Humble tokamak physicist owns generations of cosmological wankers
It’s not often I get excited about papers from the physics community. The field I used to love has turned into a dreary ghetto of noodle-theory wankers, experimental particle physics bureaucrats, cosmological mountebanks, “phenomenologists” and quantum computing charlatans. But I’m excited about this paper:
https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-021-08967-3
The author, Gerson Otto Ludwig, is a lifelong plasma physicist from Brazil; a noble profession, even if controlled nuclear fusion is unlikely as a near future energy source. Plasma physics is a fiendishly difficult field; it is both mathematically difficult and unlike the more “woo” grandiose kinds of physics hiding behind formalism, your ideas are generally testable by experiment. Maybe some cosmological wanker pissed him off, and he said “segure minha cerveja.” Maybe he just noticed something from fooling around with magnetohydrodynamic models all day. But if he’s right, he’s basically written the most dramatic single paper own of the physics and astronomy community, like ever. Assuming this paper is correct, it is a literal extinction event for thousands of wankers; a fiery asteroid across the sky, with a bunch of cud-chewing cosmological dinosaurs staring at it in dumb disbelief.
One of the things cosmologists, noodle theorists and astronomers worry a lot about is “dark matter.” When you look out in space at rotating galaxies, they appear to contain more mass than we can actually see; even weirder, the mass appears to not be in the bright centroid for some reason. Something is making those galaxies stick together and rotate in funny ways and we can’t see it. If you do a physics major and your professor isn’t incompetent, they’ll probably make you work through an example of this. I remember doing so, thinking, “huh that’s pretty weird” then proceeded to attempt a career on objects about 10^68 times smaller than a galaxy. I had always assumed that someone had worked through the General Relativity version of this calculation in detail or at least given a reason why GR doesn’t apply. But I guess nobody did. There’s a larger issue here; why do galaxies look that way at all? You can mumble a bit about angular momentum and so on, but it is kind of peculiar there are so many things out there that look like this. When you read books on Galactic dynamics, there will always be a chapter wondering why galaxies are spirals; lots of hand wavey theories are given, but it’s pretty obvious nobody has a good idea.
I never studied GR; had the opportunity to do so with the great Ezra Newman and Carlo Rovelli. Skipping that for a dumb quantum optics course or whatever my excuse was, was an error. However one picks up a smattering of these things. There are analogies to the classical Maxwell equations in GR. It’s obvious there must be a component that works like electrostatics since Newtonian gravity looks exactly like Coulomb’s law with different constants, and mass substituting for electrical charge. What isn’t obvious is that there is also a gravitomagnetic term, which looks like Ampere’s law, relating the motion of charged particles to the magnetic field. So, there is a sort of gravitational analog to the magnetic field that happens when masses flow; old idea, people think it has something to do with quasars.
You can see where this is going: plasma physicists think about Lorentz forces on gasses of charged particles all goddamned day. Professor Ludwig related all this gravitic stuff to some equations from magnetohydrodynamics, ran the numbers, and realized the weird dark matter forces are probably a consequence of the geometry of spacetime. Theoretically any ambitious grad student of the last 50 years could have thought of it. I never studied magnetohydrodynamics myself, or GR, but if I were sitting around thinking about why Galaxies look weird or dork matter, and I know there was such a thing as gravitomagnetic effects, I might be …. slightly curious about what plasma physicists have come up with. It’s not like the z-pinch effect or tokamaks are particularly secret ideas; tokamaks at least have been bellowed about for decades.
Anyway, unless I’m missing something big here, it’s all straightforward stuff; a workman like piece of physics scholarship, and it seems to give the right answer (I haven’t checked). If he’s right, it’s going to make lots of people real mad, then sad for their wasted lives. The type of people who deserve a comeuppance. I will be unspeakably happy if this is true and the man wins the Nobel for it, making fools of a field filled with fools.
