Shedding Light on Inverse-Square Laws | On Beyond Darwin - Chapter 13


Shedding Light on Inverse-Square Laws

The peculiar results of the theory of special relativity arise, I believe, because the clocks and rulers that we have to use for making measurements are themselves composed of the very thing we are measuring. There is no possibility of obtaining instruments that are independent and there is no such thing as absolute time that is independent of the clocks, or absolute space that is independent of matter. We have somehow been led to believe that we can stand outside the universe and record how it behaves, but we are actually trapped inside. All that we know about the universe and its components is gathered from inside. We have developed what must be admitted to be local physics—it is universal only insofar as local conditions elsewhere in the universe are similar to those in our own locality. Because the spectra from stars are similar to those obtained from atoms on earth, we believe that this is a reasonable assumption. Most physicists assume that there are universal laws which hold everywhere in the universe, but according to my view, this is an illusion. I would say that the same particles are elsewhere, and the environment is similar elsewhere. When we observe the spectra of distant stars, they are generally shifted toward the red end of the spectrum. This is usually explained as a Doppler shift due to their receding motion, and we conclude that our universe is expanding. But, it is possible that the value of Planck’s constant h is not the same out there as it is in our locality. Perhaps the universe is not expanding as rapidly as scientists think. We do assume that the same kind of law holds out there and this can be explained by saying that conditions out there are similar, if not exactly the same. So our explanations of our local conditions have universal usefulness just as if there were universal laws.

We have been very successful in amassing information about our universe. Many of the pieces of information have been in the form of mathematical formulas and this mathematical form has given most scientists confidence that they were true. If there were a design in the universe, they say to themselves, it would be in terms of universal laws and the laws, if made rationally, would be mathematical. I have been arguing that the appearance of having general universal laws is an illusion which stems from the fact that the universe is composed of three fundamental particles: electrons, protons, and neutrons. A question remains: If there were no detectable design, why should the descriptions of the nature of electric interaction be mathematically simple? One principal example of a mathematically simple description is that the electric field of a stationary charge—electron or proton—is an inverse-square field. This means that the size of the field, at twice the distance from the charge, is only one-quarter as large—at ten times the distance, it is only one-hundredth as much. Why this simple relationship? What is the explanation if it is not design? And there are other inverse-square laws. Hanson says this in his book Patterns of Discovery :

The great unifications of Galileo, Kepler, Newton, Maxwell, Einstein, Bohr, Schrödinger and Heisenberg were pre-eminently discoveries of terse formulae from which explanations of diverse phenomena could be generated as a matter of course; they were not discoveries of undetected regularities. It is this which now drives theorists to search for the root of all of our inverse square laws, dynamical, optical, electrical, and which spurs them on towards a formalism in quantum physics which will not be quite so productive of procedures which are, mathematically, quite ad hoc. [1]

I have already indicated how the mathematically ad hoc procedures of quantum mechanics might be explained in terms of a Brownian motion of the particles produced by the rest of the Universe. I have suggested a modification in choosing what stationary states would be allowed if we believed that electrons are really particles subject to random influences from the rest of the universe.

In this chapter, I particularly want to look at inverse-square laws. The three referred to by Hanson are “dynamical, optical, electrical.” The dynamical one is the law of universal gravitation which was proposed—or discovered—by Newton. The optical one is the law of illumination. The electrical one is the one I mentioned before about the electric field of a static charge. This last law is called Coulomb’s law.

I have called this chapter Shedding Light On Inverse-Square Laws because I believe that the key is the law of illumination. There is really no mystery to the law of illumination which is that the intensity of light from a point source decreases inversely as the square of the distance from the source. You can explain it by assuming that light travels straight out in all directions from the point source and is not lost as it goes—unless you block it. If you hold a piece of paper at a distance of one meter from the light source perpendicular to the path of the light, then at two meters the shadow of the piece of paper would cover four pieces of paper the same size. If you take the paper blocking the light away, you can see that the intensity of the light at two meters is only one-quarter what it would be at one meter. The same amount of light must spread out over an area four times as big. The electric inverse-square could be explained in a similar way. Here is Feynman:

That the field [electric field of a static charge] varies inversely as the square of the distance seems, for some people, to be “only natural,” because “that’s the way things spread out!” Take a light source with light streaming out: the amount of light that passes through a surface cut out by a cone with its apex at the source is the same no matter at what radius the surface is placed… The amount of light per unit area the intensity must vary inversely as the area cut by the cone, i.e., inversely as the square of the distance from the source… If we had a “model” of the electric field in which the electric field vector represented the direction and speed say the current of some kind of little “bullets” which were flying out, and if our model required that these bullets were conserved, that none could ever disappear once it was shot out of a charge, then we might say that we can “see” that the inverse square law is necessary… No one has succeeded in making these “bullets” do anything else but produce this one law. After that, they produce nothing but errors. That is why today we prefer to represent the electromagnetic field purely abstractly. [2]

I believe Feynman is harsh on the “bullet” model. The model was suggested by Page in his book Introduction to Electrodynamics published in 1922. In a later edition with Adams they say:

The line of force [from an electric charge]… has exactly the configuration of a stream of water issuing from a nozzle kept pointed to the right and caused to oscillate up and down. The picture is not merely approximately correct, but is an exact representation of an electromagnetic wave. In fact the entire group of Maxwell’s field equations… has been shown to be merely the kinematical equation of motion of the lines of force of the field as represented here. [3]

This model was formulated more precisely in 1960 by Lowry in an article in the Physical Review . In it the charged particle is visualized as a sphere from which emerge streams of bullets moving straight out in all directions at speed c . Lowry emphasizes that this is only a model, not to be taken literally—the bullets to him only trace the geometrical structure of the field —they are my messengers. There are two kinds of bullets: positive and negative corresponding to the sign of the source charge. The acceleration of a second charged particle, at rest relative to the first, is proportional to the number of bullets intercepted by it per second. The acceleration is along the line of travel of the bullets. We assume that the particle has a finite cross section for intercepting bullets. So far just like Feynman. But there is more!

A particle moving in a stream of bullets sees as instantaneous those bullets intersecting a plane which passes through what, in relativity, is called its world line at the moment of observation and has an orientation conjugate to the particle’s present velocity. These are the bullets which are simultaneous in the frame in which the receiving particle is at rest. This model is shown by Lowry to give the complete result of classical electromagnetism. It produces Maxwell’s laws, not just Coulomb’s law. Lowry shows in particular that electromagnetic radiation can be understood simply as the flux of bullets from an oscillating charge. Not only does this simple model explain Coulomb’s inverse-square law, but it explains, in a unified way, all of electromagnetic interaction. From my point of view, such an explanation is absolutely essential. It shows that nothing more is being emitted by a particle that is radiating electromagnetic waves than one that is just sitting at rest. For photon lovers this poses a major problem but, since I have already exorcised photons from my world view, I find no difficulty.

But, I am avoiding the third inverse-square law—the law of gravitation. It must be explained too. Are there other kinds of bullets than electrical bullets? Why have I been just ignoring gravity completely? The reason I have done this is that it could be explained as essentially electromagnetic in origin if we adopt what I will call the shadow theory of gravity. You know we can cut down illumination by placing a slightly opaque object between the source of light and the place where the illumination is being observed.

Suppose we had a situation where, instead of light spreading out in all directions from a point source, we had light coming in from all directions to a small spherical detector. If a piece of paper were held perpendicular to the light at a distance one meter from the detector, the illumination on the detector would be decreased on the side facing the piece of paper. If the same piece of paper were held at a distance of two meters, its shadow would be in the same place but this time it would only cut down one-quarter as much on the incoming light. The shadow theory of gravity is that nearby massive objects absorb some of the incoming flux of bullets from the universe. The balanced flux is what gives an object its inertia and so the object will move to maintain a balanced flux. This means it accelerates towards the object causing the shadow. The size of the acceleration will be inversely proportional to the distance squared from the shadowing object. This shadow theory of gravity is very old. Feynman has something to say about it:

