁡ Z + ln⁡ (c^2), 

so Δasym = 0 on the sheet and grows sinusoidally with curl amplitude in your model (Cauchy-type dispersion).

1. Bending law (ray):
    

dk^/ds = − ∇ ln ⁡c + (k^ ⁣⋅ ⁣∇ ln ⁡c) k^.

         2.  Red/blue shift from impedance evolution along a path C:

Δλ/λ ≈ ∫C ⁣ dℓ k^ ⋅ ∇ ln ⁡n = ∫C  ⁣dℓ k^ ⋅ ∇(1/2 ln⁡ με).

        3.  Local power exchange (radiation ↔ curl):
 

W ether = 1/2 ∣E∣^2 ε˙ + 1/2 ∣B∣^2 d/dt ⁣(1/μ), 

sign determines absorption vs release.

        4.  Impedance curvature–energy relation (coarse-grained):
 

∇^2 ln ⁡Z = β⟨uE​+uB​⟩,  

 with β a universal coupling (to be measured).

       5.  Saturation scaling (extreme density):
 

∣∇ ln⁡ c∣ ∝ ((1−Π)/Πmax⁡)^p, p>0,

predicts weakening gravity near Πmax.

From μ ∈[0.6, 2.0]×10^−6 H/m, ε∈[4,13]×10−12 F/m:

cmin⁡ ≈ 1.96×10^8 m/s, 

ceq ≈ 3.01×10^8 m/s, 

cmax⁡ ≈ 6.46×10^8 m/s.

At fixed frequency f, λ=c/f sets the wavelength ladder; at f ⁣≈ ⁣5×10^14 Hz: λeq ⁣≈ ⁣602 nm, λmax⁡ ⁣≈ ⁣1290 nm.

• Medium: μ(x,t), ε(x,t) define c = 1/sqrt (με​), Z=sqrt (μ/ε)
   
• Gravity analog: gradients of ln ⁡c (speed) and ln⁡ Z (asymmetry) steer rays and charges.
   
• Curvature: K = ∇^2 ln⁡ Z tracks coarse energy density.
   
• Exchange: generalized Poynting theorem adds ε˙ and d/dt(1/μ)  coupling—radiation ↔ curl.
   
• Saturation: Π = με approaches Πmax ⁡⇒ ∣∇ ln ⁡c∣ → 0 ⇒ weak gravity at extreme density.
   
• Energy balance sheet: μ=ε/c^2 defines equilibrium propagation.