NEWTON’S FOUNDATIONS OF MECHANICS AND THE PRINCIPLE OF RELATIVITY

V. K. Frederiks, Leningrad.

Everyone knows that the old and constant dream of physicists is to make physics into a science whose exposition would be similar to that of geometry: first a series of basic, precisely and clearly formulated definitions, then a series of axioms, whose mutual independence must be rigorously proved, and then, finally, a series of theorems leading to propositions consistent with experience and predicting new, as yet unknown phenomena. Can such a dream come true; is an “axiomatics” of physics possible?—This is a question to which, it seems to us, at the present time representatives of different currents in physics would give very different answers.

It is true that individual chapters of physics are always being presented in a form that approaches as closely as possible the geometrical form of exposition. Thus, for example, in classical thermodynamics the initial propositions are the law of energy and the law of entropy; from them all the remaining deductions and conclusions are derived. But in doing so thermodynamics makes use not only of those physical concepts which, like temperature or quantity of heat, are defined and explained within it itself, but also borrows from all branches of physics other physical concepts and relations. This circumstance alone may render doubtful the axiomatic rigor of its exposition. But the chief difficulties, it seems to us, lie in the following: first, it is required that the basic definitions of physical concepts contain no internal or mutual contradictions; second, a rigorous proof is needed of the independence of the accepted axioms; third, it is necessary that the author, unnoticed by himself and by the reader, not make use of new hidden definitions and axioms,

It is quite obvious that in physics it is especially difficult to satisfy these conditions. The basic definitions introduced into science cannot fail to depend on the general level of knowledge, and this level differs greatly in different epochs. The exact meaning of the terms used in physics depends on many accidental circumstances, to discuss which fully is sometimes by no means easy. It is enough to recall, for example, the numerous disputes devoted in earlier times to the question of defining mass or the quantity of matter, and in more recent times—to the various paradoxes concerning time and space in the special principle of relativity. How much wit and refinement of mind were required to resolve these questions, and how many vain efforts, despite all this, to make one’s reasoning entirely rigorous! The concepts of mass and energy before the appearance of the special principle of relativity were two independent concepts; afterward—they were connected with one another. There is no need even to speak of all the difficulties that quanta add to this question. We shall note only that in modern physics new and unexpected hopes—no more!—of success in this direction were brought by Einstein’s general principle of relativity. Historically, however, the first attempt was Newton’s famous work Principia Mathematica Philosophiae Naturalis1. It is set forth in the same style and spirit in which Euclid sets forth his geometry.

In the present article we should like to make several comparisons between this first attempt and, let us say, Einstein’s latest attempt to establish the foundations of physics, i.e. to substantiate mechanics.

It is customary to oppose the general principle of relativity to Newton’s classical mechanics. Who is right, Newton or Einstein? This is a question one often hears. It seems to us that to answer it in such a categorical form would not be entirely correct. Undoubtedly, the axiomatic character of Newton’s exposition, for the reasons indicated above, cannot be recognized as rigorous. The theory of relativity, although it has given hope for the possibility of axiomatics in physics, nevertheless—for the time being—no such exposition exists for it. If the axiomatics of both theories were available, then comparing them with one another would be simple. Let us compare, for example, Lobachevsky’s geometry with Euclid’s geometry. For both the one and the other we have one and the same series of definitions and axioms, except for one, which in Euclid asserts the impossibility of the intersection of two parallel straight lines and in Lobachevsky asserts precisely the opposite. As a result—two different geometries. If they could be

if it were possible to test by experiment, one could likewise express a categorical judgment about the correctness of one and the incorrectness of the other. But in the theory of Newton and Einstein there is no axiomatics, and therefore such an equally simplified judgment about their correctness is impossible. One has to compare with one another propositions whose content sometimes embraces not quite the same group of concepts and things; one cannot be certain that contradictory assertions must necessarily and completely exclude one another.

