# Implications of Gauge-Free Extended Electrodynamics

^{1}

^{2}

^{*}

## Abstract

**:**

**A**) that is contrary to classical electrodynamics (CED).

**A**(irrotational) arises in extended electrodynamics (EED) that is derivable from the Stueckelberg Lagrangian.

**A**(irrotational) implies an irrotational (gradient-driven) electrical current density, $\mathbf{J}$. Consequently, EED is gauge-free and provably unique. EED predicts a scalar field that equals the quantity usually set to zero as the Lorenz gauge, making

**A**and the scalar potential (Φ) independent and physically-measureable fields. EED predicts a scalar-longitudinal wave (SLW) that has an electric field along the direction of propagation together with the scalar field, carrying both energy and momentum. EED also predicts the scalar wave (SW) that carries energy without momentum. EED predicts that the SLW and SW are unconstrained by the skin effect, because neither wave has a magnetic field that generates dissipative eddy currents in electrical conductors. The novel concept of a “gradient-driven” current is a key feature of US Patent 9,306,527 that disclosed antennas for SLW generation and reception. Preliminary experiments have validated the SLW’s no-skin-effect constraint as a potential harbinger of new technologies, a possible explanation for poorly understood laboratory and astrophysical phenomena, and a forerunner of paradigm revolutions.

## 1. Introduction

**A**) is a measureable, physical quantity via theory and compelling experimental evidence. The curl-free (irrotational) vector potential is important in both classical and quantum domains, for example in the Aharonov-Bohm effect and the Maxwell-Lodge effect. We show that

**A**and scalar-potential (Φ) have the same physical significance as the electric (

**E**) and magnetic (

**B**) fields Section 3 further elucidates

**A**and Φ, as physically measureable and independent fields for EED theory. Section 4 discusses the EED Lagrangian and scalar field (C). [Note that

**bold**font denotes a 3-dimensional vector (e.g.,

**A**,

**B**,

**E**) and italic font denotes scalar quantities (e.g., C)] The centerpiece of this work is Section 5, which presents EED as elucidated by the second author [3]. EED is consistent with preliminary tests [3], showing that an irrotational electrical current drives the scalar longitudinal wave, and includes the missing scalar field for a provably unique EED theory [2]. Section 6 expounds on the SLW’s no-skin-effect constraint, revealing vast applications of EED in new technologies and in natural systems. Section 7 explores the possible future role that EED may play in quantum physics and general relativity. Section 8 presents our conclusions with a discussion of future prospects.

## 2. Physical Significance of the Magnetic Vector Potential

**A**), and the electric (

**E**) and magnetic (

**B**) fields.

**B**and

**E**can be expressed in terms of Φ and

**A**, as [7]:

**B**and

**E**are regarded as physical fields; Φ and

**A**are viewed as mathematical conveniences for solving Maxwell’s equations. This interpretation arises from the detectability of

**B**and

**E**through the Lorentz-force equation [7] on a point-particle at the position [

**r**(t)] with a charge (q) and mass (M).:

_{q}**B**and

**E**unchanged in Equation (1), and is termed gauge invariance [8]. Equation (3) allows an ℵ

_{2}-infinitude of choices for Λ. An analyst can choose an appropriate gauge to fix boundary conditions, find the ‘proper’

**B**and

**E**fields for known charge and current distributions, or solve an inverse problem. This convention is now engrained in CED, but restricts the potentials. For instance, the Lorenz gauge makes the Φ and

**A**inter-dependent.

**A-**wave equations are non-invertible despite their gauge ambiguity. These wave equations arise from CED via standard Fourier transforms with a unique projection operator that makes a gauge choice superfluous. Nikolova [11] notes that nonzero, radially-propagating (irrotational) potentials occur in antenna theory with zero

**B**- and

**E**-field vectors via expansion of the free-space fields in spherical harmonics. This irrotational “potential wave” is called “longitudinal,” meaning

**A**-polarization in the direction of propagation. The corresponding wave impedance is equal to that of free space. CED predicts that this solution has no radiated power, because the Poynting vector is zero. EED predicts a scalar-longitudinal wave (SLW) with a longitudinal-

