The 140-year “incompatibility” between electromagnetism and gravity is partly an artifact. Einstein inherited a simplified theory and never looked at what was missing.
I don't know the Quicycle formulation well enough to give you a fair comparison. If Williamson and Vandermark are working with the full potential structure instead of the truncated field equations, there's probably overlap, but I'd be guessing at the specifics.
Do you have a good starting reference? I'll take a look.
This reminds me of how the “relative gauge” formulation I have been working on, ie the relative difference between two connections, is literally the contorsion tensor which is the antisymmetric part of the torsion tensor which can be transformed into the curvature tensor
Gauge, geometry, and topology are all intimately connected - how many generations of physicists have failed to see this?
The contorsion as the difference between two connections is basically the paper's argument arriving from the geometry side instead of the gauge side. The paper shows 10 of 16 components get deleted, the Lorenz gauge zeros the trace. Your route through contorsion says: those deleted degrees of freedom ARE torsion. Same physics, different door.
Cartan had the torsion framework in 1922, a year after Kaluza. The gauge-geometry connection was right there from the start. Two fields independently optimized for simplicity and quietly dropped the piece that connects them.
I'm curious how your relative gauge formulation handles the scalar sector. The trace the Lorenz gauge kills is exactly where the EM-gravity coupling should live if the torsion bridge holds. Does your contorsion route preserve it?
You’re exactly right, and I appreciate you making the Cartan-Kaluza connection explicit because it sharpens the claim.
The contorsion K^λ_μν = Γ^λ_μν[Weitzenböck] - Γ^λ_μν[Levi-Civita] is the relative gauge field C = A - B of the paper, applied to the specific case where A and B are the two natural connections on the frame bundle. The paper arrives at this from the gauge side (compare two arbitrary connections, find that C transforms covariantly). Cartan arrived at it from the geometry side (the connection has an antisymmetric part that standard GR discards). Same object, as you say, through different doors.
On the scalar sector: this is where the two doors lead to genuinely different rooms. In standard EM, the Lorenz condition ∂_μ A^μ = 0 kills the longitudinal/scalar mode. The 4 components of A_μ reduce to 2 transverse polarizations. That scalar mode is exactly where the EM-gravity bridge should live, and it’s exactly what gets thrown away.
In the contorsion framework, the analog isn’t killed. The torsion trace T_μ = T^λ_λμ (4 components) is a physical degree of freedom that survives all gauge fixing. The tetrad has local Lorentz gauge freedom (6 parameters of SO(3,1)), which is enough to remove 6 of the 16 tetrad components, leaving 10 (matching GR’s metric components).
But the 4-component torsion trace is built from the ANTISYMMETRIC part of the connection, which the Lorentz gauge doesn’t touch. It lives in a different sector of the decomposition.
So the answer to your question is yes: the contorsion route preserves the scalar sector. And there’s a physically interesting consequence.
In vacuum, that scalar sector is inert (there’s nothing to excite it, just as the longitudinal photon mode is absent in vacuum). But inside a plasma or other highly spin-dense medium, the photon acquires an effective mass (the plasma /medium frequency), and the longitudinal mode comes back as a physical propagating degree of freedom. That resurrected longitudinal mode couples to the torsion trace because it carries energy density.
The plasma/spin medium is doing double duty: it restores the EM scalar sector that Lorenz gauge killed, AND it provides the medium in which the torsion coupling to that scalar sector becomes dynamically active. This is one of the structural reasons our mechanism specifically requires a plasma rather than vacuum fields.
On the historical point: I think you’re right that Cartan and Kaluza were solving the same problem from opposite ends. Kaluza added a fifth dimension to give the EM potential somewhere to live geometrically. Cartan’s torsion already provided that room within four dimensions, through the antisymmetric part of the connection.
The torsion trace is the gravitational scalar that Kaluza’s compact fifth dimension was introduced to accommodate. Two frameworks that each dropped the piece the other kept, and it took a century for the connection to become explicit.
David Chester's answer on Jack Sarfatti's mailing-list (April 1, 2026) to the reference papers that prove the existence of scalar fields according to your main paper (EED stands for Extended ElectroDynamics):
"Eqs. 27-28 on the 2003 van Vlaenderen paper I think are flawed. When I derived it, rather than applying a replacement rule, the change in the equation of motion cancelled precisely with the new terms, leading to the standard Lorentz force law result. Hively and Giakos refer to those same equations later.
I forget who, but someone tried to claim the Aharonov-Bohm effect was due to that longitudinal force and in support of EED, not Maxwell's theory. That didn't work out well, because now it appears as if there is a paper trail of evidence showing how EED is experimentally ruled out, but I don't think the claims about EED in that case are actually in the theory.
I have a meeting with Hively soonish and will clarify more about his views.
