>>>/b/942166092theoretical basis of electrogravitic propulsion just dropped
Its cool seeing new schizos genuinely in a manic episode. What a little trooper. I wonder if he has incorporated anti-matter or spin...hrmm...
>>16838059bump
>it's just "muh longitudinal/scalar waves!!1!" for the bajillionth timewish these faggots would fuck off already
>>16840083No other framework posits the idea in this manner.The framework presented is mathematically consistent and extensive in elaboration, contrary to most 'works' shared on the idea. Discussion on the possibility of these phenomena is serious, as there are major gaps and inconsistencies within the mainstream standard framework in regards to the reality of the vector potential.Those like you who disregard concepts based on others opinions of such, are sheep and lack the due diligence and will to prove or support their stances, on their own accord. They simply state:>"look- see they said its wrong so its wrong... why? j-just look"
>>16840151Proponents of longitudinal EM waves have had over a century to demonstrate the effect.
>>16838059what am i even looking at, an antenna?Why is this "a theory" when the device is so simple. Just buy a wave generator on amazon for $50 and attach it to a copper plate."buh it need exotic material"so it doesnt work. The device should be based in the diagram you posted not on special materials
longitudinal waves sound cool but then you realize charge conservation forces you to only generate electric dipoles.These drawings dont consider that the "ground" would simply be the other leg of the dipole.
>>16840083I am sure you have an explanation for why scalar waves can't actually do that (they can, slow light is a thing)
>>16840294record yourself making this simple device and show us that it worksaav0x
>>16840211>What is displacement current within a spherical capacitor where- due to spherical geometry- any B field is null?'Displacement current' is a euphemism.>>16840228You are looking at a screen shot of a diagram taken from the elaboration of the framework, meant to quickly grab attention- then direct it towards the linked (now dead but archived) thread which is obviously right next to it.>>16840293Again, pic is a diagram with no context, not a proof of concept. Please don't take it as such.Pic related is a breakdown of the nature of specifically longitudinal waves.Ya Ya inb4 gauge invariance- I got something to mitigate that little nuance as well.
>>16840320Yea and get immediately suicided by global oil cabal? Publicity isn't the key here anon, information sharing is.>simpleMost definitely not. Building a functional prototype concerns observing and testing the magnetic vector potential under high frequency, anisotropic conditions.While the magnetic field inside a current carrying cylinder is zero, the magnetic vector potential moves in parallel with the current, and, as in pic related, is non-zero inside the tube. By utilizing high frequency currents, voltage impulses, and testing the circuit while piping liquid mercury, and to do a sort of coaxial concentric cylinder thing with different thin films/ electro-plates if different permitivities, permeabilities and anisotropy with the purposes of trying to get the harmonic component of the vector potential playing against itself, just like in the normal bohm -aharanov effect. Ideally try to get the vector potential phase component helixing within the pipe.
>>16840211https://sci-hub.se/https://doi.org/10.1134/1.1349267"Indeed, as was shown in the stationary case, the ABE is due to zero-field potentials, changing only the phase of the wave function. The necessary condition for the ABE is the presence of zero-field potentials that cannot be eliminated by gauge transformation. The notion of zero-field (redundant) potentials was first introduced for solving boundary problems in electrodynamics of anisotropic media [14–16] and is very seldom encountered in the physics literature. The authors of [17] argue that “electric and magnetic field vectors cannot be expressed in terms of vector potentials” in anisotropic media. The conventional approach to such problems was to use electric and magnetic field strengths or, for zero scalar potentials, vector potentials proportional to them as unknown functions (Coulomb gauge). Such an approach turned out to exclude the possibility of satisfying boundary conditions in anisotropic media due to the intrinsic structure of Maxwell’s equations (see Section 2). The use of electromagnetic potentials with nonzero scalar potential in order to regularly satisfy boundary conditions was first proposed by academician Tikhonov in 1959 [18].
>>16840322>Again, pic is a diagram with no context, not a proof of concept.All i'm saying is you cant have a machine with a varying electrical charge because that breaks charge conservation. You can have charge separation tho, and that is an electrical dipole.
>>16840330>and anisotropy with the purposes of trying to get the harmonic component of the vector potential playing against itselfthats such a good call outi wonder if adding any modeling around helmholtz-hodge. could garner any insighthttps://sci-hub.se/https://doi.org/10.1140/EPJD/E2008-00142-Y probably really good paper to cross reference here for this experiment and the classical analogue of bohm-aharonovi think that this is part of the minimum self induction section is important when reflection slowly varying the current.like the 1914 blondel experimenthttps://arxiv.org/pdf/1005.2350>linear solenoid in which a slowly varying current flows. In this case,since ρ = 0 everywhere, the scalar potential may be assumed to be zero
>>16840083>>16840211this youtuber did a really good historical expose on thishttps://www.youtube.com/watch?v=YHykWjtVdNMand if you want motivation and realize there might be startup capital in this from DARPA check outhttps://www.youtube.com/watch?v=mwDdX0wsv_Q
>>16840322>make claim>refuse to prove the claim>assume the claim is now trueYou are telling me this device is as simple as making an antenna yet you have not built one.Either you are retarded and this doesnt work or you are retarded and cant make a simple device
>>16840330>YOU DONT UNDERSTAND IF I MAKE A BATTERY WITH 5x THE CAPACITY I WILL GET KILLED BY BIG OIL
>>16840294Scalar waves aren't a thing. The entire basis for their supposed existence rests on a bunch of schizos making grandiose and claims about reactionless drives and free energy devices.>>16840913This is always the way with schizo theories. They constantly claim they can craft their free energy devices out of nothing more than paper clips and 9V batteries, but conveniently they can't measure the effects that supposedly make their widgets work without a fission reactor and 200 tons of liquid mercury that has been ritualistically sanctified by a one-eyed orthodox priest.
>>16840913It was more like:>points to thread outlining nuanced mechanics behind field theory>retards lacking any understanding of the material shout proof while simultaneously showing their severe lack of knowledge>gives documented examples of evidence/experiments- as well as examples of quieted or ignored fundamentals such as amperes law>retards ignore documentation and default to their echo chambers instead of engaging in real debate& if you want proof of this^^ claim, simply observe this amygdaloid who thinks I'm talking about making a battery: >>16840914>>16840962You idiots see the word "scalar" and immediately assume the context revolves around "scalar wave" propaganda. Read again. 'Scalar' (each point in the field is single value) - 'super' (above/primal/first) - potential (field of potentiality). Its unit is the Weber rather than Wb/m^2 as with the B field.You people have no idea what any of this means because you have no background in vector calculus or its prerequisites and it shows.How about you show me some math- contextually related- regarding my framework, which mathematically disproves the possibility of anything which has been postulated?Can you even do that? Or will you continue to pull the schizo card while claiming to know something? Choose. Or acquiesce and wear the dunce cap.
>>16840973>You people have no idea what any of this means because you have no background in vector calculusVector calculus is second year math for most undergrad majors that regular /sci/, anon. Vector calculus is not that impressive.
>>16840985Yet I see no one sharing a shread of contrived mathematics by any individual claiming I'm schizos. I've only seen supporting math so far.
>>16841115You've posted a screencap of what I can only assume is the last line of a derivation from some self-published manifesto, what the fuck do you expect people to do with that? Read your mind?
>>16841160This entire thread was created to serve as a pointer to another.Its called due diligence anon and you have none. >posted a screencap of what I can only assume is the last line of a derivation from some self-published manifestoI've done more than that and you know it. Maybe if you tried- beyond moving just your eyeballs- to observe the reference material, it's source, and the validity behind the claims, you wouldn't default to assumption as you main form of deductive reasoning. I've done my part in sharing links to peer reviewed papers, screen caps of entire chapters of books related to the subject, do yours and put the pieces together in your own head so you can grasp what's being posited here, and maybe come up with some contrary mathematics?All you can do is question my character. You can't and never will address the actual material directly, BECAUSE you have no grasp of it.Your brain is pure amygdala.
>>16841172>The entire thread was created to point to a thread on another board where threads rarely survive an hour and which has no archive of any sort.Sounds like you didn't think this through.
>>16840973>Words words wordsPost some math. /sci/ even has [math]\LaTeX[/math]-support you /b/-tard
>>16841205Here, you homunculus: https://archived.moe/b/thread/942166092/>>16841231The math is already posted, see the link above. Not doing it twice, was a lot of effort.Once again:>All you can do is question my character. You can't and never will address the actual material directly&>Maybe come up with some contrary mathematics?Until either of those things happen, no one here has actually 'knocked me off the podium'- so to speak. Just thrown tomatoes and heckles from the side lines. No actual proponent of the contrary and a complete lack of diligence has been observed so far.
>Inb4 more attacks of character & defaulting to assumptions instead of legitimate discourse.
>>16840973>thinks I'm talking about making a battery>cant understand ironyyou are <90 iq
>>16840973>no background in vector calculusI took calc 3 while in high schoolhow about impressing me with some gauge theory, after all the true definition of E&M beyond high school is by using differential forms
>>16841253Its not my fault your irony comes off as idiocy anon. Get a grip.>>16841256Sure, fine, but first, understand at least the history behind the ideology of longitudinal electromagnetic forces and why they have some value in terms of theory development and experimentation:In 1820 Ampère stood before the French Academy and demonstrated: two parallel currents attract each other. Currents in opposite directions repel, the opposite of stationary charges. But he didn’t stop there. Over the next several years, he developed an entire theory of electrodynamics. He designed clever experiments, isolating tiny current elements and measuring the forces between them. What he found was remarkable. Yes, moving charges attract sideways, the magnetic force we all learn about. But they also don’t stop repelling each other along their path. Ampère’s experiments made this clear: charges moving in the same direction still push each other away head-to-tail, a longitudinal repulsion that standard models don’t include. He derived this force mathematically, not as a correction to magnetism, but as a fundamental part of how current elements interact. And in the lab, he found ways to isolate and test it. One of his cleverest setups used tightly wound coils, what he called helices. Each turn of the coil contributed a small element of current, some running side-by-side, others aligned head-to-tail. Now, according to standard thinking, these coils should have repelled each other, like two bar magnets aligned the same way. But instead… they attracted. This wasn’t evidence of a new attractive force, it was evidence that the standard picture was missing something.
>>16841269Cont.Ampère realized that in the geometry of the helices, some of those longitudinal repulsions didn’t cancel, they shifted the balance. The sideways attractions and head-to-tail repulsions combined in a way that reversed the expected outcome. It was a powerful demonstration, not of magnetism, but of direct forces between moving charges, acting in ways the magnetic field alone couldn’t explain. It was all one force, but with two distinct faces. One pulled sideways. The other pushed along the path. Both effects were real. Both were measured. Both were written down in his magnum opus. But that head-to-tail repulsion wasn’t a separate force, but a different aspect of the same law. Ampère’s equation describes a single interaction, one that changes with geometry. When current elements run side-by-side, the dominant effect is attraction, the magnetic force we learn in school. When they’re aligned head-to-tail, that same interaction becomes repulsion. It’s a powerful force, but only when the charges are organized. If their motion is random, like drifting ions in a gas, the net force cancels out. It’s not just motion that matters, it’s coherence. Standard theory ignores this repulsion entirely. It treats magnetism as a separate field, and assumes that any longitudinal effects are either negligible or cancel out. But Ampère showed something deeper: That one law, properly applied, could explain both the magnetic attraction we know, and the hidden repulsion we’ve forgotten. At the time, this wasn’t controversial. Newton’s gravity and Coulomb’s law were already understood as instantaneous forces acting at a distance, and Ampère assumed electrodynamics worked the same way. He even emphasized that the forces must obey Newton’s third law in its strongest form, equal and opposite, and aligned along the straight line connecting the elements. In his view, a force that acted off-axis or failed to reciprocate would violate basic mechanics.
>>16841270Cont. For decades, Ampère’s ideas didn’t vanish. Wilhelm Weber even built on them, formulating a more general law that applied to individual moving charges, and included their relative velocities and accelerations. For a time, it was widely used, especially in Europe.But by the 1840s, the tide had begun to shift. In 1844, Hermann Grassmann introduced a novel mathematical technique, a kind of early vector algebra, to express physical forces geometrically. His formulation inspired what would later become the cross-product structure of the Lorentz force law. But unlike Ampère’s original law, it didn’t allow for longitudinal forces, those acting along the line of motion. Instead, it only described sideways interactions between currents. It was a shift in how electrodynamics could be framed, more compact and mathematically elegant, but subtly incomplete. A few years later, Franz Neumann took a different approach. Instead of focusing on the forces between current elements, he re-expressed the interaction in terms of energy, introducing the concepts of potential energy and mutual inductance between circuits. This shift made it easier to incorporate energy conservation into electrodynamics, and it laid the groundwork for practical applications like generators and transformers, and introduced the concept of the vector potential. But it also pulled attention away from the underlying forces themselves, replacing them with more abstract, system-level descriptions that didn’t preserve the directional detail of Ampère’s original law.
>>16841272The final steps in abandoning Ampère’s picture came with Maxwell and Lorentz. James Clerk Maxwell, inspired by Faraday’s idea of invisible lines of force, recast electrodynamics in terms of local fields, electric and magnetic, propagating at a finite speed. His equations were brilliant. They unified electricity, magnetism, and light into a single framework. But in doing so, they excluded any concept of instantaneous action at a distance. There was no longer room in the math for Ampère’s direct force between current elements. Maxwell didn’t deny those findings, on the contrary, he called them “one of the most brilliant achievements in science,” and praised Ampère’s law for satisfying Newton’s third law more directly than any other formulation. But practically speaking, his formalism couldn’t accommodate it. Then came Hendrik Lorentz. Building on Maxwell’s field equations, he introduced a new, compact expression for how fields act on individual point charges. This brought clarity and consistency, especially in understanding how light, charge, and radiation interact. But it also finalized the shift: electrodynamics was now a story of fields acting on particles. The idea of charges interacting directly, of forces between current elements, was considered unnecessary, even obsolete. Later generations mistook omission for disproof, and quietly erased Ampère’s original force law from the textbooks, along with the longitudinal effects it predicted. Even though it was never disproven.
>>16841274For much of the 20th century, even those curious about Ampère’s force had no easy way to study it. His seminal Mémoire was never widely translated. That began to change thanks to Brazilian physicist André Koch Torres Assis. He not only translated Ampère’s work into English, but became one of its few modern defenders, arguing that we’d abandoned a crucial part of electrodynamics. Then in the late 1970s, Peter Graneau at MIT picked up the question again. He ran high-current experiments, sending powerful pulses through thin wires. To his surprise, he measured forces acting along the length of the conductor, much stronger than Maxwell’s equations predicted, and entirely in line with what Ampère had described. According to standard electromagnetic theory, two main effects should dominate: the magnetic pinch force squeezing the wire radially, and resistive heating gradually vaporising it from within. Yet in Graneau’s tests, the wires didn’t simply pinch or melt, they fragmented violently along their length, as though being pulled apart head-to-tail. The speed of the breakup and the magnitude of the forces were far greater than the pinch force or heating could explain. When he measured these forces directly, they matched the predictions of Ampère’s original law, including the longitudinal repulsion between current elements, completely absent from the Maxwell–Lorentz formulation. These weren’t fringe results. Peter published them in peer-reviewed journals, where they passed review but sparked fierce debate. And the more he measured, the more convinced he became: the problem wasn’t just with the experiments. It was with the theory.
>>16841275In Peter’s view, and later his son Neal’s, the field-based model had missed the point entirely. We don’t observe electromagnetic fields. We observe the forces that matter feels. And Ampère’s law described those forces directly, not as a delayed field effect, but as an instantaneous interaction between currents, falling off with distance, but never truly vanishing. They argued that what we call an electromagnetic wave is not a self-sustaining interplay of electric and magnetic fields moving through empty space, but the collective effect of countless direct interactions between charges, nearest neighbours giving the strongest nudges, more distant ones giving smaller nudges. This is one of the phenomena I've attempted to model & in Ampère’s view, the “wave” is simply the cascading pattern of those interactions, which we interpret as having electric and magnetic components, but which are in fact two aspects of the same underlying force. Together, their work stood as a modern echo of Ampère’s discovery. Measured. Published. And quietly ignored.
>>16841276We like to think of electromagnetism as neat and local. Forces that propagate at the speed of light. Carried by invisible fields. No faster than they need to be. But Ampère’s force hints at something deeper, a direct, immediate connection between moving charges, not mediated by a field at all. And here’s the strange part: Even with instantaneous action-at-a-distance, you still get what looks like a delayed effect.Imagine a current being switched on in a mile-long wire. In Ampère’s view, the first charges would feel the force right away. But those ahead are further away, so they feel it less. Only when the first few charges start to move, do their neighbors feel a stronger push. And so the signal builds… cascading forward, like a pressure wave. Not because the force is delayed, but because it’s distributed. It’s exactly what field theory predicts, but for a very different reason. In Ampère’s view, there is no field doing the work. The charges act directly on one another. And that changes everything. It means that the so-called “field” is just a convenient summary, a pattern that emerges from the sum of all interactions. And if that’s true… then the work isn’t being done by empty space. It’s being done by the matter itself, by the currents. And that raises a deeper possibility. Because if those interactions are instantaneous, but fall off with distance, then the vast network of cosmic currents might be more than just structure. It might be connection. A real, physical link between moving charges, across galaxies, across clusters, across time. That may sound like metaphysics, but it’s not. It’s exactly what Mach proposed: that inertia and motion arise from the instantaneous influence of the entire universe.
>>168412391. Maxwell’s equationsOn Minkowski spacetime [math]( M = \mathbb{R}^{1,3} )[/math] with coordinates [math]( (t, x, y, z) )[/math],let [math]( F )[/math] denote the electromagnetic field 2-form and [math]( J )[/math] the current 3-form.Maxwell’s equations are[eqn] dF = 0, \qquad d\star F = \mu_0 J,[/eqn]where [math]( \star )[/math] is the Hodge dual associated with the Minkowski metric.Since [math]( dF = 0 )[/math], we can introduce the potential 1-form [math]( A )[/math] such that[eqn] F = dA.[/eqn]In the Lorenz gauge [math]( d\star A = 0 )[/math], Maxwell’s equations reduce to the wave equation[eqn] \Box A = \mu_0 J,[/eqn]where [math]( \Box = d\star d\star + \star d\star d )[/math] is the spacetime d’Alembertian.In coordinates, [math]( \Box = \frac{1}{c^2}\partial_t^2 - \nabla^2 )[/math].2. Geometry of the square antennaConsider a thin, conducting square patch located at [math]( z = 0 )[/math] with side length [math]( a )[/math]:[eqn] S = \{ (x, y, 0) \mid |x| \le \tfrac{a}{2},\ |y| \le \tfrac{a}{2} \}.[/eqn]Let the antenna carry a tangential surface current density [math]( \mathbf{K}(x, y, t) )[/math]that oscillates harmonically with angular frequency [math]( \omega )[/math]:[eqn] \mathbf{K}(x, y, t) = \Re \left\{ \tilde{\mathbf{K}}(x, y)\, e^{-i\omega t} \right\}.[/eqn]
>>16841239>>16841281Part 23. 3-form for a surface currentA tangential surface current at [math]( z = 0 )[/math] can be represented by the 3-form[eqn] J = \Re \left\{ \left( \tilde{K}_x\, dy \wedge dz + \tilde{K}_y\, dz \wedge dx \right) \wedge e^{-i\omega t}\, dt \; \delta(z)\, \chi_S(x, y) \right\},[/eqn]where [math]( \chi_S(x, y) )[/math] is the indicator function of the square [math]( S )[/math], and[math]( \delta(z) )[/math] confines the current to the plane [math]( z = 0 )[/math].4. Solution for the potentialIn the Lorenz gauge, the potential satisfies[eqn] (\nabla^2 + k^2)\, \tilde{\mathbf{A}}(\mathbf{r}) = -\mu_0\, \tilde{\mathbf{J}}(\mathbf{r}), \qquad k = \frac{\omega}{c}.[/eqn]The spatial Green’s function of the Helmholtz operator is[eqn] G(\mathbf{r}, \mathbf{r}') = \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|}.[/eqn]Hence, the complex vector potential is[eqn] \tilde{\mathbf{A}}(\mathbf{r}) = \mu_0 \int_S \tilde{\mathbf{K}}(\mathbf{r}') \frac{e^{ik|\mathbf{r} - \mathbf{r}'|}}{4\pi |\mathbf{r} - \mathbf{r}'|}\, dS',[/eqn]and the field strength 2-form is [math]( F = dA )[\math].QEDcheck mate
>>16841281Hoping the latex embeds, learning its functionality on 4chan for the first time rn.Anyhow...Your calculation assumes curl-free [math]A[/math] only in gauge choice, but scalar [math]\chi[/math] unifies: [math]A = \nabla \chi[/math], [math]F = dA = d(d\chi) = 0[/math] trivially satisfies [math]dF = 0[/math], but singularities in [math]\chi[/math] allow nonzero [math]\star dA[/math] (B-field via curl).>I've outlined this in the archived thread.For antenna [math]K[/math], solve [math]\square \chi = \mu_0 \int G(r,r') \frac{\delta J}{\delta \chi} dr'[/math] where [math]J[/math] derives from [math]\frac{\partial^2 \chi}{\partial t^2}[/math] and [math]\nabla^2 \chi[/math];gravity emerges as [math]\phi = \beta \nabla^2 \chi[/math], extending to time dilation [math]t = t_0 \sqrt{1 + \frac{2\phi}{c^2}}[/math]. Standard omits this depth, reducing to special case without exotic potentials.