**E**field and radiated power (Section 5).

**A**and its corresponding physical significance in a form that is quite close to that of Faraday [13,14,15,16,17]. Namely, the vector potential is a “store” of electrodynamic-field momentum available for exchange with the kinetic momentum of charged particles in a conductor [12]. Konopinski shows that operational definitions of Φ and

**A**arise from Equation (2), in terms of the potentials in Equation (1):

**p**= M

**v**+ q

**A**) to the negative gradient of “interaction energy,” q(Φ −

**v∙A**). Konopinski used a novel gedanken-experiment to show that

**A**can be measured everywhere in space. Namely, a charged, macroscopic bead slides freely on a circular, non-conductive fiber that is concentric with the cross-section of a magnetic solenoid.

**A**has only an azimuthal component parallel to the current flow, resulting in ∇Φ = ∇●

**A**= 0. So, the right-hand side (RHS) of Equation (3) is zero, making the conjugate momentum (

**p**) a conserved quantity.

**A**is obtained by monitoring the associated change in the bead’s kinetic momentum (M

**v**) as the solenoid’s current changes. Then, q

**A**is the conjugate momentum [12] in an external field, just as qΦ is electrodynamic-field energy per unit charge. These joint properties of the imposed fields arise from their interference and determine the processes, by which fields and charges become observable. Force and power per unit charge are then expressed in terms of the transfer rates.

**A**) potentials with the complex phase of the particle’s wave function. For a particle travelling along a path (

**P**) in a region with zero magnetic field (

**B**) and

**A**≠

**0**, the magnetic phase shift is φ = (q/ħ)∫

_{P}·

**dx**●

**A**. Here, ħ is Planck’s constant divided by 2π; ʃ

**is the path integral along the particle’s trajectory. Particles with the same starting and ending points acquire a phase difference (Δφ) by travelling along two distinct paths around a multiply connected domain:**

_{P}dx**B**= ∇ ×

**A**gives the fourth form with Φ

_{B}as the enclosed magnetic flux in the fifth form. The wave nature of quantum particles allows the same particle to traverse different paths. Then, the phase difference is observable by placing a magnetic solenoid between the slits of a double-slit experiment. Tests by Tonomura et al. [20] and Osakabe et al. [21] definitively validated Equation (5). The magnetic phase shift has also been observed in micrometer-sized metal rings [22,23,24], showing that electrons keep their phase coherence despite diffusion in mesoscopic samples. Wu and Yang [25] proved that

**B**and

**E**cannot explain the ABE, which is over-described by the local value of

**A**. Rather, a path-dependent integral over

**A**around multiply-connected, topological regions yields Equation (5) [26].

**E**). The phase shift in the wave function is:

**A**and Φ play interchangeable roles. Additional experiments are described next.

**A**. Varma [31] claimed that this result is mediated by a macroscale, quantum modulation of the de Broglie matter wave along the magnetic field lines, resulting in a scattering-induced transition across electron Landau levels [32]. Shukla [33] commented that the matter wave occurs over centimeters, as a classical effect. Shukla then noted a paradox, namely that a curl-free vector potential should not affect a classical system. This apparent paradox is resolved by EED (below).

**A**) affects classical electrodynamic systems [34]. Lodge’s findings were ignored, and ascribed to deficiencies in the 19th century electrical equipment. Blondel [35] replicated this test. Rousseaux et al. [34] note that the MLE presents a fundamental paradox for CED. Figure 1 shows the test geometry for replication of Lodge’s test with modern instrumentation with a long solenoid encircled at its mid-plane by a (secondary) conducting loop [34]. Time-variation in the solenoid’s current produces a magnetic field inside the solenoid; no magnetic field occurs outside the solenoid.