Many mechanisms for gravitation have been suggested. It is interesting to consider one of these, which many people have thought of from time to time. At first, one is quite excited and happy when he “discovers” it, but he soon finds that it is not correct. It was first discovered about 1750. Suppose there were many particles [bullets] moving in space at a very high speed in all directions and being only slightly absorbed in going through matter. When they are absorbed, they give an impulse to the earth. However, since there are as many going one way as another, the impulses all balance. But, when the sun is nearby, the particles coming towards the earth through the sun are partially absorbed, so fewer of them are coming from the sun than are coming from the other side. Therefore, the earth feels a net impulse toward the sun and it does not take one long to see that it is inversely as the square of the distance… What is wrong with this machinery? It involves some new consequences which are not true. This particular idea has the following trouble: the earth, in moving around the sun, would impinge on more particles which are coming from its forward side than from the hind side (when you run in the rain, the rain in your face is stronger than on the back of your head!) Therefore, there would be more impulse given the earth from the front, and the earth would feel a resistance to motion and would be slowing up in its orbit… so this mechanism does not work. No machinery has ever been invented that “explains” gravity without also predicting some other phenomenon that does not exist. [4]

It is part of my model that the universe does affect any object and that this effect is natural—essential to survival—even at rest in an inertial frame. It is an identical effect when the object is moving at constant velocity. But not when it accelerates. The flux of bullets is not like rain.

The fact that the gravitational effect of a nearby object is related to its inertial mass is the basis of Einstein’s theory of gravity—his theory of general relativity. He noted that all objects in a gravitational field exhibit the same acceleration. An artificial gravity could be obtained in a non-inertial frame of reference that is accelerating with respect to an inertial frame. All objects uninfluenced in such a frame would exhibit the same acceleration. This is just what happens in a gravitational field—all objects no matter what their mass have the same acceleration due to the gravity. Remember the old experiment with the guinea—English coin—and the feather in a glass tube. When the tube was evacuated, the coin and the feather fell at exactly the same rate. So the artificial gravity in an accelerated frame of reference would be just like real gravity. This is Einstein’s principle of equivalence: that gravity is equivalent to an acceleration. One particular non-inertial frame is a spinning frame—the frame is accelerated toward the axis of spin. In the frame, objects are accelerated outward as if there were a force on them causing them to flee from the center. We call this a centrifugal force. There is no cause of this force as there are no nearby objects. Heisenberg says this:

Since the centrifugal forces had to be considered as due to physical properties of empty space, Einstein turned to the hypothesis that the gravitational forces are due to the properties of empty space… If therefore gravitation is connected with properties of space, these properties of space must be caused or influenced [modified] by the masses. [5]

The inertial mass of an object which resists acceleration is, I believe, the result of the effect of the rest of the universe on that object. Bondi says that Einstein believed Mach’s idea that it was the rest of the universe that produced the environment that we call an inertial environment. This is called Mach’s principle and as you see I believe it too. Newton claimed that accelerations were absolute. He said that you can tell that a bucket of water is spinning because the surface of the water takes on a curved dish shape. Mach said, “fix Newton’s bucket and rotate the heaven of fixed stars” the result would be the same dish shape of the water surface. Of course, we cannot perform this experiment. Bondi says:

Mach’s principle was perhaps put most beautifully by Einstein himself when he said that in a consequential theory of relativity there can be no inertia of matter against space, only an inertia of matter against matter.

With this formulation, Einstein clearly identified the sources of the inertial field as being material. However, there are grave difficulties in identifying and finding these sources… The fact that Newton’s theory describes the motion of the planets and satellites so very closely proves that the inertial frame is effectively one rigid frame throughout the solar system. In other words, the inertia-causing effect of the bodies in the solar system—the sun and the Earth and Jupiter and the moon must be completely negligible… If we have a law of force varying say, inversely with the distance, then certainly the very distant bodies would win hands down over the near ones because their total mass is so very much greater. With an inverse square law the effects of near and distant bodies are neatly balanced—with an inverse fourth power law the near bodies are vastly more important than distant bodies. [6]

We know that the radiation field of electric charges varies inversely as the distance from the source so would be an appropriate field to provide the inertia of bodies. The effect of the distant masses would be the dominant one. This would mean that the inertia of bodies is due to the electromagnetic interaction with the rest of the universe.

