The origin of MOND

Einstein’s Field Equations for gravity are written in tensor form, with 10-dimensional tensors that transform under the group of general covariance, that is GL(4,R). Now the representation theory of this group tells us that there are in fact two different sorts of 10-dimensional tensors, that are dual to each other. One represents the symmetric square of spacetime, the other the symmetric square of energy-momentum. If Einstein’s theory is generally covariant, which people tell me it is, then it equates two copies of the same tensor, and ignores the other one.

I have my doubts, as I think the stress-energy tensor is the symmetric square of energy-momentum, and the Ricci curvature tensor is the symmetric square of spacetime, and that Einstein equates these two things that are not equal, and hence gets a theory of gravity that is not generally covariant. But it doesn’t really matter which of these two things he does, or which of them other people have done since. What matters is that, either way, they are only using half of the degrees of freedom that the physics actually has. Hence the theory of gravity is incomplete, whichever way you do it.

Now translating to phase space makes it much clearer where these tensors come from, which is in fact the anti-symmetric cube of phase space. Phase space is 6-dimensional: 3 dimensions of space, plus 3 dimensions of momentum. Therefore the anti-symmetric cube has dimension 6x5x4/3x2x1 = 20. With the full symmetry group Sp_6(R), this tensor is irreducible. Restricting to SO(3,3), which is how GL_4(R) acts on phase space, it splits as 10a+10b, i.e. it has both of the two dual versions of the tensor used in the Einstein Field Equations. Restricting further to SO(3,C), otherwise known as the Lorentz group, it splits as 1+9+9+1.

In particular, the Lorentz group cannot distinguish between 10a and 10b, both of which have the form 1+9. Hence Einstein’s splitting of the Ricci tensor (let’s call it 10a) into the Ricci scalar (1) plus the Einstein tensor (9). But there’s another 1 in the tensor, that Einstein used for the cosmological constant. So Einstein actually used 1+9+1 out of 1+9+9+1. He called it his biggest mistake. But it wasn’t a mistake, apart from the fact that he called it a constant. Because it isn’t constant. It is a constant for the Lorentz group, so it looks like a constant to us. But it is not a constant for the bigger group SO(3,3), so different observers will measure it differently. It is a measure, in fact, of our motion relative to the whole of the rest of the universe, and is therefore an implementation of Mach’s Principle.

Now astronomers have measured the cosmological constant in different ways in different parts of the universe, and verified that in fact it is not a constant. They call this the Hubble Tension, because what they actually measure is the Hubble constant, nowadays called the Hubble parameter, since it is not constant. But it isn’t just one parameter, it is 10 parameters, forming the “other half” of 10a+10b or 1+9+9+1. So let me explain to you what these 10 parameters look like.

The splitting of the antisymmetric cube of position-momentum (as I shall call phase space from now on, since I have split it into 3 dimensions of position and 3 of momentum) into 1+9+9+1 arises from the splitting of the number of position coordinates you have, which can be 0, 1, 2 or 3. If you have two position coordinates, then you’ve got a square of the distance between them, and hence an inverse-square law of gravity in the usual Newton-Einstein sense. If you have no position coordinates at all, you could be anywhere, and that’s the cosmological constant or dark energy term. If you have three position coordinates, you have no momentum coordinates, and the rest of the universe disappears from view, and all you are left with is the rest mass term from the stress-energy tensor.

Then there’s the bit that Einstein left out, the bit with one position coordinate and two momentum coordinates. That’s the bit where you get an inverse-linear term in the law of gravity. That’s the bit that Milgrom discovered in 1983. That’s the bit that he called MOND (modified Newtonian dynamics). That’s the bit that the mainstream has still not discovered. Though they will soon be forced to do so, because their preferred alternative, Dark Matter, does not explain the observations of wide binary stars. The dynamics of wide binary stars prove beyond reasonable doubt that the inverse linear term in gravity does exist, and must be added to standard Newton-Einstein gravity in order to get a complete theory.

There have been many attempts to construct a generally covariant version of MOND, by adding various things to the 1+9 tensors that Einstein used. Typically, they add a scalar field (so that the cosmological constant can vary) or two, and a vector field or two. What they don’t do, and what they must do, is add another tensor. Scalars and vectors are not good enough, it must be a tensor. It is essential to separate 10a from 10b, or else the theory cannot be correct.

That is how to derive MOND from fundamental physical principles, that is from the fundamental principles of Hamiltonian dynamics, laid down in the 19th century. Nothing else is required. Nothing at all. Hamilton could have written down a better theory of gravity than Einstein, if he had ever considered the possibility that Newtonian gravity might be wrong. Einstein knew he had not implemented Mach’s Principle in General Relativity, and therefore regarded GR as a provisional theory. Mach’s Principle requires doubling the size of the tensors, from 10 to 10a+10b, and implies MOND.