Locklin on science 
The “GR MOND” crowd. Here’s an example of someone with no academic affiliation whacking away six years ago
Click to access 1503.07440.pdf
I emphasize *no academic affiliation* 🙂
Do you think tokamak guy is wrong? I mean, he might be!
I am not qualified to comment as I haven’t kept up. A few thoughts on human behavior:
1) This issue is rife in that things that aren’t popular aren’t explored. There’s nothing really wrong with the MOND work. It’s certainly less fucked up than string theory yet nobody built a religion around it.
2) Top down versus bottom up: another age old issue is do you invent something to fill the gap or try to explain the gap with what we already know? We didn’t throw the neutrino out because it was believed to be massless. Historically if we need something that X MeV to make a theory work we assume it’s there. It’s a bias. So is there dark matter or can the neutrino have some mass?
3) We like pretty math. If there’s a MOND theory that works really well but, a la IR divergences, we have to truncate perturbations to potentials to curb large-scale behavior that’s ugly and can’t live in the pantheon of Maxwell, Einstein, Dirac. I don’t know, I agree that arbitrary ugly things are not great, but at least be consistent with booting arbitrary ugly things.
Since today how we feel is accepted science I’ll mention how I feel: dark matter is ugly and feels wrong. String theory is pretty and feels wrong. Feels like we didn’t close the loop on some important things in the 50’s or 60’s before ambitions trumped physics.
See, I don’t think it’s MOND at all. When I studied classical mechanics as an undergrad, the professor gave us an exercise on why classical mechanics was weird in galaxies, and how it probably wasn’t a central potential MOND.
This is a vector potential; purely relativistic, and very much a real consequence of GR. You don’t get gravitational waves without it. Looks like he just plugged in the numbers and it worked. There’s all kinds of relativistic constants hanging around such that small masses, small angular momentum, nothing interesting happens.
I called it “GR MOND” above to differentiate. That’s what I call it but it’s just the extension / generalization when applied to large scales and rotation. I only scanned the paper you posted and it smacked of MOND. I should read it.
Crazy. An idea that’s probably 100 years old (Lense-Thirring) and only a few people poked at. I think Penrose and Wheeler poked some if I recall. I am not sure how to perfectly interpret those curves; IOW if there’s unexplained gaps, etc. I don’t know enough about astrophysics. It would be really interesting if plugging an chugging like this was rock solid.
Yes; Penrose and Wheeler noodled a bit. Again I have no way of checking the result against data, but the equations look right and nothing obviously fruity going on. Word from a pal in DC that cosmologist hos mad.
See, I can’t be rational here, my desire for the cosmohoes to ragequit on this one has overwhelmed my critical faculties.
Even if the schadenfraude isn’t completely kosher I’m going to indulge in it all the same.
Same, but I know chiral3 has deeper knowledge than me on this subject from before he found more virtuous things to do, so I’d probably defer to him on it if he called BS.
I clicked the link to see which mountebank you were referring to. Didn’t know her, never heard of her, and still I find myself blocked.
No great loss, as I happily accept your assessment.
And, not to sound irked that a nobody about whom I care nothing has blocked me on an app I never go to any longer, what’s that massive thing on the front of her face, above her pie hole?
What a great read, Scott. Thanks.
Daresay, I am highly aroused.
It’s like a newb. math student friend of mine in class in college in Michigan, working towards his actuary degree, working on a proof as a class assignment, and where the first five or so students attempted to prove a solution (with much impressive complexity, harvested from the semester’s learnings); my buddy went up to the chalk board when it was his turn, presented a straight-forward, simple proof. The professor acknowledged the proof was sound.
The rest of the class went “Woah” (much like /Bill and Ted’s Excellent Adventure/).
My buddy surmised that the object lesson of the class was not the math proof, but rather the nature of man to want to over-complicate shit for the sake of complexity.
Which is a metaphor for life.
it’s a serious problem in physics/math to obfuscate with mental masturbation for the theorists rather than getting at the actual physics.