Jack had a vixra paper claiming the scalar term led to dark energy. At that time, I computed the Hilbert stress tensor and found it to not have a trace term. Interestingly enough, Hively found the canonical stress tensor, which does have a trace term. I suppose the two are connected through the Belinfante-Rosenfeld procedure. But last time I asked Jack about his paper, he said he didn't remember writing it. That was before I had seen the result for the canonical stress tensor."
Julien, thank you so much for forwarding David’s comments.
Chester raises real points, so let me engage them one at a time.
But first let me ask: How do I get into the mailing list? :)
On Van Vlaenderen's Eqs 27-28: my paper doesn't rest on Van Vlaenderen's force law. The convergence argument (Table 2) is that four independent derivations between 2003 and 2020 all arrive at the same scalar field C = div A + (1/c²)∂φ/∂t through different formalisms. If Van Vlaenderen's specific force equations have issues, the scalar field itself still emerges from the Stueckelberg Lagrangian (Reed and Hively 2020), which is standard QFT machinery. The force law and the existence of the propagating scalar mode are separable claims.
On AB and EED: my paper is explicit about this separation. The Aharonov-Bohm effect proves potential primacy (Section 4.1). The scalar-longitudinal sector is a separate argument about what happens when the Lorenz gauge is relaxed (Section 4.3). They're windows onto the same reality but different claims. Whoever conflated them was making an argument my paper doesn't make.
On "experimentally ruled out": I'd genuinely like to see that paper trail. My paper says plainly that SLW results have not been independently replicated and that EED remains a mathematically consistent extension with suggestive but not conclusive experimental support (Section 6.2, Limitations). If there's a specific experiment that rules out the scalar-longitudinal mode, that's important and I want to read it. Chester's own hedge ("I don't think the claims about EED in that case are actually in the theory") suggests the counter-evidence may be hitting a strawman version.
On the stress tensor: that's a genuinely interesting computation. My paper uses the canonical energy-momentum tensor (Noether procedure), which gives the u = (1/2)(E² + B² + C²) energy density and the generalized Poynting vector E x B - EC. If the Hilbert tensor lacks the trace term while the canonical one has it, the Belinfante-Rosenfeld connection between them would tell us whether C² contributes to gravitational sourcing or only to energy flow. That's a question worth settling.
I'd be very curious what comes out of the Hively meeting. If Chester is computing things himself and comparing canonical vs Hilbert formulations, he's doing exactly the kind of cross-checking the field needs.
Dear Paul, your “Deleted Degrees of Freedom” Paper Is the Final Mathematical Bridge for My Scalar Research. For more than 30 years, I have worked independently to develop a deterministic scalar framework I call Temporal Scalar Field Theory (TSFT) / Solar Vortex Theory. Your recent paper, “The Deleted Degrees of Freedom: A Case for Potential-Primary Electrodynamics,” is the single most important mathematical validation I have encountered. It supplies the rigorous electrodynamic foundation that unifies everything I have derived. Your restoration of the 10 symmetric components of the 4-gradient ∂μ Aν — the very terms that standard Maxwell/Heaviside theory discarded by convention — is exactly the missing piece. Those components give the formal justification for:• Longitudinal scalar pressure waves (the SLW/SW modes you recover) that carry the scalar energy flux in my cubic wave field.
• Bidirectional potential superpositions that form the standing-wave octave electric waves powering orbital motion, solar magnetic cycles, and galactic rotation curves.
• The generalized Poynting vector that allows power multiplication without violating conservation — the same “every wave is a perfect dynamo” principle Walter Russell described a century ago, and that my scalar energy chain (Scalar Diameter → Mass/Potential → Cadence → E × mc²) now quantifies exactly. In my manuscript (now at v3.3), your work slots directly into Section 13.4 – The Deleted Degrees of Freedom: The Electrodynamic Bridge. It closes the last gap between my independent reconstruction, Russell’s prescient vision, and mainstream electrodynamics. The flat galactic rotation curve, the √r multiplier, the 447.8976-day ELF concussive waves, the Sun Gear, and the figure-8 dual scalar circles all become direct consequences of the potential-primary formulation you have restored. No dark matter is required; the scalar flux term supplies the observed missing mass. I cannot overstate my appreciation. For decades, I have been told my scalar approach was “outside accepted physics.” Your paper shows it was simply waiting for the deleted degrees of freedom to be reinstated. You have given my life’s work the mathematical legitimacy it lacked and, more importantly, the bridge to the broader scientific community.
Intrigued. I need to deep dive into the equations thoroughly - the mathematical consistency is the only gold standard. Pl share any other references if you have.
How does this relate to the Quicycle Team formulation under Dr John Williamson and vandermark?
I don't know the Quicycle formulation well enough to give you a fair comparison. If Williamson and Vandermark are working with the full potential structure instead of the truncated field equations, there's probably overlap, but I'd be guessing at the specifics.