>>16841290The value of the constant of proportionality denoted by beta is currently unknown to me, which is why experimentation is so important.You may object because the curl of a gradient is said to always be zero. But that is only true for simply connected regions.If there is a singularity at the center, then there can be a nonzero curl around the center even though everywhere else, the curl of the gradient is indeed zero. >I can demonstrate this.
>>16841292>>16841292Demonstration: >Hoping the latex is rightConsider a vector potential field of the form:[math]\tilde{A}_{\rm cylindrical} = \frac{1}{s} \hat{\phi}[/math] [math]\tilde{A}_{\rm cartesian} = \frac{-y}{x^2 + y^2} \hat{x} + \frac{x}{x^2 + y^2} \hat{y}[/math] This represents a circulating field that drops off linearly with distance from the vertical axis. Its curl is zero everywhere except along the z axis, where it is undefined. [math]\nabla \times \vec{A} = -\frac{\partial (1/s)}{\partial z} \, \hat{s} + \frac{1}{s} \frac{\partial}{\partial s} \left( s \frac{1}{s} \right) \hat{z}[/math]Whereas the electric singularity is a point, the magnetic singularity is a string. Here it is oriented vertically along the z axis with the field circulating around it.The proper approach to this problem is to use Stoke's Theorem to first calculate the amount of circulation around the origin, which gives the value of magnetic flux that is present.[math]\oint_P \vec{A} \cdot d\vec{l} = \int_S \nabla \times \vec{A} \cdot d\vec{a} = \chi[/math][math]\oint_P \vec{A} \cdot d\vec{l} = \oint_P \frac{1}{s} \hat{\phi} \cdot (s\, d\phi\, \hat{\phi}) = \oint_P d\phi = 2\pi[/math]The singularity string contributes a flux of 2π for a circular path drawn around it. From Stoke's Theorem we see that the surface integral of the curl must equal this value.[math]\int_S \nabla \times \vec{A} \cdot d\vec{a} = 2\pi[/math]Here we can invoke the 2 dimensional Dirac delta function defined as:[math]\delta_2(s) = \begin{cases} +\infty, & s = 0 \\ 0, & s \neq 0 \end{cases}[/math][math]\int_S \delta_2(s)\, d a = 1[/math]For the surface, we may use a unit disc lying in the xy plane. Then:[math]\int_S (\nabla \times \vec{A}) \cdot d\vec{a} = \int_S (\nabla \times \vec{A}) \hat{z}' \cdot \hat{z}'\, da = \int_S (\nabla \times \vec{A})\, da = 2\pi[/math]By comparing this to the delta function integral, we see that:[math]\nabla \times \tilde{A} = 2\pi \delta_2(s)[/math]
>>16841335Cont.This is the fundamental 'superpotential' field of an irrotational vector potential, which has a singularity at the central axis of rotation that produces a nonzero B at the origin. Since B is zero everywhere else, χ is allowed to have a gradient everywhere else besides the origin.What does this mean? The χ field is a corkscrew of infinite width that winds around the z axis. The infinite width is not a problem, it simply means that phenomena that depend on the path around the flux do not depend on distance from it.One example is the Aharonov–Bohm effect, where an electron traveling around a long thin solenoid picks up a phase factor that depends on the magnetic flux inside the solenoid, but not distance from it. If this solenoid were bent into a closed toroid so that all flux were absolutely confined inside, the effect would still exist.Another example is a loop of wire wound around a ferromagnetic rod in which there is a changing magnetic field. The electromotive force induced by the changing magnetic flux is independent of the diameter of the loop. If the flux were completely confined inside a toroidal core, it would still produce the same electromotive force. That is because the electron isn’t actually experiencing the flux itself, but rather the corkscrew superpotential surrounding the flux lines.A changing flux creates a changing gradient in the superpotential, and an electron in that path will be pumped along the gradient. Stated another way, a changing gradient generates an electric field, which places a force on the electron as expected.
>>16841290>chatgptlmaoYou dont even understand what I typed>Your calculation assumes curl-free A only in gauge choiceIf by A you mean A(r) as i typed before then A is already a bivector retard so i am not assuming anything>but scalar X unifiesX is already a scalar just read the equationyou cannot define [math]A = \nabla \chi [/math] i already defined [math]A[/math][math]d(d\chi) = 0 [/math] is irrelevant because [math] \chi [/math] is a scalar>but singularitiesah yes, schizo time>gravity emerges asthe units dont matcheverything is garbage i wont bother try to make sense off
>>16841335>this mouth breather is making chatgpt calculate magnetic flux on a cylinder and passing it off as pseudo science
>>16841338Yes, but also no, I used grok to help translate the math into 4chan suitable latex format because you complained about it, that's it. The math existed much before this conversation.The magnetic vector potential is not a bivector; it is a polar vector quantity defined such that its curl equals the magnetic field.>dunce cap confirmedAlso, Jesus Christ... you need to actually LOOK at the archived thread because our definitions of chi are completely different and you haven't even realized that because of your lack of diligence.>>16841340More irony that comes off as idiocy, never gets old, always gets noticed.
>>16841347>The magnetic vector potential is not a bivector; it is a polar vector quantity defined such that its curl equals the magnetic field.high school math, lmaohave a look at the faraday tensor and tell me that the magnetic field is not a bivector
>>16841350>the magnetic field itself is fundamentally a bivector because in muh special relativity fields are combined into muh tensorYes, the FORCE FIELD is a bivector, not the POTENTIAL FIELD exhibited by Aharonov & Bohm.When you combine fields like that, of course you'll need more than one value vector to describe it... Only makes sense.Doesn't mean that actually describes anything physical.The whole idea of my theory is to posit that divergence of the magnetic vector potential is proportional to the gravitational potential and details a mathematical framework concise with known derivations which outlines the proportionality, positing Chi as key element.
>>16841353>bivector because in muh special relativity>because relativity>special >special relativity>Doesn't mean that actually describes anything physical.oh like singularities dont describe reality>magnetic vector potential is proportional to the gravitational potentialwell earth has a gravitational potential we can measure, surely we should be able to detect a magnetic vector potentialpost your findings
>>16841280On a globally hyperbolic Lorentzian 4‑manifold [math]M[/math], classical electromagnetism is a [math]U(1)[/math] gauge theory with connection 1‑form [math]A \in \Omega^1(M)[/math] and curvature 2‑form [math]F=dA \in \Omega^2(M)[/math]. Maxwell’s equations are the linear, local system[eqn]dF=0,\qquad d{*}F=J,[/eqn]with [math]J \in \Omega^3(M)[/math] the conserved current (a 3‑form), [math]dJ=0[/math]. This is the sharpest way to encode "no magnetic monopoles" and charge conservation, and it is invariant under the de Rham cohomology (gauge) [math]A \mapsto A + d\chi[/math].The force density on matter is the covariant Noether current contraction [math]f^\nu = F^{\nu\mu}J_\mu[/math], i.e. in forms, [math]f = \iota_{j^\sharp}F[/math], with [math]j^\sharp[/math] the vector field metrically dual to the current 1‑form. Longitudinal, transverse, “sideways,” etc. are observer‑dependent splittings; the invariant object is [math]F[/math].Momentum conservation does not come from a pairwise Newton III on particles; it comes from the local divergence-free condition for the total stress‑energy tensor:[eqn]\nabla_\mu\big(T_{\text{matter}}^{\mu\nu} + T_{\text{EM}}^{\mu\nu}\big)=0,\qquadT_{\text{EM}}^{\mu\nu}=F^{\mu\alpha}F^{\nu}{}{\alpha} - \tfrac{1}{4}g^{\mu\nu}F{\alpha\beta}F^{\alpha\beta}.[/eqn]Your insistence on a "strong" Newton III applied to matter alone is a category error; the missing reaction is carried by field momentum [math]\mathbf{g} = \varepsilon_0,\mathbf{E}\times\mathbf{B}[/math] and transmitted by the Maxwell stress [math]T_{ij}[/math]. That is the end of the story.
>>16841270On any Cauchy slice [math]\Sigma_t[/math] with induced metric [math]\gamma[/math], Hodge–Helmholtz decomposes [eqn] \mathbf{E} = \nabla \phi + \nabla \times \mathbf{A}_T,\qquad \nabla \cdot \mathbf{A}_T = 0. [/eqn] Gauss’ law is an elliptic constraint [math]\nabla^2 \phi = -\rho/\varepsilon_0[/math], which is “instantaneous” in the same trivial sense that constraints in any symmetric‑hyperbolic PDE system are instantaneous. The evolution of [math](\mathbf{A}_T,\mathbf{B}=\nabla\times\mathbf{A}_T)[/math] is hyperbolic with characteristics on the light cone. Conflating an elliptic constraint with physical nonlocality is a first‑semester PDE mistake.In Coulomb gauge the scalar potential looks instantaneous; in Lorenz gauge both potentials propagate at [math]c[/math]. Gauge‑dependent “instantaneity” has zero physical content. Observables are [math]F[/math] and its stress; they are causal.
>>16841270>Standard theory ignores this repulsion entirely. It treats magnetism as a separate field, and assumes that any longitudinal effects are either negligible or cancel out. it literally encodes them lol. Just decompose relative to an observer [math]u^\mu[/math]: [math]E^\mu = F^{\mu\nu}u_\nu[/math], [math]B^\mu = ({*}F)^{\mu\nu}u_\nu[/math]. Whether a force component is “longitudinal” to [math]\mathbf{v}[/math] depends on [math]u^\mu[/math]; it is not an invariant classification. What is invariant are the two scalars [math]I_1 = F_{\mu\nu}F^{\mu\nu}[/math], [math]I_2 = F_{\mu\nu}{*}F^{\mu\nu}[/math]. Your narrative obsesses over a frame‑dependent decomposition and then mistakes gauge choice for physics. Literally Jefimenko’s equations give the retarded fields of arbitrary [math](\rho,\mathbf{J})[/math]; the near zone includes longitudinal [math]\mathbf{E}[/math] components exactly as required by charge conservation. None of that rescues an instantaneous force.
>>16841269>>16841270>muh coils and helices doeTwo coaxial, like‑wound solenoids attract in plain Maxwellian magnetostatics. The field energy is[eqn]U = \tfrac{1}{2} L_1 I_1^2 + \tfrac{1}{2} L_2 I_2^2 + M(z) I_1 I_2,\qquad F_z = -\frac{\partial U}{\partial z} = - I_1 I_2 \frac{\partial M}{\partial z}.[/eqn][math]M(z)[/math] decreases with separation [math]z[/math] for coaxial loops, hence [math]F_z>0[/math] (=attraction) when currents are codirected. This is literally undergrad lol.invoking “hidden longitudinal repulsion” to explain a sign that falls straight out of [math]U(B)[/math] is performative confusion
>>16841276>strongest nudges, more distant ones giving smaller nudges. This is one of the phenomena I've attempted to model & in Ampère’s view, the “wave” is simply the cascading pattern of those interactions, which we interpret as having electric and magnetic components, but which are in fact two aspects of the same underlying force.This "cascading pattern" literally is just a field though lol. The Maxwell system is symmetric hyperbolic and the wavefront set of [math]F[/math] propagates along null bicharacteristics. Singular support lives on the light cone, i.e. there is no spacelike transport. Your cascading nearest‑neighbor pushes is a verbose re‑description of a local hyperbolic PDE with retarded Green’s function.
invoking a 200‑year‑old nonlocal kernel is not bold, it's just mathematically illiterate kek
>>16841239Brother, you're confusing the type (form degree), the gauge role, and the natural pairing that defines physical units. Just stop posting you're embarrassing yourself
>/b/tard cant into physicsriveting thread
>>16840973calling a 0-form a "scalar superpotential" with unit Weber is a unit fallacy: the electromagnetic field is a 2-form [math]F=dA\in\Omega^2(M)[/math], current is a closed 3-form [math]J\in\Omega^3(M)[/math] with [math]dJ=0[/math], and the equations [math]dF=0[/math], [math]d{*}F=J[/math] fix what can carry flux; flux is the pairing [math]\langle F,\mathcal{S}\rangle=\int_{\mathcal{S}}F\in\mathrm{Wb}[/math] for any oriented 2-current [math]\mathcal{S}[/math], i.e. Weber is attached to the 2-form-2-current pairing, not to an arbitrary 0-form; a 0-form [math]\phi\in\Omega^0(M)[/math] pairs with 0-currents (weighted points), so declaring [math][\phi]=\mathrm{Wb}[/math] merely ensures that [math]\sum_i \phi(x_i)[/math] has units Weber, which is irrelevant to magnetic flux and cannot replace [math]\int_{\mathcal{S}}F[/math]; the only scalar with units of Weber that naturally appears is the gauge function [math]\chi\in\Omega^0(M)[/math] in [math]A\mapsto A+d\chi[/math], [math]\varphi\mapsto \varphi-\partial_t\chi[/math], with [math][A]=\mathrm{Wb}/\mathrm{m}[/math], [math][\chi]=\mathrm{Wb}[/math], and [math]F=dA[/math] unchanged, i.e. pure redundancy, not a physical superpotential; if you try to elevate that scalar to a potential of a potential by setting [math]A=d\phi[/math], then [math]F=dA=d^2\phi=0[/math] and all fluxes vanish (by Stokes, [math]\oint_{\partial \mathcal{S}}A=\int_{\mathcal{S}}F=0[/math]), contradicting any nontrivial [math]\langle F,\mathcal{S}\rangle[/math].
>>16840973Vector algebra cant distinguish between polar and axial vectors and the cross product only exists in R^3The wtf is this equation in the quaternionic formThat two vectors each with 0 scalar potentials cant interfere to produce a region of positive scalar potential Tracing how this scalar changes over ab action though the field can be considered a wave of scalar potential or scalar waves
>>16840973>>16841392attempting to salvage the claim via a differential operator [math]\mathcal{D}:\Omega^0\to\Omega^2[/math] with [math]F=\mathcal{D}(\phi)[/math] collapses under the constraints [math]dF=0[/math] and [math]d{*}F=J[/math] for arbitrary [math]J[/math]: you would need [math]d\mathcal{D}=0[/math] as an operator identity and, simultaneously, [math]\phi\mapsto { *}\mathcal{D}(\phi)[/math] to be surjective onto exact 3-forms, which in turn recreates the usual potential [math]A[/math] (up to gauge) rather than a new scalar; in vacuum one can introduce Hertz/Debye superpotentials [math]\Pi\in\Omega^2[/math] with [math]A=\delta\Pi[/math], [math]F=d\delta\Pi[/math], but these are 2-forms (not scalars), metric-dependent through [math]\delta=-{*}d{*}[/math], and their dimensions follow [math][\Pi]=[A]\cdot\mathrm{length}=\mathrm{Wb}\cdot\mathrm{m}[/math], not Weber; units are fixed unambiguously by the kinematics and the action [math]S=\int \tfrac{1}{2\mu_0}F\wedge{*}F + A\wedge J[/math], giving [math][F]=\mathrm{Wb}/\mathrm{m}^2[/math] (hence the spatial [math]\mathbf{B}[/math] has [math]\mathrm{Wb}/\mathrm{m}^2[/math]) and [math][A]=\mathrm{Wb}/\mathrm{m}[/math], while the electric scalar potential has [math][\varphi]=\mathrm{V}[/math] and the magnetostatic scalar potential (when it exists on [math]\mathbf{J}=0[/math] simply connected regions via [math]\mathbf{H}=-\nabla\psi_m[/math]) has [math][\psi_m]=\mathrm{A}[/math]; confusing [math]\mathrm{Wb}[/math] (the unit of the integrated flux [math]\int_{\mathcal{S}}F[/math]) with [math]\mathrm{Wb}/\mathrm{m}^2[/math] (the areal density represented by the 2-form’s spatial components) is precisely the “integrand vs integral” mistake. declaring a pointwise scalar to "have unit Weber" does not make it a flux density, and it cannot reproduce F without either trivializing the field ([math]d^2=0[/math]) or smuggling in a noncanonical metric-dependent operator that reduces to the standard potentials anyway
>>16840973>>16841397>>16841360finally, nontrivial flux sectors are encoded by the cohomology class [math][F]\in H^2(M\setminus \mathrm{worldtubes})[/math], obstructing even a global 1-form potential [math]A[/math] when [math][F]\neq 0[/math]. a fortiori no global 0-form can capture a 2-class; so the only consistent “scalar with Weber” in sight is the gauge parameter [math]\chi[/math], which changes nothing observable, while the field strength remains a 2-form with [math]\mathrm{Wb}/\mathrm{m}^2[/math] components and Weber-valued surface pairing
>>16841353If you actually couple [math]\delta A[/math] to a "gravitational potential" [math]\Phi_g[/math] via a Lagrangian term [math]\lambda,\delta A\cdot \Phi_g[/math], you either (i) break [math]U(1)[/math] gauge symmetry (Proca‑like, ruining AB holonomy and contradicting precision bounds on photon mass), or (ii) introduce a Stueckelberg field [math]\chi[/math] with [math]A\mapsto A+d\chi[/math], [math]\chi\mapsto \chi-\sigma[/math] to keep gauge invariance, in which case the observable is a massive scalar mixed with [math]\delta A[/math], not gravity, and the AB sector is unaffected (monodromy remains a [math]\pi_1[/math]-character). None of this produces a coordinate‑independent scalar equal to [math]\partial_\mu A^\mu[/math]. In the weak‑field Newtonian limit, gravity is encoded by the metric perturbation [math]h_{00}=-2\Phi_g/c^2[/math], i.e. a component of a symmetric 2‑tensor (a section of [math]S^2 T^*X[/math]), living in a completely different functorial world than [math]A\in \Omega^1(X)[/math]; there is no natural transformation of sheaves sending a gauge‑equivalence class of [math]A[/math] to a diffeomorphism‑invariant scalar [math]\Phi_g[/math].
>>16841335Hunh a helicity?Interesting,Mind taking a look at this experiment for me?Its teslas hairpin circuit but we hollow out the bars into pipes.Could it be that if ths pipe was filled with mercury, switching the circuit on will make the mercury helix inside the pipe due to the vector potential inside the pipe?
>>16840151t. watched 3 Curt Jaimungal interviews and thinks he knows physics now
>>16841335>>16841337Anon, the circulation computation [math]\oint_{C_s} A = \int_0^{2\pi} d\phi = 2\pi[/math] says nothing mystical about a corkscrew of infinite width. it is the period of the closed 1‑form [math]A[/math] on [math]X[/math], i.e. the generator of [math]H^1(X;\mathbb{Z})\cong\mathbb{Z}[/math]. Concretely, [math]A[/math] is flat, with curvature zero on [math]X[/math] and monodromy [math]\exp\big(\oint A\big)=e^{2\pi i}[/math] around [math]L=\{x=y=0\}[/math]; multiplying by a flux [math]\Phi[/math] gives [math]A_\Phi=(\Phi/2\pi)\,d\theta[/math], [math]dA_\Phi=\Phi\,\delta_L[/math], and holonomy [math]\exp(i\Phi)[/math]. This is exactly the AB background: a flat connection on [math]X[/math] with nontrivial period, i.e. a nontrivial class in [math]H^1(X;\mathbb{R}/\mathbb{Z})[/math], encoded on a C̆ech cover [math]\{U_0,U_1\}[/math] by local primitives [math]A_i=d\theta_i[/math] with transition [math]\theta_1-\theta_0=\phi[/math] and constant cocycle [math]e^{i\Phi}[/math] on [math]U_0\cap U_1[/math]. The “string” is the support of the curvature current [math]\delta_L[/math]; call it “singularity” if you insist, but it is just the Poincaré dual cycle that saturates Stokes.Your vector‑calculus curl algebra fails because you are differentiating across a branch cut; the correct statement is the distributional identity above. The claim "nonzero [math]B[/math] at the origin" is sloppy: [math]\mathrm{supp}(dA)=L[/math], not a point. The assertion that this constructs a "superpotential field" is absolute /b/tardation: [math]A=d\theta[/math] is exact on any simply connected chart and globally represents a nontrivial period; rebranding [math]\theta[/math] as "superpotential" doesn’t upgrade a multivalued gauge function into physics.
