The Theory

In this Age of Enlightenment no-one should still harbour the illusion of a pull 'by every particle on every other'. To satirise such glib phrases as this, Lewis Carroll parodied the proverb invoking thrift with the dictum 'look after the sounds; let the sense take care of itself'. Science (and physics in particular) is awash with examples such as this of a quaint mixture of fact and fantasy.

By it Newton sought to embellish with corroborative detail Aristotle's archaic theory, still current in his time. It spoke of an innate tendency of all things, animate and inanimate, to return to ground level, ignoring the effects of buoyancy at sea. They conform to an inverse-square law formula, without even knowing what the inverse square is of, still less its cause. So what is it that makes things fall? And would tying two weights together make them fall any faster than either on their own? The intuitive answer to that is 'no', but it took a while for anyone to find a rational answer to Galileo's question. It was Hooke who eventually tumbled to it that in free fall bodies had no weight. It was simply a measure of static resistance to an outside force preventing further fall.

Starting with Galileo's argument for a force that was not inherent, Hooke envisaged an ambient field as a physical agency providing a powerful 'flux' able to pass straight through matter, shedding some of its force in passing, illustrating it with a diagram much like the one below. And Newton initially went along with this, attributing the shedding of force to 'gumminess' in the 'flux'. Differences between them crept in a decade later when Newton claimed all the credit for inventing the inverse-square ratio.

Bodies in isolation would be in a state of equilibrium in this theory, but two, ideally spherical, bodies in proximity can offer partial mutual shielding in elements contained within the diameters of each whose extensions form tangents to the other. The result would be a drop in force between them in proportion to their density.



Elementary trigonometry shows the areas encircled on the surface of each by these diameters to be identical, the radius of which varies inversely with the distance between centres, likewise their squares. The relevant volume in each body thus depends on the square of the distance between them. There is a correlation with Kepler's law for elliptical planetary orbits that follows this pattern precisely, the arcs of sectors swept varying inversely with the radius of rotation to maintain a constant area.



This mathematical correlation that Newton claimed supported his theory of a pull, does no such thing, for correlations, however close, can be coincidental. An instance of it occurred when it was noticed that multiples of the terms of the fractional value of π corresponded to terms in the Fibonacci series, which converges on the Golden Mean, setting off a search for a mathematical link between π and φ². Even precise values of them appear to show a 5:6 ratio - to three decimal places. But it is mere coincidence as thereafter the values diverge, and the search was passed off as a bit of fun with figures. Just how irrational it was can be seen from a glance at their geometrical constructions. Many other such near-integral ratios exist among numbers, and Paul Dirac, for one, was fascinated by their appearance among very large numbers.

The inertia of a body in free fall resists the force in proportion to its mass, resulting in acceleration that is the same for all, so that can be calculated for any convenient sample mass. Only if constant density can be assumed (as for the Sun in the plane of the ecliptic) can it be considered as proportional to the total mass of the bodies involved, as seen in Newton's law of gravity; and variation of the force with distance between centres doesn't make the centre its physical source. It can be a focal point for an external force, as shown in the first diagram above.

The general case has to allow for variations in density, say with latitude due to an elongation of the dense core along the polar axis, as is the case for the Earth. Here, the force will depend on the mutual shielding from the ambient force by the average density of two bodies within the volumes identified by the inverse-square rule, but for M/d², read M/V x V/d². The relevant volume for generally spherical celestial bodies will be in the form of cones, but with average density in the equation, it can be simplified, with the same result, as part of a notional sphere with another of unit radius at its periphery.

Its volume of 4π/3[x 1³] includes the cross-sectional area of π [x 1²] to be multiplied by a third of diameter to give the volume enclosed by the diameters extended as tangents to the smaller body, which multiplied together and by average density, ρ, gives the mass in which the force is a function of specific gravity, so that:

F = d/3 x 4π/3 x ρ x SG, or

g = Gρd/3.

This translates into figures at the Earth's surface as:

9.81 m/s² = (4,240 x 10³ x 4.1888) m³ x 5,523 kg/m³ x 10^-10 N/kg

The Physics behind the Maths
It is not as though the inverse-square ratio is at all rare, it crops up all over the place, but on the strength of substituting 'd' for diameter with 'd' for distance, and a composite factor for G to hide the substitution, Newton claimed a force proportional to radius to back up a theory of a universal pull between centres. He based it on a false analogy with suction, despite Torricelli's best endeavours to explain it many years earlier, only to have to concede later that Galileo's argument ruled out an internal force. The theory was out of date before it even got into print!

