Scalar-Longitudinal Radiation in Extended Electrodynamics with Multipole Theory and a Compensated Source Model

Abstract: Extended electrodynamics (EED) leaves the Lorenz gauge condition unimposed and treats the scalar combination $$C=\nabla\cdot\mathbf{A}+c^{{-2}\partialΦ/\partial} t$$ as a dynamical field. For a conserved source with no scalar initial field, $C=0$ and the theory reduces to classical electrodynamics. A source with a nonzero local continuity anomaly has no Maxwell solution, but EED remains well posed and can support a scalar-longitudinal sector. For each radiating frequency of a localized source, the far field separates into the usual transverse Maxwell channel and a scalar-longitudinal channel with a longitudinal electric field, no magnetic field of its own, and a co-propagating $C$ field. Under the field-only energy balance used here, the time-averaged fluxes add without interference. The scalar channel depends only on $Λ=\partialρ/\partial t+\nabla\cdot\mathbf{J}$ and radiates when its moments at $k=ω/c$ are nonzero. An all-orders multipole formula is derived for this flux. Bound polarization and magnetization sources conserve charge identically and cannot excite the scalar channel. A compensated polarized carrier with a globally neutral anomalous surface layer isolates the channel and gives its dipole flux in closed form. The connection between the adopted flux and a physical stress-energy tensor remains unresolved. These results are conditional predictions of EED and do not imply a failure of charge conservation in classical electrodynamics.

• The paper presents a conditional radiation theory using extended electrodynamics.
• The source-independent two-channel decomposition separates radiation into transverse Maxwell and scalar-longitudinal channels, driven by charge continuity violations and sources .
• The exact all-order multipole formula for the scalar-longitudinal flux showed physical system constraints for the isolated channel detection.

Overview and motivation

This paper develops a conditional radiation theory for extended electrodynamics (EED), a framework in which the Lorenz gauge condition is not imposed and the scalar combination $C=\nabla\cdot\mathbf A+c^{-2}\partial_t\Phi$ is promoted to a dynamical field. The author, Natan Rentzber, is explicit that EED is nonstandard and that charge conservation is tightly constrained experimentally; the analysis is therefore framed as a set of conditional predictions: if a prescribed localized source violates local continuity, $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$, what does EED predict? For conserved sources with zero scalar initial data, $C\equiv0$ and Maxwell theory is recovered exactly.

The paper's main contributions are a source-independent two-channel decomposition of the far field, an exact all-orders spherical-multipole formula for the scalar-longitudinal flux, no-go results showing that bound polarization and magnetization sources cannot excite the channel, a reception identity for conserved test currents, and a compensated source construction that isolates the scalar channel from any transverse electromagnetic background (2608.19405).

Field equations and covariant structure

Starting from the four decoupled sourced wave equations for $(\Phi,\mathbf A)$ without imposing the Lorenz condition, the extended Gauss and Ampère laws acquire $C$-dependent corrections: $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$ and $$\nabla\times\mathbf B-c^{-2}\partial_t\mathbf E=\mu_0\mathbf J+\nabla C$$. Applying the wave operator to the definition of $C$ yields a closed scalar wave equation sourced solely by the continuity anomaly,

$$\left(\nabla^2-\frac{1}{c^2}\partial_t^2\right)C=-\mu_0\Lambda.$$

The framework admits a Lorentz-covariant Lagrangian density containing a quadratic term $-C^2/2\mu_0$, which the paper identifies (up to a total divergence) with the Fermi/Feynman-gauge field Lagrangian obtainable by eliminating a Nakanishi–Lautrup auxiliary field. The interpretive difference is that EED treats $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$0 as physical rather than gauge-fixing. A key structural result is that under a gauge transformation the interaction action changes by $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$1: arbitrary-gauge invariance of the coupling and local charge conservation fail together. When $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$2, setting $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$3 is therefore not a gauge choice for the same source-coupled problem.

The paper also derives the extended Poynting theorem with energy density including $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$4 and flux including $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$5. The author concedes plainly that whether this functional is the flux of a physical stress-energy tensor remains unresolved; all power statements below are relative to this adopted balance.

The free scalar-longitudinal wave

In vacuum, EED admits a propagating branch with longitudinal electric field, vanishing magnetic field, and co-propagating scalar with fixed ratio $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$6, traveling at $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$7. Under the adopted balance it transports energy despite $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$8. The paper argues this is the unique radiative scalar branch generated by localized sources from zero scalar initial data: configurations with $\Lambda=\partial_t\rho+\nabla\cdot\mathbf J\neq0$9 force spatially uniform, time-independent $C\equiv0$0 carrying no flux, and pure "gauge waves" carry no field energy under the adopted accounting.

Two-channel decomposition

For each monochromatic component of a localized source, the radiation zone separates into the usual transverse Maxwell channel driven by the on-shell transverse current $C\equiv0$1 and a scalar-longitudinal channel driven only by the on-shell anomaly transform $C\equiv0$2. The longitudinal electric field satisfies $C\equiv0$3 with $C\equiv0$4. Because the cross terms cancel pointwise at order $C\equiv0$5, the time-averaged powers add without interference:

$C\equiv0$6

An implication worth noting: the standard transverse nature of Maxwell radiation emerges as precisely the $C\equiv0$7 limit of the same calculation, since the radial electric field bracket is proportional to $C\equiv0$8.