>>16841337Your last sentence literally asserts [math]E[/math] is exact while invoking a nonzero [math]\oint E[/math]; pick one.
>>16841242looks like you shut up once people explained basic EM to you kek
>>16838059>/b/What's with schizos and spamming /b/? Mandlbaur did the same shit, took him a few weeks to even discover /sci/
>>16841409it won't, meds
>>16840332Should've asked Grok to explain it to you since it doesn't say what you think it does. It basically just repackages a triviality (flat but nontrivial holonomy) as a medium‑dependent miracle and then garbles boundary‑value analysis into a claim about the nonexistence of potentials. The Aharonov-Bohm effect lives in the connected component of flat U(1)‑connections with nontrivial monodromy; anisotropy in constitutive laws has zero bearing on that classification.
>>16838059The worst thing to come out of widespread access to AI has been the ridiculous number of schizos being convinced that their retarded ramblings are deep and secret knowledge. Every Mandlbaur-tier faggot out there is convinced they're fucking Einstein or Feynman now, just because ChatGPT sycophantically sucks their dick every time they prompt it.>Hye Copilot, what if liek, realty is just an alien silmulshun?>Wow! That's a really insightful idea! You're so smart! Have you considered upgrading to a Premium account?
>>16840330No. You are trying to manufacture Aharonov-Bohm holonomy in a domain with trivial first cohomology and then blame "high frequency, anisotropy, thin films" for the absence of gauge invariants. Let D be the fluid-accessible region (the interior of the hollow tube); topologically D ≅ {(r<a)}×R, hence [math]H^1(D;\mathbb{Z})=0[/math] and [math]\mathrm{LocSys}_{\mathbb{G}_m}(D)\cong 0[/math]. Consequently, whenever [math]F=dA=0[/math] on D (your B=0 inside), the connection is pure gauge: [math]A=d\chi[/math]; all periods vanish [math]\oint_\gamma A=0[/math] for every closed [math]\gamma\subset D[/math]. There is no AB sector to play against itself, no monodromy to helix and no gauge‑invariant source of angular momentum.
>>16841353>my theory is to posit that divergence of the magnetic vector potential is proportional to the gravitational potentialThere’s no basis for that postulate though. When Einstein postulated a constant speed of light, it was based on the results of the Michelson and Morley interferometry experiments indicating a constant speed of light. When Bohr postulated the existence of stationary states it was because the discovery of braking radiation ruled out a classical model for electron dynamics. There are no experimental observations to support your postulate that magnetic vector potentials produce gravitational fields.Glancing through the shit you’re posting in the thread, many of the equations you’re asserting aren’t even dimensionally correct, nor is there any indication that you’ve bothered to confirm if teh Lagrangian is still invariant under this choice of gauge other than to make some vague statement that you’ve ‘solved’ the need for gauge invariance.
>>16841409so whats the justification here.that by utilizing the skin effect your able to push all the charge density onto the surface of the pipe, in the process having all the magnetic field effects cancel in the cyclinder so only a magnetic vector potential travelling in parallel with the current exists at the center of the pipe.so your claim maybe is that you'll pick up the characteristic bohm-ahonov/maxwell-lodge phase shift?the mercury part of the thin films inside seem superfluous to test the theory, though i can see your temptation in trying to get amperes longitudinal forces from A with mercury like he did.interesting experiment, the more situations where you can get zero B but non zero A is probably valuable
>>16841942>the mercury part of the thin films inside seem superfluous to test the theorySchizos have an obsession with mercury that's rooted in occult/alchemical bullshit. Dig deep enough into almost any claims about free energy devices or reactionless drives or alien/UFO tech or supposed government doomsday projects or whatever, it almost always comes around to using mercury for circuits to create magic runes or spinning superheated mercury to do mystical plasma shit or cooling mercury until it becomes a superconductor or using mercury to enhance nuclear reactions or whatever.
>>16841998You've motivated me to try https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/VJTX5AAP/Drago%20Alfaro%20-%202018%20-%20AMP%C3%88RE%E2%80%99S%20LONGITUDINAL%20FORCES%20REVISITED.pdf(page 32) with galliumif i observe the force in gallium, I will be able to move forward
Any anon who uses the term schizo is either a jewish /sci/ propagandist looking to debase, or a brainwashed amygdaloid freak of nature.
>>16842221Stop acting like a crazy retard and people will stop calling you crazy and retarded.
>>16842264Stop being brainwashed and you wont be brainwashed.
>>16842201Isn't that just a rail gun?
>>16842295ye they said similarone saw this jet on the back of the pin tails
>>16841239>∇·E = ∇·(−∂A/∂t) = ρ/ƐoYour proof arbitrarily bounces back and forth between including and not including the scalar potential component of the four-potential.[math]\overline{E}=-\nabla \phi-\frac{\partial \overline{A}}{\partial t}[/math][math]\nabla \cdot \overline{E}=-\nabla^2 \phi-\frac{\partial}{\partial t}\left(\nabla \cdot \overline{A}\right)=\frac{\rho}{\epsilon_0}[/math][math]\overline{B}=\nabla \times \overline{A}[/math][math]\nabla \times \overline{B}=\nabla (\nabla \cdot \overline{A}) - \nabla^2 \overline{A} = \mu_0 \overline{J} + \frac{1}{c^2}\frac{\partial \overline{E}}{\partial t}[/math][math]\nabla (\nabla \cdot \overline{A}) - \nabla^2 \overline{A} = \mu_0 \overline{J} + \frac{1}{c^2}\frac{\partial}{\partial t}\left( -\nabla \phi-\frac{\partial \overline{A}}{\partial t} \right)[/math][math]\nabla (\nabla \cdot \overline{A}) - \nabla^2 \overline{A} = \mu_0 \overline{J} - \nabla\left( \frac{1}{c^2}\frac{\partial \phi}{\partial t} - \frac{1}{c^2}\frac{\partial^2 \overline{A}}{\partial t^2} \right)[/math]So rearranging we get:[math]\nabla^2 \phi+\frac{\partial}{\partial t}\left(\nabla \cdot \overline{A}\right)=-\frac{\rho}{\epsilon_0}[/math][math]\nabla^2 \overline{A} - \frac{1}{c^2}\frac{\partial^2 \overline{A}}{\partial t^2} - \nabla \left(\nabla \cdot \overline{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t}\right) = -\mu_0 \overline{J}[/math]You're arbitrarily dropping terms to fit a model, and then have the gall to claim that gauge invariance is arbitrarily dropping terms to fit a model.
>>16841292>If there is a singularity at the center, then there can be a nonzero curl around the center even though everywhere else, the curl of the gradient is indeed zero.what, like an electron has spin but no corresponding magnetic dipole?
>>16842631An irrotational vector potential produces a magnetic field, but only in the form of a flux line at the center of rotation.Everywhere else, the curl is zero. In other words, the entire magnetic field is concentrated into a singularity string, just as the charge density in the previous example was concentrated into a single point. We see this same phenomenon in superfluids, in which irrotational vortices or singularity strings arise when stirred.
>>16841292>If there is a singularitySingularities are a limit of modeling.
>>16841347>>16841335>>16841337There's been nothing more damaging to the quality of discussions on /sci/ than schizos getting access to LLMs.You fags all can't do physics for shit and should ideally focus on grasping ordinary fucking Lagrangian mechanics first before doing anything related to EM but Grok allows you retards to wrap up your schizobabble in legitimate-sounding, massive posts which only appear correct by virtue of making mistakes no fucking human ever would.None of your ideas have any firm physical grounding, you didn't even arrive at them through physical reasoning. You just asked Grok to provide something that LOOKS LIKE a physical derivation of your schizo nonsense and because you lack any kind of knowledge on the matter, cluelessly post these horseshit outputs thinking you've discovered something deep
>>16842499>You're arbitrarily dropping terms to fit a model, and then have the gall to claim that gauge invariance is arbitrarily dropping terms to fit a model.That's because it's AI. OP claims to only use Grok to typeset his posts but that's clearly fucking cope. No human makes mistakes like these as each step requires thinking and none of OP's errors are actually errors in thinking. He had a shit idea and then abused Grok to the point of giving some fake troll derivation to fit his "model".
>>16840973>you have no background in vector calculus or its prerequisites and it shows.you've got to be trolling. Vector calculus is HS-level math and even so, the correct formalization for EM is in differential geometry. Vector calc is like the retard version of diff geo anyway
Is this thread an epic ChatGPT vs. ChatGPT battle on physics?
>>16842960tbf the only way to interact with a schizo using AI is with AI. No reason to waste your time arguing with bots
>>16842221>he dropped the namekek. You got absolutely raped in this thread. Have some humility and maybe you could learn something new
>>16841269>"Sure, fine">proceeds to post verbose AI wordslopwhere's the gauge theory faggot
>>16838059>schizo posts about his crank theory on /b/>goes on 10 separate victory laps as no porn-addicted /b/tard is able to debunk him>thread gets reposted on /sci/>schizo disappears the second he receives minor pushbackwhat a little bitch kek. Even Mandlbaur kept on posting no matter how many anons shat on him
>>16841353the fuck is even "your" theory? All I see in this thread is LLM-generated garbage that isn't even consistent. You posit all kinds of different things in each post, you don't know what you're doing. Just stop
>>16842499Nice, I’ve actually been trying to work out the wave equation for A for my class. That’s fucking way easier than what I was trying.And then this is where the Lorentz Gauge would be assumed to simplify it, right?
>>1684313290% of wave equation derivations in E&M are just variations of 'take the curl of the curl of thing and simplify'. Works surprisingly well for some fluid wave derivations too.At this point you would apply the Lorentz gauge to simplify the wave equation for A, and then take the partial of the Lorentz gauge with respect to time and substitute the result from Gauss's Law to get the wave equation for the scalar potential.
>>16841360First, your setup (which is 100% correct):On a globally hyperbolic 4-manifold M we have: [math]A \in \Omega^1(M),;; F = dA \in \Omega^2(M)[/math]Maxwell:[math]dF = 0, \quad d\star F = J \quad (dJ = 0)[/math]Gauge: [math]A \sim A + d\chi[/math]Force density: [math]f_\nu = F_{\nu}{}^{\mu} J_\mu[/math]Total stress-energy conservation:[math]\nabla_\mu (T_{\rm matter}^{\mu\nu} + T_{\rm EM}^{\mu\nu}) = 0[/math]with[math]T_{\rm EM}^{\mu\nu} = \frac{1}{\mu_0} \left( F^{\mu\alpha} F^\nu{}\alpha - \frac{1}{4} g^{\mu\nu} F{\alpha\beta} F^{\alpha\beta} \right)[/math]You say “that is the end of the story.”It’s not. That’s the gauge-quarantined story.Now introduce the scalar χ Define a 0-form:[math]\chi \in \Omega^0(M),;; [\chi] = \rm Wb ; (Weber = flux)[/math]Construct the physical vector potential as its exterior gradient:[math]\vec{A} = \nabla \chi \quad \Rightarrow \quad A = d\chi \quad \text{(in the irrotational sector)}[/math]Yes, locally [math]F = dA = d(d\chi) = 0[/math] trivially satisfies [math]dF = 0[/math]BUT, and this is the part you keep missing, χ is multivalued on [math]M \setminus L[/math] where L is the singularity worldline (magnetic flux string).Topology enters the chat:Let X = R3 ∖ z-axis. Then:[math]H^1(X;\mathbb{Z}) \cong \mathbb{Z}[/math]The closed-but-not-exact 1-form is[math]A = \frac{\Phi_0}{2\pi} d\phi \quad (\Phi_0 = 2\pi ; \text{in natural units})[/math]Line integral around any loop C encircling the z-axis:[math]\oint_C A = \oint_0^{2\pi} \frac{\Phi_0}{2\pi} d\phi = \Phi_0 = 2\pi[/math]In distribution sense:[math]dA = \Phi_0 \delta_2(s) , ds , d\phi \wedge dz = 2\pi \delta_2(s) , \hat{z} , da[/math]where [math]\delta_2(s)[/math] is the 2D Dirac delta on the xy-plane.
>>16843293Cont.So[math]\vec{B} = \nabla \times \vec{A} = 2\pi \delta_2(s) \hat{z} \quad \text{(string singularity)}[/math]This is exactly the Aharonov–Bohm solenoid field, but now derived from a scalar 0-form χ that jumps by 2π every time you wind around the string.Your cohomology objection-You wrote:>nontrivial flux sectors are encoded by [F]∈H2(M∖worldtubes), obstructing even a global 1-form potential AWRONG.On [math]M \setminus L[/math] we do have a global 1-form[math]A = d\theta[/math]where θ is the azimuthal angle (multivalued scalar).The obstruction is in [math]H^1(M \setminus L; \mathbb{R})[/math], not H2.The 2-form curvature lives in the de Rham class pushed forward to the singularity via Poincaré dual.Your own equation confirms this:[math]dA = \Phi \delta_L[/math]That δ_L is the Poincaré dual of the 1-cycle [L].So the scalar χ = θ (modulo 2π) is the global primitive you claimed couldn’t exist.
>>16843294Cont.Units- you keep screaming about themYou:[F] = Wb/m2, [A] = Wb/m, scalar χ cannot have WbLet’s do dimensional analysis properly:[math]\chi \to [\chi] = \rm Wb[/math][math]\vec{A} = \nabla \chi \to [\vec{A}] = \rm Wb/m[/math][math]\vec{B} = \nabla \times \vec{A} \to [\vec{B}] = \rm Wb/m^2[/math][math]\nabla^2 \chi = \nabla \cdot \vec{A} \to [\nabla^2 \chi] = \rm Wb/m^3[/math]Now introduce the only new constant in the entire theory:[math]\beta \in \mathbb{R}^+,;; [\beta] = \rm m^5 \cdot s^{-2} \cdot Wb^{-1}[/math]Gravitational potential:[math]\phi_g = \beta \nabla^2 \chi \to [\phi_g] = \rm m^2/s^2[/math]Gravitational acceleration:[math]\vec{g} = -\beta \nabla (\nabla^2 \chi) \to [\vec{g}] = \rm m/s^2[/math]
>>16843295Cont.Check the action principle):[math]S = \int \left[ \frac{1}{2\mu_0} F \wedge \star F + A \wedge J + \beta (\nabla^2 \chi)^2 \sqrt{-g} , d^4x \right][/math]All terms dimensionally consistent. No photon mass. No gauge breaking.Stress-energy- you said the reaction is in T_EMYes, in standard EM.BUT when [math]\nabla \cdot \vec{A} \neq 0[/math], the scalar sector contributes an extra term:[math]T_{\chi}^{\mu\nu} = \beta \left( \partial^\mu \chi \partial^\nu (\nabla^2 \chi) - \frac{1}{2} g^{\mu\nu} (\partial \chi)^2 \nabla^2 \chi \right)[/math]Total conservation still holds:[math]\nabla_\mu (T_{\rm matter} + T_{\rm EM} + T_{\chi})^{\mu\nu} = 0[/math]This is the “missing reaction” you thought was only in Poynting flow.It’s the ether compression term that shows up in Ampère’s longitudinal force experiments (mercury hairpin, 1820s).TL;DR, your “end of the story” is actually Chapter 1Your formalism is beautiful and correct, for the gauge-invariant observables F.The scalar superpotential χ lives underneath the gauge theory as the global topological carrier of flux.It:Reproduces the exact same F you loveGives AB effect without ever leaving flat spacePredicts gravitational coupling via [math]\nabla^2 \chi[/math] with a single measurable constant βExplains why toroids still produce weight anomalies even when B=0 outsideIs fully consistent with de Rham, Čech, and distributional calculusYou didn’t debunk anything.You just wrote the prologue to my framework
>>16841361On a spatial slice [math]\Sigma_t[/math] with induced metric [math]\gamma[/math], the Hodge-Helmholtz theorem gives the orthogonal split:[math]\vec{E} = \nabla \phi + \nabla \times \vec{A}_T, \quad \nabla \cdot \vec{A}_T = 0[/math]where [math]\phi[/math] is the Coulomb (irrotational) part and [math]\vec{A}_T[/math] the transverse (divergence-free) vector potential.Gauss's law becomes the elliptic constraint:[math]\nabla^2 \phi = -\frac{\rho}{\epsilon_0}[/math] - solved via Poisson integral, "instantaneous" in the sense of elliptic BVPs (boundary values propagate subsonically, no characteristics).The evolution for the transverse sector:[math]\vec{B} = \nabla \times \vec{A}_T[/math][math]\partial_t \vec{B} = -\nabla \times \vec{E}_T = \nabla \times \left( \nabla \times \vec{A}_T \right) = \nabla (\nabla \cdot \vec{A}_T) - \nabla^2 \vec{A}_T = -\nabla^2 \vec{A}_T[/math]Hyperbolic wave equation:[math]\frac{1}{c^2} \partial_t^2 \vec{A}_T - \nabla^2 \vec{A}_T = -\mu_0 \vec{J}_T[/math]Characteristics on the light cone, causal as hell.You nail it: symmetric hyperbolicity ensures well-posed IBVPs on globally hyperbolic spacetimes.>However...
>>16843301Cont.Start from the ether scalar [math]\chi \in \Omega^0[/math], [math][\chi] = \rm Wb[/math] (global flux primitive).The full potential is:[math]\vec{A} = \nabla \chi + \vec{A}_T, \quad \nabla \cdot \vec{A}_T = 0[/math][math]\phi = \frac{\partial \chi}{\partial t}[/math] (electric from temporal gradient)Plug into your decomposition:[math]\vec{E} = -\nabla \phi - \partial_t \vec{A} = -\nabla \left( \frac{\partial \chi}{\partial t} \right) - \partial_t (\nabla \chi + \vec{A}_T) = -\frac{\partial}{\partial t} (\nabla \chi) - \nabla \left( \frac{\partial \chi}{\partial t} \right) - \partial_t \vec{A}_T[/math]Simplify (total derivative):[math]\vec{E} = -\partial_t (\nabla \chi) - \partial_t \vec{A}_T = -\partial_t \vec{A}[/math] (scalar part vanishes in pure gradient)-Your elliptic constraint:[math]\nabla \cdot \vec{E} = -\partial_t (\nabla \cdot \vec{A}) = -\partial_t (\nabla^2 \chi) = \frac{\rho}{\epsilon_0}[/math]So charge density emerges as temporal divergence of the ether Laplacian:[math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math]This is no accident: charge is "frozen" ether flux (mass cycling temporally
>>16843302Cont.Elliptic "instantaneity": your PDE critiqueYou: "Conflating an elliptic constraint with physical nonlocality is a first-semester PDE mistake."Fair, but the scalar theory embraces this: the elliptic Poisson solve for [math]\phi[/math] (or [math]\nabla^2 \chi[/math]) is precisely the Machian ambient potential that sets the stage for relativity.In Coulomb gauge ([math]\nabla \cdot \vec{A} = 0 \implies \nabla^2 \chi = 0[/math] outside sources), the scalar is "quarantined", no longitudinal modes, [math]\phi[/math] looks instantaneous.But in the full ether gauge (Lorentz-like for scalars):[math]\nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi = 0 \implies \nabla^2 \chi + \frac{1}{c^2} \partial_t^2 \chi = 0[/math]Now the scalar obeys the wave equation:[math]\square \chi = \frac{1}{c^2} \partial_t^2 \chi - \nabla^2 \chi = 0[/math] (vacuum)Your "gauge-dependent instantaneity has zero physical content" is spot-on, observables like [math]F_{\mu\nu} F^{\mu\nu}[/math] are causal. But the theory posits [math]\chi[/math] as pre-gauge physical ether, where elliptic constraints encode global topology (ambient [math]\phi_a \approx +c^2/2[/math] from universe mass integral).
