What is clear is that, in this model, gravity is a non-linear effect. This has some very interesting implications. While the average pressure on the two sides of a vortex ring particle near a gravitating body is the same, the non-linearity essential to this model means that this pressure translates into effective force in a non-linear way, so that the average force on the particle may not be zero, indeed is not expected to be.

This means that gravity is what is called a second-order effect, due to small and quite subtle differences of detail, and this tallies well with its being by far the weakest of the forces.

In this model we have a very limited number of mechanisms available, and are forced to look for gravity as a wave effect. We can calculate that wave energy diminishes as the inverse square of distance, and is therefore an excellent choice for gravitational mechanism. While this is true, the non-linearity postulated suggests that the inverse square law of gravity is unlikely to be perfect, and this has significant consequences.

It may, for example, provide an alternative explanation for the advance of perihelion of Mercury, where, as we shall see later, the mathematics is a lot simpler than might be expected under general relativity. Another effect of non-linearity, and a crucial one, is that two or more gravitational effects will not combine in the way we assume at present. Newton showed that an inverse square law of gravitational attraction could account for the motion of both a falling apple and the planets, within observational error of the time. After that, we make an assumption, the simplest one available, that we can add together gravitational effects in a linear manner, a conjecture known as superposition.

In current theory, we have for example two situations where gravitational attraction is zero or nearly so. One is far from the nearest star or galaxy. The other is the point between the Earth and Moon, or Earth and Sun, where the opposing attractions balance out. In the new theory this is also the case, but one situation is far more noisy than the other and so the two are not equivalent.

Where there is equilibrium, as just described, this difference is not an issue, but when there is a net attraction the non-linearity in the new model suggests that we must also take into account the *amount* of noise. In current theory we add the gravitational pulls, and take account of direction; in the new theory we do the same, but must also account for the noise, which does not cancel in this way. The new theory does not as yet tell us how to account quantitatively for this difference in noise, but it tells us that account for it we must.

The astute scientific Reader may object that we have used the current way of combining gravitational effects for centuries, and it has proved effective. The orbits of planets are affected by each other’s gravitational pull. We also accept that the gravitational pull of each of the Earth’s atoms may be combined linearly, without there being any non-linear effects.

Yet in practice we do many of these calculations in the other direction, inferring the mass of planets and the Earth’s composition from an assumed linear combination of gravitational effects. As with light pressure in the previous chapter, it is not clear that the evidence is decisively against the new model.

Where we have some sort of check on this is in the observation of nearby galaxies and groups of galaxies, and here we find that indeed their rotation is not as we currently predict. Fritz Zwicky in 1933 suggested that there was insufficient visible matter in the Coma galactic cluster to account for the mutual rotation. Vera Rubin in 1974 is acknowledged as the start of a modern upheaval, with measurements of the velocity of stars at the edges of galaxies turning out to be greater than current gravitational calculations would predict.

These observations, together with an acceptance of an ideal inverse square law and simple superposition, lead to the hypothesis of additional ‘dark’ matter. However, dark matter is invoked in different ways to account for different observations, and is not an ideal suggestion, not least because it is ad hoc, introduced for a particular set of uncooperative observations. There are therefore a number of counter proposals, including ‘modified Newtonian dynamics’, a crude and equally ad hoc adjustment to the inverse square law.

At the time of writing, there are a number of further cosmological anomalies. As Pioneer spacecraft leave our solar system, deceleration – the pull of gravity – appears greater than anticipated. The new model would appear to be an appropriate one for examining this.

Peering in the infrared toward the centre of our galaxy we see stars orbiting the centre too rapidly, and infer a black hole, but we have not taken the overall gravitational environment into account.