Model ABCD (Likes vs Opposites)

The inspiration for the next step in generalising our unified model comes from 2 places:

  • The unconstrained Dirac equation which suggests 4 kinds of electron
  • The conflicting treatments of negative mass which suggest 4 kinds of matter

By sticking with gravity as our single unified force, restricting the magnitude of all masses m = 1, yet allowing positive vs negative mass and introducing separate concepts of gravitational mass mg and inertial mass mi, we get the following:

ParticleGravitational Mass (mg)Inertial Mass (mi)

The A and B particles behave the same as before, so the interesting part comes from studying the newcomers C and D. We can think of these as opposites or mirror images of A and B, a bit like Antimatter compared to Matter.

The first thing to do is to draw up a chart of how the particles interact with each other:

→ ←
← ←
← →
→ →
→ →
← →
← ←
→ ←
← →
→ →
→ ←
← ←
← ←
→ ←
→ →
← →

We can think of the B particle as a bit like an Electron, given that they repel each other.

On the basis that opposites attract, this would make the D particle a Positron.

We could use a similar line of reasoning for Protons vs Antiprotons.

Or Muons vs Antimuons, or indeed any charged particle vs its antimatter equivalent.

Note that it doesn't matter whether we assign B or D as the positive or negative charge. This is just a convention and the maths works equally well whichever way round we choose.

This makes B and D suitable for simulating electromagnetic particles, at least in electrostatic terms.

Conversely, A and C have the opposite behaviour, where likes attract and opposites repel.

The A particle remains suitable for simulating standard gravity, whereas we expect the C particle to be its antimatter equivalent.

We note that C particles are attracted to each other, just the same as A particles. This means that if the C particle really is a good way of thinking about antimatter in gravitational terms then we would expect it to be repelled by matter.

This is the first prediction that we can make from our unified model. In gravitational terms, we expect antimatter to be repelled by matter, while still being attracted to itself. We therefore predict that the CERN experiments will confirm that Anti-Gravity is possible.

3 Binary Pairs (Cold)

In much the same way as stars frequently form binary pairs due to the attraction of gravity, we would expect a universe full of As, Bs, Cs and Ds to generate binary pairs in cases where the mutual force is attractive.

From the interaction table above we can identify 3 such cases:

Binary PairGravitational Mass mgInertial Mass mi

In each case the pair can be stable on an indefinite basis if the orbit is circular or elliptical.

We note that the BD pair is unlike AA and CC as it is neutral in gravitational terms and has a negative inertial mass.

As with the AB combination, it would manifest itself as a gravitational dipole, but this time it would be stable in-situ rather than accelerating to light speed.

On the basis that the binary pairs are stable in-situ, we refer to them as "cold".

4 Light Combinations (Hot)

As we saw with Model AB, it is possible for particles with opposite inertial masses to combine and accelerate to the speed of light. This time there are 4 cases to consider:

Binary PairGravitational Mass mgInertial Mass mi

Clearly they are all similar in the sense that the combined inertial mass is zero.

We can think of AB and CD being matter / antimatter equivalents of each other. Likewise with BC and DA.

The big difference comes when we look at the combined gravitational mass, where BA and DA have double-magnitude masses with opposite signs. We would therefore expect their behaviour to be different when interacting with other particles.

Comversely, whereas the individual components of AB and CD have the same response to an external field, we find that the components of BC and DA have opposite responses. In particular, this means that BC and DA would be unstable and hence dissociate in the presence of a strong field.

On the basis that all of the light combinations accelerate to light speed, we refer to them as "hot".

4 Symmetric Triples (Warm or Cold)

Whereas Model AB gave us the BAB triple, Model ABCD gives us another 3 cases to consider:

Symmetric TripleGravitational Mass mgInertial Mass miRating

Again we can think of BAB and DCD as matter / antimatter equivalents.

Given that the central A is repelled by the 2 Bs in the BAB triple, it is possible to replace it with a central D particle that still attracts the 2 Bs but is also attracted to them itself.

This gives rise to the BDB triple and its antimatter equivalent DBD. In these cases the components add up to a triple-magnitude inertial mass overall.

