bbachracWPAccompanying a paper of which I am very proud,
Geometrical properties of random particles and the extraction of photons from electroluminescent diodes W. B. Joyce; R. Z. Bachrach; R. W. Dixon; D. A. Sealer, J. Appl. Phys. 45, 2229–2253 (1974) https://doi.org/10.1063/1.1663571
was the accompanying foundational paper: Classical-particle description of photons and phonons W. B. Joyce, Phys. Rev. D 9, 3234 – Published 15 June, 1974 DOI: https://doi.org/10.1103/PhysRevD.9.3234. In the acknowledgement WB Boyce described that the papers were written before the author was aware of JL Synge’s earlier works and cites in Reference 89. J L. Synge, Geometrical Mechanics and de Broglie Waves (Cambridge Univ. Press, London, 1954). His final paper than related his work to that of JL Synge.
Which led me to the Introduction of
Orbits and rays in the gravitational field of a finite sphere according to the theory of A. N. Whitehead
B y J. L. Synge , F .R .S . Dublin Institute for Advanced Studies (Received 13 August 1951—Revised 2 October)
PDF URL: https://royalsocietypublishing.org/rspa/article-pdf/45604/rspa.1952.0044.pdf
The relativity theory of A. N. Whitehead permits one to calculate directly the gravitational field of a set of particles of assigned masses and arbitrary motions, and to investigate the orbits of test-particles and the paths of light rays in such a field. In this paper the hypothesis of Whitehead is extended to cover the case of a continuous distribution of matter; the field of a fixed sphere with a spherically symmetric distribution of matter is calculated and orbits and light rays discussed. Explicit formulae are obtained for advance of perihelion, angular velocity in a circular orbit, and deflexion of a light ray. The results differ only slightly from those of Einstein’s general theory of relativity by terms involving the distribution of matter in the sphere, except in the case of the deflexion of light, for which precisely the Einstein formula (depending only on total mass) is obtained
Introduction
It is not usual in papers on theoretical physics to attempt any deep discussion of the nature and scope of physical theories, since that would carry the paper out of the domain of science into the domain of philosophy, and perhaps reveal the incompetence of the writer in a professional field to which he is foreign. But critical comments of a referee have compelled me to attempt this, since otherwise the paper, although logically coherent, might be wrongly assessed in its bearing on the theory of gravitation.
The attitude of an individual physicist towards his subject is intimate and personal, and any attempt to convert him from one attitude to another is likely to lead to acrimonious controversy. Nevertheless, such attempts at conversion are being made all the time, and will continue to be made, since it is in the nature of man that he should wish to let others see what he conceives to be the light. It is only the harsh discipline of facts that has saved physics from becoming a mess of controversial attitudes.
The experimental physicist is compelled to build his apparatus out of natural objects. He cannot say: ‘Let there be a plane! ’ He must take a piece of glass and make its surface as plane as possible, knowing that, however perfect the craftmanship, the ideal is unattained and unattainable. The whole of his apparatus consists of natural objects, conforming as far as is practically possible to simplified ideals.
When the experimental physicist speaks of his experiments, his language varies; sometimes he speaks of the actual physical apparatus which he handles, and sometimes of the idealizations to which they are intended to conform. Very frequently he uses language strongly infected by theory. Thus he may speak of ‘velocity of the earth through the ether ’ when what he observes is a displacement of interference fringes, or of ‘velocity of recession of a nebula’ when what he observes is the displacement of spectral lines. Such practices are inevitable and bear bitter fruit only when theory is changed radically, and statements involving the outmoded theory have to be translated back into the sense data from which they were originally constructed.
The theoretical physicist enjoys greater freedom than the experimentalist. He has a choice between two different ways of attacking natural problems which, however similar they may sometimes appear in their finished forms, are nevertheless radically different from a conceptual point of view. These two methods we may for convenience call ‘naturalistic’ and ‘mathematical’, although this does not mean that the ‘naturalistic’ method proceeds without mathematics, nor that the ‘ mathematical ’ method proceeds without regard to natural phenomena.
The ‘naturalistic’ method starts with nature in all its complexity and proceeds by simplification. Its keyword is ‘neglect’, and its success depends on- a nice judgement as to which of the infinite complexities of nature may be ‘neglected’. The Copernican system demanded the ‘neglect’ of man as the most important thing in the universe, and although the astronomer admits the existence of men walking and rivers flowing, he neglects them in discussing the motion of the earth. Certain border-line phenomena, like ocean tides and the elasticity of the earth, are sometimes neglected and sometimes not. But always the ‘naturalistic ’ physicist regards his theory as a little vague on the edges, where neglected phenomena plead for consideration, and he must look forward to greater and greater complexity in his subject, hitherto neglected phenomena being included as his means of measurement become more and more precise and able to detect their influence.
