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.