The reality of, say, modelling something in the real world is that the complexity comes from the messy details, which are individually not complicated but interplay into complicated outcomes. However this requires real work, such as writing simulation codes.
It’s more “fun” for these intellectual perverts to just manipulate abstract symbols on a blackboard while sitting in a cosy armchair for a few hours.
Hey, it beats doing manual labor for a living.
Plasma physicists are the most hardcore and underrated of all physicists. It’s not an accident that they usually work in electrical engineering departments in the US: after all, their work isn’t “fundamental enough”.
In particular, they understand non-equilibrium statistical physics better than just about anybody, which is itself the most hardcore and underrated of all physics subfields. It is also criminally understudied.
I work in theoretical modelling of electron devices and the semiconductor physics world has benefited handsomely from the work of plasma physicists, especially the Russian theorists e.g. Klimontovich. I learned more in a chapter of Zubarev’s work on non-equilibrium physics than I would have from the entirety of Peskin and Schroeder’s famous field theory text.
Incidentally, the vast majority of work on quantum phenomena is done with a). steady state solutions of the wave equation assuming that the fastest timescale of the system is slower than the fastest quantum transition and b). at zero temperature or near equilibrium. One can only dream what a Gerson Otto Ludwig type could do with, say, quantum transport of photons…
Is there some book or reference on Zubarev you’d suggest?
The only book I’ve worked through is called non-equiibrium thermodynamics if I recall correctly
This one maybe?
http://libgen.rs/book/index.php?md5=1491567D64D32B195689A8A61A2769A2
That’s it, total classic.
But like most Russians, an excess of frivolous math in lieu of common sense in places. Still a very informative read.
I certainly don’t disagree but, looking back, there was a certain complacency and comfort. I did some work with the MHD and PiC crowd off and on for a few years. Lots of similar pubs. OK, is this better or worse than other areas? I think it was a bit worse. Those DOE grants felt like they were for perpetuity and much of the research felt too similar. Regardless of whether the perpetual “we’re 20 years away” criticism was earned or not the crowd definitely took their foot off the gas in the 90’s.
The plasma experimentalists REALLY impressed me (I was theory). Every so often they’d take something we came up with a build an instrument or calibrate the beast to take a look. Really amazing people in their ability to just fabricate and build anything. And it’s a slight to call them experimentalists because I felt their knowledge of theory was as good as it gets.
Sure, lots of mediocre theorists and incremental progress, the area is ludicrously difficult.
But if I had to bet on either the DFT, quantum Monte Carlo, lattice QCD, molecular dynamics in biophysics, or MHD/PiC/other plasma approach crowds regarding who would eventually make a major, technologically significant breakthrough, it would be the plasma guys hands down, as of the major areas of developing computational physics I can think of, they have by far the best fundamentals.
I would bet on MHD/PiC people long before I bet on DFT, MD, lattice QCD, or quantum Monte Carlo people to make some kind of major computational physics breakthrough in their respective scientific/technological domains due to the superior fundamentals.
That said these fields go through periods of stagnation. The field of quantum transport in semiconductor devices as an example has been stagnant and dominated by convoluted non-equilibrium many body theory approaches for decades which have a slew of crippling issues, but it’s still a legitimate discipline which will eventually yield technology. From my interactions with plasma physicists MHD/PiC and other competitors have fiendishly difficult theoretical and numerical problems.
I am too lazy/dumb to know whether they are “just cranks”, but there are a couple papers on my hard drive purporting to resolve the rotation curve thing with unmodified Newtonian gravity and no exotic dark matter. According to these people the problem is literally just assuming one too many spherical cows.
https://arxiv.org/abs/0803.0556
https://arxiv.org/abs/astro-ph/0309762
And this one from 2005 claims to resolve it with unmodified GR and no dark matter, so, just what Mr. Ludwig did maybe.
https://arxiv.org/abs/astro-ph/0507619
>”If he’s right, it’s going to make lots of people real mad, then sad for their wasted lives.”