That is how to correct classical Newton-Einstein gravity to get a complete theory that agrees with observations. But surely you want more than that? Would you be satisfied with a complete theory of gravity if it wasn’t quantised as well? I know I wouldn’t be. So let’s quantise it. You see, I came up with this 10a+10b tensor first of all in particle physics, not in classical gravity at all. It seemed to consist of left-handed leptons, with a particular choice of first-generation electron (i.e. normal electrons, not muons or tau particles), one copy for neutrinos and the other for electrons. But there are twice as many degrees of freedom as the leptons need. So either you can add in the other two generations, or you can add in the baryons (proton and neutron).

The model looks better, and closer to classical physics, if you add in the baryons. Then the same mathematics that is used for the Einstein Field Equations describes how the weak interaction affects neutrinos, electrons, protons and neutrons. The ordinary gravity terms, with a single factor of momentum, can therefore propagate as neutrinos (and/or anti-neutrinos). The MOND gravity terms, with two factors of momentum, arise from interference between pairs of neutrinos. The Dark Energy term arises from interference between triples of neutrinos. It’s all there in the model. In a NUT shell.

A new paper

Today I have submitted a new paper to the arXiv, but because I expect them to reject it, I am posting it here, at https://robwilson1.files.wordpress.com/2024/04/c3ine8v2.pdf. It is rather technical, and mathematical, but that is unfortunately unavoidable, since what the paper really does is provide mathematical reasons, based on fundamental physical principles, for some things I wrote three years ago, in https://robwilson1.files.wordpress.com/2021/03/lorentz5s.pdf. That paper was also rejected by the arxiv, and by the journal I sent it to, without being seriously considered in either case. I have just re-read that paper, expecting that I would find it excruciating and full of nonsense. But that is not what I found. What I found is that it says more or less exactly what the new paper says about gravity, and talks a lot of sense about physics in general, but includes very little of what the new paper says about particle physics.

The old paper was really a discussion of why the Lorentz group is inconsistent with the known properties of spin of electrons and photons, and why the group needs to be replaced by a different group. The replacement I proposed three years ago is the same replacement that I propose in the new paper. But the justification provided in the new paper is stronger, because it is shown to be consistent both with the standard model of particle physics, and with general relativity.

The paper is written in the language of E8, because I really want to persuade the E8 crowd that this is a better way to look at E8 than the myriad of other ways that they have tried. But I am still not totally convinced that E8 is necessary for this exercise. As far as I can see, what is needed is SU(3,3), a group of type A5, which already contains the Riemann Curvature Tensor. On the other hand, to get three generations of electrons I seem to need to extend to E6. And when I do extend to E6, I get the Einstein Field Equations as well. That seems to be important. Because I don’t get the Einstein Field Equations if I only have one generation of electrons.

And if I do include all three generations, then I get an extension of the Einstein Field Equations from 10 equations to 20, in twice as many variables. So extending to three generations not only reproduces General Relativity, but also provides a physical mechanism by which the gravitational field can propagate, and corrects GR for neutrino oscillations. Well, you know, I’ve said all this before, and I know I will say it again, but I’m getting tired of casting pearls before swine (not you, obviously, you know which swine I mean).

The Total Perspective Vortex

You know, of course, about the Total Perspective Vortex, as described by Douglas Adams. But did you know that it is not fiction, but fact, based on sound physical principles – Mach’s Principle, to be precise? It is actually possible to get real information about the larger universe out of a small piece of fairy cake. And it has been done, sort of. Let me explain.

The experiments are usually done with a glass of water, rather than a fairy cake, but the principle is the same. Every molecule of water feels the rotation of the Earth, knows which way the axis is pointing, knows where the Sun is, and how many days there are in a year. Preposterous! you might say. How do you know the molecule of water knows all that? How do I know? I know because I asked the glass of water, and it told me. It gave me two numbers, the mass ratios of the electron, proton and neutron. And from those two numbers I calculated the number of days in a year to be 363, and the angle of tilt of the axis to be 23.44 degrees.

Preposterous! They say. Absurd! These mass ratios have nothing to do with the motion of the Earth! They are universal constants! They say. I say: prove it. Take your glass of water to the Moon, and do the experiment on the Moon. Then come back and tell me the result. Then we can discuss it. But if you refuse to go and do the experiment on the Moon, then I refuse to accept your argument. The “universality” of these “constants” is no more than a hypothesis, and I adopt the opposite hypothesis, that they are not constants.

One particularly egregious troll suggested my idea was absurd, because it would give a supposedly “ridiculous” answer on Uranus. Well, troll, go to Uranus and do the experiment. Otherwise you are not doing physics, but metaphysics, and your opinions are worth nothing.