Do you have a good starting reference? I'll take a look.
This reminds me of how the “relative gauge” formulation I have been working on, ie the relative difference between two connections, is literally the contorsion tensor which is the antisymmetric part of the torsion tensor which can be transformed into the curvature tensor
Gauge, geometry, and topology are all intimately connected - how many generations of physicists have failed to see this?
The contorsion as the difference between two connections is basically the paper's argument arriving from the geometry side instead of the gauge side. The paper shows 10 of 16 components get deleted, the Lorenz gauge zeros the trace. Your route through contorsion says: those deleted degrees of freedom ARE torsion. Same physics, different door.
Cartan had the torsion framework in 1922, a year after Kaluza. The gauge-geometry connection was right there from the start. Two fields independently optimized for simplicity and quietly dropped the piece that connects them.
I'm curious how your relative gauge formulation handles the scalar sector. The trace the Lorenz gauge kills is exactly where the EM-gravity coupling should live if the torsion bridge holds. Does your contorsion route preserve it?
You’re exactly right, and I appreciate you making the Cartan-Kaluza connection explicit because it sharpens the claim.
The contorsion K^λ_μν = Γ^λ_μν[Weitzenböck] - Γ^λ_μν[Levi-Civita] is the relative gauge field C = A - B of the paper, applied to the specific case where A and B are the two natural connections on the frame bundle. The paper arrives at this from the gauge side (compare two arbitrary connections, find that C transforms covariantly). Cartan arrived at it from the geometry side (the connection has an antisymmetric part that standard GR discards). Same object, as you say, through different doors.
On the scalar sector: this is where the two doors lead to genuinely different rooms. In standard EM, the Lorenz condition ∂_μ A^μ = 0 kills the longitudinal/scalar mode. The 4 components of A_μ reduce to 2 transverse polarizations. That scalar mode is exactly where the EM-gravity bridge should live, and it’s exactly what gets thrown away.
In the contorsion framework, the analog isn’t killed. The torsion trace T_μ = T^λ_λμ (4 components) is a physical degree of freedom that survives all gauge fixing. The tetrad has local Lorentz gauge freedom (6 parameters of SO(3,1)), which is enough to remove 6 of the 16 tetrad components, leaving 10 (matching GR’s metric components).
But the 4-component torsion trace is built from the ANTISYMMETRIC part of the connection, which the Lorentz gauge doesn’t touch. It lives in a different sector of the decomposition.
So the answer to your question is yes: the contorsion route preserves the scalar sector. And there’s a physically interesting consequence.
In vacuum, that scalar sector is inert (there’s nothing to excite it, just as the longitudinal photon mode is absent in vacuum). But inside a plasma or other highly spin-dense medium, the photon acquires an effective mass (the plasma /medium frequency), and the longitudinal mode comes back as a physical propagating degree of freedom. That resurrected longitudinal mode couples to the torsion trace because it carries energy density.
The plasma/spin medium is doing double duty: it restores the EM scalar sector that Lorenz gauge killed, AND it provides the medium in which the torsion coupling to that scalar sector becomes dynamically active. This is one of the structural reasons our mechanism specifically requires a plasma rather than vacuum fields.
On the historical point: I think you’re right that Cartan and Kaluza were solving the same problem from opposite ends. Kaluza added a fifth dimension to give the EM potential somewhere to live geometrically. Cartan’s torsion already provided that room within four dimensions, through the antisymmetric part of the connection.
The torsion trace is the gravitational scalar that Kaluza’s compact fifth dimension was introduced to accommodate. Two frameworks that each dropped the piece the other kept, and it took a century for the connection to become explicit.
David Chester's answer on Jack Sarfatti's mailing-list (April 1, 2026) to the reference papers that prove the existence of scalar fields according to your main paper (EED stands for Extended ElectroDynamics):
"Eqs. 27-28 on the 2003 van Vlaenderen paper I think are flawed. When I derived it, rather than applying a replacement rule, the change in the equation of motion cancelled precisely with the new terms, leading to the standard Lorentz force law result. Hively and Giakos refer to those same equations later.
I forget who, but someone tried to claim the Aharonov-Bohm effect was due to that longitudinal force and in support of EED, not Maxwell's theory. That didn't work out well, because now it appears as if there is a paper trail of evidence showing how EED is experimentally ruled out, but I don't think the claims about EED in that case are actually in the theory.
I have a meeting with Hively soonish and will clarify more about his views.
Jack had a vixra paper claiming the scalar term led to dark energy. At that time, I computed the Hilbert stress tensor and found it to not have a trace term. Interestingly enough, Hively found the canonical stress tensor, which does have a trace term. I suppose the two are connected through the Belinfante-Rosenfeld procedure. But last time I asked Jack about his paper, he said he didn't remember writing it. That was before I had seen the result for the canonical stress tensor."