You’re a lot less pessimistic about the community than I am. I cordially doubt that “dramatic single-paper owns” are at all possible. The days of the Solvay conferences when everyone could thrash things out and really make progress are ancient history. There is such an ocean of noise now, tidal waves of LaTeX piss, the true goods might be somewhere in there but society will never know it, because science is no longer “agreement-capable” on its own. Now ‘the facts’ can only come by fiat from political power: ‘the science is settled on climate change’. Sort of like the early Christian bishops needing Constantine and his successors to enforce their theology.
https://duckduckgo.com/?q=dark+matter+site%3Acrev.info&ia=web
I’d laugh even harder if the dude was a creationist somehow, *and* he was right. What do the creationists have against dark matter? Other than generally realizing most modern day scientists are just completely full of shit.
This. Once you’ve been lied to about one thing from a group of people, you start looking under the hood. You do this. You just posted a paper that gives an explanation for a phenomenon without the religious claims of the dark matter cult. From the abstract:
To paraphrase Ockham, “the simplest explanation that accounts for all the evidence is most likely to be true.”
Got it. As I said, the math on this guys paper looks legit to me. No real way of testing it, but it ought to make other testable predictions.
I’ll admit that cosmology is way out of my league, but I always felt that dark matter and dark energy were the 20th century equivalent of epicycles to justify a failed model. I tried to inquire online if something as simple as electromagnetic forces or a grossly underestimated number of black holes, neutron stars, etc. could account for the apparent deficit but I was not so politely dismissed as a rube. If the dark matter/dark energy acolytes object without a valid counter argument, I shall know Dr. Ludwig is on the right track.
[…] Source link […]
Haven’t had a chance to review the paper/math, but it’s in my queue now.
Here are some favorite “heretical papers” of mine:
Essen, Hanno, and Miguel C. N. Fiolhais. “Meissner Effect, Diamagnetism, and Classical Physics – a Review.” American Journal of Physics 80, no. 2 (February 2012): 164–69. https://doi.org/10.1119/1.3662027.
Consa, Oliver. “Something Is Rotten in the State of QED,” 2020, 12.
– This one is interesting: I’ve been a little skeptical about claims of “32/100+/some-ridiculous-number decimals of agreement” in the determination of the vacuum-polarization-component of the electron magnetic moment. *What laboratory insturment* could ever produce 32 decimals of anything repeatably? What other knowledge do we have about the world that is remotely that precise, and how could we build such precision from our imperfect knowledge? Claiming that a theory is in agreement with measurement is one thing, making such an outlandish claim of precision and accuracy is fishy.
Getting anything like that out of a process that requires anything as questionable as renormalization is astonishing. The story Consa is telling may indicate that my skepticism is warranted.
Lamb, Jr., W. E.; Appl. Phys. B 60, 77-84 (1995): What are we really talking about when we talk about photons?
Jaynes, Edwin, and Frank Cummings. “Comparison of Quantum and Semiclassical Radiation Theories with Application to the Beam Maser,” n.d.
– Yes, it’s actually possible to build a reasonable semi-classical picture that gets most of laser physics right. In fact, that’s the mental picture that was used by the inventors of the laser, not cumbersome quantum field theory. It’s important to understand the actual mental models used to arrive at an invention, so as not to drown in a complicated and cumbersome formalism when not needed.
Jaynes, Edwin T. “The Gibbs Paradox.” Maximum Entropy and Bayesian Methods, 1992, 1–22.
(and I have Gibbs paper floating around somewhere, but it’s not showing up in my indexing program)
Nice list; I totally forgot to read the Consa paper when it came out (other things were keeping me busy); look forward to fixing that. Remember anti-photon in grad school. Will look at the others.