Anyway, as I explained in the previous post, neutrinos are the only particles we know about that could in principle enforce Mach’s Principle in the entire universe. So Occam’s Razor says that until proved otherwise, or until some other potential enforcers are found, we can suppose that the neutrinos do enforce Mach’s Principle. Now, when a neutron decays into a proton and an electron, it sends out an antineutrino to tell the rest of the universe what has just happened. This antineutrino joins all the rest of the neutrinos and antineutrinos in an army of law-enforcers to ensure that Mach’s Law is not violated. And notice also that this antineutrino conveys not the mass of the electron, but its total energy. This is the information that General Relativity needs to know in order to enforce the Law of Gravity.

The neutrinos, in other words, are the secret agents of General Relativity. And I mean that completely literally: they are secret, because they are almost impossible to detect, and they are agents, because they act, that is, they do the work. If you want to know how they work, ask the KGB. Oh, no, that won’t be any good – of course, the KGB know how neutrinos work, but they are not going to tell you.

There is no rest for the wicked

Neutrinos are wicked. Of that there can be no doubt. They simply refuse to obey the rules that physicists have laid down for them. Physicists pray for them at mass three times a day, but they refuse to be seen with mass. They refuse to rest, but are always gadding about at (almost) the speed of light, going straight through (almost) everything without stopping, and are very hard to catch. If you want to catch a neutrino, you have to build a huge tank of dry-cleaning fluid (or something else containing lots of chlorine) underground, and wait a long time for one to show up, among all the zillions that whizz straight through the tank without even saying hello. Even then, you never see the neutrino, all you see is the reaction of one of the chlorine atoms when it gets hit.

Neutrinos are produced in nuclear reactions, and physicists have worked out exactly how the nuclear reactions in the sun work, and so they worked out exactly how many neutrinos the sun should produce, and exactly how many they should detect in their tanks of chlorine (or whatever it was – it may have been xenon). And how many did they find? One-third as many as they predicted. Oops. Experiment says theory is wrong. Theory says, wait a minute, we’ll change the theory. Hmmm. Dangerous move, that. But they changed the theory.

You see, neutrinos come in three versions, like triplets, and just so you can tell them apart, one of them wears a red jumper, one wears a blue jumper and one wears a green jumper. The neutrinos produced in the sun should all have been wearing red jumpers, so the experiment looked for the red jumpers. But it turns out that the neutrinos that showed up were wearing any old jumpers, they didn’t care whether they were red, green or blue. So the theory had to cook up a way to make the jumpers change colour. Well, I’m not going into the details of that, but the only way that physicists could think of explaining this was to say that the neutrinos have some rest mass. Originally they were supposed to be massless, like photons of light, but this strange phenomenon of the jumpers changing colour could apparently only be explained if the neutrinos had mass, and the different coloured jumpers had different masses.

Well, now, any time you invent a new theory like this, you have to test it. That is the first rule of science. If you have an idea, you have to test it. No ifs, no buts, no excuses. You have to test it. And you are not allowed to say the idea is correct until it has been confirmed by experiment. That is the second rule of science. So they tested it. Did they confirm it? No. Not yet, anyway. So they are not allowed to say that neutrinos have mass, are they? Of course not. This hypothesis has not been confirmed by experiment, and therefore it is against the rules of science to say that it is “true”, or “known”, or “a fact”.

Practically every physicist on the planet, almost without exception, says that it is “known” that neutrinos have non-zero mass. This is simply FALSE. Yet when I point this out to them, they do not change their tune at all. The statement that neutrinos have a non-zero rest mass (which is equivalent to the statement that they have a state of rest) is a hypothesis, that has been neither confirmed nor refuted by experiment. It is not a “fact”, it is not “known”, it is not “true”. It is a hypothesis.

I hypothesise the opposite: I hypothesise that there is no rest for the wicked neutrinos. My hypothesis is not a “fact”, it is not “known”, it is not “true”. But until experiment provides a conclusive answer, my hypothesis is just as good as the other one. Provided, of course, that I can come up with an alternative explanation for why the jumpers appear to change colour. Which I have done: the jumpers are multi-coloured, and the colour that you see depends on which way you look at it. What does this mean? It means the colour depends on which direction is up. And since the Earth in NOT flat, despite what many physicists seem to think, the colour depends on where you are. And since the Sun is also NOT flat, the colour also depends on where the neutrino came from.

You cannot define mass without defining rest. You cannot define rest without taking into account the entire universe. Nobody has ever caught a neutrino and made it sit still. Nobody ever will. Neutrinos do not have a “rest” state. They do not have a “mass”. Yes, they fall “down” in a gravitational field, but so do photons, and nobody pretends that photons have a “mass” or a “rest”. We detect photons that have been travelling for 13 billion years without ever taking a rest. We detect neutrinos that have been travelling in convoy with photons for millions of years, and arrive within seconds of each other – we know they set out within seconds of each other, but we don’t know the precise timing.

And what exactly is the purpose of these wicked neutrinos in the grand scheme of things? Why did God put them there? What use are they? I’ll tell you – they are there to tell you where the rest of the universe is, and where it is going. They are there to implement Mach’s Principle. They are there to tell you exactly how you are moving with respect to the rest of the universe, and exactly how much resistance the universe is going give you if you try to move differently. They are there, in other words, to define inertia. They are there, in other words, to define rest. They do not have inertia, they define inertia. They do not have rest, they define rest. They do not have mass, they define mass, in exactly the same way that photons do not have electric charge, but they define electric charge.