‘Naturalistic’ physics would have died of its own weight were it not for simple natural laws, of which Newton’s inverse square law of attraction is one of the most striking. Although the ‘naturalistic’ physicist regards the physics in which he theorizes as a patch cleared in the jungle of reality by the skillful neglect of what appears negligible, he has tended to regard the simple laws of nature as absolute and exact and applicable universally. In fact, he incorporates them into his picture of nature and they take on the reality of natural objects. He asserts that water does not flow uphill, less as a conclusion from experience, than because such a phenomenon would contradict a law of nature.
When a fabric made of wide experience of nature interwoven with faith in natural laws is confronted with a denial of the validity of certain of those laws, the denial is first met with incredulity, followed by a grudging consent to attempt to think differently. But the change of thought is very slow, and it is safe to say that at the present time there are very few astronomers indeed who frame their thoughts about gravitation along the lines of the general theory of relativity. They ‘neglect’ it, holding that, within the limits of observational accuracy, its conclusions are indistinguishable from those of Newtonian theory except in certain critical cases.
Thus to the neglect of natural objects there has been added the neglect of natural laws, and to some the prospect is not displeasing. They exercise their skill not only in deciding which objects are to be included in the discussion, but also in deciding which of two rival laws of nature is to be used.
The ‘mathematical’ approach to theoretical physics, on the other hand, is much concerned with clarity of thought. It proceeds, not by clearing a patch in the jungle of nature, but rather by taking seeds from the jungle and planting them in the mind. These seeds are the axioms of deductive science, and they can grow to maturity only through the techniques of mathematics, by which is meant a logically coherent pattern involving (as a matter of convenience) the notation commonly called mathematical.
It is essential to note that the words ‘right’ and ‘wrong’, ‘true ’ and ‘false’, ‘possible’ and ‘impossible’, are fraught with confusion unless one distinguishes between the ‘naturalistic’ and the ‘mathematical’ points of view. To the ‘naturalistic’ physicist, a theory is ‘right’ or ‘true ’ if it leads to conclusions in agreement with observation. A theoretical conclusion is ‘ possible ’ only if the phenomena predicted could actually occur in this physical universe as we know it. The ‘naturalistic’ physicist who plays with imagined worlds unlike our own (for example, a Newtonian world with attraction replaced by repulsion), who, in fact, deals with the‘ impossible’, is only amusing himself. His earnest concern is with ‘possible’ things only.
To the ‘mathematical’ physicist, on the other hand, the words ‘wrong’, ‘false’ or ‘impossible’ are to be applied with grave emphasis only when logical laws have been violated. Worlds ‘impossible’ to the ‘naturalistic’ physicist are a matter of interest to him, because he feels that be can gain understanding of the world as it is only by considering it as it might be. In fact, the ‘ mathematical ’ physicist builds in his mind mathematical models, each reasonably coherent from a logical standpoint, and these models he compares with reality. However, in point of historical fact, the ‘ mathematical ’ physicist as here depicted rarely chooses his own axioms and builds from them; he is much more likely to take a body of physical thought from a ‘naturalistic’ physicist, and reduce it to a strictly logical form in which axioms are separated from conclusions and different alternative axioms suggested. The ‘naturalistic’ physicist is perhaps inclined to describe such work as ‘mathematics ’ and not ‘physics ’, and, indeed, the choice of words is a private option. But the ‘mathematical’ physicist, working in the spirit described above, is as old as physics, and is likely to be more conspicuous in the future, now that the upsetting of old natural laws has created in physics a state of logical chaos unpleasing to the tidy mind. The impact of a modern Lagrange or Hamilton on present-day physics would be interesting to see.
However, while the ‘mathematical’ physicist insists on his inalienable right to freedom of thought in creating logically ‘possible’ but realistically ‘impossible’ universes, his work attains dignity and recognition only in so far as it lies close to reality. We might think of nature, as it appears without scientific sifting, as a photograph full of intricate detail. The ‘naturalistic’ physicist rubs out a great deal of detail (neglects it), leaving a ‘studio portrait’. The ‘mathematical’ physicist, on the other hand, glances at the photograph, and taking a blank sheet of paper sketches on it a nose and ears. It is a cartoon. And as a cartoon it derives its merit from resemblance to the original, but in a different way from the studio portrait. In the latter the absence of a mole on the cheek may be criticized; it has been neglected without due reason. This criticism is beside the point in the case of the cartoon, because the cartoonist’s aim was the capture of some characteristic trait, and not the reproduction of all details which (he admits as readily as anyone) were actually present in the original.