Presently reading “the physicists” about the American physics community. Only up to 1918.
edit add: Consa seems real uncomfortable with renormalization group theory. It doesn’t bother me at all. Maybe I was brainwashed by Itzykson and Zuber (jeebers I am so old this is a dover reprint now, rather than the ludicrously expensive hardcover I own), but I’m perfectly OK with how perturbation theory works. Works just fine in optics also! The specific fudge factor incidents he mentions (not Schwinger’s) are indeed mostly embarrassing. As I recall Millikan’s estimate for electron mass was fairly embarrassingly off, and it took a long time for people to get it right. Mostly out of cowardice. Even the giants were just people.
Photon one, my old boss at LBNL showed me I think. A classic.
The GEM formulation G.O. Ludwig is using in this paper is actually just general relativity (GR) in a perturbative expansion with the lowest order terms kept. It is nothing to do with MOND at all (unless you think GR is MOND!). If you do that (start from Einstein’s equations in a global background intertial frame and perturb it due to the presence of gravitating source and the source is a slowly (wrt light-speed) moving ideal matter) you can derive the GEM metric in which all terms O(c^-4) are ignored. All this is legit and you can do in a few lines of algebra yourself.
Clearly, GEM will not work in strong source limit and it also is not Lorentz invariant. However, neither is the Vlasov-Poisson equation (collisionless Boltzmann equation) used to describe self-gravitating systems (see Binney and Tremaine text book). In fact, to explain rotation curves using dark matter we only need classical physics + postulate the existing of classically gravitating matter that is otherwise dark.
(Incidentally, Scott, this paper does not really say anything about cosmology but about galactic dynamics. Most galactic dynamics work is done in the classical limit and on global scales [I mean outside of local black-holes and extreme objects] GR effects are need to be accounted for in large-scale structure. I don’t think this paper disproves the existence of dark matter, but merely that it may not be needed to explain galactic rotation curves).
Once you have the GEM equations (which look nearly identical to fluid + Lorentz force + Maxwell equations) you can use standard manipulations which every plasma physicist can do in his sleep to derive axisymmetric equilibrium equation. So Ludwig’s Eq 3.15 have the term “Grad-Shafranov” and “polodial equilibrium” “flux-function” appearing which are absolutely the most important things in fusion physics of tokamaks. So it is not surprising that Ludwig see this more easily that others as I am sure he has internalized these equations completely. Incidentally, he talks a lot about “free boundary” problem. Again, “free boundary” is a term you will hear a lot in plasma physics.
I have not re-derived all his expression but I am pretty certain his equations are correct, though one should certainly cross-check. In fact, the basic idea that rotation curves flatten as one relaxes the assumption of Keplerian circular orbits and centrifugal + central force balance is the key to understanding galactic rotation curves anyway. What Ludwig is proposing is an alternate mechanism (other than dark matter) to do this, i.e. GR effects in weak-field perturbative sense. There is nothing obviously wrong unless one can show that these equations are not valid in the galactic geometry (thin rotating disk with central core halo).
BTW: plasma physicist have a long history of making contributions to GR and many basic fields of physics. For example, the well-know Kruskal coordinates (that remove the coordinate singularity at the event horizon around a Schwarzschild black-hole) were invented by Martin Krusal while at Princeton’s plasma physics laboratory. Legend has it that he presented his idea at a seminar at that lab and was not really interested in writing up the paper on it, but in fact, John Wheeler was the one who forced/wrote it for him!
Other major contributions of plasma physicists besides non-equilibrium stat mech (which is very highly developed in plasma physics) are in perturbation theory, Lagrangian and Hamiltonian mechanics, sympletic methods, study of instabilities (which is extremely highly developed in plasma physics), kinetic theory, very sophisticated numerical methods, etc. Recently, plasma astrophysics has really taken off (after the work in magnetorotational instabilities which basically brought the role of weak magnetic fields into astrophysics) and specially plasma physics around compact object is becoming very critical to understanding energetic processes around black-holes, neutron stars etc.