One may also note the following: when reading the introduction and the instructive passages of Newton in Volume I of the Principia of Natural Philosophy, it is hard not to think that his thought proceeds along exactly the same path as Einstein’s. The level of mathematical and physical knowledge is now not what it was in Newton’s time. If the logical course of thought, proceeding along the same path as that followed by Newton, therefore now leads to a different result, then what should be considered the true content of his theory: the path indicated by him, or the result at which he arrived? The formal advantage belongs to the latter. The former entails the danger of an arbitrary interpretation of Newton’s intentions, combined with the impossibility of detaching oneself from subjectively modern views of things. Nevertheless, it seems to us that in essence it is the more just.

We shall not undertake a formal comparison of Newton’s and Einstein’s formulas. So much has been said and written about the conclusions of the principle of relativity and about the difference between them and classical mechanics that a new repetition of the same thing cannot be of interest. We shall confine ourselves to comparing the basic concepts: 1) space and time and 2) mass and force. The comparison will be far from complete and will not exhaust all aspects of the question, since such a comparison could be given only by a special work of a historian of physics.

1. Space and Time.

Newton’s definition of absolute space and absolute time is well known, and we do not consider it necessary to repeat it here. Absolute, continuously flowing time has one dimension; continuous space has three. Euclidean geometry describes its properties. Space and time are independent of one another. In the introduction to the first edition of his book and, chiefly, in the instructive remarks to the chapter on the axioms or laws of motion, Newton explains his definitions. These explanations seem to us very interesting, and we shall cite some of them. According to Newton, one must distinguish the place occupied by a body from the body itself. “Place is a part of space occupied by a body” (p. 30). The question of how practically to separate the “place” from the body is very difficult. We read (p. 32): “However, it is completely impos-

it can neither be seen nor in any other way distinguished, by means of our senses, in its separate parts, one from another, and instead one has to resort to measurements accessible to the senses”... It may turn out that in reality there exists no body at rest to which the places and motions of the others can be referred. “From the consideration of the question follows the necessity of distinguishing relative and absolute motions. One of the conclusions (p. 33): “Absolute motion is entirely independent of those relations by which relative motion is determined.” Further (p. 33): “The phenomena by which absolute and relative motion are distinguished consist in the forces of endeavor to recede from the axis of rotational motion, for in purely relative rotational motion these forces are equal to zero, while in true or absolute motion they are greater or less, in proportion to the quantity of motion”¹). Thus dynamics—specifically centrifugal force—must make it possible to find absolute space. Newton returns once more to the difficulty of this question (p. 35): “The recognition of the true motions of individual bodies and their exact demarcation from apparent ones is very difficult, for the parts of that immovable space of which we have spoken, and in which the true motions of bodies are performed, are not perceived by our senses. Yet this matter is not wholly hopeless. Grounds for judgments may be borrowed partly from the apparent motions, which represent differences of the true ones, and partly from the forces, which represent the causes and manifestations of the true motions.” We must know the absolute and true motions of bodies: “The finding of the true motions of bodies from the causes producing them, from their manifestations, or from the differences of apparent motions and, conversely, the finding, from true or apparent motions, of their causes and manifestations is set forth in detail in what follows; indeed, with precisely this aim the proposed work has been composed.” Thus these explanations lead approximately to the following: however difficult it may be to conceive absolute space, inaccessible to the senses, the manifestations of certain forces of nature nevertheless compel us to acknowledge it; let us note that the manifestations of the forces of nature precisely with respect to it have a certain quite special meaning, to which we shall return below. As for the geometry of absolute space, Newton says already in the introduction (p. 2): “Thus, geometry is founded upon mechanical practice and is nothing other than that part of ‘general mechanics’ in which the art of exact measurement is set forth and proved.” This geometry, grounded in mechanical practice, i.e. in experience, is naturally identified by Newton with Euclid’s geometry. At the present time the question of the nature of geometrical axioms and propositions—

¹) Spacing in the original.

...propositions in geometry, regarded as an abstract mathematical discipline, and of the role played by experience in establishing them, is developed with exhaustive completeness.