A piece of physical theory created by a ‘ mathematical ’ physicist consists of an ordered mathematical argument, with a beginning and an end. The order cannot, in general, be reversed, although reversible arguments (necessary and sufficient conditions) are highly prized. Words borrowed from ‘naturalistic’ physics (mass, velocity, gravitational constant, and so on) occur throughout the argument, and appear to link it to reality, since the ‘naturalistic ’ physicist, reading them, is at once ready to assign his own meanings to those terms. Indeed, most physicists are mixed in their own natures, being neither purely ‘naturalistic ’ nor purely ‘mathematical ’, and the ‘naturalistic’ infects the ‘mathematical’ to such an extent that ‘mathematical’ physicists are inclined to accept without question the ‘naturalistic’ meanings of the physical words which they use.
If one sought the maximum of clarity, one would prefer to see the arguments of ‘ mathematical ’ physics presented without the inclusion of these physical words— perhaps with special words coined for the purpose and devoid of meaning outside their immediate context. So developed, the argument would lie in a void of interest, in a no-man’s-land between mathematics and physics. The next step would be to supply bonds with physical reality.* The ‘naturalistic’ physicist would like to see the theory tied to reality from beginning to end, with every concept and step in close correlation with natural phenomena (cf. Kelvin’s desire to make models). But he may have to be satisfied with much less—a bond here and a bond there. He is particularly anxious to have such bonds at the conclusion of the theory; he wants the ‘ mathematical ’ physicist to commit himself to physical predictions which can be checked, for it is only when such predictions have been made and checked that the ‘naturalistic’ physicist can write ‘tru e ’ or ‘false’ (in his own meaning) against the theory.
It is perhaps wise to repeat that the attitude of the individual physicist to physics is intimate and personal. Two different attitudes have been described above. Between them there can exist tolerance, but not real sympathy, since, although both engaged in the common task of understanding nature, their private interpretations of the word ‘understanding’ are at variance. The most striking example of this tolerance is to be found in work in classical fields of applied mathematics during the past decade, ‘mathematical’ physicists (indeed pure mathematicians) being linked to ‘ naturalistic ’ physicists of the engineering type by a common interest in problems of hydrodynamics, elasticity and electromagnetism. I t may be that tolerance is easier here than in the fields lying nearer the centre of modern physics just because the laws governing these ‘engineering’ phenomena are clearly recognized as manmade approximations, sometimes grossly violated under experimental test, whereas in the core of physics men cling to the idea that the laws are definite, exact and eternal, and so not to be varied in order.to satisfy the curiosity of those who would like to explore the properties of universes which might be different from our own, or which on the other hand might prove to be so like it as to be indistinguishable.
(* I believe that I am borrowing here from a conversation with Professor L. Infeld some ten years ago.)
The present paper has been written in the spirit of ‘mathematical’ physics, as described above. I t stands essentially on the hypotheses put forward by Whitehead (1922), but they are not stated quite as he stated them, and they are extended to cover a continuous distribution of matter instead of a discrete one. I do not know at all whether Whitehead would have approved of this extension, nor whether he would have approved of the analysis of physical thought attempted above. However, the intention of this paper is not historical. There is no attempt whatever to recreate Whitehead’s viewpoint or notation. What is definitely taken from Whitehead is a formula (or its equivalent), this formula expressing the gravitational field due to a particle, and so providing a theory of gravitation within the framework of the special theory of relativity. This should make it clear that criticism to the effect that the theory now presented differs in some essential way from Whitehead’s theory is not valid criticism. The present theory should be judged on its own merits as a contribution to ‘mathematical’ physics, with due respect to the memory of Whitehead if the judgement is good and without reflection on him if it is bad. The mechanics of Newton, including his theory of gravitation, the concept of force, and the equality of action and reaction, formed for ,a long time a common ground for the ‘naturalistic’ and the ‘mathematical’ physicist. The theory was extremely powerful, in the sense that nearly all mechanical problems could be formulated, and in most cases the solutions reduced to techniques which might be tedious but nevertheless sure. The word ‘insoluble’, if applied for example to the famous three-body problem, meant that the solution could not be expressed in closed form; step-by-step integration was always available to obtain a numerical solution to any desired degree of accuracy.
To the ‘mathematical’ physicist, Newtonian mechanics remains exactly as it was, a consistent logical structure. But to the ‘naturalistic’ physicist its laws are ‘ wrong ’ because they violate the principle of relativity, and they are to be used only when the violation may be ‘neglected’. The ‘right’ laws were formulated by Einstein in his general theory of relativity, a theory which in the mind of the ‘mathematical’ physicist is placed beside Newton’s theory as a not-quite-so consistent logical structure. It is not the purpose of the present paper to enter in detail into why it is ‘ not-quite-so-consistent ’; suffice it to say that it is no small task to reduce to complete logical order a theory based on a set of non-linear partial differential equations in a space of four dimensions.
