“Scalar waves” is one of those terms that makes physicists wince and enthusiasts lean in. The term carries 30 years of accumulated noise: devices that don’t work, claims that aren’t testable, marketing labels on standard coils.
But the physics underneath is real. Four independent derivations from two research programs recovered scalar-longitudinal solutions from the standard electromagnetic Lagrangian between 2003 and 2020. The math is peer-reviewed. The predictions are testable. The question isn’t whether these solutions exist in the equations. They do. The question is whether nature uses them.
This is Part 1 of 4. Twenty-five foundational questions. Equations where they help, plain language where they’re enough.
⬅️ Previous: Standard electrodynamics is a single point in a two-dimensional parameter space. Both axes lead to the same hidden physics.
1. What is a scalar-longitudinal wave?
An electromagnetic wave mode where the electric field oscillates along the direction of propagation, not perpendicular to it. It carries a scalar field C and longitudinal 𝐄, but no magnetic field 𝐁. Standard electromagnetic radiation is transverse: 𝐄 and 𝐁 perpendicular to propagation. The scalar-longitudinal mode is its complement: longitudinal 𝐄, scalar C, zero 𝐁.
2. Where does it come from mathematically?
From the Stueckelberg Lagrangian, which is standard quantum field theory. When the gauge-fixing parameter is promoted from a mathematical device to a physical term (γ = 1), the Lorenz divergence C = ∂_μA^μ acquires its own wave equation: □C = sources. This field C was always in the equations. The Lorenz gauge set it to zero. Relaxing the gauge lets it propagate.
3. Is this new physics?
No. The Stueckelberg mechanism dates to 1938. The Lorenz gauge condition dates to 1867. What’s new is the synthesis: four independent derivations between 2003 and 2020 showed that relaxing the Lorenz constraint recovers a scalar-longitudinal sector with specific, testable signatures. The theory is old. The explicit prediction set is new.
4. How fast does it travel?
At c. The wave equation is □C = 0, which is the same massless wave equation that governs electromagnetic radiation. No superluminal propagation. Reports of faster-than-light scalar waves in the older literature typically confuse phase velocity in waveguides with group velocity in free space. Guided waves routinely exceed c in phase velocity. The group velocity stays at or below c.
5. How is it different from a sound wave?
Sound is a longitudinal pressure wave in a material medium: air, water, metal. The scalar-longitudinal mode is an electromagnetic wave in vacuum. No material medium required. It propagates at c, not at the sound speed. It’s coupled to charges and currents through the electromagnetic Lagrangian, not through pressure gradients. The only similarity is the longitudinal polarization direction.
6. Why doesn’t standard EM include it?
Because the Lorenz gauge condition ∂_μA^μ = 0 forces the scalar field C to vanish. This constraint was adopted for mathematical convenience: it decouples the wave equations for the scalar and vector potentials, making them independently solvable. The decoupling is real. But so is the information it discards.
7. What did Maxwell’s original theory say about it?
Maxwell’s original 1873 formulation used quaternions. When the nabla operator acts on the vector potential in quaternion algebra, the result has both a scalar part and a vector part. The vector part gives 𝐁 = ∇×𝐀. The scalar part, which is related to ∇·𝐀, has no counterpart in the four modern equations. Heaviside and Gibbs split this quaternion product into two independent operations and suppressed the scalar part. Not because it was wrong. Because it complicated the equations.
8. What’s the relationship between C and Maxwell’s seventh component?
Maxwell’s “seventh component” T = -(1/c)∂φ/∂t + ∇·𝐀 is the scalar part of the quaternion electric field. The EED scalar field C = ∂_μA^μ = ∇·𝐀 + (1/c²)∂φ/∂t is the Lorenz four-divergence. They’re related by T = -cC: same physics, different sign convention and a factor of c from unit conversion. Both vanish under the Lorenz gauge. Both become dynamical when the gauge is relaxed.
9. Is the scalar-longitudinal mode the same thing as a “scalar wave”?
Not exactly. “Scalar wave” is a community term that covers three distinct things: (1) the coupled scalar-longitudinal mode (longitudinal 𝐄 plus scalar C, zero 𝐁), which is what most practical discussions refer to; (2) a pure scalar wave (C only, energy without momentum), which is theoretically possible but harder to generate; and (3) marketing labels on devices that may not produce any of the above. Precision matters. “Scalar-longitudinal” is the correct term for the mode that EED predicts and that experimental protocols can test.