Does Newton apply Euclid’s geometry as the only possible one, or does he regard the geometry that is the only one possible for him as the result of observations and experience? It seems to us that the quotation given leaves no doubt on this point. If Newton dwells little on this question, it is because the “exact mechanics” of his time provides only Euclid’s geometry and can provide nothing more, since other geometries do not yet exist. The modern physicist is in an entirely different position. He must, he is obliged to give a detailed explanation of what he understands by a geometry founded on “mechanical practice,” and precisely this explanation constitutes a very essential part of the principle of relativity. But whatever the results of such an explanation may be, the fundamental demand made of Newton’s geometry and of the theory of relativity is one and the same. According to the theory of relativity, geometry is introduced into physics by means of experience in approximately the following way. First of all, it is indicated how the things and objects of the physical world are numbered. To a definite physical phenomenon—for example, a stone lying on the road, the intersection of two spider’s threads in an astronomical telescope, and so on—there is assigned the geometrical concept of a point, and to this, now already physical, point three numbers are ascribed; in the example with the stone these three numbers may be the latitude and longitude of the place and the distance from the center of the earth. This process of numbering is the introduction of a certain coordinate system. Next, to some other physical phenomenon—for example, a light ray going from one physical point to another, or, if someone prefers it, a taut string between the same points—there is assigned the concept of a geometrical straight line. The light ray or the string, or something else, becomes a physical straight line. In exactly the same way, to other physical phenomena there is assigned a further series of basic geometrical concepts: area, angle, length, and so forth; and thus there arise physical areas, angles, lengths, etc. It is quite obvious that the physical geometry that has thus arisen will depend on the way in which the fundamental correspondences have been made, and on the laws governing the physical phenomena, i.e. on experience. It is also obvious that this does not contradict Newton’s fundamental idea.

Let us suppose that some geometry has been chosen on the basis of experience. The next question will then be the question of absolute space. Physical space is composed of physical points. According to Newton, it will be relative space. Is the absolute needed? Is it necessary to separate the body from the space occupied by it?

places? The necessity of this follows, according to Newton, from the study of relative motion, from the existence of centrifugal force. Dynamics answers the question posed. If the answer, as in Newton, is affirmative, then the existence of absolute space may be put forward as an axiom. But if the answer is negative, if absolute space, “being invisible and inaccessible to our senses in itself,” also proves inaccessible to our instruments, then it loses the meaning of its existence.

It should be added that what the modern physicist does for space, he must also do for time and, reasoning consistently, must give a physical definition of time. This circumstance in itself likewise does not contradict Newton, who is fully aware that in practice he uses sidereal time: “it is possible that there is no such uniform motion by which time could be measured with absolute exactness” (p. 31). The difference between the principle of relativity and Newton here is that in the former the definition of time corresponds not to one physical phenomenon, but is divided into the establishment of physical time at one place and the establishment of simultaneity at different places (i.e. the choice of some definite physical process is required, by means of which clocks in different places are to be compared). This partition in the process of introducing time into physics has special significance. It precisely shows that time and space are not completely separable from one another. Indeed, the distance measured between two points with the aid of a ruler has meaning only if the points and the ruler do not move, or if the measurement is made at both ends simultaneously. The question of what form the connection between time and space has must again be decided by experiment and observation. Direct geometrical experiments, however high the “art of exact measurement” may stand, and experiments with clocks give, within the limits of the observational errors proper to them, rather crude results. In order to obtain more exact results, the principle of relativity, just like Newton in proving the existence of absolute space, is compelled to turn to dynamics.

2. On Masses and Forces.

After the introduction of the concepts of time and space and the establishment of physical geometry, the next step in mechanics consists in the introduction of the concepts of mass and force. To define a physical quantity means, above all, to indicate the method of its measurement. This is what Newton does, and the principle of relativity, of course, in this respect does not