**A**is time-varying both inside and outside the solenoid and is parallel to the current. Rousseaux et al. measured a voltage in the secondary coil from

**A**outside the solenoid via Equation (1),

**E**= −∂

**A**/∂t. Rousseaux et al. [34] note that the gauge indeterminacy in the vector potential is negated by the boundary conditions, showing the physical significance of

**A**.

**B**=

**0**. Several secondary coils were passed through the hollow core of the vector-potential coil. The coiled coil (primary) was driven with alternating current that induced voltages in the secondary coils that were path independent and occurred, when enclosed by a (super)conducting material. Daibo [37] patented the novel vector-potential transformer (VPT) and related devices, involving high precision measurements in: living organisms (e.g., medical diagnostics), sea water, nuclear-power-plant reactor pressure vessels, and AC electric-field generation without bare electrodes in corrosive media.

## 3. Road to Re-Structure: Independence of Non-Gauged Potentials

**E**- and

**B**-fields (and their six 3-space components) is then presumptuous and counter-productive, in sharp disagreement with quantum electrodynamics. QED uses the covariant 4-vector potential that describes the effective momentum-energy state of an electromagnetic system (e.g., ABE). The result is contradicting views between CED and QED.

**A**to physically-measureable field in CED. Indeed, Faraday [13,14,15,16,17] and Maxwell [38,39,40] originally proposed this idea.

**A**and Φ) are physically measureable quantities at all spatio-temporal scales, eliminating the unnatural separation of CED and QED. Second, the potentials are independent of one another, resulting in a provably-unique, gauge-free electrodynamics, as discussed in Section 5. This proposal follows the Occam’s razor test, which posits that the simplest model with the fewest assumptions is best. Consequently, we propose an extension of Maxwell’s equations without any requirement for gauged potentials. These two extensions of CED imply that the Minkowski electromagnetic 4-potential, A

_{μ}= (Φ/c,

**A**), is a single unit, as advocated by Majumdar and Ray [9,10]. These authors note the absence of a fully (special) relativistically, covariant electrodynamic formulation in many excellent texts, despite its implicit presence in CED as discovered by Einstein in 1905 [41]. Specifically, the CED formulation in terms of the

**E**- and

**B**-fields has no obvious relativistic invariance in different inertial frames. When the equations are written in terms of Φ and

**A,**CED is manifestly invariant under Lorentz boosts and spatial rotations. The latter formulation is therefore preferable for explicit display of the special-relativistic symmetry that is intrinsic to CED.

## 4. Derivation of the Form for the Scalar Field

**U**) is uniquely decomposable as irrotational (∇V) and solenoidal (∇×

**W**) parts:

**U**= ∇V + ∇×

**W**. Here, V and

**W**are scalar and vector space-time functions, respectively. Woodside [42,43] proved a unique decomposition of a smooth Minkowski 4-vector field (3 spatial dimensions plus time), into 4-irrotational and 4-solenoidal parts with normal and tangential components on the bounding surface. Woodside [2] then used the Stueckelberg Lagrangian [44]:

^{μv}is the Maxwell field tensor; c is the speed of light (not necessarily vacuum); J

_{μ}= (ρc,

**J**) is the 4-current; the 4-potential is A

_{μ}= (Φ/c,

**A**); the Compton wave number for a photon with mass (m) is k = 2πmc/h; and h is Planck’s constant. The source terms are the electric charge density (ρ) and the electrical current density (

**J**); ε and μ are the electrical permittivity and magnetic permeability, respectively (not necessarily vacuum). The fully-relativistic Stueckelberg Lagrangian includes both

**A**and Φ and resolves many issues with previous CED Lagrangians. For γ = 0 and m > 0, Equation (7) yields the Maxwell-Proca theory, for which a 2012 test [45] measured m ≤ 10

^{−54}kg (equivalent to ≤10

^{−18}eV), consistent with massless photons. For γ = 1 and m = 0, Equation (7) is [43]:

^{μ}(∂

_{μ}A

^{μ}= 0) and is called 4-solenoidal. The other has zero 4-curl of A

_{μ}, F

^{μν}= ${\partial}^{\mu}{A}^{\nu}-{\partial}^{\nu}{A}^{\mu}=0$(four-irrotational vector field), if and only if the last term in Equation (8) is included. The form inside the parentheses of the last term is the Lorenz gauge when set to zero [8], restricting Φ and