Multipole formula and long-wavelength limits

Expanding the outgoing Green function in spherical harmonics gives the exact all-orders result

$C\equiv0$9

with moments $(\Phi,\mathbf A)$0. In the long-wavelength limit each multipole is suppressed by $(\Phi,\mathbf A)$1 relative to comparable lower orders. Two limiting cases are worked out explicitly: the monopole power $(\Phi,\mathbf A)$2 requires oscillation of total charge and is excluded by Maxwell theory outright; the dipole power is $(\Phi,\mathbf A)$3, where the anomaly dipole obeys $(\Phi,\mathbf A)$4-related identity $(\Phi,\mathbf A)$5. Notably, the scalar dipole pattern is $(\Phi,\mathbf A)$6, strongest along the dipole axis — complementary to the $(\Phi,\mathbf A)$7 pattern of an ordinary electric dipole.

No-go results and detector corollaries

Three corollaries sharply delimit the channel. First, the bound pair $(\Phi,\mathbf A)$8, $(\Phi,\mathbf A)$9 is identically conserved, distributionally even across discontinuous interfaces, so bound sources cannot drive $C$0; the auxiliary fields $C$1 and $C$2 gain no independent radiative degree of freedom. Second, magnetization currents are solenoidal and provide no anomaly, so there is an intrinsic electric–magnetic asymmetry: a magnetic counterpart would require magnetic charge or a dual scalar sector absent from the model. Third, a conserved test current extracts zero cycle-averaged power from a pure incident scalar-longitudinal wave, $C$3, because the overlap integral reduces to a term proportional to the detector's own anomaly. This is a strong and consequential claim: ordinary charge-conserving matter cannot absorb energy from the scalar channel under the adopted balance, so detection would require anomalous or nonstandard coupling. The author flags this as a test-current result, not a complete receiver theory — active, nonlinear, and back-reacting detectors require separate models.

Compensated source model

To isolate the channel, the paper constructs a sphere of radius $C$4 with time-harmonic uniform polarization plus a free compensating surface charge and volume current chosen so that the total carrier source vanishes as a distribution: $C$5, $C$6, hence exactly zero transverse radiation. Onto the compensating layer is placed a globally neutral, dipolar surface anomaly $C$7 with no associated current, which violates local continuity while conserving total charge. Since its on-shell transverse amplitude vanishes, the configuration emits only through the scalar channel.

The anomaly dipole is $C$8, giving the long-wavelength power

$C$9

and, because the shell remains a single $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$0 multipole at every $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$1, the exact result $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$2. The zeros of $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$3 render the ideal infinitesimally thin shell radiation-silent at discrete frequencies, but the author notes these nulls are not universal: finite thickness or phase variation shifts them. As a benchmark, with $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$4 cm, 1 GHz, and reference current density $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$5 A/m², the model gives $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$6 W and an on-axis longitudinal field of about $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$7 V/m at one meter; $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$8 corresponds to a few µV/m and roughly 40 fW. These numbers normalize the prescribed model and do not establish detectability.

A direct comparison with Minotti and Modanese's symmetric tensor treatment is instructive: for the same currentless anomaly dipole, their tensor flux equals the adopted-balance flux in magnitude but with opposite sign (inward rather than outward). The paper states there is no mathematical contradiction since the expressions come from different currents, but they cannot both represent the same physical radiated power, and the paper does not decide which does.

Limitations and open questions

The paper is careful about scope. All results are conditional on EED being physically realized; nothing implies charge conservation fails in nature, and process-specific bounds such as the electron-lifetime limit $$\nabla\cdot\mathbf E=\rho/\epsilon_0-\partial C/\partial t$$9 yr do not directly constrain the coherent anomaly ratio without a microscopic mapping. Three limitations bear directly on the results. First, the relation between the adopted Poynting functional, the canonical energy (whose plane-wave representative can be negative), and a Hilbert stress-energy tensor remains unresolved, so the positive fluxes derived here are not by themselves proofs of positive physical energy. Second, whether any physical system produces an effective $$\nabla\times\mathbf B-c^{-2}\partial_t\mathbf E=\mu_0\mathbf J+\nabla C$$0 — e.g., via reduced mesoscopic transport descriptions with modified continuity equations — is outside the work and constitutes an extra phenomenological assumption when invoked. Third, the detector corollary shows conserved matter cannot receive the channel, leaving open what coupling could mediate detection; turning a null experimental result into a bound would require a calibrated source-to-multipole transfer function, a detector model, and background analysis, none attempted here.

Conclusion

The paper provides a self-contained radiation theory for the scalar-longitudinal sector of EED: an orthogonal two-channel decomposition, an exact multipole power formula, no-go results confining the channel to anomalous free-source sectors, and a compensated source that isolates it with closed-form power. Within the framework, a radiative anomaly produces a longitudinal electric field traveling at $$\nabla\times\mathbf B-c^{-2}\partial_t\mathbf E=\mu_0\mathbf J+\nabla C$$1 with no magnetic field of its own and a $$\nabla\times\mathbf B-c^{-2}\partial_t\mathbf E=\mu_0\mathbf J+\nabla C$$2 dipole pattern, while conserved matter generates none of these fields. The physical status of both the energy balance and the required anomalous source remains open, making the entire construction a well-defined conditional prediction rather than established physics.