A new version of my recent paper is now available on the arXiv, at https://arxiv.org/abs/2404.18938, with more discussion of gravity, both GR and MOND. For those who are only interested in gravity, and not in the particle physics side of things, the paper can be read as a derivation of MOND from basic symmetry principles. Einstein based his theory of general relativity on the symmetry principle of general covariance – this means the theory should not depend on what coordinates you use for spacetime, so that any observer can use coordinates appropriate to their own gravitational environment, and the physics works out the same whatever the coordinates. My principle is that the theory of gravity should be not only generally covariant, but also covariant under the symplectic symmetries of phase space (three dimensions of relative position, and three of relative momentum).
So first I do a bit of group theory, to work out how these two symmetry groups work together. Then I do a bit of representation theory, to find out where all the tensors used in GR live. Then I see that the field-strength tensor is more or less OK, except that it must be complex rather than real. The Riemann curvature tensor is OK too, and can be interpreted as enforcing symplectic covariance. But the stress-energy tensor is not so good. In GR, this tensor has 10 degrees of freedom, but for symplectic covariance as well we need 20. And this time it is not good enough just to take 10 complex dimensions. Under Lorentz symmetry, this 20 breaks up as 1+9+9+1, so you can see how one might get the idea that 1+9=9+1, and accidentally throw away half the tensor.
But that would be a mistake, because the pieces 1+9+9+1 propagate differently. They have different powers of r in them, effectively 0, -1, -2, and -3 respectively. The -2 term gives us Newton’s inverse-square law of gravity, and the -3 term gives us inertia – thereby showing that gravity and inertia transform differently, so that the Equivalence Principle (equivalence of gravity and inertia) does not hold in a model that has symplectic covariance. The 0 term is Einstein’s cosmological constant (nowadays called dark energy, and no longer considered to be constant). That leaves the -1 term, which does not appear in GR, but does appear characteristically in MOND. It follows, therefore, that MOND can be derived directly from the principle of symplectic covariance, combined only with Einstein’s principle of general covariance.
Well, perhaps I am over-simplifying. The numbers 0,1,2,3 are not really power laws for a single distance r, but are products of distances in different directions. So the number 2 corresponds to an area, and encodes Kepler’s law, that a planet in an elliptical orbit sweeps out equal areas in equal times. The fact that this is equivalent to an inverse-square law of attraction was Newton’s contribution. The number 3 corresponds to a volume. So in a 3-dimensional environment, things have inertia. But in a 2-dimensional environment, such as the edge of a really flat spiral galaxy, inertia fades away – as the Modified Inertia (MI) version of MOND describes. Or perhaps what it is really saying is that in order for the volumes to stay the same, when things get really flat, the areas have to increase over and above what Kepler says.
The number 1 corresponds to a single distance, or to the derivative of an area. Here there is nothing to complicate the issue, and we really do get an inverse linear law of gravity directly from the mathematics. This is equivalent to the Modified Gravity (MG) version of MOND. So it looks to me as though we need a total of THREE modifications to Newtonian gravity. We need both MI and MG – it is not a choice of one or the other – and we need the scalar field that generalises the dark energy/cosmological constant concept.
But I’ve only done the group theory and the representation theory, I haven’t done the physics. In particular, I haven’t calculated the scales on which the four terms take over from each other in dominating the dynamics. The characteristic scale of MOND is usually expressed as an acceleration, but this can be multiplied by powers of the speed of light to convert it to an inverse time or an inverse distance. Similarly, the cosmological constant is usually expressed as an inverse square distance or time. The MOND critical acceleration is 1.2 x 10^-10 m/s^2, and the speed of light is 3 x 10^8 m/s, which gives an inverse time of 4 x 10^-19 s^-1, or a time of 2.5 x 10^18 s or about 8 x 10^10 years. The cosmological constant is about 10^-35 s^-2, which gives a time of about 3 x 10^17 seconds or 10^10 years.
Should we expect these numbers to be equal? Not necessarily, since they are timescales relevant to different directions in space. Should we expect them to be of the same order of magnitude? Probably, because their ratio should be some trigonometric function of some angle that describes our situation in the wider universe, perhaps something like the inclination of the plane of the Solar System to the disk of the Milky Way, or perhaps something on an even larger scale. These times or inverse times relate the 0, 1 and 2 parts of the tensor to each other. There should be a third time or inverse time relating the 2 and 3 parts, but this is another (MI) version of the MOND critical acceleration, and is not currently distinguished from the MG version. Anyway, I can’t really add anything to the existing speculation about these numbers, except to say that there really should be a geometrical relationship between the MOND critical acceleration and the cosmological constant, although I can’t say what this geometrical relationship should be.