changes. Further, among the concepts introduced, certain definite relations are established. This is done either directly with the aid of experiment, or else with the aid of some considerations, with the intention subsequently—through the conclusions that follow from them—of subjecting them to experimental verification. But mass, determined by weighing; force, measured by a dynamometer; and in general all experiments, always take place in a setting which the physicist very often cannot regard as accidental. Thus, for example, when undertaking weighing for the first time, the physicist may fall into doubt as to whether at different latitudes and longitudes he will obtain the same result or different ones; whether the comparison of two forces by a dynamometer will give the same numerical ratio on the moon as on the earth, or not, and so forth. A physicist who wishes to foresee and predict events can in no way be satisfied with relations in which he has not freed himself from the contingencies of experiment noticed by him. If the question concerns the influence of latitude and longitude on the determination of mass by weighing, then everything is, of course, very simple: one can move from one place on the earth’s surface to another and verify this circumstance. But if, in order to remove doubts, it is necessary to move to the moon, then the matter is far worse. Hence arises the necessity of ascribing to certain physical quantities or phenomena properties which do not follow directly from experiment and which can be verified only indirectly. It seems to us that Newton, by identifying mass with the “quantity” of matter, thereby precisely elevates the physical quantity, mass, to the rank of something that has, so to speak, an absolute significance independent of the accompanying circumstances of measurement. In modern terminology we would say the same thing by asserting that mass is a scalar or an invariant. Let us note that in this more rigorous modern definition of the property of this quantity there is contained a whole series of additional ideas and indications which not only are absent, but also could not be present, in Newton.

The same may be said of forces. According to Newton, one must distinguish the “true” or absolute force from that which arises as a consequence of the fact that things in our relative measurements appear to us not as they are. The basic relation between “true” force, the “quantity” of matter, and motion in absolute space may, in the terminology familiar to us, be written as follows:

\[ m\vec{g}=\vec{f}, \]

where \(m\) is the mass, \(\vec{g}\) the acceleration vector, and \(\vec{f}\) the force vector. If we are in relative space, then only in rare cases will the equation written above retain its form; generally speaking, its right-hand side will be

another. If from what has been found for the relative motion of the right-hand part one subtracts the “true” force \(\vec f\), then the remaining difference will be an apparent force. The simplest example of such an “apparent” force we have in the centrifugal force. If there is no true force, then a free body is in a state of rectilinear and uniform motion. Hence follows the law of inertia and the existence of the so-called “inertial systems,” with respect to which Newton’s fundamental law retains its form—the so-called Newton–Galileo principle of relativity.

If, for Newton, in order to express a law of nature having general significance, absolute space, true force, and absolute mass in the form of a “quantity” of matter are necessary, then the modern physicist can go toward the same goal by an entirely different path. Mathematics now gives him other means for this. Let experiment and observation attach him to a definite place and time. If he wants to free himself from the contingencies of their choice, i.e. from the kind of motion in which he participates, then he can turn to the theory of groups and point transformations and, with their help, try to express the results of his measurements. This is precisely what the theory of relativity does. The geometry at which, on the basis of experience, he must arrive must necessarily depend on the distribution and magnitude of masses and on their motion. Time, not fully separable from spatial measurements, must also depend on them. In the motion of mass, spatial elements are combined with elements of time. Conventionally one may speak of time as a fourth coordinate and introduce the concept of a four-dimensional world, as distinct from three-dimensional space. One may say conventionally: the physical geometry of the world depends on the distribution of masses in it. The theories of invariants and point transformations teach that a whole series of properties of such a geometry does not depend on the choice of a special coordinate system taken for its description. A coordinate system in the four-dimensional world means an ordinary three-dimensional coordinate system moving in ordinary three-dimensional space. The expression of the “absolute” in the laws of physics may be sought not in relation to a hypothetical absolute space, but in the invariant properties of the geometry of the four-dimensional world. The “quantity” of matter is the “rest mass,” a scalar, or invariant, independent of the choice of coordinate system in this world. Instead of a “true” force with respect to absolute space, we have here a covariant or contravariant definition by means of a tensor of the corresponding rank, i.e. again a liberation from the contingencies of the choice of coordinate system. If the fundamental proposition of physics, in the form of a relation among mass, acceleration, and force, is to be preserved, then this relation must be written for an invariant