**A**to be dependent on each other. However, EED allows for a non-zero value of this scalar expression, making the potentials completely independent of one another. The scalar field arises from the third term on the right-hand side of Equation (7) for γ = 1 and m = 0. Equation (8) shows clearly how this term arises by writing the Lagrangian density in terms of Φ and

**A**for a massless 4-vector field (A

^{μ}). Thus, the relationship between Φ and

**A**reveals the missing scalar field and its key role in EED. The result is a provably-unique, gauge-free formulation of extended electrodynamics (EED) with a new scalar component (C) as a space-time function [2]:

**B**and

**E**does arise when a scalar field is incorporated in the model. Williamson [49,50,51,52] used Clifford algebra to formulate an extended model of elementary particles with the scalar field and a topological structure for photons and electron-positron pair production/annihilation.

## 5. EED and the Scalar Lonitudinal Wave

**J**is the current density vector. Equation (13) explicitly decomposes

**J**into solenoidal (∇×

**B**) and irrotational (∇C) parts, in accord with the Helmholtz theorem. The new terms in Equations (13) and (14) change only the irrotational (longitudinal) electrodynamics, as elucidated by Keller and Hively [82,83]. These irrotational components sre gauged away in CED by Equation (3). Thus, the solenoidal (transverse) electrodynamics remain unchanged, as described by CED.

**E**- and

**B**-fields are unchanged in terms of the classical potentials (

**A**and Φ) in Equation (1). Third, EED yields the same wave equations for

**E**and

**B**as CED. Fourth, the

**A**-wave equation is obtained by replacing

**B**, C, and

**E**in Equation (13) with Equations (10)–(12); using the vector calculus identity [90], ∇ × ∇ ×

**A**= ∇(∇●

**A**) − ∇

^{2}

**A**; and noting that ∂∇C/∂t − ∇∂C/∂t = 0. Fifth, the Φ-wave equation arises by: substituting $E$ and $C$ from Equations (10) and (12) into Equation (14); and noting that ∂∇●

**A**/∂t − ∇●∂

**A**/∂t = 0. Sixth, the

**A**- and Φ-wave equations are derived without a gauge condition, making EED gauge free. Seventh, EED predicts a scalar-longitudinal wave (SLW), consisting of a

**E**-field along the direction of propagation (longitudinal) and the scalar field (C); the SLW carries energy and momentum. Eighth, EED predicts a scalar wave (SW) that carries only energy. Ninth, both the SLW and SW are unconstrained by the skin effect, because they lack a

**B**-field that induces eddy currents. Tenth, recent SLW tests [3] are consistent with previous work by Tesla [91], Monstein and Wesley [92], and Meyl [93]. Eleventh, the

**A**-,

**B**-, C-,

**E**- and Φ-wave equations are unchanged under time reversal, so that reciprocity holds. A SLW transmitter can then be used as a receiver. Twelfth, EED predicts relations among the longitudinal field components:

**B**= ∇ ×

**A**

^{L}= ∇ × ∇α =

**0**) and vice versa.

**B**= 0 [90]:

**J**= 0 on the right-hand side (RHS) of Equation (16), in accord with long-standing experiments that validate classical charge balance [94]. The lower bound on electron lifetime for charge balance has been measured as ≥6.6 × 10