10. Can it penetrate a Faraday cage?
The EED prediction is yes. Standard electromagnetic radiation is blocked by Faraday cages because the oscillating 𝐁 field induces eddy currents in the conductor, which generate canceling fields. The scalar-longitudinal mode has no 𝐁 field. No eddy current induction. No cancellation mechanism. US Patent 9,306,527 describes transmission through Faraday cages using vector potential topology. This is one of the three binary discriminator tests for the mode.
11. What antenna receives it?
A monopolar antenna. Standard dipole antennas are sensitive to transverse 𝐄 fields. A monopolar antenna, which measures scalar potential gradients rather than transverse field components, is sensitive to the longitudinal mode. If a signal is receivable by a monopolar antenna but not a dipole, that’s a mode-selectivity test. This is the second binary discriminator.
12. How does it attenuate with distance?
As 1/r², not 1/r. Standard far-field electromagnetic radiation attenuates as 1/r in amplitude (1/r² in power). The scalar-longitudinal mode attenuates as 1/r² in amplitude because it lacks the self-sustaining 𝐄×𝐁 cross-coupling that maintains transverse waves in the far field. This distance law is the third binary discriminator.
13. What is the three-test discriminator protocol?
Three binary tests in one apparatus. (1) Faraday penetration: does the signal pass through a Faraday cage? (2) Monopolar reception: does a monopolar antenna receive while a dipole doesn’t? (3) Distance law: does attenuation follow 1/r² instead of 1/r? All three positive together rule out standard near-field effects, which can individually mimic each signature but cannot produce all three simultaneously.
14. Has anyone detected the mode experimentally?
The NASA Breakthrough Propulsion Physics program documented transmission of longitudinal electrostatic waves through solid dielectrics (glass, Plexiglas) with less dispersion than through air, and detection through closed wooden doors at several meters distance. The VPT patent describes through-barrier transmission. These results are consistent with EED predictions but have not been independently replicated with the full three-test discriminator protocol. The scalar-longitudinal mode is predicted and has suggestive experimental support. It does not yet have definitive confirmation.
15. What source geometry produces it?
The Vector Potential Transformer (VPT) uses a coiled-coil toroidal geometry designed to produce 𝐁 = 0 in the exterior region while maintaining nonzero 𝐀. The toroidal topology is essential: it’s the simplest multiply connected geometry that creates field-free regions with non-trivial potential structure. A straight solenoid confines 𝐁 to the interior but the topology is simply connected. The torus creates the topological condition for scalar-longitudinal emission.
16. Does it carry energy?
Yes. The generalized Poynting vector in EED contains the term -𝐄S, where S is the scalar field. For pure scalar-longitudinal waves, the standard 𝐄×𝐁 Poynting contribution vanishes identically because 𝐁 = 0. The entire energy flow is carried through the scalar channel -𝐄S. This has its own energy density proportional to S² and its own source coupling. The scalar-longitudinal mode is not a field disturbance without energy. It’s a distinct energy transport mechanism.
17. Does a massive photon produce the same mode?
Yes and no. A massive photon (Proca theory) acquires a third polarization: the longitudinal mode. This is the mass axis of the Stueckelberg parameter space. The EED scalar-longitudinal mode comes from the gauge axis: promoting the gauge-fixing term to a dynamical term at zero mass. Different mechanism, same sector. The Stueckelberg Lagrangian unifies both approaches in a single parameter space.
18. Is the gauge-fixing parameter γ physical?
In standard QFT: no. γ is chosen for mathematical convenience (Feynman gauge, Landau gauge) and physical observables are independent of the choice. In EED: γ = 1 is promoted from a convention to a specific theory. The gauge-fixing term becomes a kinetic term. There is no residual gauge symmetry. C cannot be gauged away because there is no gauge freedom left to fix. Whether this promotion is physically realized is the experimental question the three-test discriminator addresses.