masses, for the force-tensor, for the co- or contravariant equivalent of the acceleration vector. Suppose that such a relation has been written; can one then speak of true forces and apparent forces? Let us take, for example, the force of gravitation and the centrifugal force. According to Newton, the motion of masses under the action of the force of gravitation depends on the magnitude of this force, i.e., on the same masses. In the principle of relativity, geometry and time in their totality likewise depend on masses and their distribution. But physical geometry and time are determined by means of observations of masses and their motions. Thus, in studying physical geometry and time, we thereby study the dependence between the motions of masses and their distribution, i.e., we do precisely the same thing that Newton does with the help of the force of gravitation. It goes without saying that the result obtained in the process of this study and in dependence on the chosen coordinate system must be obtained by us in such a form that, within the limits of the accuracy with which Newton’s conclusions are confirmed by experience, it coincides with his results for the same coordinate system. But since Einstein’s equations are written in invariant, i.e., system-independent, form, and for this very reason are the general law of nature, there is no ground to give preference to any one of them and to say that the expression which Newton calls a force is in one case a true force and in another an apparent one. Therefore the difference in the classification of the centrifugal force and the force of gravitation disappears. It could be preserved only in the case if the Newtonian mass causing gravitation and the mass influencing the character of physical space and time were not one and the same. It can be shown that the mass influencing the character of geometry and the mass entering into the expression for centrifugal force, i.e., the mass creating inertia and entering as the coefficient of acceleration in expression (1), coincide with one another. Newton himself, on the basis of experiment, asserts their identity. All subsequent experiments carried out with extremely great accuracy have confirmed his observation. The postulative assertion of the identity of both masses constitutes Einstein’s principle of equivalence, which he placed at the foundation of his theory.

If the content of a law of nature does not depend on the accidents of place and time, i.e., on the choice of the coordinate system, and in this sense has absolute significance, then its content can be described and verified by experiment only with the help of a special, particular coordinate system. To such a special system, not to absolute space, also belong, according to Newton, the results of observations. The formulas practically applied by both theories, even if they may differ from one another, differ only by quantities smaller than those within which the old theory has hitherto been valid. More precise observations, in individual cases already made—

...for example, for the motion of Mercury’s perihelion, may point to a difference between the two theories. In other words: to the greater rigor of one in comparison with the other. There is hardly any need to repeat that all the predictions of the theory of relativity have proved correct in experiment.

Let us note the following as well: absolute space, without which Newton cannot express the general laws of nature, if it has any real physical meaning at all, must be included among those spaces in which one can introduce an invariant expression of the general laws of the general principle of relativity. If the conclusions of the principle of relativity with respect to gravitational phenomena, i.e. chiefly celestial mechanics, were to prove incorrect, this could in no way be attributed to the fact of the existence or non-existence of absolute space and time and Euclidean geometry, and so forth, but would have to be attributed to the fact that the geometry “founded on mechanical practice” is founded incorrectly, i.e. is not in agreement with experiment. In this sense what may be refuted by experiment—for example by the Dayton–Miller experiment¹—are not the fundamental propositions of the principle of relativity, but only the form of connection between geometrical quantities by means of which certain definite physical phenomena are interpreted, and masses and, perhaps, still other physical quantities not at present taken into account. Precisely for this reason one also cannot oppose Newton’s mechanics to the principle of relativity; the latter is merely the natural and logical development of the ideas laid down in the former, which in our day has become possible thanks to broader mathematical knowledge. Nor should the principle of relativity be considered complete in its development. Quite the contrary—this is only the first stage on a still very long road. To bring physical geometry into agreement with “mechanical practice” is by no means so simple as soon as one has to go beyond quite crude measurements. In recent years, in connection with the works of Born, Heisenberg, and Jordan on the one hand, and Schrödinger² on the other, it has once again become necessary to revise the concepts of space and time. In this direction very little has so far been done; in essence it has only been indicated that the comparison of geometrical concepts with physical ones

¹ Incidentally, let us mention that in the November issue of the Proceedings of the Washington Academy of Sciences (vol. 12, No. 11), 1926, there was published a paper by Kennedy, who checked the work of Dayton–Miller and did not confirm his results: Kennedy was unable to detect any ether wind.