^{28}years [95] (decay into two γ-rays, each at with an energy of m

_{e}c

^{2}/2 = 0.256 MeV; m

_{e}= electron mass). Nevertheless, long-time charge conservation is not inconsistent with charge non-conservation over short-time scales, ΔT ≤ Δt, per the Heisenberg uncertainty relation, ΔEΔt ≥ ħ/2. Here, ΔE is the charged-quantum-fluctuation energy; and $\u0127$ is Planck’s constant divided by 2π. Equation (16) can be interpreted as charge non-conservation (particle-antiparticle fluctuations [PAPF]) driving C, and vice versa, not unlike energy-fluctuations driving mass-fluctuations in quantum theory and vice versa [3,67]. Equivalently, the RHS can be interpreted as PAPF over some non-local region (Δx), consistent with the Heisenberg uncertainty relation, ΔpΔx ≥ ħ/2. Here, Δp is the charged-quantum-fluctuation momentum over the non-local region, Δx. (While this interpretation is novel, it is consistent with PAPF according to the Heisenberg uncertainty principle.) Thus, Equation (16) predicts charge conservation on long time-scales (consistent with CED), and exchange of energy between C and quantum fluctuations for ΔT ≤ Δt. Confirmation of these quantum charge fluctuations involves tests, consistent with the Heisenberg uncertainty relation. One possible test could use the electron [ΔE (electron) = m

_{e}c

^{2}= 0.511 MeV] corresponding to a time, Δt~6 × 10

^{−22}s. Subzeptosecond dynamics have been measured [96], so a direct measurement of this prediction is feasible. Moreover, quantum fluctuations can control charge quantization [97], in accord with Equation (16).

^{2}C = ∇●∇C and using the divergence theorem in the limit of zero pill-box height yields continuity in the normal component (“n”) of $\nabla C/\mu $ for long times [3,67]:

**A**-wave equations are [3,67]:

**J**

_{A}and ρ

**are the interface surface-current density and surface-charge density, respectively; $\stackrel{\u2322}{n}$ is the unit-vector that is normal to the interface; ε and μ are the permittivity and permeability, respectively (not necessarily vacuum). Validation tests are clearly needed for these predictions that are inconsistent with CED. CED and EED have the same tangential interface matching conditions.**

_{A}_{o}is the scalar field amplitude; j = √−1; the wave number k is (2π/λ) for a wavelength (λ); ω = 2πf for a frequency (f); and r is the spherical radius. Boundary conditions for Equation (20), include C (r→∞)→0, which is trivially satisfied. Equation (25) predicts that the energy density of the C-field is (C

^{2}/2μ), yielding a constant energy, 4πr

^{2}(C

^{2}/2μ), through a spherical boundary around a source in arbitrary media, as required [3,67]. The scalar field then can be interpreted as an electrodynamic pressure, similar to that in acoustics and hydrodynamics, which expands and contracts radially. (By contrast, CED forbids a spherically symmetric transverse wave.) Substitution of Equation (15) into Equation (13) yields a separation into transverse (T) and longitudinal (L) forms via Equations (21a) and (21b), respectively:

**E**

^{L}, and

**J**

^{L}are related by Equation (21b) with [98] σ = ε

_{o}ε″ω and ε = ε

_{o}(ε′ − jε″) for a SLW impedance, Z:

**E**

^{L}=

**J**

^{L}/σ = ∇κ/σ, where σ is the electrical conductivity. The unit vector in the radial direction is $\widehat{r};$ ε

_{o}and μ

_{o}are the free-space permittivity and permeability, respectively; ε′ and μ′ are the relative permittivity and permeability (not necessarily vacuum), respectively; tan(δ

_{ε}) = ε″/ε′. Here, the same definitions are used for k, r, t, and ω as above. From Equation (13), C has the same dimensions as

**B**= μ

**H**. Consequently, Equation (22a) uses |

**E**|/(C/μ) to obtain the SLW impedance, like the CED form, Z = |

**E**|/|

**H**|. The C- and

**E**

^{L}-field energies from Equation (25), 4πr

^{2}(εE

^{2}/2) and 4πr

^{2}(C

^{2}/2μ), are constant through a spherical boundary and C(r→ ∞)→0 and |E

^{L}(r→∞)|→0. Equation (22a) predicts Z =√μ

_{o}/ε

_{o}in free-space (ε′ = μ′= 1 and ε″ = μ″ = 0). The SLW then consists of C and

**E**

^{L}propagating together. A SLW monopole antenna radiates power (

**P**

_{OUT}) isotopically in free space [3]:

**E**

^{L}=

**0**with no SLW power output or reception.

**B**/∂t =

**0**), implying no (circulating) eddy currents from Faraday’s law. Thus, EED predicts that the SLW is unconstrained by the skin effect. Preliminary experimental results [3] are consistent with this prediction (propagation through thousands of skin-depths of solid copper), using standard electronic instrumentation. This no-skin-effect feature cannot be explained by CED, and has practical applications in wireless communications and wireless transmission of electrical power.