General relativity from quantum mechanics?

I may have been a bit hasty to dismiss GR in my most recent paper, posted here a couple of weeks ago at https://robwilson1.files.wordpress.com/2024/04/camps3.pdf. What happens in that paper is that I work with 3×3 matrices over the split quaternions H’, in order to implement phase space, and I find that they split as 1+14+21, which is compatible with QM, and not as 1+15+20, as would be required for GR. However, when I tried to embed the model into E8, in order to try and get the E8 crowd interested, I found that the embedding I wanted was A1+A2+A5 (which I’ve written about before, but not managed to get onto the arxiv), and that I had got the wrong real form of A5.

I’m not sure exactly how many real forms A5 has, but it is at least seven, of which two occur in the semisplit magic square that underlies most of the E8 models that people consider, including our “octions” model https://arxiv.org/abs/2204.05310. The real form I was considering occurs in the split magic square only. The 3×3 matrices over ordinary (non-split) quaternions were studied in some detail in the octions paper, as they formed one of the main stepping stones to get to E8. But I am now convinced that the octions paper got it wrong, and should have transposed the magic square, and used the other real form, which is SU(3,3). And I also got it wrong, and should have used SU(3,3).

The reason is that SU(3,3) not only contains Sp_6(R), which is the symmetry group of phase space in Hamiltonian mechanics, and is, as I have recently shown, the formalism that underlies the Dirac algebra in (relativistic) quantum mechanics, but also contains SO(3,3), which is how the group of general covariance in GR acts on phase space. In other words, we can combine the two provided we complexify phase space. This is not even a crazy idea, it was already done by Maxwell in the 19th century. Complex phase space allows one to treat momentum and current at the same time. If you can’t treat momentum and current at the same time, then you can’t do electrodynamics at all! And it was done by Dirac in 1928 in the context of QM. So it is known to be necessary, and you may well ask, why I thought I could get away without it? Good question, to which I do not have an answer.

Anyway, now that I know it is necessary, I know that I have to study the action of SU(3,3) on complex phase space. This group has got everything in it that is needed for physics. It’s got the Dirac algebra for QM, it’s got general covariance and the Riemann curvature tensor for GR, it’s got twistors if you want to follow Penrose, it’s got Hamiltonian symmetries of phase space if you want to follow Bohm/Hiley/de Gosson, and its centralizer in E8 is SU(3) x SL(2,R), which is the correct real form for the gauge group of the Standard Model of Particle Physics. What more could you want? The Moon?

Maybe you want to see the Riemann curvature tensor, because that’s the bit that no-one believes can be there in quantum mechanics. As a representation of SO(3,3), the RCT is usually constructed by first converting to compact SO(6), then taking the symmetric 6×6 matrices and subtracting off the identity matrix. If you want to do this with SU(3,3), then what you get is a 21-dimensional irreducible complex representation, in place of the real 1+20. In other words, we get a significant generalisation of the RCT, and a corresponding extension or correction to GR.

On the other hand, it may be worth noting that SU(3,3) has got a real 20-dimensional irreducible representation, that is constructed in quite a different way, as the anti-symmetric cube of the natural 6-dimensional representation (instead of the symmetric square). Is this what the RCT is really trying to be? I have no idea. BUT this representation plays a prominent role in the E8 model, and represents (among other things) the left-handed leptons. There are two copies of this representation, one for neutrinos, one for electrons. So does this mean that the RCT is trying to describe neutrinos? Can this possibly make sense? Instead of working in the fourth power of spacetime, we work in the cube of phase space? The cube of phase space breaks up as 1+9+9+1 if we separate position from momentum, whereas the RCT breaks up as 1+9+10 if we restrict to SO(3,1). So we get the Einstein tensor coming out, but not the (spin 2) Weyl tensor. So we get something like the field equations, but not the spin 2 graviton. Instead of the spin 2 graviton we get three directions of momentum times three directions of distance-squared. Well, if that isn’t quantum gravity travelling at the speed of light then I’ll eat my hat!

But, did you notice that there are also 3 dimensions of momentum squared times 3 dimensions of distance? That means that, in addition to the 1/r^2 ordinary Newtonian gravity, there is a 1/r type of gravity, which is what is needed for MOND. Moreover, there is a transition between the two that occurs when the momentum/distance relation swaps over. Why does this occur empirically at a particular acceleration scale? I have no idea. But one thing is for sure, this isn’t caused by Dark Matter. It is caused by neutrinos interacting with each other, because it has a momentum-squared term – one momentum for each neutrino.