Julien, thank you so much for forwarding David’s comments.
Chester raises real points, so let me engage them one at a time.
But first let me ask: How do I get into the mailing list? :)
On Van Vlaenderen's Eqs 27-28: my paper doesn't rest on Van Vlaenderen's force law. The convergence argument (Table 2) is that four independent derivations between 2003 and 2020 all arrive at the same scalar field C = div A + (1/c²)∂φ/∂t through different formalisms. If Van Vlaenderen's specific force equations have issues, the scalar field itself still emerges from the Stueckelberg Lagrangian (Reed and Hively 2020), which is standard QFT machinery. The force law and the existence of the propagating scalar mode are separable claims.
On AB and EED: my paper is explicit about this separation. The Aharonov-Bohm effect proves potential primacy (Section 4.1). The scalar-longitudinal sector is a separate argument about what happens when the Lorenz gauge is relaxed (Section 4.3). They're windows onto the same reality but different claims. Whoever conflated them was making an argument my paper doesn't make.
On "experimentally ruled out": I'd genuinely like to see that paper trail. My paper says plainly that SLW results have not been independently replicated and that EED remains a mathematically consistent extension with suggestive but not conclusive experimental support (Section 6.2, Limitations). If there's a specific experiment that rules out the scalar-longitudinal mode, that's important and I want to read it. Chester's own hedge ("I don't think the claims about EED in that case are actually in the theory") suggests the counter-evidence may be hitting a strawman version.
On the stress tensor: that's a genuinely interesting computation. My paper uses the canonical energy-momentum tensor (Noether procedure), which gives the u = (1/2)(E² + B² + C²) energy density and the generalized Poynting vector E x B - EC. If the Hilbert tensor lacks the trace term while the canonical one has it, the Belinfante-Rosenfeld connection between them would tell us whether C² contributes to gravitational sourcing or only to energy flow. That's a question worth settling.
I'd be very curious what comes out of the Hively meeting. If Chester is computing things himself and comparing canonical vs Hilbert formulations, he's doing exactly the kind of cross-checking the field needs.
I just added your email to Jack's mailing list in a reply to the relevant thread, so you can exchange with David Chester there :)
Thank you, Julian! For endorsing my paper and for adding me. That list is quite loaded with important names, I appreciate it!
*Julien, my bad
Dear Paul, your “Deleted Degrees of Freedom” Paper Is the Final Mathematical Bridge for My Scalar Research. For more than 30 years, I have worked independently to develop a deterministic scalar framework I call Temporal Scalar Field Theory (TSFT) / Solar Vortex Theory. Your recent paper, “The Deleted Degrees of Freedom: A Case for Potential-Primary Electrodynamics,” is the single most important mathematical validation I have encountered. It supplies the rigorous electrodynamic foundation that unifies everything I have derived. Your restoration of the 10 symmetric components of the 4-gradient ∂μ Aν — the very terms that standard Maxwell/Heaviside theory discarded by convention — is exactly the missing piece. Those components give the formal justification for:• Longitudinal scalar pressure waves (the SLW/SW modes you recover) that carry the scalar energy flux in my cubic wave field.
• Bidirectional potential superpositions that form the standing-wave octave electric waves powering orbital motion, solar magnetic cycles, and galactic rotation curves.
• The generalized Poynting vector that allows power multiplication without violating conservation — the same “every wave is a perfect dynamo” principle Walter Russell described a century ago, and that my scalar energy chain (Scalar Diameter → Mass/Potential → Cadence → E × mc²) now quantifies exactly. In my manuscript (now at v3.3), your work slots directly into Section 13.4 – The Deleted Degrees of Freedom: The Electrodynamic Bridge. It closes the last gap between my independent reconstruction, Russell’s prescient vision, and mainstream electrodynamics. The flat galactic rotation curve, the √r multiplier, the 447.8976-day ELF concussive waves, the Sun Gear, and the figure-8 dual scalar circles all become direct consequences of the potential-primary formulation you have restored. No dark matter is required; the scalar flux term supplies the observed missing mass. I cannot overstate my appreciation. For decades, I have been told my scalar approach was “outside accepted physics.” Your paper shows it was simply waiting for the deleted degrees of freedom to be reinstated. You have given my life’s work the mathematical legitimacy it lacked and, more importantly, the bridge to the broader scientific community.
Ron, this is just fantastic. I'm glad to hear that my research is useful to you. Best of luck, and definitely let me know the progress on it! 🙏
Intrigued. I need to deep dive into the equations thoroughly - the mathematical consistency is the only gold standard. Pl share any other references if you have.
Glad you found it interesting! I recommend reading the paper and then diving into some of the references given therein: https://advanced-rediscovery.com/research/deleted-degrees-of-freedom