We have assumed that the principle of superposition holds for electric interaction, but our shadow theory of gravity requires that the nearby masses absorb some of the effect. So what gravitation is to me is the failure of the principle of superposition for electromagnetic interaction. Einstein’s general theory of relativity differs from Newton’s gravitation in form. It is very complicated mathematically but Bondi points out that there are only two basic differences:

I have mentioned the fact that general relativity differs from Newton’s gravitational theory in practice only in some minor observational matters… there are basically two such circumstances: (1) when the gravitational potential energy is large [compared to the rest energy mc2 ]… (2) When fast motions are involved [because the interaction is not instantaneous, the interaction speed is presumed to be the same as the speed of electromagnetic interaction]. [7]

So Einstein’s theory basically agrees with what I have put forward. The fact that the speed of gravitational interaction is assumed to be the same as electric interaction follows immediately in the shadow theory. The difference between Newton and Einstein with large gravitational potential has to do with the failure of the principle of superposition for gravity. Feynman says this:

Although the principle of superposition applies exactly for electrical forces, it is not exact for gravity if the field is too strong, and Newton’s equation is only approximate, according to Einstein’s gravitational theory. [8]

I do not agree that the principle of superposition holds for electromagnetic interaction—to me gravity is the evidence that it does not. It is no surprise if it does not hold for the shadowing. There is no reason for it to be strictly additive if objects shadow each other.

Presumably we could compute the gravitational constant if we knew about the charge distribution in the universe. Bondi says:

In as far as we can give any real meaning to the constant of gravitation, it does express the relation between the gravitational and the inertial properties of matter. The gravitational ones, we have every reason to believe, are directly connected with the local sources… on any theory of Mach’s principle, the inertial properties are connected with the distant sources. If we have an evolving universe—which of course, we might not have—then the structure and layout of the distant sources will be changing in time. One must then contemplate, to say the least, that the constant of gravitation itself will change in the course of time. [9]

In this chapter, I have drawn several very important inferences. Because certain physical laws are mathematically simple, I must seek an explanation. This has led me to adopt as an appropriate explanation of electromagnetic interaction the “bullet” theory introduced by Page and elaborated by Lowry. These bullets are not particles of matter. Neither are they photons. Photons are associated with a particular frequency of radiation. For me, all electromagnetic waves are just patterns travelling out as the bullets travel out. They are like information travelling on a carrier—the carrier is a stream of bullets. Photons are part of a model that I cannot accept and for me, can be forgotten about.

Because the gravitational mass and the inertial mass of an object are identical there is a strong suggestion that gravity is of the same nature as what gives rise to inertial mass. Bondi argues quite convincingly that inertia must be the result of the electromagnetic influence of the rest of the universe. Inertia is not a gravitational effect. So if inertia is not gravitational then, if inertial and gravitational mass are identical, gravity must be an electromagnetic effect of some sort. I cannot stand coincidences—they seem too much like design. So this all leads me to adopt the shadow theory of gravity and identify the gravitational effect as evidence of the failure of the principle of superposition for electromagnetic effects. The shadow theory also neatly explains the inverse-square character of gravity. It was about gravity that Newton said, “Hypotheses non fingo.” Perhaps it is time to “fingo” an hypothesis. The shadow theory, I believe, is generally consistent with Einstein’s theory of gravity, his general relativity. It is somewhat simpler too.

Bondi says that we cannot think about laws for the cosmos because there is only one:

But in the case of [the] universe we have got precisely this one example. It is bound to affect our whole outlook enormously… We have got to take the motion of the universe, and not its law of motion. It is boring to describe separately the motion of the apple and of the moon and so on. But, if there is nothing but one apple falling, then you would be silly if you did anything but describe that motion. So the best that we can hope to provide about the motion of the universe is a description, not a law of motion. [10]

My whole thesis is that there are no laws, only descriptions of the behavior of specific objects and their behavior in the presence of each other. It just happens that there are many natural recurrences of some few fundamental objects. An apple and the moon can be described by the same laws because they are both made of these fundamental objects and both in the same environment, of the rest of the universe.


1. The simplicity of the mathematical formula for the interaction between two charged particles must, for me, be explained in the same kind of way as the law of illumination is explained. This leads me to an interest in the “bullet” model of electromagnetic interaction proposed by Page.

2. The inertial effect of the rest of the universe on any particle must be an electromagnetic effect.

3. Because of the equivalence of gravitational and inertial mass I believe that gravity is an electromagnetic effect.

4. The shadow theory of gravity explains the inverse-square law and the connection with electromagnetism.

5. If a nearby object shadows other objects from the effect of the rest of the universe, the principle of superposition cannot be precisely true for electromagnetic interaction.

6. Gravity is the result of the fact that the principle of superposition is only approximately true for electromagnetic interaction. Gravity is not a separate kind of force.

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