**A**

^{T},

**B**, and

**E**

^{T}) in the same way as CED, while adding longitudinal (irrotational) components (

**A**

^{L},

**E**

^{L}, and

**J**

^{L}) and the scalar field that CED gauges away. Table 1 shows the specific, quantifiable EED predictions, which are discussed next in an item-by item fashion. Items 1–9 involve the scalar-longitudinal wave (SLW, also called the longitudinal-scalar wave or electroscalar wave). The SLW driver is an irrotational current (

**J**, Item 1 in Table 1), or an irrotational vector potential (

^{L}**A**

^{L}), or an irrotational electric field (

**E**

^{L}); such a driver creates no magnetic field (

**B**=

**0**). The SLW consists of a scalar field (C)—Item 2, together with an irrotational electric field (

**E**

^{L})—Item 3. The corresponding SLW power (C

**E**/μ.) and momentum densities (−C

**E**/μ.)—Items 4–5 are comparable to a transverse electromagnetic (TEM) wave—Item 6. The SLW is unconstrained by the classical skin effect—Item 7, and has a free-space attenuation over distance (r) as 1/r

^{2}—Item 8. SLW creation by a monopole transmitter is isotropic—Item 9. The equation numbers in Table 1 correspond to the equations in this paper, as adapted from [3]. The scalar wave (SW) arises from the conditions of Item 1 with the additional constraint of Item 10. The SW has pressure (∇C

^{2}/2μ) and energy (C

^{2}/2μ) densities—Items 11–12. The scalar wave/field is a pseudo-scalar—Item 13 and can be driven by charge fluctuations (e.g., particle-antiparticle fluctuations)—Item 14. CED cannot explain the SLW/SW, because the longitudinal components (

**A**

^{L},

**E**

^{L}, and

**J**

^{L}) are gauged away. EED predicts a material-interface matching condition for C—Item 15, for the normal component of an electric field from a surface charge density—Item 16, and for the normal component of a vector potential from a surface current—Item 17. Item 15 is new; Items 16–17 are inconsistent with CED. EED also predicts a new, mixed mode force density—Item 18, a new longitudinal force—Item 19, and a new power source term—Item 20. These EED predictions clearly need careful validation.

**B**=

**0**), generating of an irrotational current for transmission (and reception) of the SLW. This coil has zero inductance (counter-going electrical currents in adjacent turns), zero capacitance (same electric charge density in adjacent turns), as a 2-dimensional monopole that accumulates positive and negative charge over each half of a sinusoidal cycle for a coil length <λ/10. The antennas in Figure 4 are significant, because CED cannot explain the SLW.

## 6. Evidence for the SLW: Both Inanimate and Biological

^{2}/2μ), SLW energy (C

**E**/μ), and a power source (ρC/εμ).

**E**/μ), a mixed mode flux [(∇×

**B**C)/μ], a force (

**J**C) parallel to the current density, and scalar-field pressure (∇C

^{2}/2μ). The right-most term in Equation (25) is the divergence of the CED Maxwell-stress tensor.

^{2}. This prediction is consistent with tests [101]: a force on Ampere’s bridge, tension that ruptures current-carrying wires, the Graneau-Hering submarine, Hering’s mercury pump, and an oscillating force in a current-carrying mercury wedge. These tests need replication.

**E**/μ) with concomitant power gain (+C

**E**/μ) in Equation (24), and vice versa. This sign difference means that SLW emission (energy loss) drives a momentum gain in a massive object that emits the SLW.

^{12}electrons/cm

^{2}was more than ten-fold greater than typical tribo-charging systems. A 2010 experiment measured the angular distribution of X-rays during tape peeling [103] that is inconsistent with the CED prediction for transverse Bremsstrahlung (Figure 6). Specifically, ordinary Bremsstrahlung bounds the angular distribution from below, while polarizational Bremsstrahlung bounds the angular distribution from above. Neither model predicts the 20% peak between 80° to 100°. EED predicts that the SLW is unconstrained by the skin effect and is more penetrating in matter. We conjecture that the SLW-Bremsstrahlung might explain the angular distribution in Figure 6, which CED cannot explain. Careful replication tests are needed.