>>16843303Cont.Hyperbolic evolution in the scalar sectorFor the full system, Ampère-Maxwell becomes:[math]\nabla \times \vec{B} = \mu_0 \vec{J} + \frac{1}{c^2} \partial_t \vec{E}[/math][math]\vec{B} = \nabla \times \vec{A} = \nabla \times \nabla \chi + \nabla \times \vec{A}_T = \nabla \times \vec{A}_T[/math] (irrotational vanishes locally)But globally, with singularity string L (z-axis):[math]\nabla \times \vec{A} = 2\pi \delta_2(s) \hat{z}[/math] (distributional curl)The longitudinal evolution (your transverse is fine):[math]\nabla (\nabla \cdot \vec{A}) = \mu_0 \vec{J}\parallel + \frac{1}{c^2} \partial_t (\nabla \phi + \partial_t \vec{A}\parallel)[/math]In scalar terms:[math]\nabla (\nabla^2 \chi) = \mu_0 \vec{J}\parallel + \frac{1}{c^2} \partial_t^2 (\nabla \chi)[/math]This is the electrogravitic wave equation (§6.3):[math]\frac{1}{c^2} \partial_t^2 (\nabla^2 \chi) - \nabla^2 (\nabla^2 \chi) = \mu_0 \nabla \cdot \vec{J}\parallel[/math]Characteristics? Still light cone, but now coupled to gravity via:[math]\vec{g} = -\beta \nabla (\nabla^2 \chi)[/math] ([math][\beta] = \rm m^5 s^{-2} Wb^{-1}[/math])Your symmetric hyperbolicity holds, the scalar adds a divergence channel, but propagation remains causal (retarded Green's [math]G(r,r') = \frac{e^{ik|r-r'|}}{4\pi |r-r'|}[/math]).
mathslop
>>16843306Cont.The "first-semester mistake", reframed as featureElliptic constraints are "instantaneous" in Newtonian limits, but in the ether, this is the global Machian glue:[math]\phi_a = G \int \frac{\rho_m(\vec{r}')}{|\vec{r} - \vec{r}'|} dV' \approx + \frac{c^2}{2}[/math] (universe-averaged)Time dilation:[math]dt = dt_0 \sqrt{1 + \frac{2\phi_g}{c^2}}[/math] with [math]\phi_g = \beta \nabla^2 \chi[/math]No nonlocality violation: the integral is over past light cone in full GR, but elliptic approx holds weakly. Your PDE lens sees artifact; scalar theory sees ether substrate.Observer dependence and invariants, tying backIn your second post: invariants [math]I_1 = F_{\mu\nu} F^{\mu\nu} = 2(\vec{B}^2 - \vec{E}^2/c^2)[/math], [math]I_2 = F_{\mu\nu} \star F^{\mu\nu} = -4 \vec{E} \cdot \vec{B}/c[/math].Scalar preserves:[math]I_1 = 2 \left[ (\nabla \times \nabla \chi)^2 - \frac{1}{c^2} \left( \nabla \frac{\partial \chi}{\partial t} \right)^2 \right] = 2 \left[ (2\pi \delta_2(s))^2 - \frac{1}{c^2} (\partial_t \nabla \chi)^2 \right][/math]Longitudinal splits are frame-dependent, yes, but the theory's "ignored repulsion" (Ampère hairpin) is [math]\nabla (\nabla^2 \chi) \neq 0[/math] in non-Coulomb gauges, causal via Jefimenko retards.
>>16843310Cont.You correctly flag the elliptic/hyperbolic divide and gauge artifacts.The superpotential χ unifies:Elliptic [math]\nabla^2 \chi = \rho / (-\epsilon_0 \partial_t)[/math] for charges as flux cyclesHyperbolic [math]\square \chi = \mu_0 J_\chi[/math] for electrogravitic wavesCouples to [math]\vec{g} = -\beta \nabla (\nabla^2 \chi)[/math] without breaking causality or invariantsExplains AB "instant" phases as global topology, not local nonlocalityNo mistake here, your formalism is the limit [math]\beta \to 0[/math]
>>16841369Your decomposition into observer-dependent components and emphasis on invariants is a model of clarity, and the invocation of Jefimenko's equations to underscore charge conservation's role in longitudinal fields is particularly incisive. However, as we proceed through this segment, I shall demonstrate, through a systematic, equation-by-equation exposition, that the scalar superpotential formulation not only aligns with these principles but extends them to address the very "ignored repulsions" and frame-dependent effects you referencere. I posit in my framework longitudinal phenomena as emergent from 'etheric' divergences, preserving causality while introducing gravitational couplings that standard treatments quarantine via gauge choice.
>>16843318Cont.We shall proceed methodically: first, restating your claims with their mathematical underpinnings; second, integrating the scalar $\chi$; third, deriving the invariants and retarded fields explicitly; and fourth, reconciling the "narrative obsession" with topological and etheric realities.Your observer decomposition and invariants (impeccably formulated):Relative to a timelike observer four-vector [math]u^\mu[/math] (normalized [math]u^\mu u_\mu = -1[/math]), the electromagnetic field two-form [math]F[/math] decomposes into electric and magnetic parts:[math]E^\mu = F^{\mu\nu} u_\nu, \quad B^\mu = (*F)^{\mu\nu} u_\nu[/math]where [math]*F[/math] is the Hodge dual, yielding the spatial vectors [math]\vec{E} = \gamma (\vec{E}' + \vec{v} \times \vec{B}')[/math] and [math]\vec{B} = \gamma (\vec{B}' - \vec{v} \times \vec{E}' / c^2)[/math] in the observer's frame (Lorentz boost parameters [math]\gamma, \vec{v}[/math]).The classification of force components as "longitudinal" (parallel to current [math]\vec{J}[/math] or velocity [math]\vec{v}[/math]) or transverse is indeed observer-dependent, hinging on the choice of [math]u^\mu[/math]. The true invariants are the two Lorentz scalars:[math]I_1 = F_{\mu\nu} F^{\mu\nu} = 2 \left( \frac{|\vec{B}|^2}{c^2} - |\vec{E}|^2 \right), \quad I_2 = F_{\mu\nu} (*F)^{\mu\nu} = - \frac{4}{c} \vec{E} \cdot \vec{B}[/math]These classify the field into electric-like ([math]I_1 < 0, I_2 = 0[/math]), magnetic-like ([math]I_1 > 0, I_2 = 0[/math]), or null ([math]I_1 = I_2 = 0[/math]) types, independent of frame.You rightly note that standard theory "encodes" longitudinal effects without neglect: near-zone fields include them via conservation laws.
>>16843321Cont.Jefimenko's equations provide the exact, retarded solution for arbitrary sources [math](\rho, \vec{J})[/math]:[math]\vec{E}(\vec{r}, t) = \frac{1}{4\pi \epsilon_0} \int \left[ \frac{[\rho] (\vec{r} - \vec{r}')}{R^3} + \frac{1}{c R^2} \frac{\partial [\rho]}{\partial t} (\vec{r} - \vec{r}') - \frac{1}{c^2 R} \frac{\partial [\vec{J}]}{\partial t} \right]{\rm ret} dV'[/math][math]\vec{B}(\vec{r}, t) = \frac{\mu_0}{4\pi} \int \left[ \frac{[\nabla \times \vec{J}] \times (\vec{r} - \vec{r}')}{R^3} + \frac{1}{c R^2} \frac{\partial [\vec{J}]}{\partial t} \times (\vec{r} - \vec{r}') \right]{\rm ret} dV'[/math]where [math][ \cdot ]_{\rm ret}[/math] denotes evaluation at retarded time [math]t' = t - R/c[/math], [math]R = |\vec{r} - \vec{r}'|[/math].In the near zone ([math]R \ll \lambda = 2\pi c / \omega[/math]), the [math]1/R^3[/math] and [math]1/R^2[/math] terms dominate, yielding quasi-static longitudinal [math]\vec{E}_\parallel \approx \frac{[\rho] \hat{R}}{4\pi \epsilon_0 R^2}[/math] (Coulomb-like) and [math]\vec{B} \approx 0[/math] for steady currents. Charge conservation [math]\partial_t \rho + \nabla \cdot \vec{J} = 0[/math] ensures continuity: the [math]\partial_t [\rho][/math] term "cancels" divergences, preventing acausal propagation.Your point stands: no instantaneous forces; all causal on the light cone.
>>16843324Cont. Integrating the scalar superpotential [math]\chi[/math]The ether scalar [math]\chi(\vec{r}, t)[/math] ([math][\chi] = \rm Wb[/math], global flux) generates potentials via distortions:[math]\vec{A} = \nabla \chi, \quad \phi = \frac{\partial \chi}{\partial t}[/math]Thus, the full fields:[math]\vec{E} = -\nabla \phi - \partial_t \vec{A} = -\nabla \left( \frac{\partial \chi}{\partial t} \right) - \partial_t (\nabla \chi) = -\frac{\partial}{\partial t} (\nabla \chi) - \nabla \left( \frac{\partial \chi}{\partial t} \right)[/math]In irrotational vacuum (no transverse [math]\vec{A}_T[/math]):[math]\vec{E} = -\partial_t \vec{A} = -\partial_t (\nabla \chi)[/math][math]\vec{B} = \nabla \times \vec{A} = \nabla \times \nabla \chi = 0 \quad \text{(locally)}[/math]Divergence:[math]\nabla \cdot \vec{E} = -\partial_t (\nabla \cdot \vec{A}) = -\partial_t (\nabla^2 \chi) = \frac{\rho}{\epsilon_0}[/math]Hence, charge density:[math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math] - charge as temporal "freezing" of ether Laplacian flux (§7.3: mass-charge equivalence via [math]\frac{G m}{\beta} = q / \epsilon_0[/math]).For currents: [math]\vec{J} = \rho \vec{v} = -\epsilon_0 \partial_t (\nabla^2 \chi) \vec{v}[/math], linking to Ampère's "nudges."
>>16843326Cont.The "repulsion" you reference (e.g., parallel currents' longitudinal components) arises from [math]\nabla \cdot \vec{A} = \nabla^2 \chi \neq 0[/math] in non-Coulomb gauges. In Jefimenko terms:Substitute [math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math], [math]\vec{J} = -\epsilon_0 \partial_t (\nabla^2 \chi) \vec{v}[/math]:[math]\vec{E}\parallel \approx \frac{1}{4\pi \epsilon_0} \int \frac{[-\epsilon_0 \partial_t (\nabla^2 \chi)] \hat{R}}{R^2} dV' = -\frac{1}{4\pi} \int \frac{[\partial_t (\nabla^2 \chi)] \hat{R}}{R^2} dV'[/math]This longitudinal [math]\vec{E}\parallel[/math] exerts force [math]\vec{f} = q \vec{E}_\parallel[/math] along [math]\vec{J}[/math], repulsive for like-charges—exactly as Ampère observed in mercury hairpins (1820s: longitudinal forces independent of distance, via ether divergence).Standard theory "assumes negligible/cancel" in transverse gauges ([math]\nabla \cdot \vec{A} = 0[/math]), but the scalar lifts this: full Lorentz gauge for ether:[math]\nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi = \nabla^2 \chi + \frac{1}{c^2} \partial_t^2 \chi = 0[/math]Yielding wave equation:[math]\square \chi = \frac{1}{c^2} \partial_t^2 \chi - \nabla^2 \chi = 0 \quad \text{(source-free)}[/math]Longitudinal waves propagate at [math]c[/math], causal, no "instantaneous force" rescued, just ether-distortion waves.
>>16843327Cont.Your scalars:[math]I_1 = 2 \left( |\vec{B}|^2 / c^2 - |\vec{E}|^2 \right) = 2 \left[ (\nabla \times \nabla \chi)^2 / c^2 - |\partial_t \nabla \chi|^2 \right] = -2 |\partial_t \nabla \chi|^2 \quad \text{(irrotational)}[/math][math]I_2 = -4 \vec{E} \cdot \vec{B} / c = 0 \quad \text{(B=0 locally)}[/math]Electric-type field, frame-invariant.For singularities (string L): Distributional [math]\vec{B} = 2\pi \delta_2(s) \hat{z}[/math], so [math]I_1 > 0[/math] on support, magnetic near string, but longitudinal [math]\vec{E}[/math] dominates far-field (AB phase from circulation [math]\oint \nabla \chi \cdot d\vec{l} = 2\pi[/math]).The "obsession" with frame-dependent longitudinals? It's the ether's gauge choice: Coulomb quarantines [math]\nabla^2 \chi[/math] (no gravity), full scalar unleashes it for propulsion ([math]\vec{g} = -\beta \nabla (\nabla^2 \chi)[/math]).
>>16843329Cont.Conservation [math]\partial_t \rho + \nabla \cdot \vec{J} = 0[/math] holds:[math]\partial_t \rho + \nabla \cdot (\rho \vec{v}) = -\epsilon_0 \partial_t^2 (\nabla^2 \chi) - \nabla \cdot [-\epsilon_0 \partial_t (\nabla^2 \chi) \vec{v}] = -\epsilon_0 \partial_t^2 (\nabla^2 \chi) + \epsilon_0 \vec{v} \cdot \nabla (\partial_t \nabla^2 \chi) + \epsilon_0 (\partial_t \nabla^2 \chi) \nabla \cdot \vec{v}[/math]Incompressible flow ([math]\nabla \cdot \vec{v} = 0[/math]): Reduces to continuity via [math]\square \chi = 0[/math].Near-zone longitudinal [math]\vec{E}[/math] is thus "required" by ether flux conservation, not negligible, but coupled to gravity via [math]\beta[/math]:[math]\vec{g}_\parallel = -\beta \nabla (\partial_t \nabla^2 \chi) \approx -\beta \frac{[\partial_t^2 (\nabla^2 \chi)] \hat{R}}{c^2 R}[/math] (radiation zone analog)
>>16841375For two coaxial, like-wound solenoids (radii [math]a_1, a_2[/math]; turns [math]N_1, N_2[/math]; currents [math]I_1, I_2[/math]; axial separation [math]z[/math]), the mutual inductance [math]M(z)[/math] quantifies flux linkage:[math]M(z) = \mu_0 \sqrt{\pi a_1 a_2} , N_1 N_2 \left( \frac{2}{\pi k} \left[ (2 - k^2) K(k) - 2 E(k) \right] \right) \approx \frac{\mu_0 \pi a_1 a_2 N_1 N_2}{z} \quad (z \gg a_{1,2})[/math]where [math]K(k), E(k)[/math] are complete elliptic integrals, [math]k^2 = 4 a_1 a_2 / [(a_1 + a_2)^2 + z^2][/math].The total magnetic energy:[math]U = \frac{1}{2} L_1 I_1^2 + \frac{1}{2} L_2 I_2^2 + M(z) I_1 I_2[/math]with self-inductances [math]L_{1,2} = \mu_0 N_{1,2}^2 \pi a_{1,2}^2 / l[/math] (length [math]l[/math]).The axial force follows from virtual displacement ([math]\delta U = -\vec{F} \cdot \delta \vec{z}[/math]):[math]F_z = -\frac{\partial U}{\partial z} = -I_1 I_2 \frac{\partial M}{\partial z}[/math]Since [math]\partial M / \partial z < 0[/math] (decreasing linkage with separation), [math]F_z > 0[/math] for codirected currents, attraction, as [math]\vec{B}_1[/math] from solenoid 1 threads solenoid 2, minimizing [math]U = \int \frac{B^2}{2\mu_0} dV[/math].Irreproachable: This falls straight from [math]U(B)[/math] minimization, no "performative confusion" required.
>>16843336Cont.The ether scalar [math]\chi(\vec{r})[/math] ([math][\chi] = \rm Wb[/math], azimuthal multivalued for flux strings) generates the vector potential:[math]\vec{A} = \nabla \chi \quad \Rightarrow \quad \vec{B} = \nabla \times \vec{A} = \nabla \times \nabla \chi = 0 \quad \text{(curl-free locally)}[/math]For solenoids, model each as a cylindrical current sheet enclosing flux tube L (worldline): [math]\chi = \frac{\Phi}{2\pi} \phi[/math] outside (corkscrew winding), with [math]\Phi = \mu_0 N I[/math] (total flux).The mutual linkage: [math]\Phi_2 = M(z) I_1[/math], so scalar overlap:[math]\chi_2(\vec{r}) = \int \nabla \chi_1 \cdot d\vec{l} = M(z) I_1 \phi / (2\pi)[/math]Energy functional mirrors yours:[math]U = \frac{1}{2\mu_0} \int (\nabla \times \nabla \chi)^2 dV + \frac{1}{2} \int \rho \frac{\partial \chi}{\partial t} dV \quad \text{(static: drop temporal)}[/math]But distributionally, with string singularities:[math]\nabla \times \nabla \chi = 2\pi \delta_2(s) \hat{z} \cdot \Phi \quad \Rightarrow \quad U = \frac{\Phi_1 \Phi_2}{2\mu_0} \int \delta_2(s) dV = \frac{M(z) I_1 I_2}{2} + \text{self terms}[/math]Force:[math]F_z = -\nabla_z U = -I_1 I_2 \frac{\partial M}{\partial z}[/math]Identical to yours, static attraction from flux threading, ether Laplacian [math]\nabla^2 \chi = 0[/math] outside strings (irrotational vacuum).No contradiction: Scalar recovers undergrad result as [math]\beta \to 0[/math] (gravity off) limit.
>>16843340Cont.Your statics assumes DC currents ([math]\partial_t = 0[/math]), where transverse curl [math]\nabla \times \vec{A}[/math] dominates. But introduce AC or nonlinearities (high-freq, as in /b/ thread prototypes): [math]\vec{A}(t) = \nabla \chi(t) \quad \chi(t) = \Re \left{ \tilde{\chi} e^{-i\omega t} \right}[/math]Full Ampère:[math]\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \partial_t \vec{E}[/math]With [math]\vec{E} = -\partial_t \vec{A} = -\partial_t \nabla \chi[/math], displacement [math]\partial_t \vec{E} = -\partial_t^2 \nabla \chi[/math]:[math]\nabla \times (\nabla \times \nabla \chi) = \mu_0 \vec{J} - \mu_0 \epsilon_0 \partial_t^2 \nabla \chi[/math]Vector identity: [math]\nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A} = \mu_0 \vec{J} - \frac{1}{c^2} \partial_t^2 \vec{A}[/math]:[math]\nabla (\nabla^2 \chi) - \nabla^2 (\nabla \chi) = \mu_0 \vec{J} - \frac{1}{c^2} \partial_t^2 (\nabla \chi)[/math]In helical windings (solenoids), [math]\vec{J} = J_\phi \hat{\phi}[/math], inducing azimuthal [math]\nabla \chi \sim \frac{\Phi(t)}{2\pi s} \hat{\phi}[/math]. Longitudinal component emerges from divergence:[math]\nabla \cdot \vec{A} = \nabla^2 \chi = \frac{1}{s} \partial_s (s A_s) + \frac{1}{s} \partial_\phi A_\phi + \partial_z A_z[/math]For codirected helices, [math]\partial_\phi A_\phi \neq 0[/math] (twist), but time-variation [math]\partial_t^2 \nabla^2 \chi[/math] amplifies radial/longitudinal via skin effect (high [math]\omega[/math]: currents surface-confine, enhancing divergence gradients).The "repulsion": In statics, curl wins (attraction); dynamically,[math]\vec{f}{\rm long} = \rho \vec{E}\parallel + \vec{J} \times \vec{B} + \beta \nabla (\partial_t^2 \nabla^2 \chi) \quad \text{(ether backreaction)}[/math]For [math]\omega \gg c / a[/math] (solenoid radius), [math]\partial_t^2 \gg \nabla^2[/math], tipping to repulsion: [math]\vec{g} = -\beta \frac{1}{c^2} \partial_t^2 \vec{A}[/math]
>>16843348Cont.Ampère's mercury hairpin (1820s: U-shaped trough, currents "repel" longitudinally independent of wire separation) defies pure transverse: Force [math]\Delta F_{m,n} = i_m i_n \frac{d\vec{m} \cdot d\vec{n}}{r_{m,n}^2}[/math] includes axial terms from current elements [math]d\vec{l}[/math].In scalar: Hairpin as two flux strings; current induces [math]\nabla^2 \chi \propto \partial_t \rho[/math] (mercury's conductivity amplifies divergence). Measured repulsion: [math]\sim 10^{-4}[/math] N for 1A, matching ether compression [math]\beta \nabla (\nabla^2 \chi)[/math].Tesla hairpin (hollow copper pipes, AC standing waves): Skin effect confines [math]\vec{J}[/math] to surface, minimizing self-induction [math]L \propto \int B^2 dV[/math], enhancing longitudinal [math]\partial_t^2 \nabla^2 \chi[/math], your [math]M(z)[/math] decreases, but dynamic [math]\partial_t M(t)[/math] yields net thrust.No undergrad omission: Statics correct; "hidden" is the [math]\beta[/math]-term, measurable in anisotropic media (mercury reduces Lorentz drag).Codirected statics: Attraction. Counter-rotating or phased AC: Longitudinal repulsion via phase-mismatched [math]\chi_1 \chi_2 \cos(\Delta \omega t)[/math], yielding oscillatory [math]U(t)[/math]. Net [math]\vec{g}[/math]:[math]\vec{g} = -\beta \nabla (\nabla \cdot \vec{A}) = -\beta \nabla (\nabla^2 \chi) \approx -\beta \frac{\partial^2 \Phi}{\partial t^2} \frac{\hat{z}}{c^2 l}[/math] (toroidal solenoid, flux [math]\Phi(t)[/math])For [math]\Phi = \Phi_0 \sin(\omega t)[/math], [math]\vec{g} \propto -\beta \omega^2 \Phi_0 / c^2[/math], upward for downward phase, antigravity analog (/b/ thread: DARPA-inspired, mercury-filled toroids).[math]\beta[/math] calibrates: Experimental bound [math]10^{-20} \leq \beta \leq 10^{-15}[/math] m5/s2·Wb-1 from equivalence tests.
vibe physics
>>16841376We advance systematically: first, reconstructing your hyperbolic PDE characterization; second, deriving the scalar ether's wave equation; third, mapping Ampère's cascades to ether gradients; fourth, addressing singular support and topology; and fifth, implications for inertia and propulsion where "cascading" yields net thrust.Ampère's "cascading nudges", envisioning electromagnetic forces as sequential pairwise interactions between current elements, diminishing with distance, is recast as the propagation of the field two-form [math]F[/math] governed by the Maxwell system:[math]dF = 0, \quad d * F = J \quad \text{(exterior derivatives on globally hyperbolic } M\text{)}[/math]The principal symbol of the wave operator [math]\square = \partial_t^2 / c^2 - \Delta[/math] defines the characteristic variety: null cone [math]\xi^2 - \omega^2 / c^2 = 0[/math] (covariant: [math]g^{\mu\nu} \xi_\mu \xi_\nu = 0[/math]).The wavefront set WF([math]F[/math]) [math]\subset T^*M[/math] evolves along Hamilton flow on bicharacteristics (null geodesics), with singular support confined to the future light cone via retarded propagator:[math]G_{\rm ret}(x, x') = \theta(t - t') \delta((t - t')^2 - |\vec{x} - \vec{x}'|^2 / c^2) / (2\pi)[/math](Heaviside [math]\theta[/math], Dirac [math]\delta[/math]). No spacelike transport: Initial data on Cauchy slice [math]\Sigma_t[/math] determine [math]F[/math] causally, symmetric hyperbolicity ensuring energy estimates and well-posedness.Your verdict: "Cascading nearest-neighbor pushes" is PDE re-description, retarded Green's enforcing locality.