But, did you also notice that there is a momentum-cubed term? Independent of distance? Is that what Dark Energy really is? Interactions of three neutrinos at once?! No wonder the cosmological constant is so small! No wonder it is 120 orders of magnitude smaller than particle physicists think it is!

So, are we any nearer to a resolution of this conundrum? Is Einstein’s GR like Eric Morecambe’s piano concerto? All the right notes, but not necessarily in the right order?

Zombie physics

The main theories of fundamental physics died several decades ago, and have been wandering around like the undead ever since. General relativity, Einstein’s beloved theory of gravity, died in the 1980s, but this fact has still not been noticed by relativists, who still follow it like zombies, unable to smell the rotting flesh, or see the bones sticking out. The 1980s was the time when measurements of galaxies became accurate enough to demonstrate pretty conclusively that Einstein’s theory could not explain the observed structure of spiral galaxies. What did the theorists do, when faced with incontrovertible proof that the theory was wrong? They went into denial, and said the experiments must be wrong. And when the experimenters said, no, there’s nothing wrong with the experiments, the theorists said, well then, nature must be wrong. So they invented something called Dark Matter, to try and rescue their theory from certain death.

Of course, this strategy didn’t work, but it didn’t stop armies of zombies from believing in it. They “know” Dark Matter exists, despite the fact that 40 years of diligent searching has turned up precisely no evidence for it. Now, with the latest tests of gravity on widely-separated binary stars, even Dark Matter cannot rescue the theory. Now, everyone can see that the theory is well and truly dead. Well, you would think so, wouldn’t you? Unfortunately not. Believers in the magical properties of Dark Matter simply say it must be even more magical than we thought. Poppycock!

Why is GR wrong, and what can we do about it? The diagnosis I came up with ten years ago, and which still appears to be the correct diagnosis, is that there is no consistent definition of mass in physics. That is why I looked at all the experimental evidence for what mass really is, at all scales from a single neutrino to the entire visible universe. And then I looked at the theories, and analysed where the inconsistencies in the theories lie, buried deep in the mathematics.

So when I heard Basil Hiley, in his talk a few weeks ago, bemoan the fact that there is no (consistent) definition of mass in particle physics, and highlight this as one of the central problems, I pricked up my ears. And strangely enough, I found that he had given me the clue I needed, and that I had neglected, namely to look at the problem from a strictly Hamiltonian viewpoint. So to cut a long story short, after a few weeks’ work, I found a consistent definition of mass. It isn’t very difficult. Just take the symmetry group of Hamiltonian physics, as a group of 3×3 anti-Hermitian quaternion matrices, make it act on the Hermitian matrices, which represent fermions (matter) and form a Jordan algebra under the Poisson bracket {A,B}=AB+BA, and project onto the identity element of this algebra.

When I say I found a consistent definition of mass, what I really mean is that I found the consistent definition of mass – this is the only definition that is consistent with the Hamiltonian formulation of mechanics. In particular, it does not satisfy the Dirac equation. Which is hardly surprising, since the Dirac equation is based on Einstein’s mass equation, which is based on the theory of special relativity, which is inconsistent with the symmetries of Hamiltonian mechanics. Unfortunately, there is another herd of zombies following the undead Dirac equation, who will slaughter any messenger who has the temerity to deliver the long overdue message that the Dirac equation is dead.

It is hard to deliver a message that says the basic definition is wrong – zombies will say, it’s a definition, definitions can’t be wrong, calculations can be wrong, proofs can be wrong, theorems can be wrong, but definitions can’t be wrong because I can define mass however I like. Up to a point, that is true. But definitions can be useful, or they can be not useful. The Einstein/Dirac definition of mass has outlived its usefulness. It is no longer useful, and needs to be replaced.

There is no denying the inertia of the zombies who don’t want to change the definition, but I don’t think they appreciate the true gravity of the situation.

Wu meets Mach

The Wu experiment conducted in 1956 is arguably the most important physics experiment conducted in the whole of the 20th century. If you read about it in wikipedia, for example, you will get a very clear idea of its importance, even if you don’t understand all the details (which I don’t). You will then be surprised to learn that Wu did not get a Nobel Prize for this work. When you discover that the two men who suggested the idea got the Nobel Prize, whereas the woman who designed, built and ran the experiment did not, you will no longer be surprised. You will be outraged.

Anyway, I don’t want to get into the politics of the Nobel Prizes, or the extreme irony of some of the awards (for example, the Nobel War Prizes). I want to explain how the interpretation of the results of the Wu experiment led the field of physics into a dead end, from which it cannot extricate itself. The interpretation of the results as being internal properties of the cobalt 60 atom, completely isolated from its surroundings, led to the Standard Model of Particle Physics that we have today. But you cannot isolate any experiment from gravity, and therefore the interpretation of the atom as being isolated from its surroundings is physically absurd.