On the basis that all of the symmetric triples are stable in-situ, we might refer to them as "cold". However, there is a scenario where BAB and DCD will decompose and release "hot" particles as we will see next. Hence we refer to them as "warm".

Simulation of Electron-Positron Annihilation

At this point, it is tempting to compare our generalised version of gravity with electromagnetism. Although we now have candidates for electrons and positrons (B and D respectively) on the grounds that likes repel and opposites attract, what happens when they encounter each other?

In the real world, we get a phenomenon known as Electron-Positron Annihilation, whereas in our model it seems that B and D simply orbit each other to form a BD pair.

If we look a little closer though, we find that electrons and positrons do indeed start to orbit each other in a configuration known as Positronium. In fact they never appear to get closer than the size of the atomic ground state in neutral hydrogen (56,000 times the diameter of a proton), yet after 125 picoseconds they emit 2 gamma ray photons (or 3 photons after 142 nanoseconds) totalling the sum of their mass energies according to Einstein's formula E = mc2.

Whilst we don't see this behaviour with just B and D, we can see something similar if we look at the interaction of BAB with DCD:

BAB + DCD → AB + CD + BD

Step 1Step 2
BAB vs DCD Step 1 BAB vs DCD Step 2
Step 3Step 4
BAB vs DCD Step 3 BAB vs DCD Step 4

Effectively, the reaction generates the 2 light combinations AB and CD, which start to accelerate in opposite directions because the A and C repel each other. It also generates the neutral BD pair.

If BAB and DCD are analogous to the electron and positron and AB and CD are similar to photons, then what do we make of BD?

This leads to the second prediction from our unified gravitational model. In the case of electron-positron annihilation, we expect a third (neutral) particle to be generated by the reaction. In contrast to the photons which accelerate to the speed of light, the neutral particle remains in-situ in the original frame of reference.

In particular, this prediction is consistent with the analysis from Don Hotson, based on conservation of angular momentum. In his 3rd paper, he refers to the electron-positron pair or "epo" as part of the quantum vacuum. The epo is equivalent to the neutral composite BD particle in our model.

We might further speculate that singletons or neutral pairs are undetectable with current technology and hence part of the quantum vacuum. With this line of reasoning, only charged composites, heavy composites or light combinations would be detectable.

Dark Matter and WIMPs

If we take a closer look at the BD particle, we can characterise it as:

  • Neutral (gravitational mass = 0)
  • Massive (inertial mass = -2)
  • Cold (rotates in-situ)
  • Unmagnetised (contributions from B and D cancel)
  • Weakly Interacting (rotating gravitational / electrostatic dipole)

We note that this fits the description of a Weakly Interacting Massive Particle (WIMP), which is one of the favourite candidates for Cold Dark Matter (CDM).

Whereas BD is neutral, AA and CC are doubly-charged (gravitational mass = +2 or -2) and so would be expected to strongly interact. Hence they are not dark matter candidates.

Conversely, we might consider the light combinations AB and CD as possible candidates for Hot Dark Matter.

Whereas AB and CD are neutral and therefore fit the description, BC and DA are doubly-charged and so again are stongly interacting.

Furthermore, we can imagine a sea of BD particles as part of a quantum vacuum. Occasionally an interaction between 2 of them could temporarily generate a charged pair via the following reaction:

BD + BD ↔ BDB + D

Both of the resulting charged particles are classified as cold and hence equilbrium with the original neutral state seems to be the most likely outcome. Is this a plausible mechanism for the large-scale emergence of gravity?

Comparison with Cygnus A

The strongest radio source outside of the Milky Way is a radio galaxy known as Cygnus A.

Discovered by Grote Reber in 1939, it appears unremarkable in visible light but has an astonishing structure when viewed with a radio telescope:

Cygnus A
Image courtesy of the National Radio Astronomy Observatory (NRAO) and Associated Universities Inc (AUI)

It has 2 near-light-speed jets (assumed to be electrons) travelling in opposite directions from a massive, compact central object (assumed to be a black hole). At the ends of the jets are 2 lobes which are themselves strong radio sources where the jets collide with the intergalactic medium. The whole structure is truly enormous, nearly 100,000 light years across.