19. How does this relate to the Aharonov-Bohm effect?
The Aharonov-Bohm effect demonstrates that the vector potential 𝐀 produces measurable physical effects (quantum phase shifts) in regions where 𝐄 = 0 and 𝐁 = 0. This proves that potentials carry physical content beyond what fields represent. The scalar-longitudinal mode extends this: not only does 𝐀 have physical content in field-free regions, but the divergence of 𝐀, ∂_μA^μ, is itself a dynamical field with its own propagation and energy transport. AB proves potentials are real. EED shows what they can do when unconstrained.
20. Where can I read the full technical case?
My paper “The Deleted Degrees of Freedom: A Case for Potential-Primary Electrodynamics” covers the complete argument: the tensor decomposition, the three acts of deletion, eight independent lines of evidence, the Stueckelberg parameter space, and the engineering implications. Free to read at advanced-rediscovery.com/research/deleted-degrees-of-freedom.
21. What is the Helmholtz decomposition and why does it matter?
Any vector field can be split into a curl-free (irrotational) part and a divergence-free (solenoidal) part. Extended to four dimensions via the Hodge decomposition, the antisymmetric tensor F_μν captures the solenoidal, transverse sector. The symmetric part S_μν captures the irrotational, longitudinal sector. Standard EM keeps only the solenoidal part. The Hodge decomposition adds a further insight: a harmonic component that’s topologically conserved. This is what gauge-invariant quantities like the Aharonov-Bohm phase encode.
22. What’s the difference between kinematic components and dynamical degrees of freedom?
The 16-component decomposition of ∂_μA_ν is kinematic: it identifies the mathematical content available in the tensor. The number of propagating degrees of freedom is a separate, dynamical question that requires a Lagrangian and constraint analysis. The four-potential A_μ allows at most 3 propagating modes: four components minus one gauge parameter. Standard EM (Lorenz gauge) further reduces to 2. EED (γ = 1) keeps all 3: two transverse plus one scalar-longitudinal.
23. What does geometric algebra say about the deletion?
In Hestenes’ spacetime algebra, the geometric product ∇𝐀 = ∇·𝐀 + ∇∧𝐀 produces both the divergence (scalar part) and the exterior product (bivector field F, the EM field) as grades of a single operation. They’re inseparable by construction. Heaviside and Gibbs split this into two independent operations and suppressed the scalar grade. The deletion is algebraically visible: projecting out one grade of a multivector.
24. What’s the Lorenz gauge in plain language?
A rule that says the scalar field C = ∂_μA^μ must equal zero everywhere, always. It’s imposed as a mathematical convenience because it decouples the wave equations for φ and 𝐀, making them independently solvable. The cost: any independent physical content carried by ∇·𝐀 is suppressed. The rule isn’t derived from a physical principle. It’s chosen for computational ease.
25. How many independent derivations of the scalar-longitudinal sector exist?
Four, between 2003 and 2020. Van Vlaenderen (2003): gauge-free classical ED. Hively and Giakos (2012): Stueckelberg mechanism. Reed and Hively (2020): systematic gauge relaxation with Woodside’s uniqueness theorems. The Stueckelberg/Proca literature: photon mass parameter. Different starting points, same destination. The parameter space diagram (Part 3, Question 56) explains why this convergence is mathematically necessary.
Part 1 of 4. 25 foundations down, 75 to go.
https://advanced-rediscovery.com/research/deleted-degrees-of-freedom
I wrote a field guide that takes the paper and distills it into what you need at the bench — the three-test discriminator, the experiment design, the annotated reading list. Paid subscribers can download it: The EED Playbook.
⏭️ Next: Part 2: The Evidence. Aharonov-Bohm, superconductors, the dynamical Casimir effect, and what a definitive confirmation experiment looks like. Friday.
What is this non-printable character preceding the C in equation □C = 0?
Some of the mathematics is unfortunately not rendered correctly.
https://pubs.aip.org/aip/jap/article-abstract/114/4/044907/373024/Reception-of-longitudinal-vector-potential?redirectedFrom=fulltext
This paper describes the simplest setup I've seen for transmitting vector potential using a monopole antenna and receiving it line-of-sight by using an aligned NE2 neon bulb...