^{3}H,

^{22}Na,

^{36}Cl,

^{44}Ti,

^{54}Mn,

^{60}Co,

^{85}Kr,

^{90}Sr,

^{108m}Ag,

^{133}Ba,

^{152}Eu,

^{154}Eu,

^{222}Rn,

^{236}Ra, and

^{239}Pu [108,109,110,111,112,113], and references therein. Periods in the decay rate are: one day, 12.08/year (solar rotation rate), one year, and ~12 years (sun-spot cycle). Low-energy nuclear theory [114] cannot explain these observations. An improved radioactive-decay theory is needed, for which we propose that EED is appropriate, based on the Stueckelberg Lagrangian density [44] in Equation (8), for which the corresponding Hamiltonian is [57]:

## 7. EED Implications for Quantum Mechanics and General Relativity

**A**,

**E**, and

**J**; (d) a new scalar field (C); (e) scalar-longitudinal waves whose properties are consistent with preliminary experiments [3]; (f) free-space scalar waves; and (g) inclusion of the CED transverse waves. Quantum-EED would then be non-local and gauge free.

**E**- and

**B**-field lines that are topologically interleaved (analytically and by simulation). Other work formulated a framework [127] for construction of divergence-free, knotted, vector fields, and scalar fields as found in EED. Kedia et al. [137] showed that an initially knotted light field will stay knotted, if and only if the Poynting-vector-flow field is shear free. This work [122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137] provides a firm basis for electrodynamic topological solutions (knots, braids, and Mobius-strips). Furthermore, an explicit analogy exists between CED and the equations for fluid dynamics, as shown (for example) by Martins and Pinheiro [138]. Two recent experimental papers [139,140] showed evidence for knotted vortices (trefoil knots and linked pairs of rings) in fluid-dynamics experiments. Spirichev [141] proposed a version of EED (by analogy to elasticity theory), finding: (i) A

^{μ}is a physically measurable field, as discussed above; (ii) C and the SLW exist; and (iii) PAPF create a particle-antiparticle plasma, which is compressible (elastic), facilitating SLW propagation. Modanese [69,70,71,72,73,74,75,76,77,78] has explored non-local, quantum wave-functions via violation of local charge conservation in condensed matter. A high-frequency dipole oscillation at 10 GHz leads to double-retarded integrals [69] from EED; the

**E**

^{L}magnitude then exceeds the

**E**

^{T}magnitude by a factor of 10

^{2}–10

^{3}in the far-field. This theoretical, simulation, and experimental evidence provides a compelling basis for a first-principles, topological approach to EED-quantum and particle physics.

_{μv}) of Equation (27), which is consistent with Saa [143]. Moreover, the T

^{oo}component of the stress-energy tensor becomes T

^{oo}= (B

^{2}/2μ + C

^{2}/2μ +εE

^{2}/2)/c

^{2}, with B = |

**B**|, E = |

**E**|, and C as the scalar-field magnitude. Li et al. [144] studied the non-local current density in various nano-devices, as a negative gradient of a scalar function, consistent with EED.

**E**

^{L}-fields do not cancel, as in QED but rather form the SLW; (ii) a dispersion relation arises for

**E**

^{L}-waves in terms of the Hubble constant (H

_{o}) and the

**E**

^{L}wave-vector amplitude (q

_{o}); and (iii) this approach is a first-principles path to an EED-quantum theory that includes acceleration and gravity. The work [83] is important, because: (iv) C and γ (scale factor in Equation (7)) allow consistency between

**E**

^{L}and A

^{μ}; (v) γC arises from superposition of longitudinal fields on the energy shell, |ω| = cq; and (vi) free-space SLWs can be emitted by nonlinear, electrodynamic mixing. A quantum-gravity simulation of a spherically-symmetric, lattice with a stationary metric produces polarized regions of positive and negative curvature [71]. These vacuum fluctuations are markedly different from Wheeler’s space-time foam.