We can speculate that Cygnus A may be driven by the electron-positron annihilation mechanism, as above. This would explain the 2 jets travelling in opposite directions, as well as the acceleration to light speed. It would then lead to the prediction that one of the jets is formed from matter, the other from antimatter, although in the case of photons it isn't clear what this distinction would mean in practice.

Electron Spin and g-Factor

Up to this point, we have modelled things in purely classical (i.e. continuous) gravitational terms, albeit with a non-standard adaptation for negative mass and a variation between gravitational and inertial mass. We have made comparisons with electromagnetism, purely on the basis of attraction vs repulsion, without getting into Maxwell's field theories or any quantum mechanics.

Our first foray into the quantum world will be to consider the case of Electron Spin.

In particular, the building blocks in our unified model are point masses with no angular momentum, which makes them "spinless". Therefore single A, B, C and D are by themselves not candidates for electrons.

The challenge is to see if we can model particles with quantised spin according to the known experimental results. The natural way to do this is via composite particles, where the spin is introduced as a net rotation about the center point.

Drawing on the results from electron / positron annihilation, we will start with the BAB composite triple.

Without getting distracted by units, we will assume that the B particles are a distance of 1 from the central A and are travelling in a circle with speed 1. This gives them 1 unit of angular momentum each (or -1 depending on how we want to define it).

Note: Actually we can calculate a realistic speed based on the force of gravity, but that isn't important for the purposes of this discussion.

The view from above, in a slowly-rotating frame is as follows:

BAB (Rotating)

Effectively the 2 orbiting Bs form a current loop, without the A taking part. If we assume that the gravitational mass is taking the place of electric charge, then we can use the Biot-Savart Law to calculate the magnetic moment at the center-point:

B = μ0 I / R

In our unified model, the magnetic constant μ0 = 1, the loop radius R = 1 and the current I = 2 because there are 2 B particles and hence 2 units of charge.

The net effect of this is that we find for the BAB particle:

  • Angular Momentum = 2
  • Magnetic Moment = 2

This gives us a ratio (known as the g-factor) of 1.

This should come as no surprise because in our model the distribution of charge (i.e. gravitational mass) and the distribution of mass (i.e. inertial mass) is the same.

Yet, here we have a problem because in the real-world the electron is known to have a g-factor close to 2 (actually 2.002319304361).

There are potentially a number of ways of resolving this problem:

  • A more complex model of the electron based on the existing A,B,C and D
  • Building blocks with different gravitational vs inertial mass ratios
  • Building blocks with built-in spin
  • Introduction of a simple rule

It turns out we can keep the simple spinless A,B,C and D building blocks and the simple BAB electron model (which also does a good job of simulating electron-positron annihilation), if we opt for the latter approach. The rule is as follows:

In the context of composite particles, gravitational mass (a.k.a charge) remains in-situ while inertial mass (a.k.a mass) levels out as far as possible.

For the BAB model, this means that the +1 inertial mass associated with the central A levels out -1 of the inertial mass from the surrounding Bs, in equal proportion. This leaves an inertial mass of 0 with the A particle and a remainder of -0.5 associated with each B:

Gravitational Mass mg-1+1-1
Inertial Mass mi-1+1-1
Residual Inertial Mass ml-0.50-0.5

If we now redo the calculations based on the residual inertial mass, we find:

  • Angular Momentum = 1
  • Magnetic Moment = 2

This gives us a semi-classical picture of an electron with a g-factor of 2.

We note that this is as close as the Dirac Equation gets, whilst acknowledging that Quantum Electrodynamics still has the edge in predicting a fully-accurate value.


Again we can simulate a Model ABCD universe with a computer. There is certainly a lot more going on compared to Model AB. After extensive analysis we come to much the same conclusion though, i.e. that it's too simple to simulate everything that goes on in the real universe.

The sticking point this time surrounds particles with differing masses. Although we might take the approach that heavier particles (such as the proton and neutron) are composite particles made from large numbers of smaller ones (such as the electron and positron), we have no way of explaining why the proton and neutron in particular are stable, while pretty much everything else isn't.

At this point, it might be tempting to go looking for the meaning of life in the bottom of a beer glass. What we need is another bit of inspiration...

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