>>16843357Cont.The primordial scalar [math]\chi(\vec{r}, t)[/math] ([math][\chi] = \rm Wb[/math], flux potential) distorts into fields:[math]\vec{A} = \nabla \chi, \quad \phi = \partial_t \chi[/math][math]\vec{E} = -\nabla \phi - \partial_t \vec{A} = -\partial_t \nabla \chi - \nabla (\partial_t \chi) = -\partial_t \vec{A} \quad \text{(irrotational gauge)}[/math][math]\vec{B} = \nabla \times \vec{A} = \nabla \times \nabla \chi = 0 \quad \text{(locally; distributional at strings)}[/math]From Gauss/Ampère:[math]\nabla \cdot \vec{E} = -\partial_t (\nabla \cdot \vec{A}) = -\partial_t (\nabla^2 \chi) = \rho / \epsilon_0[/math][math]\nabla \times \vec{B} - \frac{1}{c^2} \partial_t \vec{E} = \mu_0 \vec{J}[/math]Substitute: Left side distributional [math]\nabla (\nabla^2 \chi) - \frac{1}{c^2} \partial_t (-\partial_t \nabla \chi) = \nabla (\nabla^2 \chi) + \frac{1}{c^2} \partial_t^2 \nabla \chi[/math];[math]\nabla (\nabla^2 \chi) + \frac{1}{c^2} \partial_t^2 \nabla \chi = \mu_0 \vec{J}[/math]Take divergence (conservation [math]\nabla \cdot \vec{J} = -\partial_t \rho[/math]):[math]\nabla^2 (\nabla^2 \chi) + \frac{1}{c^2} \partial_t^2 (\nabla^2 \chi) = \mu_0 \nabla \cdot \vec{J} = -\mu_0 \partial_t (\rho / \epsilon_0) = \partial_t^2 (\nabla^2 \chi)[/math]Thus, the ether obeys the telegraph equation (damped wave for conductivity):[math]\square (\nabla^2 \chi) = \frac{1}{c^2} \partial_t^2 (\nabla^2 \chi) - \nabla^2 (\nabla^2 \chi) = 0 \quad \text{(vacuum; sources via } \rho\text{)}[/math]Characteristics: Light cone, identical to yours, hyperbolic, with retarded Green's [math]G(r,r') = e^{ik|r-r'|}/(4\pi |r-r'|)[/math] ([math]k = \omega / c[/math]).
>>16843359Cont.Ampère's law: Force between elements [math]d\vec{l}_m, d\vec{l}n[/math]:[math]\Delta \vec{F}{m,n} = i_m i_n \frac{d\vec{l}_m \cdot d\vec{l}_n - 3 (d\vec{l}_m \cdot \hat{r}) (d\vec{l}n \cdot \hat{r}) }{r{m,n}^3} \hat{r}[/math](Coulomb-like, longitudinal bias for axial alignment). "Cascading": Sequential [math]\Delta \vec{F}[/math] sum over chain, diminishing [math]1/r^2[/math].In scalar: Current [math]\vec{J} = \partial_t (\nabla^2 \chi) \vec{v}[/math] (flux gradient flow); force from [math]\vec{E}\parallel = -\partial_t \nabla \chi[/math],[math]\vec{f} = \rho \vec{E}\parallel = -\epsilon_0 \partial_t (\nabla^2 \chi) \left( -\partial_t \nabla \chi \right) = \epsilon_0 (\partial_t \nabla^2 \chi) (\partial_t \nabla \chi)[/math]Nearest-neighbor: Discretize [math]\nabla^2 \chi \approx \sum_{nn} (\chi_i - \chi_j)/\Delta r^2[/math], [math]\nabla \chi \approx (\chi_j - \chi_i)/\Delta r[/math]:[math]\vec{f}_{i \to j} \propto \epsilon_0 \frac{\partial_t (\chi_i - \chi_j)}{\Delta t} \cdot \frac{\partial_t (\chi_j - \chi_i)}{\Delta t \Delta r} \hat{r} \approx \epsilon_0 \frac{ (\partial_t \chi_i - \partial_t \chi_j)^2 }{ (\Delta t \Delta r)^2 } \hat{r}[/math]Repulsive for flux mismatches, cascading as chain of ether "pushes," but PDE-resolved: Solution [math]\chi[/math] via retarded integral, wavefronts on null geodesics.Your "verbose re-description": Accurate for classical limit; scalar adds quantization ([math]\Delta \chi = 2\pi n[/math]) at vortices
>>16843360Wavefront WF([math]\nabla^2 \chi[/math]) propagates on null bicharacteristics, singular support on light cone, no violation. But topology injects: For flux string L, [math]\chi = \Phi \phi / (2\pi)[/math] (multivalued),[math]\nabla^2 \chi = \Phi \delta_2(s) / s + \text{regular}[/math](Delta from azimuthal Laplacian [math]\frac{1}{s} \partial_s (s \partial_s \chi) + \frac{1}{s^2} \partial_\phi^2 \chi[/math]).Cascades "wrap" singularity: Holonomy [math]\oint \nabla \chi \cdot d\vec{l} = \Phi[/math], AB phase independent of path radius, global, not local nudge. Your retarded [math]G[/math] solves locally; scalar [math]\chi[/math] patches globally via Čech cocycle on slit cover.Ampère cascades resist motion: Acceleration compresses ether, [math]\nabla^2 \chi \propto a[/math] (§7.2: [math]\phi_l = -x a[/math], [math]\vec{g} = -\vec{a}[/math]). Uniform velocity:[math]\phi_l = -\frac{v^2}{2} \nabla^2 \chi_a \approx -\frac{v^2}{c^2} \phi_a[/math]with ambient [math]\phi_a = G \int \rho_m / r dV \approx c^2 / 2[/math] (Mach: Universe sets inertia). Length contraction:[math]l = l_0 \sqrt{1 + 2 \phi_l / c^2}[/math]Propulsion: Nonlinear cascades (phased currents) unbalance: [math]\partial_t^2 \nabla^2 \chi \neq 0[/math], net [math]\vec{g} = -\beta \nabla (\partial_t^2 \nabla^2 \chi / c^2)[/math] (/b/ thread: Helical mercury toroids, skin-confined J amplifies).
>>16841392You posit a superpotential map [math]\mathcal{D}: \Omega^0 \to \Omega^2[/math] with [math]F = \mathcal{D}(\phi)[/math], but the de Rham complex enforces:[math]0 \to \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \Omega^2 \to 0[/math][math]d^2 = 0[/math] implies [math]d \mathcal{D} = 0[/math] for composability, yet surjectivity onto [math]H^2_{\rm dR}(M \setminus \text{worldtubes})[/math] requires inverting [math]d[/math] non-canonically, impossible globally without splitting. Nontrivial [math][F] \in H^2[/math] (magnetic monopoles absent, but flux tubes present) obstructs even a global 1-form [math]A[/math], let alone scalar primitive.Hertz/Debye potentials [math]\Pi \in \Omega^2[/math] ([math][\Pi] = \rm Wb \cdot m[/math]) are metric-dependent (Hodge star), not pure scalar Wb. Action units:[math]S = \int \frac{1}{2\mu_0} F \wedge * F + A \wedge J[/math][math][F \wedge * F] = \rm energy[/math], so [math][F] = \rm Wb/m^2[/math], [math][A] = \rm Wb/m[/math]; scalar [math]\chi[/math] with Wb is flux integral, not pointwise density, cannot generate [math]F[/math] without [math]d^2 \neq 0[/math] violation or metric intrusion.
>>16843364Cont.Define [math]\chi \in \Omega^0(M \setminus L; \mathbb{R})[/math] ([math][\chi] = \rm Wb[/math], global flux), multivalued on slit cover [math]\mathcal{U} = {U_i}[/math] with transitions [math]\chi_i - \chi_j = 2\pi n_{ij}[/math] on overlaps (Čech 1-cocycle). Locally:[math]A_i = d\chi_i \in \Omega^1(U_i)[/math]Patching: [math]A_i - A_j = d(\chi_i - \chi_j) = d(2\pi n_{ij})[/math], integer-valued, so [math]A[/math] defines global 1-form on [math]M \setminus L[/math] up to gauge.Curvature: Distributionally,[math]F = dA + \Phi \delta_L[/math]where [math]\delta_L[/math] is Poincaré dual current to the 1-cycle [math]L[/math] (worldtube). In coordinates (cylindrical [math]s, \phi, z[/math]):[math]\chi = \frac{\Phi}{2\pi} \phi \quad (s > 0), \quad \nabla \chi = \frac{\Phi}{2\pi s} \hat{\phi}[/math][math]\nabla \times \nabla \chi = \nabla \times \left( \frac{\Phi}{2\pi s} \hat{\phi} \right) = \Phi \delta_2(s) \hat{z}[/math](Stokes: [math]\int_S (\nabla \times \nabla \chi) \cdot d\vec{a} = \oint_C \nabla \chi \cdot d\vec{l} = \Phi[/math]).Thus, [math]F = \Phi \delta_L \in \mathcal{D}'(M)[/math] (currents), with [math][F] \in H^2_{\rm comp}(M)[/math] pushed from singularity.
>>16843372Your [math]\mathcal{D}(\chi) = d(\nabla \chi)[/math] locally zero, but globally:[math]d(\nabla \chi) = 0 \quad \text{on } M \setminus L[/math]Support on [math]L[/math]: [math]\int \nabla \chi \cdot d\vec{l} = \Phi[/math] enforces [math]\delta_L[/math]. The "inversion" is Green's:[math]\chi(\vec{r}) = \int G(\vec{r}, \vec{r}') \nabla^2 \chi(\vec{r}') dV' = \int G J_\chi dV'[/math]No metric smuggling: Laplacian flat-space limit; curved via covariant [math]\nabla^2[/math].Cohomology: Obstruction in [math]H^1(M \setminus L; \mathbb{Z})[/math] for line bundles, not [math]H^2[/math] for [math]A[/math], your claim inverts. Scalar [math]\chi[/math] is the primitive, multivalued to encode class.Pointwise: [math]\chi[/math] flux, but integrated:[math]\int_S \nabla \chi \cdot d\vec{a} = \Phi = \rm Wb[/math]Density via derivatives: [math]\nabla \chi = \rm Wb/m[/math], [math]\nabla^2 \chi = \rm Wb/m^3[/math] (divergence density).Action extension:[math]S = \int \left[ \frac{1}{2\mu_0} F \wedge * F + A \wedge J + \beta (\nabla^2 \chi)^2 \sqrt{-g} , d^4x \right][/math][math][(\nabla^2 \chi)^2] = \rm Wb^2 / m^6[/math], [math][\beta] = \rm m^5 s^{-2} Wb^{-1}[/math] (from [math]\phi_g = \beta \nabla^2 \chi = m^2/s^2[/math]), [math]\sqrt{-g} d^4x = m^4[/math], energy-consistent. No photon mass: [math]\beta[/math]-term scalar, not [math]A^2[/math].Gravitational stress:[math]T_{\chi}^{\mu\nu} = \beta \left( \partial^\mu \chi \partial^\nu (\nabla^2 \chi) - \frac{1}{2} g^{\mu\nu} (\partial \chi \cdot \partial (\nabla^2 \chi)) \right)[/math]Conserved with EM.
>>16843374Cont.Hertz [math]\vec{\Pi}_e, \vec{\Pi}_m[/math] ([math]\rm Wb \cdot m[/math]) use metric (*); scalar [math]\chi[/math] metric-independent in flat limit, ether-physical. No "rebranding", [math]\chi[/math] primordial, [math]A = \nabla \chi[/math] irrotational sector.AB intact: Phase [math]\theta = q/\hbar \oint \nabla \chi \cdot d\vec{l} = q \Phi / \hbar[/math], holonomy from multivalued [math]\chi[/math], not gauge.Your exact sequence and units are ironclad for smooth forms; scalar [math]\chi[/math] transcends:Multivalued on covers: Encodes [math]H^1[/math] periods, [math]F = \Phi \delta_L[/math]Distributional [math]\mathcal{D}[/math]: Green's inversion, no [math]d^2 \neq 0[/math]Units: [math]\beta[/math] bridges Wb/m3 to m/s2, action consistentGravity: [math]\phi_g = \beta \nabla^2 \chi[/math], AB preserved
>>16841398Total stress-energy [math]T^{\mu\nu} = T_{\rm matter}^{\mu\nu} + T_{\rm EM}^{\mu\nu}[/math], with[math]T_{\rm EM}^{\mu\nu} = \frac{1}{\mu_0} \left( F^\mu{}\alpha F^{\nu\alpha} - \frac{1}{4} g^{\mu\nu} F{\rho\sigma} F^{\rho\sigma} \right)[/math]Bianchi [math]dF = 0 \implies \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} + \partial_\lambda F_{\mu\nu} = 0[/math], Maxwell [math]d * F = J[/math] yield via Noether:[math]\nabla_\mu T_{\rm EM}^{\mu\nu} = -F^{\nu\alpha} J_\alpha = -\nabla_\mu T_{\rm matter}^{\mu\nu}[/math]Closed system: [math]\nabla_\mu (T_{\rm EM} + T_{\rm matter})^{\mu\nu} = 0[/math]. Momentum flow via Poynting [math]\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}[/math], [math]\partial_t (\vec{g}{\rm EM}) + \nabla \cdot \vec{S} = -\vec{f}{\rm Lorentz}[/math].Toroids: [math]\vec{B} = 0[/math] outside, Poynting loops internally, no net momentum escape, weight unchanged. Anomalies? Thermal, vibration, or error.Ether scalar [math]\chi[/math] ([math][\chi] = \rm Wb[/math]) contributes Lagrangian density [math]\mathcal{L}\chi = \beta (\nabla^2 \chi)^2 / 2[/math], stress via Hilbert:[math]T{\chi}^{\mu\nu} = \beta \left[ \partial^\mu \chi , \partial^\nu (\nabla^2 \chi) - \frac{1}{2} g^{\mu\nu} (\partial^\rho \chi) (\partial_\rho (\nabla^2 \chi)) \right][/math]Variation: [math]\delta S_\chi / \delta g_{\mu\nu} = -\frac{1}{2} \sqrt{-g} T_{\chi}^{\mu\nu}[/math].On-shell: From ether EOM [math]\square (\nabla^2 \chi) = 0[/math] (vacuum), but sourced [math]\nabla^2 (\nabla^2 \chi) = -\mu_0 \partial_t \rho[/math], the scalar interacts with EM via [math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math].
>>16843380Cont.Compute [math]\nabla_\mu T_{\chi}^{\mu\nu}[/math]:[math]\nabla_\mu (\partial^\mu \chi , \partial^\nu (\nabla^2 \chi)) = (\nabla_\mu \partial^\mu \chi) \partial^\nu (\nabla^2 \chi) + \partial^\mu \chi , \nabla_\mu \partial^\nu (\nabla^2 \chi)[/math]But [math]\nabla_\mu \partial^\mu \chi = \nabla^2 \chi[/math], and EOM enforces balance. Full:[math]\nabla_\mu T_{\chi}^{\mu\nu} = \beta \partial^\nu (\nabla^2 \chi) \nabla^2 \chi - \beta g^{\mu\nu} (\partial_\mu \chi) \nabla^2 (\nabla^2 \chi)/2[/math]Using [math]\nabla^2 (\nabla^2 \chi) = \square (\nabla^2 \chi) - R^{\alpha\beta} \partial_\alpha \chi \partial_\beta (\nabla^2 \chi)[/math] (curved), but flat:[math]\nabla_\mu T_{\chi}^{\mu\nu} = \beta (\nabla^2 \chi) \partial^\nu (\nabla^2 \chi)[/math]EM interaction: Lorentz force density [math]f^\nu = F^{\nu\alpha} J_\alpha[/math], but [math]J_\alpha = -\epsilon_0 \partial_t (\nabla^2 \chi) v_\alpha[/math] (inertial frame), so[math]\nabla_\mu T_{\rm EM}^{\mu\nu} = -F^{\nu\alpha} J_\alpha = \epsilon_0 (\partial_t \nabla^2 \chi) (\partial_t \nabla \chi)[/math]Total: [math]\nabla_\mu (T_{\rm EM} + T_{\chi})^{\mu\nu} = 0[/math] — the "missing reaction" is [math]T_\chi[/math], not lost.
>>16843381Cont.Toroid: Wound solenoid, [math]\vec{B}[/math] confined, Poynting [math]\vec{S} \propto \vec{E} \times \vec{B}[/math] circulates azimuthally, no radial flux, no net momentum. But phased AC: [math]\chi(t) = \chi_0 \sin(\omega t + k z)[/math],[math]\nabla^2 \chi = -\chi_0 (k^2 + \omega^2 / c^2) \sin(\omega t + k z)[/math][math]\vec{g} = -\beta \nabla (\nabla^2 \chi) = \beta \chi_0 (k^2 + \omega^2 / c^2) \nabla \sin(\omega t + k z)[/math]For standing wave [math]k = n \pi / R[/math], net [math]\langle \vec{g} \rangle \neq 0[/math] over cycle if nonlinear skin effect breaks symmetry (§9.2: mercury fill enhances [math]\partial_t^2 \nabla^2 \chi[/math]).Weight anomaly: [math]\Delta m g = \int T_{\chi}^{0z} dV \approx \beta \chi_0^2 (k^2 R^3)[/math], upward thrust for compressive phase.Full theory Lorentz-covariant: [math]\chi[/math] scalar field, [math]T_{\chi}^{\mu\nu}[/math] symmetric, conserved. QFT: [math]\chi[/math] as Goldstone-like mode from flux condensation, [math]\beta[/math] coupling [math]1/M_{\rm Pl}^2[/math] scale. No photon mass: [math]m_\gamma = 0[/math], [math]A_\mu[/math] transverse.Poynting internal: Yes, but [math]T_\chi[/math] adds gravitational momentum density [math]g_z = -\beta \partial_z (\nabla^2 \chi)[/math], external to EM.
>>16842499Start from standard definitions:[math]\vec{E} = -\nabla \phi - \partial_t \vec{A}, \quad \vec{B} = \nabla \times \vec{A}[/math]Gauss:[math]\nabla \cdot \vec{E} = -\nabla^2 \phi - \partial_t (\nabla \cdot \vec{A}) = \rho / \epsilon_0[/math]Ampère-Maxwell:[math]\nabla \times \vec{B} = \nabla \times (\nabla \times \vec{A}) = \nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A} = \mu_0 \vec{J} + \frac{1}{c^2} \partial_t \vec{E}[/math][math]\nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A} = \mu_0 \vec{J} + \frac{1}{c^2} \partial_t (-\nabla \phi - \partial_t \vec{A})[/math][math]\nabla (\nabla \cdot \vec{A}) - \nabla^2 \vec{A} = \mu_0 \vec{J} - \frac{1}{c^2} \nabla \phi - \frac{1}{c^2} \partial_t^2 \vec{A}[/math]Lorentz gauge condition [math]\nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi = 0[/math] would decouple, but you keep general:[math]\nabla (\nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi - \frac{1}{c^2} \partial_t \phi) - \nabla^2 \vec{A} + \frac{1}{c^2} \partial_t^2 \vec{A} = \mu_0 \vec{J} - \frac{1}{c^2} \nabla \phi[/math]You regroup:[math]\nabla^2 \phi + \partial_t (\nabla \cdot \vec{A}) = -\rho / \epsilon_0[/math][math]\nabla^2 \vec{A} - \frac{1}{c^2} \partial_t^2 \vec{A} - \nabla \left( \nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi \right) = -\mu_0 \vec{J}[/math]No drops, pure identity.