If you think the atom is spinning in isolation from the rest of the universe, then you are in violation of Mach’s Principle. So let us see what happens if we analyse the Wu experiment with Mach’s Principle in mind, so that we couple the weak interaction to gravity, and interpret the results as being influenced by gravity. This will lead in a completely different direction of interpretation, not only for this experiment, but for the whole of the Standard Model. Mach’s Principle says that the atom can detect the rotation of the Earth around it. Wu’s experiment says that the atom does detect the rotation of the Earth around it.

That is the real significance of the Wu experiment. It underlines the importance of Mach’s Principle at the quantum level. Mach’s Principle is completely ignored in particle physics, and almost completely ignored in relativity. Until Mach’s Principle is incorporated into the foundations of quantum mechanics, no further progress in this area is possible.

The cobalt 60 atom, or the weak force in general, detects not only the rotation of the Earth on its own axis, but also detects the tidal forces of the Sun and the Moon. Because the theory has to be “renormalized” – which is code for using the symmetries of Hamilton’s equations to transform your coordinates to a different (usually hypothetical) “observer” – you don’t detect the sizes of anything, you just detect the directions they come from. So you detect the tilt of the Earth’s axis, and you detect the angle of inclination of the Moon’s orbit. That is all. You cannot detect anything else. Oh, apart from the latitude of the experiment, which tells you the direction of the centre of the Earth.

The Standard Model has been built to be (mostly) independent of the latitude, so we have two other angles to deal with. They have to be incorporated into the SM, if you refuse to couple the weak force to gravity. So they are. In fact, they are incorporated in very strange (pun intended) ways. The inclination of the Moon’s orbit appears in the fact that the mass ratio of charged to neutral kaons is the (average) cosine of the angle. Strange, but true. The kaons have strange quarks in them. No, I have not taken leave of my senses. I am not hallucinating. All I am doing is applying Mach’s Principle to the Standard Model. Somehow, the mainstream prefer to think that they can decouple their experiments completely from gravity, an assumption that is known to be false, rather than grapple with the confusing and counter-intuitive consequences of Mach’s Principle, rigorously applied.

The other angle appears in various places. For example, there is an angle called the “CP-violating phase”, which is also derived from properties of kaons, specifically the property that there are two orthogonal “states” which are distinguished by whether they decay into an even or odd number of pions. The experiment that detected this property sent some kaons, carefully cleaned to be purely odd, across the lab, and found some kaons had broken the rules and decayed into two pions instead of three. Naughty, naughty kaons! They must be punished! This was in the old days, before CP was outlawed, so you can imagine what happened. Nowadays, it’s just a phase they’re going through. Anyway, the actual angle that made its way into the SM (oh, dear, I’m going to get into trouble with these acronyms, aren’t I?) is the complement of the angle of tilt of the Earth’s axis. You see, the experiment cannot be isolated from gravity, so Mach’s Principle says it can detect the motion of the Earth. And it did. What it actually measured was the change in direction of the centre of the Earth. The angle of tilt arose from the theory by adding in the tides, that had been subtracted off by the interpretation of the Wu experiment.

Ah yes, you wanted to know where the electro-weak mixing angle came from, didn’t you? It’s the sum of the angle of tilt of the Earth’s axis, and the angle of inclination of the Moon’s orbit. Well, not exactly, because the particle experiments are more difficult to do than the astronomical ones, but it’s pretty close. So you see, the two angles that you must see in particle physics, simply from applying Mach’s Principle, do appear in the SM, individually and together. So why beat yourself up with SM and CP, just to avoid the Hamiltonian experience of putting yourself in someones else’s shoes?

Physics on a Flat Earth

Today I want to explain what physics looks like on a Flat Earth. It looks, in fact, exactly like the Standard Model of Particle Physics. Particle physicists are finding more and more anomalies, that prove beyond reasonable doubt that in fact the Earth is Round. I have tried for ten years to explain to them that the Earth is Round, like the Emperor’s New Balls, but they just say “Emperor’s N.E.W. Bollocks!” and ignore me. They do not consider it possible that a 12-year-old child can see things they can’t see.

So, let’s begin by pointing out the obvious: the Large Hadron Collider is a Large Horizontal Collider. It is flat. Almost all other experiments are horizontal. More or less the only ones that are not are the neutrino experiments, that measure neutrinos fired through the Earth from one side to another. All such neutrino experiments produce anomalies, called neutrino oscillations. The neutrinos don’t come out the same as they went in. This anomaly contradicted the Standard Model at the time, so they added neutrino masses to the model, and pretended everything was hunky-dory. But it’s not, because they can’t measure these predicted masses.

But I’m jumping the gun here, let’s go back to 1887, and the Michelson-Morley experiment. This experiment is usually explained as a measurement of the speed of light, but actually it was a search for Dark Matter (luminiferous aether). Like all other dark matter searches since then, they didn’t find any. They were expecting to see a difference between the speed of light measured in different directions at different times of year, but didn’t find any. So after that people started to assume that the speed of light was the same in all directions at all times and in all places. But they haven’t tested that properly, and in fact it is false, as detections of gravitational lensing of light from distant galaxies demonstrates.