In any case, I guess one should perhaps study the paper more carefully. I am sure it will be completely ignored by the people who do galactic N-body simulations. That field, TBH, is very sophisticated but the numerics are not completely correct and much of the stuff they do to get any simulations done at all is very hokey (like modify gravity at close range to prevent captures and very strong sling-shots of start). Hence, I think N-body simulations are actually and literally based on a form of MOND in which binaries can’t form! This paper seems pretty solid but again one needs to study it carefully, in particular, to see of the GEM formulation of perturbative GR is valid in the galactic case or not.
GEM isn’t Lorenz invariant? I thought the “magnetism” terms corresponding to any force came out of making the resulting dynamics consistent with special relativity. Your force vectors end up becoming anti-symmetric 2nd rank tensors inducing curvature of the worldline, and magnetism are 3 of the components that show up when tilting the 2-form into a moving frame.
Right, GEM is not Lorentz invariant as the mass and mass-currents do not form a 4-vector. GEM is an approximation and not a full theory of gravity. In fact, in the paper by Ludwig he uses the non-relativistic cold fluid equations coupled to the gravitoelectromangetic field which clearly are (by definition) not Lorentz invariant. However, again, this is probably okay as the collisionless Boltzmann equations are not Lorentz invariant either and speeds are v << c.
Actually, I think there may be a way to derive GEM-like Lorentz invariant equations. I am not sure. So instead of applying perturbations directly to the Einstein's field equations one should instead perturb the Lagrangian in a manner that one is left with relativistically correct weak-field equation term-by-term. Perhaps has been done already. I don't know.
BTW: The approximations that lead to the GEM equations are described for example in Section III.B of Branginsky, Caves and Thorne Phys Rev D vol 15, Number 8 page 2047 and used for many problems in an another of Kip Thorne's paper. Not sure links can be inserted here but search for "Gravitomagnetism, jets in quasars, and the Stanford Gyroscope Experiment" on Google and it is the first hit.
One does not even need to assume a global inertial frames. Just curry a deduction on the first order perturbation of metric and one gets linear equations for perturbations on the background of arbitrary metrics. This is useful for analytical modelling gravitational waves of black holes.
Right, GEM is not Lorentz invariant as the mass and mass-currents do not form a 4-vector. GEM is an approximation and not a full theory of gravity. In fact, in the paper by Ludwig he uses the non-relativistic cold fluid equations coupled to the gravitoelectromangetic field which clearly are (by definition) not Lorentz invariant. However, again, this is probably okay as the collisionless Boltzmann equations are not Lorentz invariant either and speeds are v << c.
Actually, I think there may be a way to derive GEM-like Lorentz invariant equations. I am not sure. So instead of applying perturbations directly to the Einstein's field equations one should instead perturb the Lagrangian in a manner that one is left with relativistically correct weak-field equation term-by-term. Perhaps has been done already. I don't know.
BTW: The approximations that lead to the GEM equations are described for example in Section III.B of Branginsky, Caves and Thorne Phys Rev D vol 15, Number 8 page 2047 and used for many problems in an another of Kip Thorne's paper. Not sure links can be inserted here but search for "Gravitomagnetism, jets in quasars, and the Stanford Gyroscope Experiment" on Google and it is the first hit.
Omg. I love the lack of sanctimony in this blog. Spot on, and a great post 🙂 Looking forwards to reading the paper.
Scott, you type “dork” so frequently that it came out in place of “dark” above 🙂
On purpose.
Even Wikipedia on rotational curves sites papers about using proper General Relativity to describe galaxies, like https://www.worldscientific.com/doi/abs/10.1142/S0217751X0703666X and https://arxiv.org/abs/1810.04445 that show that current assumption that one can use Newtonian mechanics on Galaxy scale at the very least is just wrong. The papers uses very different approaches to modeling, but the general conclusion is that the GR alone is enough to describe the rotational curves.
