It has been obvious for the best part of half a century that there is something wrong with General Relativity, Einstein’s theory of gravity that is supposed to explain how the universe fits together. I won’t rehearse the evidence here – you can find lots of it on tritonstation or darkmattercrisis (see blogroll). But the difficult question is *what* is wrong with it? And how do we put it right? Well, I’m glad you asked…

There are many things wrong with GR, but the most basic and most important is that it is based on the principle of conservation of mass: the principle that the total mass of an object stays the same (though if it falls apart, burns or explodes, you might have some trouble accounting for all the little bits of it). But we now know that mass is not conserved, for example in radioactive decay on Earth or nuclear fusion in the Sun. We can hardly blame Einstein for this, because these experiments were decades in the future at the time he devised the theory.

You might also say, does it matter? These changes in mass are small details, and if you account for all the *energy* lost in the process, surely everything will be all right? Unfortunately, it is not a small detail, it is a fundamental principle, and it is wrong. It means that the symmetry group of the theory is wrong, because it does not take account of the fact that mass can change. Einstein used the Lorentz group SO(3,1) under which mass is both conserved and invariant. He extended to “general covariance”, which means you can use any coordinates you like for spacetime, and still get the same answer. That means you can use any coordinates you like for momentum and energy, but you are not allowed to change your mass coordinates.

That is why it doesn’t work properly: in the real universe, you have to be able to change your mass coordinates. Your theory has to be covariant under SO(4,1), not generally covariant, which means covariant under GL(4,R).

Which brings me to another problem. Despite the advertisements, GR is not a theory of gravity. Let me explain. Newton’s theory of gravity was a theory of matter: how matter moves relative to other matter. It was not a theory of how matter moves relative to space. This is important, because it means you do not need a physical “space” in which matter moves. In any case, the existence of such a “space” (called “aether”) was already long discredited by the time of Einstein’s GR. But strange to tell, Einstein’s theory is a theory of spacetime. It is a theory of how spacetime moves relative to matter. But there is no such thing as spacetime, so how can it move?

Well, you may say it’s just a mathematical abstraction that is useful in the equations, and that doesn’t mean it has a physical reality. That is the same way physicists try to explain away the wave-functions in quantum mechanics, and it fails for the same reason: reality itself disappears, and we all just become figments of the physicists’ fevered imaginations. A theory of gravity must be a theory of how matter moves relative to other matter. For that we need the concepts of mass, momentum and energy. Nothing else. Einstein’s mass equation tells us the symmetry group here is SO(4,1).

Two clumps of matter are each described by 5 coordinates: one for mass, one for energy and three for momentum. The force between them (if we assume it is instantaneous, and Newton’s third law applies) is an antisymmetric tensor in these coordinates, so is 10-dimensional in total. That is equal to the 10 dimensions in the Einstein field equations, but they are not the same. I’m going to have to spell this out in detail, I’m afraid. Hang on to your hat.

Newton had one of these 10 terms, namely m1.m2 (the mass of the first object times the mass of the second object). Einstein generalised this by adding three mass x momentum terms, three momentum x momentum in the same direction, and three momentum x momentum in perpendicular directions. All for the sake of changing m1.m2 to E1.E2, in other words using total energy instead of rest mass. The result of this is simply to change the Lorentz group SO(3,1) to SO(4), which was a lot of effort to go to in order to make no progress at all.

Now if we use the correct group SO(4,1), and the anti-symmetric tensor instead of Einstein’s symmetric tensor, then we don’t get any m1.m2 terms or E1.E2 terms, what we get instead is m1.E2 – E1.m2. Interesting, wouldn’t you say? If there are no momentum terms, then this is all there is. In Special Relativity, this collapses to zero, because the masses are constant and equal to the energies. But the reality is more complicated. The masses are not constant, and because the force propagates at the speed of light, the masses of the two objects are measured at different times, and this difference in mass is what causes the force of gravity. In Newtonian terms, m1 and m2 are the inertial masses, and E1 and E2 are the (active) gravitational masses. But it is the time delay due to the finite speed of light that causes the gravitational constant G to be non-zero.

Now consider the gravity of the Sun. It takes 8 minutes for this gravity to reach us. During those 8 minutes the Sun has burnt a lot of hydrogen to make helium, and has lost a significant amount of mass. So we think the Sun is more massive than it “really” is. Where has that mass gone? It has gone into neutrinos. Lots of them. Where have those neutrinos gone? Through the Earth. Did the Earth notice? Yes, it did. Not much, but a little. What did the Earth do when it noticed? It fell a little bit further towards the Sun. In other words, the Earth is measuring the rate of decrease in the mass of the Sun. Isn’t that clever? That is how gravity works. You heard it here first.

August 14, 2022 at 9:09 am |

So far I have only considered the Newtonian term m1.E2 – m2.E1 in the tensor, which gives a good approximation to gravity when the momentum is small. But there comes a point where the momentum can no longer be considered small, and the other terms in the tensor need to be taken into account. Examples may include (a) a spacecraft on a high-speed flyby of Earth to pick up speed, or (b) a spacecraft travelling at high speed out of the Solar System. In both cases, GR gives incorrect predictions, although in case (b) this has been covered-up by increasing the alleged experimental uncertainty. In case (a) correction for the finite speed of propagation apparently fixes the problem, taking into account the speed of travel of the spacecraft, and the rotation of the Earth. My tensor does much the same thing, by including the momentum terms.

At what point do the momentum terms become important? In my model, they become important as the velocity increases towards the speed of light. But that is because my model uses the change in mass, rather than the mass itself. If we use the change in momentum that my model requires, and compare with the standard Newtonian mass instead of the rate of change of mass, then the critical point becomes an acceleration rather than a velocity. That is what MOND has observed.