Essentially, this demonstrates a failure of the scientific method, because the Michelson-Morley experiment predicted this phenomenon, which means it cannot also be used to test it. This applies not only to the original experiment, but to every subsequent experiment that tests essentially the same thing. And since the Michelson-Morley experiment was a horizontal experiment, you must at the very least test it vertically as well. The same applies to particle physics experiments, like the kaon decay experiments, the muon gyromagnetic ratio experiments, and the rest. You must test your conclusions vertically as well as horizontally if you want to use them vertically as well as horizontally. You don’t test your car on a road and then assume it can fly, do you? So why do particle physicists assume their cars can fly?

So let’s get back to Michelson-Morley, and the theoretical analysis of it. Let’s suppose that it actually has been tested well enough horizontally, and that it makes sense to assume the speed of light is the same North-South as East-West, and anywhere in between. Then you get Lorentz to work out the transformations and you build a symmetry group SO(2,1) on the two horizontal directions and time, and everything works out nicely. And then you find that you get a dual action of SO(2,1) on momentum and energy, and then you write SO(2,1) as 2×2 matrices acting by conjugation on another 2×2 matrix, and then there’s a scalar matrix which represents the mass, and the mass is the same for all Flatland observers, so mass is nice, and it looks like the square root of a determinant in the matrices, and the square root of a metric on the SO(2,1) spacetime Flatland, so you think these things aer equivalent, and off you go.

Then you pretend you know what happens in the third dimension as well, even though you really shouldn’t do this without testing it first. Now you have a problem: if you work with the metric, then you extend from SO(2,1) to SO(3,1); but if you work with the matrices, the you extend from GL(2,R) to GL(3,R). You cannot translate from one to the other, because there is no translation. You need to square root the metric, but cube root the determinant. Which one is right? This is a serious question, and a serious difficulty. The problem, though, is that nobody ever asked this question. They just assumed the metric was correct and went with that. Unfortunately, the metric is not correct. The determinant is correct.

Well, they pretended to test the model vertically. The measured muons falling through the atmosphere, at very high speed, and said, aha, these muons live much longer than the ones we make in the laboratory, that proves it. Not really. These muons weigh much less up in the atmosphere, and they are essentially in freefall, hitting very little on the way down, and carrying a huge amount of energy so they just smash everything out of the way anyway. It is only when they sit still that they lose energy and decay. If you use the correct GL(3,R) model, you have to work with the 3-dimensional weight vector in order to calculate the mass, and you get a completely different answer for vertical muons than you get for horizontal muons. That’s a fact, and it has been detected in the muon g-2 experiments, where their muons are almost perfectly horizontal – but the vertical direction changes from one side of the experiment to the other, and you cannot keep a perfect circle perfectly horizontal unless you live on a Flat Earth. That’s the experimental anomaly – their prediction is correct on a Flat Earth, but nature abhors a flat earth.

Einstein used the metric to devise his theory of gravity. He did not use the determinant. Therefore his theory of gravity is a Flatlander’s theory of gravity. It may work quite well sometimes, but it is based on the assumption that the Earth is Flat, so it will fail at some point. And when it fails, it will fail spectacularly. Or to put it more accurately, when it failed it failed spectacularly. Unfortunately, not many people accept that it has failed at all. A very large number of Republicans refuse to accept that Trump lost the 2020 election. The Democrats (particle physicists) of course would be very happy if Trump lost, and particle physics could rule the world. But the political system is so bizarre, that the particle physicists and the relativists are locked in continual combat, so that no progress can ever be made. Relativists continue to defend the indefensible, string theorists invent ever more ridiculous theories of aliens, gods pushing the Sun and Moon around, etc etc,

But this isn’t a political debate, and the particle physicists are just as bad – they also live on a Flat Earth. Let us see how they describe the decay of a neutron. They start off with a group U(2). When I was a student, U2 was a boy band, not a rock group, and certainly not a Lie group. Anyway, U(2) is just the wrong name for the group they actually use, which is GL(2,R). Exactly the same as the group that should be used for special relativity in 2 dimensions. At this point they go to extreme lengths to ensure that they cannot confuse these two copies of GL(2,R), one of which they mangle into SO(3,1) and call SL(2,C), and the other of which they call U(2). But actually, if you do the mathematics properly in Hamiltonian phase space, they are in fact the same group. How do I know that? I read up the experiments. The Wu experiment that put radioactive cobalt-60 in a magnetic field at about 1/1000 of a degree above absolute zero detected that the electrons came out in a specific direction relative to the magnetic field. Awesome. That means that the direction of the gravitational field that defines the group GL(2,R) for light and electromagnetic fields is the same direction that defines the group GL(2,R) for beta decay.

