Clearly in order to use a unified model of gravity as an explanation for everything that goes on in the universe, we will need both attractive and repulsive elements.

It turns out that it's a relatively simple matter to achieve this, starting with the always-attractive Model A and relaxing the constraint about masses being positive.

We can stick with Newton's law of gravitation and his second law of motion:

F = - G m_{g1} m_{g2} ⁄ r^{2}

F = m_{i} a

To keep things simple, we can still constrain the magnitude of the masses to be 1, but this time the sign can be either positive or negative.

We can keep the label A for particles with mass +1 and introduce the label B for particles with mass -1.

Clearly the 1-body model for a particle of type A is the same as before and is always attractive.

Note that if we do the maths, we find that particles of both types A and B respond in the same way to the presence of a gravitational field and are attracted, so there is nothing more to say about the 1-body model for type A.

In contrast, the 1-body model for a particle of type B has the opposite sign and is always repulsive:

Because particles of both types A and B respond the same way, everything is repelled by a B particle.

*We note that B particles repel each other according to an inverse square law, so we can imagine them as Electrons, to a first approximation.*

Things start to get more interesting when we consider the 2-body problem. Because there are now 2 distinct fundamental types of particle, there are 4 possible 2-body interactions for us to analyse. If we do the maths, we get:

Interaction | A | B |
---|---|---|

A | Attract → ← | Combine ← ← |

B | Combine → → | Repel ← → |

The AA pair is the same as before, with straight-line, circular, elliptical, parabola and hyperbola orbits.

The BB pair can be analysed with essentially the same mathematics and we find that the circular and elliptical orbits no longer apply. Instead, the particles move in one of:

- Straight line (directly towards or away from each other)
- Parabola
- Hyperbola

In particular, these are mathematically the same parabolic and hyperbolic solutions as before, but we're now using the negative part of the curves whereas previously we were restricted to just the positive part.

In the general case, because a pair of B particles are likely to be moving away from each other, it makes sense to view the gravitational field in a non-rotating frame:

We note that there is an unstable centre point where other particles may be temporarily located.

For cases AB and BA (which are just mirror images of each other), we find a new type of behaviour.

If we do the maths, we find that B is attracted to A at the same time as A is repelled by B. Essentially, they accelerate at a constant speed in a straight line, always remaining the same distance apart.

This continues until Special Relativity starts to take effect at high speeds and the particles approach the speed of light, without ever quite getting there.

Likewise, Length Contraction causes them to move closer together at high speed, as seen by a stationary observer.

Again, it makes sense to view the gravitational field around an AB pair in a non-rotating frame:

In terms of relating this to known phenomena, clearly we don't see this behaviour at the macroscopic scale. Real-world objects simply don't pair up and hare off at the speed of light...

We can make the following observatioms though:

- The combined mass of A and B is zero
- They are unlikely to be observed travelling at anything other than the speed of light
- They have a characteristic length governed by their initial separation, which is preserved (subject to length contraction)
- The long-range gravitational effect on anything else is effectively zero
- The short-range effect manifests itself as a gravitational dipole, in the direction of motion
- This can be thought-of as a quantum of a longitudinal gravitational wave
- It is likely that A will be in the lead followed by B (i.e. has a characteristic polarity)

There is little of interest to say about the 3-body problem for exclusively type A (which is the same as before) or exclusively type B (where everything repels everything else and the whole thing explodes).

This does, however, give us the basis of a model that can explain different areas of the universe:

- Attractive (i.e. dominated by particles of type A)
- Repulsive (i.e. dominated by particles of type B)

Conversely, if we start with an AB combination, then we have 2 cases to analyse depending on whether the 3rd particle is of type A or B:

- AB + A → A + AB
- AB + A → AA + B
- AB + A → A + A + B

We note that the 2nd and 3rd cases seem unlikely because there will be a high tendency for the B to combine with one of the As

Therefore the most likely scenario would appear to be the 1st case where the "reaction" products are the same before and after.

- AB + B → B + AB
- AB + B → A + B + B
- AB + B → BAB

Again the most likely scenario appears to be the first case where AB is effectively preserved.

The second case shows what happens when the AB combination is ripped apart by an incoming B with sufficient speed.

The 3rd case presents a very interesting new scenario, where the 2 B particles are in a stable orbit around the central A. By looking at the gravitational field in a non-rotating frame:

We note that the BAB triple behaves as a composite particle with a rotational symmetry of 180^{0}.

We can see that at long-range, it looks very much like a single B particle, whereas from close-range the attractive vs repulsive aspects start to take effect. On average we see an increased repulsive effect at short range before the attractive effect becomes apparent at very close range.

*Is this a plausible explanation for the experimental result that electron repulsion increases by roughly 10% at very close range, compared to the inverse square law? This would imply that contrary to commonly-accepted wisdom, the electron is a composite particle after all.*

*Is it also possible that this kind of situation could provide an explanation for the Strong and Weak forces? Although all interactions may be subject to the same unified force, the existence of composite particles with varying components could make it appear as though one force takes effect at long range with another one at short range.*

Although the central A is repelled by the pair of orbiting Bs, it is effectively sandwiched in the middle and so unable to go anywhere. To a first approximation, it makes little difference whether the A is attracted to or repelled by the Bs.

We can think of this as a bit like a pair of electrons orbiting around an atomic nucleus. Because the electrons repel each other, they act to keep the nucleus in the middle and to ward off any intruders.

Picking up on this theme, we can take a closer look at what happens when the masses vary. For a central A particle with the mass of a helium nucleus and a single B particle with the mass of an electron, in a rotating frame:

We can clearly see that there is a point M1 near which a second B particle would be stable. This starts to give us an idea of how a pair of electrons could behave in an atomic orbital, in purely inverse-square-law (i.e. electrostatic) terms.

In particular, this is the opposite scenario compared to the Lagrange Points with the Sun and Earth, because the 3rd particle is repelled by rather than attracted to the 2nd one.

Model AB certainly allows us to simulate much more of the universe compared to Model A. In particular, the repulsive element introduced by the negative-mass B particle gives us a plausible mechanism for explaining Dark Energy.

We indentified 3 cases where the individual particles can behave together as a composite particle:

Composite Particle | Gravitational Mass m_{g} | Inertial Mass m_{i} | Location |
---|---|---|---|

AA | +2 | +2 | Stable |

AB | 0 | 0 | Light Speed |

BAB | -1 | -1 | Stable |

The AA pair behaves like a binary star and is stable in either a circular or elliptical orbit.

If we were to start speculating a little, we could say that the properties of the AB composite particle make it look a bit like:

- Gravitational Wave (hypothetical, has never been observed in practice)
- Neutrino (except that we haven't accounted for the spin-half property)
- Photon (except that a photon has a transverse electromagnetic component)

Furthermore, whichever interpretation we go with, we can imagine AB travelling at the speed of light through the quantum vacuum. On occasions where A or B particles arise from the vacuum, AB can react with them, but typically still carries on going, even though its consituents have changed places.

In the case of the BAB composite particle, we note that its overall properties appear to be identical to a single B particle. We can think of it as a bit like a B that has absorbed an AB.

There are still many aspects of the real universe that Model AB fails to deal with though. In particular, from the laws of electromagnetism:

- Opposites Attract
- Likes Repel

Neither of these can be catered for with Model AB (where opposites combine and only Bs repel).

In particular, Protons were discovered by Ernest Rutherford in 1920 and are known to repel each other while being attracted to electrons.

This means that at this stage, our unified approach is unable to model electrons and protons at the same time, even to a first approximation.

Clearly we need something extra.

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