But the ‘mathematical’ physicist is well aware of the fact that the number of problems which can be completely formulated (let alone solved) in the general theory of relativity is very small indeed. The simple geophysical problem of determining the change in period of a pendulum due to local variation in the earth’s density, for example, seems impossible to formulate in terms of the general theory of relativity, and the problem of the tides presents the same difficulty. To the ‘naturalistic’ physicist this causes no intellectual unrest; he will say that, in such phenomena, relativity may be ‘neglected’, or possibly allowed for by a semiempirical correction. But the ‘mathematical’ physicist may see a defect of the general theory of relativity in its limited range of applicability; he would like to calculate the errors which the ‘naturalistic’ physicist is prepared to neglect, if only to see that they are negligible and perhaps to find critical circumstances under which they are not.
The theory of Whitehead seems to offer something between the two extremes of Newtonian theory on the one hand and the general theory of relativity on the other. It conforms to the requirement of Lorentz invariance (thus overcoming the major criticism against the Newtonian theory), but it does not reinstate the concept of force, with the equality of action and reaction, so that its range of applicability remains much lower than that of Newtonian mechanics. However, it does free the theorist from the nearly impossible task of solving a set of non-linear partial differential equations whenever he seeks a gravitational field. I t is not a field theory, in the sense commonly understood, but a theory involving action at a distance (propagated with the fundamental velocity c). To a ‘naturalistic’ physicist who feels that action at a distance is not ‘the way in which nature works ’, the theory of Whitehead is repellent from the outset, but the ‘ mathematical ’ physicist is glad to add it to his collection of theories of gravitation, provided it is logically consistent, which it is.
However, the facts of observation must be given consideration. First, if we omit minutiae, all three theories (Newton, Einstein, Whitehead) predict the planetary orbits correctly. The observational data connected with the three well-known minor effects (rotation of perihelion, deflexion of light ray, red-shift in sun’s spectrum) are much more open to discussion by experts than appears from the brief references which appear in text-books on relativity. The ‘ mathematical ’ physicist may well prefer to leave their discussion to experts, and keep his mind open to two possibilities: (a) the findings of the general theory of relativity are confirmed by observation, and (6) they are not confirmed.
If (a), then the ‘naturalistic’ physicist is prepared to accept Einstein’s theory as true (and this is in fact the present position). might just as well accept Whitehead’s theory, since its predictions in the case of the first two of these phenomena are the same as Einstein’s {or practically so), while in the case of the third the same red-shift as Einstein’s is given by an obvious interpretation of Whitehead’s theory.
If, on the other hand, (6) holds, then both Einstein and Whitehead must be dismissed (‘naturalistic’ judgement). Newton, however, cannot be restored, since the violation of Lorentz invariance is too gross. Then the way lies open for a new theory.
The essential thing to note is that the theories of Einstein and Whitehead, although completely different in their formulations, are tied very closely together as far as the prediction of phenomena in the solar field is concerned. But they are not one and the same theory in two different forms. They are different theories, the results of which happen to coincide under special circumstances of symmetry.
The present paper has been written with a dual purpose. The first purpose, fulfilled in § 1, is to write down the basic hypotheses of Whitehead as briefly and clearly as possible in a convenient notation, without ‘deriving’ them as he does. This is done for the sake of those who would like to know what Whitehead’s theory is in a nutshell. The second purpose is to work out a problem which seems to rank next in importance to the determination of the gravitational field of a particle— the determination of the gravitational field of a sphere. According to the theory of the present paper the field of the sphere depends only on the proper masses of the particles composing it and on their world-lines. The idea that the stress in the sphere causes a gravitational field is not present.
The sphere is taken at rest in some Galilean frame, and it is assumed that the density distribution is spherically symmetric. The external gravitational field is expressed by the form (2*14). It will be observed that the coefficients depend not only on the total mass of the sphere but also on the distribution of density.
This provides us with a comparison with Einstein’s theory. In his theory the whole class of spherically symmetric statical solutions of the field equations depends on one parameter only (commonly called the mass of the gravitating sphere). The whole class of solutions (2-14) is much wider, since the coefficients are not functions of a single parameter m, but functionals of a function expressing the density distribution. The formula (4*14) for the rotation of the perihelion of a planet also includes the functional il/4. The term involving it is very small, it is true, and so perhaps ‘negligible’ to the ‘naturalistic’ physicist. But to the ‘mathematical’ physicist the exact comparison of the two theories is of interest. In the case of the deflexion of light (6*8), the density distribution disappears and we get the Einstein formula. I t is true that the mass m of the sphere and the distance r0 from its centre do not have precisely the same meanings as in Einstein’s theory, but in an astronomer’s interpretation any such discrepancy will make no observable difference in an effect which is very small.
I am much indebted to the referee whose comments have led to the foregoing analysis, which it is hoped clarifies some basic points of view, and also to some improvements in the body of the paper.