Well, not quite. It’s a bit more complicated if you work in 3-dimensional space rather than just 2 dimensions. Both groups sit inside GL(3,R), in a related way. That means that between them they break the symmetry between the three directions in space. This is what particle physicists call chirality. But what it means in terms of the directions up/down, North/South and East/West is that if you turn yourself upside down, then either you face in the opposite direction from before, or you’ve swapped your left with your right. Or if you want to swap left with right, you either have to turn round and go backwards, or turn upside down. But the particle physicists’ problem is, how do you tell left from right in an absolute sense? They think that beta decay answers this question. But they are wrong, because they have forgotten to couple to the gravitational field. The masses of the proton and neutron are almost but not quite equal. You can therefore detect the chirality of beta decay because it couples to the chirality of the tidal forces of the Sun and the Moon caused by the rotation of the Earth.

There’s much more where this came from. I could write a book about it. Perhaps I will one day. Perhaps I already have. But the Sun is shining, and I want to go outside and feel its electromagnetic field as well as its gravitational field.

Complete Algebraic Model of Physics

I’ve adopted a slightly less provocative title for the paper itself, that is, Clifford Algebra Model in Phase Space, but I am sure you can read between the lines. It’s not ready to go to the arXiv yet, so you can read it here (https://robwilson1.files.wordpress.com/2024/04/camps.pdf) first. Carry on CAMPing, happy CAMPers!

Update: There is now an arXiv version (somewhat surprisingly) at https://arxiv.org/abs/2404.04278/ and a further update here at https://robwilson1.files.wordpress.com/2024/04/camps3.pdf. I’m still working on it, so there may be further updates in due course.

Mach’s Principle, or the Equivalence Principle, that is the question.

You remember Mach’s Principle? It basically says you can tell when you are rotating relative to everything else. We know it’s true, because you get dizzy when you are spinning. That is something that an 8-year-old definitely understands. Meteorologists understand it, because the Earth gets dizzy when it spins, and that dizziness is what we call weather systems, driven by what physicists call the Coriolis force. Physicists say this force is “fictitious”, but put them in the path of a hurricane, and I think they might change their tune. It may be fictitious in their minds, but you can damn well feel it!

You remember Einstein’s Equivalence Principle? It basically says you can’t tell the difference between acceleration and gravity. Which means, you can’t tell the difference between inertia and gravity. And it means you can’t tell the Earth is rotating except by looking at the Sun or the Moon or the stars. So although it is often said that Mach’s Principle is part of Einstein’s General Theory of Relativity, this is actually not true. Strip away all the obfuscation, and Einstein’s Equivalence Principle says you can’t tell that the Earth is rotating if you don’t look at the sky. Bottom line. Mach’s Principle say the exact opposite, it says you can always tell whether something is rotating by seeing if it gets dizzy or not.

So don’t believe any of this crap that says Einstein based his theory on Mach’s Principle, it’s a load of bollocks. He said the exact opposite. He actually tried to apply Mach’s Principle to quantum mechanics in a wonderful paper written in 1919. This paper has been written out of history, “cancelled” in the modern vernacular, because it is not politically correct. Never mind that it is physically correct, like all the rest of Einstein’s work that I have read. But even he became convinced by the woke response, and abandoned this line of enquiry. If he had pursued it, he would have eventually realised he had got GR wrong, and developed a better theory of gravity. But he didn’t. His work was “cancelled”.

I first applied Mach’s Principle to quantum mechanics in 2014, and by January 2015 I had dug up enough experimental evidence to prove beyond reasonable doubt that Mach’s Principle does apply to quantum mechanics. You can tell the Earth is rotating by looking inside a hydrogen atom. If the Earth wasn’t embedded in the Solar System, you’d never detect a fine structure “constant”, you’d never detect a “Lamb shift”, no hyperfine structure, it just wouldn’t be there. You can tell the Earth is rotating by measuring the mass ratio of electron to proton, and you can detect the existence of the Sun by measuring the mass ratio of proton to neutron. You can detect the existence of the Moon by measuring the mass ratio of charged to neutral pions, and if you want to know where the Moon is, you can measure the mass ratio of charged to neutral kaons. If you want to know how big the Earth is, you can count the change in the ratio of odd to even pion decays of neutral kaons as the kaons travel across the surface of the Earth. Or you can measure the gyromagnetic ratio of the muon, although that is hard. What about the W mass anomaly? It means there is something in the rotation of the Earth that has been detected by this experiment. I don’t know what exactly it is, but I think it involves both the Sun and the Moon, and the latitude on the Earth’s surface where the experiment was done.

It’s not as though I’ve pointed out one numerical coincidence that you can ignore. Everywhere I look, experiment proves that Mach’s Principle holds, and Einstein’s Equivalence Principle does not. The evidence is literally everywhere. But the mainstream doesn’t understand Mach’s Principle, so they ignore it. And they “cancel” any experimental evidence that supports it. Mach’s Principle is apparently not politically correct, because it contradicts the Equivalence Principle. But I am afraid that some Principles are more equivalent than others, whatever you might think.