There is no denying that as a rule-of-thumb Newton's 'law' has some practical use, but is it not time this embellishment of Aristotle's 2300-year-old theory was abandoned, along with the string of anomalies it has spawned? The alternative equation applies equally to cosmic expansion as to global compaction under gravity, as opposite reactions to the same force. There is no reason to believe that the two are fortuitously balanced on a knife-edge with disaster looming if either should predominate, as some would have us believe.

Only a marginal imbalance in the force has so far been measured. It is akin to a pressure gradient in the atmosphere resulting in a gentle breeze without regard to the overall pressure of some eleven tonnes per square metre. What remains to be discovered is where the force gets all its energy from to maintain a powerful universal field. Perhaps we can now look to CERN to finally resolve this issue.

Bohr favoured neutrinos for the availability of their penetrative power as the output of nuclear fusion on countless stars, and wondered what other overlapping functions they might perform in addition, and by what means. He saw this as the usual way Nature most economically maintains an overall balance of forces. The question remained what would enable them to interact with other particles, seeing how rare is physical contact. Diamagnetic inductance seems a prime candidate for maintaining a gravitational field, and as a potential source of energy to keep radiation, after the initial impulse, radiating in perpetuity - or several billion years anyway.

A consequence of such interdependence between the two would mean that as one varied so would the other, meaning that light passing close to a body boosting the strength of the field, could give the speed of light a boost too, making its source reappear from behind another sooner than expected, rather than 'bent space'. Eddington had to make a host of adjustments to the raw observation to make it even approximate to Einstein's prediction. Yet theories being as much a matter of fashion as of fact, the orthodox majority in their world of make-believe prefer to insist, like TS Eliot's chorus of diehards, 'that things remain as we have always imagined them to be'!

I said at the start physics is a quaint mixture of fact and fantasy; on second thoughts, make that 'fallacy'. It is the result of reverse engineering physical theories out of simplified equations, which by themselves give no indication of cause and effect in the data employed. There is no punctuation in them to sort out the kind of absurdities in plain language that Lynne Truss made famous in 'Eats, Shoots and Leaves'. The omission of a comma in other contexts can distort the sense just as disastrously.

Algebraic symbols may conceal in simplified statements more than they reveal, and only by monitoring them with dimensional analysis can the full story come out. Even regrouping symbols can change the interpretation of them, as can be seen in Einstein's famous E = mc², which turned round to read c² = E/m, shows the rate of spherical expansion of a light wave to be a function of the ratio between energy and mass in the field traversed. Conversion of one into the other is inconsistent with a constant value of c, which could, though, be affected by conversion between different forms of energy.

The value C.V.Boys put on Newton's gravitational constant G gave no indication that it was a composite figure, part physical, part geometrical ratio. Subsequent use of G then needs an accompanying factor to take it out again. Moreover, this ratio implied a belief in the force of gravity emanating from the centre of a body, ignoring the fact that it was weightless until brought to rest and so relied for weight on an outside force; as also does acceleration in free fall. The equipment Boys used was a refinement of that used by Cavendish to determine the mass of the Earth by measuring its mean density in relation to its volume. Omitting this crucial variable from his calculation then led to the strange units in which G is stated to accommodate the concept of mass per unit area.

One might imagine that it is no more than coincidence that the physical component of G, 10‾¹º N/kg was at all related to the diameter of the Earth's orbit of 1000 light-seconds, but it is as a result of the choice of an integrated system in ancient antiquity of units for time, force and linear measure, coupled with a fortuitous value of π within a whisker of √10, that this is the case. A period of swing under gravity of a Rod, 5 metres in length, determined the second as a significant fraction of the time of Earth's orbit, which has led to this result.

The advances made by theoretical Greek and Arabic mathematicians have blinded us to the fact that experimental geometry preceded algebra by thousands of years, and it may yet yield the simple answers that Feynman, speaking as a physicist, believed would, if found, render mathematical analysis redundant; a view diametrically opposed to Dirac's who declared himself in awe of the mathematical genius of the Creator, though he would scarcely have needed mathematics to analyse his own creation. This is the anthropomorphic fallacy.