Relational Mechanics

Andre K.T. Assis

(Apeiron, Montreal, 1999)



The criticism that in his epoch-making history of mechanics Ernst Mach levelled at the basic concepts of Newtonian mechanics is more or less familiar to most physicists, thanks to the references to Mach that Albert Einstein and other relativists interspersed in numberless papers and books. However, as is typical of the way the historical heritage is transmitted and absorbed in contemporary scientific education, the main use that of Machian ideas is usually made is to motivate some of Einstein's steps in the construction of general relativity (GR), and also, of course, to give an example of a 'prophecy' seemingly vindicated and fulfilled by subsequent more rigorous studies. More precisely, it is to GR that the honour is attributed of having transformed some vague philosophical intuitions concerning inertia and gravitation into the gem of a mathematical equation (Einstein's field equation).

However, among specialists in the foundations of mechanics, as opposed to popularizers, there is much less agreement as to which extent Einstein implemented Mach's ideas on space and time in his theory. An amusing indication of the variety of opinions about the 'Machian' quality of general relativity is provided by two opinion polls made at a conference on the Mach's principle (the proceedings are in the book Mach's Principle, edited by J. Barbour and H. Pfister, Birkhäuser, 1995). In the entrance poll the question "Is general relativity perfectly Machian?" got 2 'yes' and 30 'no', while at the exit poll there were 3 'yes' against 21 'no'. Notice that one of the organizers, Julian Barbour, contributed a paper entitled "General Relativity as a Perfect Machian Theory", so there is no mystery as to where he cast his vote. On the other hand, both his contribution at that conference and the book under review make it sufficiently clear that Assis shared - rightly, in my opinion - the majority view.

Relational mechanics is no doubt a less controversial way than GR of implementing Mach's programme in the foundations of mechanics. To understand what the stakes are, let us remember Newton's viewpoint. As is well known Newton postulated, in the "Scholium" to the "Definitions" of the 1st book of the Principia, both "absolute space" and "absolute time", but he did not conceive them just as metaphysical entities. In fact, in order to justify "absolute space", Newton offered what became a very famous physical argument. Take a bucket filled with water, and set it into rotation (for instance, by attaching it by a rope to the ceiling, twisting the rope and then letting it unwind); you shall see that as soon as the motion of the bucket is communicated to the water, the surface of the liquid will become curved (as a paraboloid), and curved it will remain if the bucket is stopped all of a sudden. This means that the water 'feels' the rotation independently of its relative motion with respect to the bucket. So this rotation must be regarded, in Newton's opinion, as relative to absolute space. Mach's objection, as it had been more or less Berkeley's, was that the fact that the relative motion of the water and the bucket is not causally responsible for the surface's curvature, does not imply that the motion with respect to other bodies cannot be so. In particular, one may conjecture that it is motion with respect to the distribution of matter at large in the universe (shortly described as the 'heaven of fixed stars' - we should say 'galaxies' instead of 'stars') which causes the deviation from flatness of the water's surface.

To give a mathematical formulation of the idea that inertial effects, such as the one just described, do not prove the existence of absolute space, but only the interaction of matter with other matter has remained a challenge for positivistic-minded physicists. Einstein accepted the challenge and tried to construct a gravitational theory which would satisfy this requirement, which he named Mach's principle, though ultimately he became rather more sceptical both on the importance of the principle and on the degree it was embodied in his theory. The book under review follows a different path, reaching back to W. Weber's theory of action at a distance, to which the author had already devoted, among other writings, also a book in English (Weber's Electrodynamics, Kluwer 1994).

In a short review such as this one I cannot deal adequately with the rich content of this book. I shall limit myself to describing very briefly its main features and make some general comments. The book is divided in two parts, according to the following scheme:

I. Old World (pp. 13-160)

1. Newtonian Mechanics

2. Applications of Newtonian Mechanics

3. Non-Inertial Frames of Reference

4. Gravitational Paradox

5. Leibniz and Berkeley

6. Mach and Newton's Mechanics

7. Einstein's Theory of Relativity

II. New World (pp. 161-258)

8. Relational Mechanics

9. Applications of Relational Mechanics

10. Beyond Newton

11. History of Relational Mechanics

12. Conclusion

Weber's idea in his electromagnetic theory (1848) was to use a potential which depends on the distance and the relative velocity (or, to be more precise, on the rate of change of the distance) of two electric charges. This results in a force depending explicitly on the second derivative of the distance between the charges, and which is always directed along the straight line joining the two charges. Assis's proposal for the basic force between two particles is to use the same expression, with charges substituted by masses (and the proportionality constant suitably adjusted), in the same way as Newton's attraction law can be obtained (in inverted historical order, of course) from Coulomb's law. The "basic (or primitive) concepts" of his theory are: "(1) gravitational mass, (2) electrical charge, (3) distance between material bodies, (4) time between physical events, and (5) force or interaction between material bodies" (p. 163); the postulates of the theory are three:

"(A) Force is a vectorial quantity describing the interaction between material bodies.