² I have in mind the new quantum mechanics: the matrix theory of the first three authors and the “wave” mechanics of the last. For readers unfamiliar with these questions we recommend: Foundations of the New Quantum Mechanics. A collection of articles edited by Academician A. Ioffe. State Publishing House, 1927.

phenomena is made in the principle of relativity only for macroscopic phenomena, i.e. for those which necessarily include a large number of atoms, molecules. Therefore the physical concepts of length, time, and, perhaps, mass and charge, are a kind of average concepts. To apply them to microscopic phenomena, i.e. intramolecular ones, would be a logical error. In this sense, it seems to us, one must understand Bohr’s thought that in intra-atomic processes the principle of causality may, strictly speaking, not be fulfilled, and that to construct models of atoms does not, for this reason, have much meaning. But it seems to us that there is no need for so radical a change in our views of things. The work begun by Newton and continued by Einstein can be completed and brought to its logical conclusion. The introduction of the electromagnetic field into four-dimensional geometry in the manner of Weyl–Mie, Eddington, Einstein himself, and others represents the first, still very imperfect attempts in this direction. One may hope that, proceeding along this path, it will be possible to solve the very problem which is so successfully solved by an approach—one may say—from the diametrically opposite end in the new quantum mechanics.

Let us return once more to Newton’s absolute space. This space, or “place,” separated from the body occupying it, as well as the force of gravitation in its classical form,

\[ f = k \frac{m m'}{r^2} \]

turn out, according to the theory of relativity, to be unnecessary. They are not needed in essence; in the everyday practice of the physicist, however, in his terminology, with the appropriate reservations, they will naturally remain and may even be very useful. The principle of relativity discards what is unnecessary in essence. The concept of absolute space is often associated with the concept of the ether. If the ether is needed only as a framework for absolute space, then for Einstein it is likewise unnecessary, just as absolute space is. If, however, the ether is to serve as a framework for the electromagnetic and gravitational fields, then the question changes somewhat. The gravitational field exists also in Newton. The difference lies in the speed of propagation and in the fact that in Einstein the energy of the field is equivalent to mass, whereas in Newton it is not. If one were to invent such a fluid—an ether—which would be needed for physics only, and only to the same extent as the fields mentioned, this would in no way change the actual level of our knowledge. Roger Cotes, the editor of Newton’s Principia, in the preface to the second edition, objecting to contemporary supporters of the ether, says (p. 18): “A fluid of this kind can in no way be distinguished from empty space, and the whole dispute will concern words, and not the essence of the matter,” except

(page 17): “should not, therefore, such a hypothesis, which is utterly devoid of justification, which cannot in the slightest degree serve to explain the phenomena of nature, be recognized as most absurd and wholly unworthy of a philosopher.” Of course, Cotes does not distinguish the void from the gravitational field; for Einstein, complete void in general has no physical meaning. If one places an identity sign between the electromagnetic and gravitational fields and the word ether, then this word can in principle be retained in the theory of relativity. But ether, as a fluid of quite special properties with its motions, is as unnecessary to Einstein as it was to Newton. The former, like the latter, could have said “hypotheses non fingo,” while adding the reservation inevitable for the modern physicist: “as far as possible.”

We have seen that both theories strive to follow one and the same path. To go farther does not mean to take another road. It is easier for a modern aviator to break the distance record than it was for a traveler of the seventeenth century; and if one recalls the state of science at that time, it is hard not to repeat the delighted admiration of the English poet: “Nature and Nature’s laws hid in Night, God said: ‘Let Newton be,’ and all was Light”¹).

¹) “Nature and her laws were covered by the gloom of night; and God said: ‘Let Newton be,’ and all was illumined with light.”

1. In Russian: Mathematical Principles of Natural Philosophy. 2 vols. Translated by A. N. Krylov. Published by the Naval Academy. Petersburg, 1915. All subsequent references to pages and quotations refer to this edition. ↩