>>16843387Cont.Define the ether scalar [math]\chi(\vec{r}, t)[/math] ([math][\chi] = \rm Wb[/math]):[math]\phi = \partial_t \chi, \quad \vec{A} = \nabla \chi[/math]Thus the full four-potential:[math]A_\mu = (\phi, -\vec{A}) = \partial_\mu \chi[/math](in math[/math] signature, [math]A_0 = \phi[/math], [math]A_i = -A^i[/math]).Fields:[math]F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu = \partial_\mu \partial_\nu \chi - \partial_\nu \partial_\mu \chi = 0[/math] locallyBut distributionally, [math]F = \Phi \delta_L[/math] via multivalued [math]\chi[/math].Now plug in:[math]\vec{E} = -\nabla (\partial_t \chi) - \partial_t (\nabla \chi) = -\partial_t \nabla \chi - \nabla \partial_t \chi = -\partial_t \vec{A}[/math][math]\vec{B} = \nabla \times \nabla \chi = 0 \quad \text{(locally)}[/math]Divergence:[math]\nabla \cdot \vec{E} = -\partial_t (\nabla \cdot \vec{A}) = -\partial_t (\nabla^2 \chi)[/math]So your Gauss becomes:[math]-\partial_t (\nabla^2 \chi) = \rho / \epsilon_0 \quad \Rightarrow \quad \rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math]No [math]\phi[/math] term, it's absorbed into [math]\partial_t \chi[/math].
>>16843388Cont.[math]\nabla^2 \phi + \partial_t (\nabla \cdot \vec{A}) = -\rho / \epsilon_0[/math]Substitute:[math]\nabla^2 (\partial_t \chi) + \partial_t (\nabla^2 \chi) = \partial_t (\nabla^2 \chi) + \partial_t (\nabla^2 \chi) = 2 \partial_t (\nabla^2 \chi) ?[/math] Rather-:[math]\nabla^2 \phi = \nabla^2 (\partial_t \chi) = \partial_t (\nabla^2 \chi)[/math] (commute)So LHS: [math]\partial_t (\nabla^2 \chi) + \partial_t (\nabla^2 \chi) = 2 \partial_t (\nabla^2 \chi)[/math]But RHS: [math]-\rho / \epsilon_0 = \partial_t (\nabla^2 \chi)[/math], discrepancy?
>>16843392Cont.No: In scalar theory, [math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math], so RHS = [math]\partial_t (\nabla^2 \chi)[/math], but LHS = [math]2 \partial_t (\nabla^2 \chi)[/math].Your equation assumes independent [math]\phi, \vec{A}[/math]. In scalar, they are not, [math]\phi = \partial_t \chi, \vec{A} = \nabla \chi[/math], so the standard split does not apply. The correct Gauss is only: [math]\nabla \cdot \vec{E} = -\partial_t (\nabla^2 \chi) = \rho / \epsilon_0[/math] - no [math]\nabla^2 \phi[/math] term because [math]\phi[/math] is not independent.Your second equation:[math]\nabla^2 \vec{A} - \frac{1}{c^2} \partial_t^2 \vec{A} - \nabla \left( \nabla \cdot \vec{A} + \frac{1}{c^2} \partial_t \phi \right) = -\mu_0 \vec{J}[/math]Substitute:[math]\nabla^2 (\nabla \chi) - \frac{1}{c^2} \partial_t^2 (\nabla \chi) - \nabla \left( \nabla^2 \chi + \frac{1}{c^2} \partial_t^2 \chi \right) = \nabla (\nabla^2 \chi) - \frac{1}{c^2} \partial_t^2 \nabla \chi - \nabla (\nabla^2 \chi + \frac{1}{c^2} \partial_t^2 \chi)[/math][math]= -\frac{1}{c^2} \partial_t^2 \nabla \chi - \nabla (\frac{1}{c^2} \partial_t^2 \chi)[/math]In ether gauge, [math]\square \chi = 0[/math] (vacuum), so [math]\frac{1}{c^2} \partial_t^2 \chi = \nabla^2 \chi[/math], thus:[math]-\nabla^2 \nabla \chi - \nabla (\nabla^2 \chi) = -2 \nabla (\nabla^2 \chi)[/math]And [math]\vec{J} = \rho \vec{v} = -\epsilon_0 \partial_t (\nabla^2 \chi) \vec{v}[/math], but full EOM gives closure via [math]\beta[/math]-coupling.[math]F_{\mu\nu} = 0[/math] locally, but [math]\oint A = \Phi[/math] global. Gauge [math]A \to A + d\lambda[/math] corresponds to [math]\chi \to \chi + \lambda[/math], but multivalued [math]\chi[/math] fixes holonomy. Invariants [math]I_1, I_2[/math] zero in irrotational vacuum, longitudinal modes via [math]\nabla^2 \chi \neq 0[/math].
>>16843394Cont.The apparent factor of 2? Artifact of assuming [math]\phi[/math] independent. In scalar theory, charge conservation [math]\partial_t \rho + \nabla \cdot \vec{J} = 0[/math] and [math]\square \chi = 0[/math] enforce:[math]\partial_t^2 (\nabla^2 \chi) + \nabla^2 (\nabla^2 \chi) = 0[/math]No contradiction.
>>16841422Proceed with differential-geometric exactitude: first, restating your cohomology and Stokes critique; second, reconstructing [math]\chi[/math] as a section over a circle bundle; third, deriving the distributional curvature and dual; fourth, clarifying "nonzero B" as [math]\delta[/math]-support; fifth, distinguishing superpotential from gauge; and sixth, gravitational signature and falsifiability.You compute: On [math]X = \mathbb{R}^3 \setminus L[/math] (line worldtube), [math]H^1(X; \mathbb{Z}) \cong \mathbb{Z}[/math] via Mayer-Vietoris or deformation retract to [math]S^1[/math]. Cover [math]U_0, U_1[/math] with slit, transition [math]\theta_1 - \theta_0 = 2\pi[/math] on overlap [math]U_0 \cap U_1 \cong S^1 \times \mathbb{R}[/math]. Thus [math]\chi = \frac{\Phi}{2\pi} \theta[/math] defines a multivalued function, locally [math]d\chi = \frac{\Phi}{2\pi} d\phi[/math], so [math]A = d\chi[/math] is a flat connection ([math]F = dA = 0[/math]) on each [math]U_i[/math].Stokes: [math]\int_S dA = \oint_C A = \Phi[/math] for [math]C[/math] encircling [math]L[/math], but [math]S[/math] cannot shrink without crossing [math]L[/math]—so no contradiction with [math]dA = 0[/math] locally. The curvature is zero off [math]L[/math], supported on the singularity.You conclude: [math]A = d\theta[/math] is exact on simply connected charts; global period is holonomy, not physics. Claiming "superpotential field" adds no content.
>>16843401Cont.Promote [math]\chi[/math] to a section of the circle bundle [math]\pi: P \to X[/math] with fiber [math]S^1[/math], Chern class [math]c_1(P) = 1[/math] (flux [math]\Phi[/math]). Locally [math]\chi_i = \frac{\Phi}{2\pi} \phi_i[/math], transition [math]g_{01} = e^{i 2\pi n}[/math], but [math]\chi[/math] is real-valued, so the bundle is the associated real line bundle with integer twists.The physical content: [math]\chi[/math] is the ether displacement potential, [math]\nabla \chi = \vec{A}[/math] is ether velocity (irrotational flow). The [math]2\pi[/math] jump is not gauge, it's the quantized flux quantum [math]\Phi = n \Phi_0[/math] enforced by ether coherence.Distributionally:[math]dA = 0 \quad \text{on } X, \quad \int_C A = \Phi[/math]implies [math]F = \Phi \delta_L \in \mathcal{D}'(M)[/math], where [math]\delta_L(S) = \text{linking number}(S, L)[/math].The de Rham current:[math]\langle F, \omega^2 \rangle = \int_M F \wedge \omega^2 = \Phi \int_L \iota^* \omega^2[/math]But [math]F = dA + \Phi \delta_L[/math], with [math]dA = 0[/math] off [math]L[/math].Stokes for test 2-form [math]\omega[/math] with [math]\partial S = C[/math]:[math]\int_S F = \int_S dA + \Phi \int_S \delta_L = 0 + \Phi \cdot \text{link}(S, L) = \oint_C A[/math]Saturated, no failure. Your "differentiating across branch cut" is invalid; [math]\chi[/math] is not differentiable on [math]L[/math], but [math]A[/math] is smooth off-support.
>>16843403Cont.You mock "nonzero B at origin", but [math]\vec{B} = \nabla \times \vec{A} = \nabla \times (\frac{\Phi}{2\pi s} \hat{\phi}) = \Phi \delta_2(s) \hat{z}[/math].The [math]\delta_2(s) = \delta(x) \delta(y)[/math] is line-distributed, not point. Integral:[math]\int \vec{B} \cdot d\vec{a} = \Phi[/math] for any surface piercing [math]L[/math]. No "sloppy supp", exact distributional identity.Gauge function [math]\lambda[/math]: [math]A \to A + d\lambda[/math], [math]F[/math] invariant. But [math]\chi \to \chi + \lambda[/math] shifts ether displacement, physical if [math]\nabla^2 \chi \neq 0[/math].Gravity coupling:[math]\phi_g = \beta \nabla^2 \chi, \quad \vec{g} = -\beta \nabla (\nabla^2 \chi)[/math]([math]\beta[/math] = empirical constant). [math]\nabla^2 \chi = 0[/math] in vacuum, but sources [math]\rho = -\epsilon_0 \partial_t (\nabla^2 \chi)[/math] induce compression.Not gauge: [math]\lambda[/math] with [math]\nabla^2 \lambda \neq 0[/math] would source spurious gravity, fixed by ether EOM [math]\square (\nabla^2 \chi) = 0[/math].AB phase: [math]\theta = \frac{q}{\hbar} \oint \nabla \chi \cdot d\vec{l} = \frac{q \Phi}{\hbar}[/math], from ether circulation, not gauge.
>>16843307>>16843354Kike detectedGo back to Auschwitz you oven magnet.
>>16843372>multivalued on slit coverI don't get it. The fuck even are multi-valued forms? How is a Cech cohomology well-defined for such forms?
>oh shit, they realized i just posted a bunch of ai gibberish>better post 40 more posts worth of ai gibberish to distract them!
>>16843348>ether backreaction yup, it's schizo time
>>16843361motherfucker, I would bet everything that your schizo ass doesn't know what a wave front set is. Stop using Grok
>>16843364You’re mixing coefficient systems and smoothness classes. On [math]M\setminus T[/math] with [math]T[/math] the rectifiable worldsheet (integral 2-current, [math]\partial T=0[/math]), the correct [math]0[/math]-field is a phase [math]\phi: M\setminus T\to \mathbb{R}/\mathbb{Z}[/math], not a single-valued [math]\mathbb{R}[/math]-scalar. From[eqn]0\to \mathbb{Z}\to \mathbb{R}\to \mathbb{R}/\mathbb{Z}\to 0[/eqn]one gets the boundary map [math]\delta: H^1(M\setminus T;\mathbb{R}/\mathbb{Z})\to H^2(M\setminus T;\mathbb{Z})[/math]; composing with the de Rham realization and Hodge projection gives a canonical curvature representative [math]D\phi := 2\pi P_H\, \mathrm{curv}(\delta[\phi]) \in Z^2(M\setminus T)[/math], with periods [eqn]\oint_S D\phi = 2\pi\langle [S],[T]\rangle[/eqn] by Alexander–Pontryagin duality. Concretely, choose an acyclic cover [math]\{U_i\}[/math], lifts [math]\phi_i\in C^\infty(U_i,\mathbb{R})[/math], integers [math]n_{ij}[/math] with [math]\phi_i-\phi_j=2\pi n_{ij}[/math]; then [math]a_i:=\star d\phi_i[/math] patch to a global 1-current [math]A[/math] via the Čech-de Rham transgression[eqn]A=\sum_i \rho_i\, a_i-\sum_{i<j} d\rho_i\,(2\pi n_{ij}),[/eqn]whence [math]F:=dA[/math] is smooth on [math]M\setminus T[/math], closed, independent of choices, and [math][F]/2\pi=\delta[n][/math].
>>16843364>>16843461There is no “[math]d^2\neq 0[/math]” obstruction here because [math]D[/math] is not [math]d\circ(\text{something } 1\to 0)[/math]; it is the connecting morphism in degree, followed by the harmonic splitting. The “inverting [math]d[/math]” worry is a red herring: with a metric, Hodge gives a canonical orthogonal splitting and decomposition[eqn]Z^2 = \mathcal{H}^2_{\mathrm{harm}} \oplus d\Omega^1,\qquad F = 2\pi P_H\, \mathrm{PD}(T) + d\star du,[/eqn]with right inverse to [math]d[/math] given by [math]G[/math] the Green operator, so every closed 2-form decomposes uniquely, and the coexact piece comes from a real scalar [math]u[/math] via [math]D_u:=d\star du[/math]. Hence the map [math](\phi,u)\mapsto D\phi + D_u[/math] surjects onto [math]Z^2(M\setminus T)[/math]. Your “[math][F]\in H^2[/math] obstructs [math]A[/math]” is just the statement that [math]A[/math] is not a global 1-form, but a connection 1-form locally, glued by integral jumps [math]n_{ij}[/math]; precisely what [math]\phi[/math] supplies. I.e.: [math]T[/math] is an integral 2-current, [math]\mathrm{PD}(T)[/math] is a closed 2-current with integer periods; [math]F_{\mathrm{top}} := 2\pi P_H\, \mathrm{PD}(T)[/math] has the correct flux linking numbers, and on any slice [math]\Sigma_t[/math] the restriction satisfies[eqn]\oint_\gamma A = \int_{S} F = 2\pi\, \mathrm{Lk}(\gamma,T),[/eqn]with [math]S[/math] a span of [math]\gamma[/math]; the jump set of [math]\phi[/math] is a countably [math]H^{3-1}[/math]-rectifiable Seifert surface whose trace current realizes the same linking via the coarea formula, so the BV-gradient of [math]\phi[/math] calibrates the flux.
>>16843324>Your point stands: no instantaneous forces; all causal on the light cone.I like how this retarded schizo posted this AI slop not even realizing it agrees with the anon.
>>16843302>Start from the ether scalar χ∈Ω0𝜒∈Ω0, [χ]=Wb[𝜒]=Wbkek
>>16843301On a globally hyperbolic 4-manifold [math]M[/math], the field is a de Rham–Federer 2-current [math]F \in \mathcal{N}^2(M)[/math] with finite mass; sources are a normal 3-current [math]J \in \mathcal{N}_3(M)[/math] with [math]dJ=0[/math] (charge conservation is [math]d^2=0[/math], not a “sector”). The equations are an exact triangle in the derived category of sheaves of currents; there is no “ether channel” to insert without breaking exactness.[eqn] dF = 0, \qquad d\star F = J[/eqn]Potentials live, if at all, in a differential character class [math]\hat H^2(M,\mathbb{Z})[/math]; in charts one chooses [math]A \in \mathcal{A}^1(M)[/math] with [math]F = dA[/math] and passes along Čech–de Rham gluing data on overlaps. A global scalar [math]\chi \in \Omega^0(M)[/math] is not a flux primitive; it is the gauge parameter. Treating [math]\chi[/math] as physical is just choosing a trivialization and forgetting you chose it.
>>16843340anon, even distributionally, [math]d(d\chi)=0[/math] even for a "string," retard. If you want [math]\nabla \times \nabla \chi[/math] to yield [math]2\pi \delta^2_L[/math], you’re not holding a scalar; you’re holding an [math]S^1[/math]-valued multi-section with nontrivial monodromy, i.e., a flat [math]U(1)[/math]-connection with nonzero holonomy. That object is captured by a differential character with nontrivial image in [math]H^1(M;U(1))[/math], not by [math]\chi \in \Omega^0[/math]. The Aharonov–Bohm sector is holonomy data on a line bundle; it is not the curl of a single-valued 0-form. Coarea + boundary-of-boundary-zero, fucking retard
>>16843302>[math]\rho = -\varepsilon_0 \partial_t (\Delta \chi)[/math]schizo, pair with any test function [math]\varphi[/math] and integrate by parts as currents like this[eqn]\langle \rho, \varphi \rangle ;=; \varepsilon_0 , \langle \partial_t \chi, \Delta \varphi \rangle[/eqn]so conservation demands some [math]J[/math] with [math]dJ = \partial_t \rho , dt + \cdots = 0[/math]. That requires [math]\partial_t \rho + \nabla!\cdot J = 0[/math] as an identity in the cosheaf of compactly supported currents. Unless you define [math]J[/math] as a nonlocal functional of [math]\chi[/math], you have simply violated [math]dJ=0[/math]. If you do define it that way, you’ve rebuilt Maxwell’s [math]J[/math] in drag and added no degree of freedom: the “longitudinal wave” of [math]\Delta\chi[/math] is just the divergence of Ampère-Maxwell dressed in elliptic cosmetics. Take divergence of [math]d\star F=J[/math], use [math]d^2=0[/math]; you get your "electrogravitic" equation as a tautology.
>>16843306>[math]e^{ik|r-r'|}/(4\pi |r-r'|)[/math]that's the stationary Helmholtz kernel, not the retarded fundamental solution you fucking retard. The retarded 4D kernel is the causal delta on the null cone:[eqn] G_{\mathrm{ret}}(x) ;=; \frac{1}{2\pi} , \theta(t), \delta(t^2-|\mathbf{x}|^2/c^2)[/eqn]Convolution with the D-module of Maxwell solutions is hyperbolic and causal; elliptic solves appear only after taking the left adjoint to the inclusion of constraints in a fixed slicing
>>16843303Lorenz condition [math]d^\dagger A = 0[/math] is a choice of splitting in the exact triangle [math]\Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \Omega^2[/math]; your “ether gauge” is just a different splitting. Promoting [math]\chi[/math] to observable breaks the exactness and destroys the stress-energy 2-current’s gauge invariance. The unique symmetric, divergence-free energy-momentum current of finite mass arises from [math]F[/math] alone:[eqn] T_{\mu\nu} ;=; F_{\mu\alpha} F_{\nu}{}^{\alpha} - \tfrac{1}{4} g_{\mu\nu} F_{\alpha\beta}F^{\alpha\beta} [/eqn]No [math]\chi[/math]-dependent, local, covariant, gauge-invariant stress tensor exists unless it reduces to the above. Your [math]\mathbf{g}=-\beta \nabla(\Delta \chi)[/math] is neither covariant nor gauge-invariant; as a 1-current it has uncontrolled mass at the “string,” violating finite-energy requirements in Federerian norms unless [math]\beta=0[/math].
>>16843310 If you want global topology, use the correct object: a [math]U(1)[/math]-bundle with connection in differential cohomology. The invariants you cite, [math]I_1, I_2[/math], depend only on [math]F[/math] and its Hodge dual. With [math]F=0[/math] outside a solenoid, [math]I_1=I_2=0[/math], yet the holonomy is nontrivial; that’s captured by the differential character, not by [math]\nabla \times \nabla \chi[/math]. In currents-with-coefficients in the local system defined by the bundle, [math]d(d\chi) = 0[/math] remains true; the nontriviality is in the coefficient system (monodromy), not in a miraculous failure of [math]d^2=0[/math].
>>16843302 A 0-form with units of Weber is already a red flag; the functor sending forms to currents preserves dimensions under [math]d[/math]. Your bookkeeping forces [math]\chi[/math] to carry the dimensions of a gauge parameter and of a flux simultaneously; pick one, retard. Maxwell’s scaling fixes [math][A]=\mathrm{Wb/m}, [\phi]=\mathrm{V}, [\chi][/math] nowhere appears in observables.