## 8. Conclusions and Prospects

**A**,

**E**, and

**J**together with the scalar field (C). EED includes these irrotational (longitudinal) components, providing fresh understanding of scalar-longitudinal waves, energy, and momentum. The examples above have the common feature of irrotational currents: (a) seismic precursor signals due to fracturing in the Earth’s crust [99], (b) peeling of adhesive tape [102,103], and (c) SLW irradiation by the TESLAR chip [106,107]. These examples generalize to any phenomena involving the breaking/alteration of chemical bonds in inanimate or biological systems and/or irrotational currents (e.g., radial oscillations of solar plasma). The gradient-driven current and the SLW then elucidate currently unexplained electrodynamic phenomena, both laboratory-based [101,102,103,104,105,106,107] and astrophysical [66,100].

**A**) physical reality. Experimental evidence from the VPT [36,37] validates the Maxwell-Lodge effect [34] with

**A**as a physical field in the classical domain, along with validation by the Aharonov-Bohm effect in the quantum domain. The invention (and experimental validation) of SLW antennas [68] provides a springboard for disruptive technologies in wireless sources of energy generation and power conversion. EED has the potential to produce a revolution in electricity generation and distribution.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Circuits for the Maxwell-Lodge effect representation [34].

**Figure 2.**VPT with secondary circuit coil configurations [36].

**Figure 5.**Possible registration of solar SLW radiation [100].

**Figure 6.**Spectral output of radiation as a function of tape angle [103].

Item | Brief Description of Testable Prediction | Reference |
---|---|---|

1 | The SLW drivers are: A^{L}= ∇α⇔E^{L}= ∇e⇔J^{L}= ∇κ⇔B = 0. | Equation (15) * |

2 | The SLW has a dynamical scalar field, C = ∇∙A + εμ∂Φ/∂t. | Equation (12) |

3 | The SLW has a longitudinal E-field. | Equation (22b) |

4 | The SLW has a power density vector of CE/μ. | Equation (24) |

5 | The SLW has a momentum density of −CE/μ. | Equation (25) |

6 | The SLW has a power comparable to the TEM wave. | Equation (22a) * |

7 | The SLW is unconstrained by the skin effect. | Section 5 * |

8 | The SLW free-space attenuation goes like 1/r^{2}. | Equation (23) * |

9 | The SLW monopole radiation is isotropic. | Equation (23) * |

10 | The scalar wave arises from Φ= −∂α/∂t, plus Item 1. | [3] |

11 | The scalar-wave has a pressure density of ∇C^{2}/2μ. | Equation (25) |

12 | The scalar-wave has an energy density of C^{2}/2μ. | Equation (24) |

13 | C is a pseudo-scalar. | Equation (12) |

14 | The scalar wave is also charge-fluctuation driven. | Equation (16) |

15 | The interface matching condition for C is… | Equation (17) |

16 | The interface matching condition for ρ_{A} is… | Equation (18) |

17 | The interface matching condition for J_{A} is… | Equation (19) |

18 | Momentum balance has a mixed-mode term, ∇ × BC/μ. | Equation (25) |

19 | Momentum balance also has source term, JC. | Equation (25) |

20 | Energy balance has a new source, ρC/εμ. | Equation (24) |

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Reed, D.; Hively, L.M.
Implications of Gauge-Free Extended Electrodynamics. *Symmetry* **2020**, *12*, 2110.
https://doi.org/10.3390/sym12122110

**AMA Style**

Reed D, Hively LM.
Implications of Gauge-Free Extended Electrodynamics. *Symmetry*. 2020; 12(12):2110.
https://doi.org/10.3390/sym12122110

**Chicago/Turabian Style**

Reed, Donald, and Lee M. Hively.
2020. "Implications of Gauge-Free Extended Electrodynamics" *Symmetry* 12, no. 12: 2110.
https://doi.org/10.3390/sym12122110