One of the big problems in MOND is to explain this critical acceleration, which appears to be a universal constant, approximately equal to the speed of light divided by the age of the universe. My model suggests that it is not a universal constant, but a calibration of mass against change in mass in order to derive Newtonian gravity from a more general model. Calibration of Newtonian gravity on a Solar System scale neglects our acceleration towards the centre of the galaxy, and therefore this is the point at which Newtonian gravity departs from reality. It has indeed been verified that this is of the same order of magnitude as the critical acceleration in MOND. In other words, my model explains this parameter, and shows that the age of the universe has nothing to do with it.

August 14, 2022 at 10:09 am |

Oh, you probably wanted to know where the 1/r term in MOND comes from. Well, if you replace one of the mass terms in Newton’s law by a momentum term, then you’ve multiplied by a velocity, so a distance/time. This means the 1/r^2 becomes 1/rt, where t is a timescale. Although t can in principle vary, it is very large (~ the age of the universe) and for many purposes can be treated as constant. That at least is what MOND does.

But it is not obvious that, when you change from considering galaxy rotation curves to considering galaxy clusters or even larger structures, this timescale does not change. My back-of-an-envelope calculation says it should be nearer to 1000 times the age of the universe, which would be very interesting if true!

Moreover, the tensor also includes momentum x momentum terms, which should be “independent” of distance and dependent only on time. Such terms would normally be called “dark energy” or “cosmological constant”, but they are far from constant (cf. the Hubble tension). One curious fact that comes immediately out of the group theory is that these terms have the opposite sign from the Newtonian mass-energy term, so that these terms describe a repulsive force, or an accelerating expansion of the universe.

BUT this may be an illusion caused by our choice of coordinate system. Can we change our coordinate system to remove any local momentum, so that the dark energy terms do not appear? Watch this space.

August 14, 2022 at 12:36 pm |

Well, I think the point is that gravitational spacetime is Eucildean, whereas electromagnetic spacetime is Lorentzian. Peter Woit seems to think the same thing (if I interpret what he says correctly). In Euclidean spacetime you can convert all the momentum into mass and get rid of the dark energy terms, but I don’t think you can do it in Minkowski spacetime.

So at the end of the day you have a choice of gauge: Lorentzian or Euclidean. If you choose Lorentzian, in order to get electromagnetism looking nice, then (a) you get a horrible mess mixing the weak force with electromagnetism, and (b) gravity is awful. If you choose Euclidean, then (a) the mass-changes in the quantised weak force correspond to macroscopic mass-changes in the theory of gravity, (b) automatic electro-weak unification, and (c) no dark matter and no dark energy. Oh, and quantised gravity using only the Dirac matrices – with gamma_0 and gamma_5 interchanged!

I know which I prefer.

August 14, 2022 at 12:37 pm |

Or to put it another way, what you see (with electromagnetism) is not what you get (with gravity).

August 14, 2022 at 1:37 pm |

What you see is not what you get. For those who are familiar with Dirac’s model of quantum mechanics, you will know that the Lorentz group is generated by . The gravitational equivalent is SO(4) generated by . Therefore they agree about the concept of space, but they do not agree about the concept of time.

In the old days, time was measured with gravitational clocks. Nowadays, time is measured with electromagnetic clocks. Therefore you have to be very careful which kind of clock you are talking about when you are discussing fundamental physics. No-one in mainstream physics even considers the possibility that gravitational clocks could tell a different time from electromagnetic clocks. But they do, and many people outside the mainstream are very well aware of this.

In the early scientific age, time was measured with some very clever devices (pendulum clocks, spring watches) that combined gravitational elements (weight) with electromagnetic (mechanical) elements to measure particular compromises between gravitational and electromagnetic time. Therefore a practical concept of time grew up, which is a compromise. My researches indicate that it was approximately 22% gravitational and 78% electromagnetic. BUT because of the practical nature of the devices, the exact proportion of gravitational to electromagnetic time changed slightly from year to year.

This effect caused serious problems in physics in the 1950s and 1960s, when experiments became accurate enough to be able to detect the difference between gravitational and electromagnetic time. The solution that was adopted was to define time as electromagnetic time, and completely ignore gravitational time. Now look what a fine mess that got us into!

August 14, 2022 at 2:27 pm |

So, shall we try and quantise gravity now? We’ve done all the hard work, so let’s just get on with it. Electromagnetism is quantised by identifying the adjoint representation of SO(3,1) with photons in two polarisations (helicities, if you like). Gravity is quantised by identifying the adjoint representation of SO(4) – with what? Well, they are things that change mass, and they live in two copies of SU(2), so what else could they possibly be, but neutrinos and antineutrinos?

What is the common ground between the two? A pair of photons with opposite polarisations is equivalent to a pair of a neutrino and an antineutrino. What about the three generations of neutrinos? Well, we know from experiment that the generation of a neutrino depends on its environment, so has no fundamental meaning, so we can ignore that. Now what have we got? 9 dimensions out of 10. Where is the 10th dimension?

In electromagnetism it is pure energy, in gravity it is pure mass, in particle physics it is the Higgs boson – I say tomato, you say tomaido, let’s call the whole thing off. It is a concept alien to physics, but central to metaphysics – it is the here-and-now.

August 15, 2022 at 12:18 am |

I am very pleased to see that the basic ideas here, sound like they can be turned into field equations that researchers in “modified gravity” would be able to parse.

For comparison, I will point out https://arxiv.org/abs/hep-th/0009153, a theory of gravity based on gauging SO(4,1). The philosophy as well as the details would be quite different to yours, e.g. I see nothing about gauging mass (if adopting a gauge is indeed what’s involved in the selection of mass coordinate that you envisage).