Then during modelling on computers even Newtonian mechanics is not used. To avoid significant complications with loss of precision when stars are close to each other things like reflection radius are introduced without any mathematical justifications that this avoids global artifacts.
I had a buddy in the physics department, Carlo Rovelli student, who once tried to explain to me the way cosmologists look at the universe is all borked up; aka the idea that things that are farther away are moving faster is an optical illusion. Since I never studied GR I couldn’t understand him, but I believe it anyway. He got a job in tech after grad school and never published it, if it was an original idea.
For what it’s worth I am not sure studying GR helps with cosmology, prima facie. A bunch of time is spent just getting up to solving the linearized Einstein equations and then even more time is spent on exact solutions, like spherically or cylindrically symmetric fields. Whatever time is left is probably taking all the math you learned and cranking on different metrics, homogenous or otherwise, that have come up through the years. Cranking on those calcs rivals quantum field theory in mechanical tedium. That’s a far cry from the messy numerical stuff and the massive zoo of data that constitutes cosmology but maybe the syllabi have changed over the past couple of decades.
I think Sean Carroll responds to the Ludwig article here: https://twitter.com/seanmcarroll/status/1371141258236764160
According to him, there are a lot of other things that only Dark Matter currently explains
Thanks for that. I don’t know who that is, and I was always waaaay out of touch with cosmological wanking since I don’t believe in much if any of it, but I’m pretty sure his statement is false. Nobody would have thought of dork matter without the galaxies thing; that’s a very obvious effect you can see in a small telescope. The fact that you can make cosmic background anisotropies consistent with dark matter is … practically no evidence at all.
Anyway time will tell.
I’m pretty sure this paper contains an elementary error at the very start that calls all the results into question. His equations looks exactly like Maxwell’s equations, but the gravitomagnetism equations actually always differ from Maxwell’s equations by some factor, usually a factor of 1/2, 2, or 4. You can re-scale the fields to move the extra factor around but you can never make them exactly match Maxwell’s equations. The Wikipedia article on gravitoelectromagnetism mentions this, and it’s also pretty easy to derive. And I’m pretty sure it’s not just a typo – there’s a sentence that acknowledges the convention he is using, but not the part about not being able to exactly the square the analogy.
I played around with his definitions and if you rescale his fields to match the form given in a more standard work you can get it in a form where the fields obey all the right equations except you have $div E = 2 * 4\pi G* \rho$. I think this means that he is modeling gravity correctly except that mass density sources twice as much gravity as it is supposed to, and that’s pretty much exactly what dark matter does! Even if I’m wrong about him mistakenly slipping in dark matter the confusion on the field scaling definitions is enough to make me highly skeptical.
I think it more likely you borked it, anon. Looked right to me.
Gravity being 2x per mass density is definitely not what dork matter is supposed to do.
If it’s right then you should be able to rescale the E and B fields to change Ludwig’s equations into the ones this source https://arxiv.org/pdf/gr-qc/0311030.pdf, or the ones in on Wikpedia https://en.wikipedia.org/wiki/Gravitoelectromagnetism.
You can rescale the fields in that paper by Mashhoon to get the Wikipedia equations by doing E –> -E, and B –> -2c B, so those sets of equations are equivalent.
The equations for E and B in Ludwig already match the equations in Wikipedia, except for the force equation. In Wipedia it’s F = m(E + v x (4B)) but Ludwig has it as F = m(E + v x B). If you try to rescale B by 1/4 to make the force equations match it makes the equations for E and B match. You can try and also include a rescaling of the mass density function \rho, but it still leads to a contradiction. Are you seeing a way to transform Ludwig’s equations into a more standard form that I’m missing?
I don’t know if my interpretation about generating too much gravity is the right one, but his equations definitely do not match other sources. And that v x B term is the one that he says makes all the difference, so it’s especially important that it be right.
Maybe you should email him instead of me if you actually think you’re correct. I don’t think you are.