(B) The force that a point particle A exerts on a point particle B is equal and opposite to the force that B exerts on A, and is directed along the straight line connecting A to B.

(C) The sum of all forces of any nature (gravitational, electric, magnetic, elastic, nuclear, etc.) acting on any body is always zero in all frames of reference" (p. 164).

The main result reached by this approach is that the so-called 'fictitious forces' can be seen as the effect of the matter at large in the universe (described, as far as the derivation of the inertial forces is concerned, as a spherically isotropic matter shell). The proportionality between inertial and gravitational mass need not be postulated independently on experimental grounds, but can be seen as a consequence of interpreting Newton's second law in terms of relational mechanics. More strikingly, three basic constants of physics, namely, Newton's gravitational constant, Hubble constant and the average density of the universe turn out to be linked by an equation which was noticed as a numerical coincidence (and then expanded in a wider theoretical hypothesis) in the 1930's by Dirac.

I find these results (which in different forms have been found also by other authors - see chapter 11) fascinating, and I think that they should be more widely known. They do not involve mathematics beyond the level of a (good) first degree in physics or applied mathematics, so an attempt might be done to introduce them at a fairly basic stage.

In order to make his proposal more plausible than what is its main competitor, Assis advances several criticisms against both special and general relativity. I do not know how much weight he places on this pars destruens, but in my opinion the relativists will hardly be convinced by it.

For instance, as regards the twin paradox Assis does not deal with Hafele and Keating's celebrated experiment (the replication of which is, in my opinion, much overdue), and for the mesons he favours the hypothesis that "the half-lives of the mesons depend on their high velocity relative to the distant material universe" (p. 133). However, should Hafele and Keating type experiments be confirmed, it would not be clear how to make room for the concept of 'proper time' in relational mechanics. Against the constancy of the velocity of light, he contrasts Einstein's postulate with the fact that all velocities we know depend on the relative motion of the observer either to the source (like bulletts) or to the medium (like ordinary sound waves). Indeed, the speed of light is an anomaly in Einstein's universe, and Assis is right in insisting that "light is a physical entity which carries momentum and energy [...] is affected by the medium in which it propagates [...] it acts on bodies" (p. 139), so its unique status is not a priori plausible. But the relativists would reply either that it is only a posteriori, i. e. by experimental evidence, that we are led to a notion of light having a special status, or that what is essential to special relativity is not so much that the limiting signal speed be identified with the speed of light, as that a finite limiting signal speed exists at all and is close to c (this option is favoured by J.-P. Vigier and J.-M. Lévy-Leblond, for instance). Moreover, to Assis the idea that there can be a length contraction without an aether is unacceptable (p. 146), but this kind of philosophical opinions should be argued much more in depth in order to bear the burden of justifying rejection of a physical theory apparently in good agreement with experiments (if one admits that this is the case).

But are, vice versa, the philosophical grounds of relational mechanics very sound? One can have doubts, not for any fault in Assis's approach, but for general reasons which affect the whole Machian programme. For instance, special and general relativity have accustomed us not to lend an absolute meaning to simultaneity and distance between simultaneous events; however, while formally denying absolute space and time, Assis has no qualms whatsoever in endorsing absolute simultaneity and distances. What is the rationale for this different treatment of absoluteness? A second general objection pertains to action at a distance. We know that for Leibniz (and Galileo as well) this concept was the height of philosophical absurdity, and from his first letter to Bentley we know that also Newton disliked it. As far as I can see, it is not clear why a Machian should not find anything objectionable about it. Finally, why should the constant c enter into the general expression for the force between two particles if the velocity of light has nothing to do with interactions, these being instantaneous?

These remarks, of course, are not meant as a refutation of the Machian programme. They only show that a contemporary Machian physics today cannot avoid to come to terms again with certain philosophical issues which could be dismissed out of hand more easily in late Nineteenth century than it is desirable, or even legitimate, today. A new philosophy of nature has to be built in which our theoretical preferences would be given as ample, explicit, and argued foundations as possible.

Notwithstanding these critical remarks, Relational Mechanics is an interesting book, rich in information, and with abundant quotations from the classical works of Berkeley, Leibniz, Newton and, of course, Mach. It has a 18-pages long bibliography listing 244 full references (i. e. with title and both first and last page, so often missing, unfortunately, in physical articles and books), for which the reader wishing to pursue the topic will be grateful to the author. I also appreciated the author's awareness that in order to make fundamental theoretical advances in physics it is very important to look back and re-examine the history of one's subject. I think that science will only progress if more people will stop accepting theories as articles of faith and begin to question the received opinions with the necessary persistence and boldness (both intellectual and professional). In this the author has been totally successful, and deserves the appreciation of all who think that the end of science, and of physics in particular, is by no means in sight, and that science can only thrive by relentless questioning and proliferation of alternative approaches.

Review of André K. T. Assis, Relational Mechanics

Montreal, Apeiron, 1999, pp. 285, US$ 25


(Marco Mamone Capria)