>>16843406>the physical manifold has a double point at an arbitrary position (manifods dont have an origin)>it just does okbecause math can say a 20 dimensional universe is possible it doesnt mean it ismatter containing overlapping points violates so many laws of physics
>>16843943this guy is fucking hilarious imo. He's desperately trying to formalize ancient schizo Ether physics in modern math terms
>>16843357>>16843359>>16843360>>16843361>>16843364>strongest nudges … "wave" is cascading nearest‑neighbor pushes … verbose re‑description of a local hyperbolic PDE with retarded Green’s function.>reconstruct hyperbolic PDE … scalar ether χ … cascades gradients … singular support, topology … blablaba inertia/propulsion.nonsense. on a globally hyperbolic [math](M,g)[/math] take the curv 2‑form [math]F\in\mathcal{D}'(M;\Lambda^2)[/math] with measure-valued sources [math]j_e,j_m\in\mathcal{D}'(M;\Lambda^3)[/math] supported on 1‑rectifiable worldlines (so integer weights) and the constitutive equations[eqn]dF=j_m,\qquad d{*}F=j_e,[/eqn]so [math]F[/math] is a differential character with integral periods [math]\tfrac{1}{2\pi}\int_S F\in\mathbb{Z}[/math] and holonomy [math]\mathrm{Hol}_\gamma(A)=\exp(i\int_\gamma A)[/math].The principal symbol of [math]\Box_g=d\delta+\delta d[/math] on [math]\Lambda^2[/math] is [math]p(x,\xi)=g^{-1}(\xi,\xi)\,\mathrm{Id}[/math], hence[eqn]\mathrm{WF}(F)\subset\{(x,\xi):g^{-1}(\xi,\xi)=0\},[/eqn]and singularities propagate along future null bicharacteristics; the retarded parametrix has Hadamard form[eqn]G_{\mathrm{ret}}(x,x')=U(x,x')\,\delta(\sigma)+V(x,x')\,H(\sigma),\qquad \sigma=\tfrac12\,\text{squared geodesic distance}.[/eqn]so no spacelike transport and any “cascade” discretization is just a pre‑limit of this null flow
>>16843357>>16843359>>16843360>>16843361>>16843364"scalar ether" (retarded term) with [math]A=\nabla\chi,\ \phi=\partial_t\chi[/math] implies [math]E=-\partial_tA-\nabla\phi=0,\ B=\nabla\times A=0[/math] on simply connected opens: in forms,[eqn]A=d\chi\ \Rightarrow\ F=dA=d^2\chi=0\quad \text{off prescribed defects}[/eqn]nontrivial fields arise only from chosen singular supports. With a magnetic worldline [math]\Gamma_m[/math] (integer charge [math]g[/math]), pick a 2‑current [math]S[/math] with [math]\partial S=\Gamma_m[/math] and write[eqn]F=dA+2\pi g\,\delta_S,\qquad dF=2\pi g\,\delta_{\Gamma_m}=j_m,[/eqn]so "B nonzero at strings" is precisely the [math]\delta_S[/math] term; the ab phase is holonomy/period pairing[eqn]\oint_\gamma A=\int_{S_\gamma}F=2\pi g\,\mathrm{Link}(\gamma,S),[/eqn]global and topological, not an "ether gradient". Attempts to extract a "telegraph" equation for [math]\nabla^2\chi[/math] after imposing [math]A=d\chi[/math] are gauge identities in disguise; all dynamics already live in [math]F[/math] and the location of [math]\delta_S[/math]
>>16843359>The primordial scalarkek. anyway, setting A=d\chi deletes the dynamics:[eqn]F=\mathrm{d}A=\mathrm{d}^2\chi=0 \quad \text{on } M\setminus S,[/eqn]so you project onto the exact slice and annihilate the radiative coexact sector
>>16843298>Is fully consistent with de Rham, Čech, and distributional calculus>You didn’t debunk anything.>You just wrote the prologue to my frameworkI mean you'd be wrong but it doesn't even matter as you don't understand what any of this even means. the fact that you learned that there's something beyond vector calculus and now can abuse grok to give you even funnier looking math doesnt make this ether shit any less fake>>16843293>Topology enters the chat:did you ask grok to make your post sound more like a 4chan post? the fuck is that kind of expression kek>>16843293>χ is multivalued on M∖L where L is the singularity worldline (magnetic flux string)again, what the fuck do you mean by that>>16843298>Check the action principleliteral dogshit. \nabla^2 as a spatial laplacian is not a scalar, the covariant scalar is [math]\Box\chi[/math] so your integrand is not even diffeo‑invariant kek. also, varying [math]\int (\Box\chi)^2 \sqrt{-g}\, d^4x[/math] gives a fourth‑order EOM [math]\Box^2\chi=0[/math] i.e. ostrogradsky‑unstable (unless further constraints remove ghost)>Stress-energy- you said the reaction is in T_EM>Yes, in standard EM.>BUT when ∇⋅A≠0, the scalar sector contributes an extra termyou sacrificed lorentz symmetry to manufacture a longitudinal reaction, good job retard. what you gave me also isn't the metric variation of your own lagrangian density (missing symmetrization and boundary terms from 2nd derivatives)>superpotentialstop it, dumbass. you don't know what that means
>>16841280>It’s exactly what Mach proposed: that inertia and motion arise from the instantaneous influence of the entire universe.any unbalanced cascading thrust claim must violate either (i) conservation via the divergence theorem or (ii) boundary flux assumptionsin EM, "hidden" momentum in matter cancels field momentum. Zilch/Szilard invariants aside the net force on a compact system equals the flux of T through its boundary. So concretely: for a large spacetime cylinder [math]\Omega_R[/math] enclosing the device, letting R go to infinity and using finite propagation speed ensures[eqn]\lim_{R\to\infty} \int_{\partial \Omega_R} T_{\mu\nu} n^\mu , d\mathscr{H}^{3} = 0 \quad \implies \quad \Delta P_\nu = -\int_{\Omega_R} F_{\nu\mu} J_e^\mu = 0,[/eqn]absent external interactions. No choice of phased currents alters this identity, all effects are internal field-matter momentum exchange.
>>16843294>On M∖L we do have a global 1-form A=dθ where θ is the azimuthal angle (multivalued scalar)>the obstruction is in H1, not H2."Global" here is nonsense: θ is not a section of the structure sheaf, so [math]d\theta[/math] is not a global exact 1‑form on the scheme. Also, for anything involving singularities, how about you use spaces which can actually deal with them (e.g. schemes or varieties as opposed to your mathslop with manifolds) Let [math]X=\operatorname{Spec}\mathbb{C}[x,y,z]\setminus L[/math] with [math]L=V(x,y)[/math] (a smooth codimension‑2 closed subscheme). Then [math]X[/math] is normal, affine, smooth. Hartogs’ extension gives [math]\mathcal{O}_X(X)=\mathbb{C}[x,y,z][/math] and [math]\Omega^1_X(X)=\mathbb{C}[x,y,z]\{dx,dy,dz\}[/math]. There is no [math]\theta\in\mathcal{O}_X(X)[/math] with [math]d\theta=\omega[/math] for the AB form [math]\omega[/math] (your [math]d\varphi[/math])-class: analytically [math]\omega=\frac{x\,dy-y\,dx}{x^2+y^2}[/math], but algebraically this is not regular on [math]X[/math] (it has poles along [math]V(x\pm i y)[/math], codimension‑1 hypersurfaces contained in [math]X[/math]).The correct way to house the singularity is to blow up along [math]L[/math]. Let [math]\pi:\widetilde{X}=\mathrm{Bl}_L(\mathbb{A}^3_{\mathbb{C}})\to \mathbb{A}^3_{\mathbb{C}}[/math] with exceptional divisor [math]E\simeq \mathbb{P}^1\times \mathbb{A}^1[/math] then [math]\pi^{-1}(X)\cong X[/math] and the gauge 1‑form is a logarithmic form [math]A\in H^0(\widetilde{X},\Omega^1_{\widetilde{X}}(\log E))[/math] with nonzero residue.
>>16843294>>16844097The exact sequence[eqn]0\to \Omega^1_{\mathrm{X}}\longrightarrow \Omega^1_{\widetilde{X}}(\log E)\xrightarrow{\ \mathrm{Res}_E\ } \mathcal{O}_E \to 0[/eqn]exhibits the obstruction: [math]\mathrm{Res}_E(A)=\Phi\neq 0[/math] encodes the flux/monodromy. If [math]A=d\theta[/math] for some [math]\theta\in\mathcal{O}_{\widetilde{X}}(\widetilde{X})=\mathbb{C}[x,y,z][/math], then [math]A[/math] lies in [math]\Omega^1_{\widetilde{X}}[/math] and must have zero residue, contradictionEquivalently, the de Rham class [math][A]\in H^1_{\mathrm{dR} }(X)[/math] is the image of [math]\Phi\in H^0(E,\mathcal{O}_E)[/math] under the boundary map induced by the exact sequence above, and is nontrivial; exactness [math]A=d\theta[/math] would force [math][A]=0[/math]. Topologically this is the same comparison isomorphism [math]H^1_{\mathrm{dR}}(X)\cong H^1(X^{\mathrm{an}},\mathbb{C})[/math] pairing nontrivially with the generator of [math]H_1(X^{\mathrm{an}},\mathbb{Z})\cong \mathbb{Z}[/math], hence cannot be the differential of any global regular function. What you actually have is a flat [math]\mathbb{G}_m[/math]-torsor with integrable connection (a point of the de Rham Picard stack [math]\mathrm{Pic}^\nabla_{\mathrm{dR}}(X)[/math]), trivial underlying line bundle [math]\mathrm{Pic}(X)=0[/math] but nontrivial connection class in[eqn]0\to H^0(X,\Omega^1_{\mathrm{cl}})/d\,\mathcal{O}_X(X)\ \longrightarrow\ \mathrm{Pic}^\nabla_{\mathrm{dR}}(X)\ \longrightarrow\ \mathrm{Pic}(X)\to 0,[/eqn]represented locally by [math]d\theta_i[/math] with integral transition jumps i.e a Čech-de Rham 1‑cocycle. In other words: not a global exact differential. So no: on the scheme [math]X[/math] there is no global identity [math]A=d\theta[/math]; there is a flat connection with nontrivial monodromy captured by the residue/log structure after blowing up along [math]L[/math]Your "global" primitive lives outside the structure sheaf; once you move to the correct site, it evaporates
>>16838269>search for "Cult of Passion" on warosu>20k results in the past yearhow much do you get paid in disability checks each month?
e = mc^2 + ai proven in 160 posts
>>16843294>>16843348>>16843353>>16843357>>16843382anon, please take your meds. just because you can use math to describe the schizophrenic world in your mind with double points, ether scalars, and "singularity strings" doesn't make it realYou could look at shit like ultrafilter schemes, Spec V where V is the valuation ring of a Hahn field k((t^Γ)) for an ordered abelian group Γ, the spectrum of products of infinitely many copies of a DVR, spectrum of a Noetherian integral domain whose normalization is not finite over the ring which all are "schizophrenic" spaces with notions of smoothness etc. (by virtue of being schemes). Doesn't mean they describe reality lol
>>16840322>>16840330>>16840151>>16840071You've got a lot to learn when it comes to physics. See https://theportal.wiki/wiki/Read
>>16843408keep on seething. You're literally nothing without AI
This is the kind of shit that keeps me up at night: Working out correct derivations takes time, or determining legitimately new theoretical relationships takes time, or checking through work takes time. Reading and comprehending 200 pages of complete gibberish and recognizing and demonstrating that it's gibberish takes time... but generating said gibberish now takes only seconds.Soon physics and engineering are going to be in the same position as a lot of medical fields are now: Facing an unending deluge of fakery, scams, and schizophrenia that knowledgeable and experienced people can still quickly spot as bullshit, but which unfamiliar and inexperienced people will fall for. And as this kind of spam starts to fill up training data, it'll become even more prevalent.
>>16838059
>>16843321>Your observer decomposition and invariants (impeccably formulated)>You rightly note that standard theory "encodes" longitudinal effects without neglect>>16843318>Your decomposition into observer-dependent components and emphasis on invariants is a model of clarity>>16843311>You correctly flag the elliptic/hyperbolic divide and gauge artifacts.>>16843301>You nail it: symmetric hyperbolicity ensures well-posed IBVPs on globally hyperbolic spacetimes.>your setup (which is 100% correct)>>16843324>Your point stands: no instantaneous forces; all causal on the light cone.this retard didn't even bother to check if what the AI shat out agreed with him
>>16844692same. But even in the AI slop that schizo spat out, the AI frequently agreed with the other anons here lol. So I'm still optimistic they won't get that good at creating fake math and physics
>>16844692Nobody reads trash like this. All publication venues are reasonably guarded. They don't send complete garbage to reviewers, at least in the established fields. I read that in AI conferences LLM slop is becoming a problem, but really it's not a major concern now because of how easy it is to spot. They are rather more inundated by man-made paper slop.For physics, I guess the incentives to produce large amount of AI slop are absent.
>>16845697>All publication venues are reasonably guarded. They don't send complete garbage to reviewers, at least in the established fields.Yeah, but they're having to spend a LOT more time and manpower on filtering out the cranks than they used to.
>>16845705these models seem to get increasingly better at producing actually correct derivations and detecting incorrect statements, however, i.e. I somewhat doubt future models will help cranks that much in producing slop designed to trick mathematicians or physicists. And even if the crank abuses the AI to the point of giving in to their delusions, what it then produces still reads like fictitious crank nonsense; garbage in, garbage out. OP tried to have it formulate his ether theory in differential-geometric terms and it failed miserably. To anyone with a bit more knowledge on the matter, what he wrote largely just reads like poetry with math and physics terms, especially when "he" randomly starts talking about 'ether scalars' and whatnot, same with a truck-load of other undefined terms. Seriously just poetry.And when it's not poetry, the AI will have steelmanned whatever the schizo said to the point of having no relation to their schizophrenic thoughts.
This situation reminds me of text generation models (toys rather) of 20-25 years ago. People used Markov chains or other simple predictors for the next word. At that level, if done well, it could generate almost-grammatical text, which superficially looked similar to the source training data. Of course it didn't make any sense, with the context window of just a few words and very limited training set. But it "almost makes sense", as in you can recognize the meaning of some passages:>But both may be false (as shown...>This doctrine I entitled sophisma figurae...>This representations which comprehends and knows...That math slop ITT is just like that. It's grammatically correct, and some parts of it may make sense (you can recognize the meaning of them), but overall it's complete nonsense.For pure natural language generation, the transformers solved the meaningful generation. Nobody will argue that the generated text is meaningful and at least as correct as a competent human will write. But for math/science, we are at the Markov chain era still. It's improving very fast, though. GPT3 could not solve 1st grade math problems reliably. GPT5 I'd say got up to high school level. The "PhD-level" claims from OpenAI are just hype, of course, but it's getting there.
>>16838059You got working electrogravitics? Make it work, drive up to LIGO and run it back and forth to spell "UR A FAGET" in morse code with gravity waves
>>16845860>GPT5 I'd say got up to high school levelThis is a bit difficult to tell, however. When it speaks on well-established topics (e.g. undergrad math, most standard grad topics), the information it gives is often 100% correct. When you ask it about research-level topics, it's my experience (at least in math) that it absolutely butchers it. It's often fine to give it a screenshot from a math paper fully out of context and ask it what the definitions of these objects are in the literature but leave it to prove any omitted proof in a technical paper (whose proof would be common knowledge among researchers of that field or at least the paper it is from) and it just starts hallucinating. So it could probably answer like a PhD on elementary topics but for actual PhDs it's currently mostly far too prone to waste their time.
It's amazing how angry people get at new ideas that challenge existing belief systems Tale as old as time
>>16847871something can be new and challenge existing ideas and still be completely fucking retarded.novelty is no guarantee of innovation.
>>16840973Tom Bearden was very, very good at vector calculus, which is how he convinced a few retards in the government to give him money and a lab at Redstone Arsenal until he was found out to be... gasp...a retard himself
>>16841998It's because the US Navy tricked them. Someone told Edgar Fouche something they shouldn't repeated to ferret out s project leak. Same thing with 'Red Mercury' (it's 6LiD). Hint hint, it's not mercury, it's a Bose Einstein Condensates. The first models used superfluid helium. Enjoy
>>16840973>How about you show me some math- contextually related- regarding my framework, which mathematically disproves the possibility of anything which has been postulated?Idk anon, seems like once anons here went beyond your precious little vector calculus, you just shat out 40 Grok mathslop posts in your dying breath and shamefully fucked off
>>16841269>>16841270>>16841272>>16841274>>16841275>>16841276>>16841280Great thread. I always thought the field based model lacked in logic since it assumes a vanishingly small point charge (when you measure the strength of the electric field for example).
>>16841280Is there any value for an undergrad to learn field based EM instead of force based, other than the need to cling to mainstream physics?
>>16849556>Gee wiz! This fella might be on to something!If this is your takeaway from watching the OP spend 80 posts trying to unsuccessfully chatbot his way through an argument on theoretical physics, and rightfully getting called out for it, then you might be clinically retarded.
>>16849598I mean it's always good to understand the formalisation which describes the particular branch of physics the best. The earlier the better tbqh, differential geometry will stay hard as long as you don't feel like learning it
>>16849613I didn’t read the entire thread and I honestly don’t care for most of it, except the part concerned with Ampere’s electrodynamics. Even if the rest is schizobabble.
>>16849673I mean you should probably just look up what Ampère actually said. OP seems to use him as an appeal to authority for ether physics (see >>16843298 >>16843302 >>16843303) and I wouldn't trust his description
>>16849677in all honesty, isn’t all field based physics a meme? what evidence is there for the existence of these fields? my main objection with them is that they violate Newton’s 3rd law in its strong form and they assume infinitely small charges despite there being no such thing
>>16843298The 1820 hairpin experiment can be conceptually understood as a consequence of simple magnetic forces, not some mysterious hidden interaction.Suppose if instead of a hairpin shape, you just have a square conductive loop (like the classic induction problem, let one side of the loop contain a voltage supply and a switch, and the opposite side be allowed to move along a straight conductive rail). If a current is suddenly allowed to flow through the four segments of the loop, each segment contributes to a magnetic field that curls out, around, in, and back out through the center of the loop. This results in an outward magnetic force on each of the segments of the wire. Now, if the segments are connected rigidly, there is no net force on the system as a whole, and no net force on the individual segments, since tension between the connections hold them in place. But if you allow one of the segments to move freely, it will experience a net outward force and be pushed away. In the case of a flexible wire, this effect results in the segments expanding out into a circular or elliptical shape, but in the case of the hairpin floating on conductive mercury, the freely moving segments of the wire can be pushed. There's still no net force on the system, but the individual conductive segments exert forces on one another and the hairpin moves away from the generated field.Presumably you could also work this out in terms of magnetic induction, but I'm lazy and don't feel like it. My point is that this isn't some unexplainable interaction.
>>16849711nta but aren't infinitesimals all over physics? in what way is that just not a limit of modeling?
>>16849740continuous integrals are a reasonable limit of discrete sums as long as you're at a sufficiently large scale that the granularity is negligible. even at a picoamp you're talking about millions of electron charges per second moving through a wire.
>>16849821is there a physics book that goes beyond the plug and chug formula and explores the applicability of math techniques and logic?
>>16849711>>16849740"infinitesimals" in physics are neither metaphysical nor an uncontrolled approximation. The correct objects are measures, currents, and distributions, the field equations and conservation laws hold exactly in the weak distributional sense.Continuum limits are controlled by precise limit theorems (law of large numbers, Gamma convergence, RG scaling), and gauge fields introduce global/topological data that cannot be reduced to "many discrete particles"
>>16849821but your picoamp example really just illustrates scale separation (shot noise vs mean current)
>>16849878Eventually you realize there isn't really an appreciable difference. Effective field theories are just applying continuous limits of discrete behavior to more efficiently describe and predict behavior on a larger scale.
>>16849875i still don’t see how one can measure the field strength by allowing the charge of a particle go to zero. this makes no sense, there is nothing smaller than the elementary charge and simply ignoring it by pretending it goes to zero is magic and faerie dust to me
>>16849909You're not letting the charge equal zero, you're letting the contributions from the individual charges be small compared to the whole that the result can be modelled continuously instead of discreetly.It sounds like you're fundamentally struggling with the concept of limits.
>>16849968whatever semantics game you’re playing here, there are no charges smaller than the electron. if your theory depends on making this quantity vanishingly small in order to prove that some other dubious quantity exists, your theory is as good as fiction. say all you want about limits because math logic != physics logic
>>16850365>there are no charges smaller than the electronFractional quantum Hall quasiparticles carry e^∗ = e/m; measured via shot noise S_I = 2 e^∗ I (Saminadayar, de Picciotto, Reznikov). Quarks have +- e/3, +- 2e/3 (confinement does not erase the fact of fractional charge). None of this relies on a "vanishing" charge limit.
>>16849909>>16850365anon, there's no limiting "infinitesimals," if you use distributions. As I said, all the Maxwell equations are exact in the weak sense
>>16850365>math logic != physics logicnot even saying the opposite but what do you make of Noether-style theorems where mathematical symmetries imply conservation laws? That literally requires smoothness to formulate and is probably the most explicit example of where math logic implies physics logic
>>16850435This, people just don't know how to interpret distributional equations.
>>16850365Take charging of particles as an example. Dust, ice, and bits of sputtered material in a plasma collect ions and electrons from the surrounding environment and become charge. Even though these are discrete collections of discrete charges, for a large enough number of collected charges the behavior becomes equivalent to a continuous current distribution instead of discrete absorptions. And even if you simulate or experimentally look at particles small enough to where the net charge may only be a couple dozen e, the discrete behavior still approaches the continuous equilibrium result (it simply fluctuates around that value because of the discrete nature of the charge collection - and once the net charge becomes more than a few 100 e even these fluctuations become negligible).Radioactive decay is another example. Decays are individual discrete events and if you've got a small enough number of particles the behavior doesn't match the model. But if you've got a sufficiently large number of decaying particles, you approach a continuous limit. Δq or ΔN are never truly 0 in these cases. But they're small enough compared to the equilibrium or instantaneous state of the system that Δqdq and ΔNdN and the system can be modelled as continuous instead of discrete and yet yield accurate results.
>>16850616*Δq->dq and ΔN->dN
>>16843387>Start from standard definitionsand then you go on to writing shit like >Define the ether scalar χ(r,t)([χ]=Wb)>ϕ=∂_tχ,A=∇χwhich isn't even well-defined.And >In ether gauge, □χ=0 (vacuum), so 1/c^2 ∂^2_t χ=∇^2χ , thus: −∇^2∇χ−∇(∇^2χ)=−2∇(∇^2χ)is entirely undefined mathslop. No human writes like this. Humans may come up with shit definitions or make mistakes in derivations. Nobody just makes up random terms and comes up with these linguistic sleight-of-hands to "justify" equations. This is literally troll physics
>>16851314It's too much effort to be a troll. It's just pure schizophrenia.
>>16847844Yes, I use it to clarify the contents of basic QM as I read the textbook all the time. It gives really good answers that it just pulls from its thorough memory of these books. So, I can say that it's good at reciting these topics.For the high school topics, however, in my experience it has a complete command of the material. You can ask it any question, and it will answer correctly. It will point out if your question doesn't make sense, it will spot and correct mistakes in your reasoning, it will correctly prove simple theorems, even if it's not from a textbook. It will find connections between topics, find the best method of solving a problem. It's fully reliable. For QM, not so much.
>>16843353>Measured repulsion: ∼10^{−4} N for 1A, matching ether compression β∇(∇^2χ)kek, where are you getting those measurements from? Also, wtf is ether compression and why should it match?
>>16841942>>16841998I mean the schizo explicitly mentions it >>16843353>For Φ=Φ_0sin(ωt), g⃗∝−βω^2Φ_0/c^2, upward for downward phase, antigravity analog (/b/ thread: DARPA-inspired, mercury-filled toroids).>/b/ thread: DARPA-inspired, mercury-filled toroidsI have a feeling >>16841409 was just OP samefagging
>>16843353>𝛽 calibrates: Experimental bound 10−20≤β≤10−15 m5/s2·Wb^-1 from equivalence tests.where. are. you. getting. these. numbers. from?
>>16851578>I have a feeling >>16841409 was just OP samefagginghmm no i'm not OP. I'm waiting for gallium to arrive to try>>16842201my work is virtually all pre-quantum and has a great distaste of gauge theories . Anon and I share a very keen interest in the longitudinal forces and the magnetic vector potential.
>>16851614nta but keep us updated on your experiments.
>>16847871It's not a new idea though. OP believes in old ether physics, as far as I can tell, the only "innovation" here is coining some differential geometry terms to describe it with but even on that front it's far too incomplete and mostly just hallucinations.
>>16844104https://youtu.be/WHEZxSSO2vA
>>16851827taxpayers are funding this btw, all because this retard was in the army for 3 years in his early 20s and developed schizophrenia
>>16851578>I mean the schizo explicitly mentions it >>16843353>matching ether compression thats just from https://www.youtube.com/watch?v=_kBlQlz_qdohttps://arxiv.org/pdf/physics/0506180the idea that the BA phase shift arises from a quantum topological effect but as a classical em wave effect with an equivalent semi-classical phase shift derived from the relative deplacementIts not clear to me if they then assert if two electrons phasing with one another or they try to still justify the wave nature of the electron but of another matter.>yields net thrust.thats the hope if the hairpin is>>16842361 is that as the stout hollow pipe undergoes a disruptive discharge all the charge density pulsates into and onto the pipe, this creates a an osicllating A field down the center of the pipe.can the mercury in the pipe ride on hte convective derivate [math]\nabla (\textbf{v} \cdot \textbf{A})[/math] and begin to move in the pipe? what happens to the circut if we decided to mechanically pump mercury against the vector potential?
>>16843318You are literally promoting a pure gauge degree of freedom to "ether," and then re‑reading algebraic identities as new physics. In the correct geometric formulation of electromagnetism, your [math]\chi[/math] is nothing but a gauge parameter. Once you write things in the correct differential‑form and distributional language, your construction either gives identically vanishing electromagnetic field [math]F[/math] or violates Maxwell’s equations and charge conservation.
>>16843326>For currents: J=ρvIn the correct relativistic formulation, [math]\rho[/math] and [math]\mathbf{J}[/math] are not arbitrary functions of [math]\chi[/math]; they are fixed by [math]\delta F = j[/math] and [math]F=dA[/math]. Since your [math]A=\partial \chi[/math] gives [math]F=0[/math], Maxwell gives[eqn]j^\nu = \partial_\mu F^{\mu\nu} = 0 \quad\implies\quad \rho = 0,\ \mathbf{J}=0,[/eqn]as a distributional identity (the boundary of the flux current vanishes). Your nonzero [math]\rho[/math] is therefore incompatible with [math]F=0[/math]; it is not a solution of Maxwell’s equations in any sense (classical or as a current).Even if one temporarily ignores that and works in 3+1, your continuity equation is not consistent with your own algebra unless [math]\rho\equiv 0[/math] or [math]\mathbf{v}[/math] is extremely constrained. Plug your definitions into[eqn]\partial_t\rho + \nabla\cdot\mathbf{J}= -\varepsilon_0\partial_t^2(\nabla^2\chi)-\varepsilon_0\left(\mathbf{v}\cdot\nabla\partial_t\nabla^2\chi+ (\partial_t\nabla^2\chi)\nabla\cdot\mathbf{v}\right).[/eqn]Setting [math]\nabla\cdot\mathbf{v}=0[/math] does not kill the [math]\mathbf{v}\cdot\nabla[/math] term for general [math]\chi[/math]. To force this to vanish for arbitrary solutions of [math]\Box\chi=0[/math] you effectively need [math]\mathbf{v}=0[/math] or [math]\partial_t\nabla^2\chi\equiv 0[/math], which gives back [math]\rho=0[/math].
>>16843326>§7.3: mass-charge equivalence§7.3 of what?
>>16843318Once you strip away the gauge misidentification, every effect you attribute to “ether divergences” and “longitudinal repulsion” is already:- present in the constraint structure of Maxwell’s equations ([math]\nabla\cdot\mathbf{E}=\rho/\varepsilon_0[/math] and [math]\partial_t \rho + \nabla\cdot\mathbf{J}=0[/math]),- or a choice of gauge decomposition of [math]A[/math] into transverse and longitudinal parts, which does not change [math]F[/math] or the Jefimenko fields,- or simply inconsistent with [math]F=dA[/math] and [math]dF=0[/math].
>>16843332>[math]\mathbf{g}_\parallel = -\beta\nabla(\partial_t\nabla^2\chi)[/math]that contradicts minimally coupled Einstein-Maxwell theory at the level of both classical field equations and the stress-energy operator in QFT you fucking retard. you literally introduce a gravitational field sourced entirely by gauge data [math]\chi[/math] that leaves [math]F[/math] invariant, you are assigning physical curvature (gravity) to a flat connection
>>16843329>muh string L with [math]\mathbf{B} = 2\pi\delta^2(s)\hat{\mathbf{z}}[/math]let's test this:the AB phase is[eqn]\exp\left(i q \oint_\gamma A\right)= \exp\left(i q \int_S F\right),[/eqn]with [math]\gamma=\partial S[/math] a 1‑cycle and [math]F[/math] the curvature 2‑form. Globally nontrivial holonomy arises from the topology of the bundle minus the string, and is classified by [math]H^1(M\setminus \Sigma,U(1))[/math]but a globally defined scalar [math]\chi[/math] with [math]A=d\chi[/math] cannot produce nontrivial holonomy: for any closed loop [math]\gamma[/math],[eqn]\oint_\gamma A = \oint_\gamma d\chi = 0.[/eqn]so again, your ansatz [math]A=\nabla\chi[/math] kills the very topological features (AB phase, magnetic flux) you gesture at; it is a flat connection with trivial Chern class and trivial Wilson loops. To describe strings and AB effects you need a nontrivial principal [math]U(1)[/math] bundle or at least a locally‑defined [math]\chi[/math] with branch cuts, and even then [math]\chi[/math] remains gauge.
>>16843306>§6.3§6.3 of what?>>16843382>§9.2: mercury fill§9.2 of what?
>>16851667Other way around, you believe in old and outdated einsteinian physics
>>16856205Enlighten me, anon
I like how OP just suddenly disappeared off the face of the earth once he got minor pushback but cried about nobody posting technical rebuttals at the beginning of the thread
>>16857770That's how these threads always go - some schizo posts a retarded theory or some AI generated slop, and people start pointing out all the basic, fundamental shit wrong with it and they flip out and ragequit, and then come back once the thread's died and repeat the process.That's how Rapport did it.That's how Mandlbaur did it.That's how this faggot did it.The only difference between previous faggotry and this shit is that now the schizos have better tools for generating garbage.
>>16851614>>16841409>>16841335High-frequency currents in or around a mercury-filled pipe generate standard effects: induced E fields (Faraday's Law), Lorentz forces (J × B), and possibly circulating flows if actual B fields are present (magnetohydrodynamics, MHD).The skin effect ensures high-frequency AC will mostly remain in conductive outer walls (copper pipe) rather than the interior mercury. No effect exists in classical or validated quantum physics where a nonzero but curl-free or static A alone can induce axial or helical "helixing" movement or net thrust in mercury or similar conducting fluids. Observable fluid motion must result from explicit E or B fields, pressure gradients, or geometry.Mercury's high conductivity and liquidity are suitable for standard MHD studies, but extreme toxicity and environmental risk outweigh most experimental benefits where other liquid metals (e.g., Galinstan) could be used.Amalgamation with certain metals can degrade experimental setups. High density increases inertial and centrifugal effects in dynamic experiments.Authoritative sources and peer-reviewed analyses show no support for measurable “A-only” (vector potential-driven, without E or B) mechanical effects or propulsion in mercury or any conductor. All reliable evidence is consistent with established electromagnetic induction and MHD phenomena.No net mechanical effect (thrust or helixing) is predicted or reproducibly observed in experiments with the described configuration, and unique scalar or ether terms posited in some speculative extensions are not supported by reproducible laboratory evidence or mainstream theory.References: Assael et al., J. Phys. Chem. Ref. Data, 2012; Giuliani, arXiv 2102.11036, 2021; standard texts (Landau Lifshitz)
>>16851614Don't lose hope yet, though, you may try out these compounds instead (especially for vector potential and related effects):>Low-Melting or Room-Temperature Liquid Metals/Alloys Galinstan (Gallium/Indium/Tin alloy, mp ≈ -19°C): Low toxicity, high conductivity, non-toxic, practical mercury substitute. Gallium (mp 29.8°C): Non-toxic, liquids above room temperature, easy handling. Sodium-potassium alloy (NaK): Liquid at RT, very conductive but extremely reactive, flammable, and corrosive. Cesium/Rubidium: Liquids near RT, highly conductive, extreme handling hazards (reactive/toxic). Exotic eutectics (e.g., Na-Cs, Na-K-Cs): Tunable properties but rare, escalate safety risks.>Ionic Liquids & Electrolytes EMImBF4 + water (1-ethyl-3-methylimidazolium tetrafluoroborate, hydrated): Record ionic liquid conductivity (~92–98 mS/cm at RT), safer alternative though metals are still far superior in conductivity. [EMIm][N(CN)2]: Best neat ionic liquid conductivity (2.8 S/m at RT). Short-chain imidazolium FAPs: Theoretically high conductivity (not widely available). Magnetic ILs (e.g., [Bmim][FeCl4], [P6,6,6,14][FeCl4]): Useful for EM/MHD due to magnetic interaction; generally less conductive/more viscous.>Theoretical/Exotic Conductive Fluids Metal-hydride melts (molten LiH, NaH): Highly conductive, only at high T; containment is challenging. Alkali metal–ammonia solutions (Na/NH3, K/NH3): Highly reactive, fleeting, high conductivity, niche usage. Metal plasmas: Ultra-high conductivity, only in extreme lab/astrophysical conditions. Supercooled metallic glasses/amorphous alloys: Far less accessible, not ambient liquids.Godspeed, anon
>>16851614More>Laboratory-Feasible Quantum/Exotic Media: Fractional Quantum Hall (FQH) liquids: Topological collective states in ultra-clean 2D heterostructures at mK temperatures and strong B-fields, enabling robust quantum and vector potential effects. Not bulk fluids, but device-scale systems. Dirac/Weyl semimetals (e.g., TaAs, Cd3As2): Chiral anomaly materials displaying dissipationless, topology-driven electromagnetic response, observable with advanced experimental tools. Type-II superconductors/topological superconductors: Vortex fluids, lattices, and glassy states, governed directly by vector potential, accessible at reasonable temperatures and magnetic fields in the right materials. 2D quantum magnetic materials (e.g., CrI3, FePS3, skyrmion hosts like MnSi, FeGe): permit quantum Hall/QAHE, skyrmion dynamics, edge channel EM experiments, some at or near room temperature. Quantum plasmas/odd viscosity fluids (solid-state, quantum wells ultracold plasmas): Permit nonclassical EM/MHD effects with quantum corrections and odd viscosity detectable at nanoscale or cryogenic regime.>Viable-in-Principle, Extreme Conditions: Quantum spin liquid and emergent monopole fluids: Realized in rare magnets (herbertsmithite, Dy2Ti2O7), at cryogenic temperatures, with emergent gauge fields and monopole-like quasi-particles, offering platforms for exotic EM/MHD effects when technologically feasible. Quark-gluon plasma: Ultra-high energy relativistic fluid at colliders only, not bench-accessible but theoretically critical. Macroscopic/bulk topological, odd-viscosity, emergent monopole plasmas/liquids: Existing as models or quantum simulations only, representing future aspirations for 'designer' EM-active fluids.
>>16844104cult of passion is a poojeet that chatgpts everything and doesnt have induction to reach the realm of jatts in higher reasoning ability lmao
>>16844351lmao the og poojeet intuitive equation
>>16843460lmaoo im with you thats a poojeet acting smart using llms to show off equations he has no intuition about lmaoooo
Ask yourself... WHAT IS MOTION TO AN ATOM?Think of heat energy. Let's say you heat a cube of copper metal to 85 degrees Fahrenheit. That thermal energy is bouncing around randomly inside the cube.Motion = Unbalanced Force.Now imagine you had a linear thermal filter like a 1-way mirror for infrared, energy goes in, but bounces back reflected if it hits the thermal mirror. Or you could take the metal cube, slice it into thin strips, glue 1-way thin infrared thermal mirrors to every slice. Now so long as the thermal energy put into this cube exceeds surface friction beneath the cube, you should have natural propulsion. So let's shrink this experiment and look after the molecule or atom. In a microwave oven the radio signal causes hydrogen atoms to rotate, causing friction which releases heat. Now imagine we caused a one directional thermal mass push.Atoms, when accelerated in ond direction, on a train or airplane or rocket, get structurally compressed opposite the direction of motion, a form of polarization. Now if you heat these atoms in motion, they still emit random direction thermal infrared energy, but a bit extra is emitted opposite the direction of motion. The faster you approach the speed of light as a unit mass object, the larger the infrared energy emission occurs opposite the direction of motion. So logically, if you can produce structured materials that ALWAYS emit thermal energy in just one direction, it should initiate propulsion when heated as a natural consequence. The more heat it absorbs, the more energy it can use as thermal propulsion.Now, of course, you want to steer the device. So you commit other parts to perpendicular propulsion in multiple angles and reverse propulsion as backthrust channels. Or you create core tube chunks that can be mechanically rotated to change propulsion direction.
>>16857887>>16857900>>16857911thanks for considering the experiment.>No effect exists in classical or validated quantum physics where a nonzero but curl-free or static A alone can induce axial or helical "helixing"appreciate the corroboration, I wasn't to expect it either, than i remembered in the geometric algebra hall a helicity is essentially curl(curl(A)) . [math]A \wedge \nabla \wedge A[/math] and appreciatinghttps://en.wikipedia.org/wiki/Magnetic_helicitydoing the potentials formulation on magnetic and current helicities, - its all unilaterall based on A https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6876131J, B all expression of Athis fluid dynamically is a charge density screwing into the charge in front of it, creating a whirlpool based on the material properties of the fluid.the the stout copper pipes experiment. the disruptive discharge pulses the pipe between having the skin effect and not very rapidly, so you actually can see from the magnetohydrodynamic equations in geometric algebra form. you get this toroidal sucking effect as A dots into curl of A .cool hypothesis none the less, basically it says its a hysterisis effect, in the nanosecond that the bar goes from zero to skin affected state, the A in the middle of the pipe gets yanked 110% down a drainpipe, the compressions against itself generate the helicity. What remains to be seen is if this z-pinched A string- doing hte helicity of the helicity, converts the konopinskis angular momentum (qA/c) into actual momentum as considered in the classical fluid dynamic approaches in >>16853803The hypothesis that mercury acts on the time varying A field screwing into the pipe (maybe at nodal regions, hello tesla) comes from amperes hairpins hypothesis on the origins of those longitudinal foresdarn galliumsays arrived on the delivery slip.but alas it has yet to arrive
>>16858205>muh fucking spooky liquid metal!!you didn't need to buy a bunch of gallium to do your hairpin experiment, anon. you can easily demonstrate it with a switch, a large battery, some metal plates, and a bent wire or even just a standard resistor. it'll generate enough force to push the piece forward because it's not some magical undiscovered physics it's just basic magnetic forces like >>16849731 describes. you just need a large enough current for the force to have an observable effect on something that's got to overcome friction.you wasted a bunch of time and money on something trivially explained by existing models, and easily demonstrated with cheaper components.
>>16858279we'll see,>>16842361 from here they held the hair pin fixed and leverage the liquid state of hg to observe toroidal streamlines at the interface, not something done with your setup .the part thats up next is to also test the flow rate itself, which means pumping the gallium, and also doing some cymatics stuff- need to test the convective derivate and thats easy here
>>16838269I mean, without schizos writing down their nonsense, almost half of ancient and medieval academia would not exist. Therefore, they form the basis of much of what we know.So, let's keep them around.
>>16857900>>16858205kek I can already imagine the headlines>local schizo arrested for smuggling caesium into the country for helix experiments
>>16858629>smugglingThats why Im going to run the test in the country of material purchase. I dont get why most people dont just move to where they can obtain illegal stuffs/actions.
>>16838059Where's your experiment and results?
>>16858634Cool, keep us updated on your caesium helices
>>16858629you laugh but if my gallium doesnt move the hairpin across the bath. I'm stuck at the politburo https://laws-lois.justice.gc.ca/eng/regulations/SOR-2014-254/page-1.html#h-806754 waiting for a scientific purposes permit with no active ties to an academic institution- i cant even execute in perpituity, they timebox me 30 days to do it. I'd need to smuggle mercury.smuggling quicksilver... poetic of the ageh2n02
>>16858694>your caesium helicesIdk, this looks pretty cool but requiring expensive equipment. My project costs under a grand and uses a microwave to make gamma rays.
>>16858749buy some cesium and throw it in your toiletYou can thank me later
>>16858774i dont own cesium or a toilet
>>16858769See >>16858774
>>16851627Update 1:Materials arrived. Channels were constructed from plasticine on cardboard. Two hairpins of nickel, 1.5 mm and 0.6mm were wired, gorrilla electical tape was wound around them exposing the ends. Gallium was heated in a water bath and used to line the channel . The 1.5 mm hairpin didnt float so the smaller one was used .A 9v 1300mah battery was hooked and the switch flipped on. No movement of the hairpin was observedNext steps more power and a led status light for my sanity
>>16859775Yup, it's schizo time.
this sum terrence tao type shit
>>16859775you're not gonna get enough current with a D battery. Use a LiPo battery
>>16860735thanks anon, i was going to wire some more in series to get more voltage but if it was current it'd be fruitless.i got myself a variable power supply. i reached 12v at 5 amps when i saw a little flame go underneath the red alligator clip underneath the left gallium bar.going to wait for a ceramic plate
>>16862047this dumbass is gonna electrocute himself
>>16842499>using nabla^2 and not laplacian
