Classical physics

“ If it is found that k is different from unity we shall be forced to the conclusion that either our conception of mass in the classical mechanics or our conception of charge in the classical electrical theory is in need of essential modification. ”

“ Section 21 - In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory? We have already stated several times that classical mechanics starts out from the following law: Material particles sufficiently far removed from other material particles continue to move uniformly in a straight line or continue in a state of rest. We have also repeatedly emphasised that this fundamental law can only be valid for bodies of reference which possess certain unique states of motion, and which are in uniform translational motion relative to each other. ”

“ In the second place, classical mechanics indicates a limitation which directly demands an extension of the principle of relativity to spaces of reference which are not in uniform motion relatively to each other. The ratio of the masses of two bodies is defined in mechanics in two ways which differ from each other fundamentally ”

“ I should like to make it clear that the limitation of the scope of physics to pointer readings and the like is not a philosophical craze of my own but is essentially the current scientific doctrine. It is the outcome of a tendency discernible far back in the last century but only formulated comprehensively with the advent of the relativity theory. The vocabulary of the physicist comprises a number of words such as length, angle, velocity, force, potential, current, etc., which we call “physical quantities”. ”

“ As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. ”

“ In the light of the new investigations, the mechanics of the atom would stand on an entirely different footing from that of classical or even of relativistic science. Just as classical mechanics appears as a first approximation holding good for low velocities, but breaking down for high velocities, in which case the more embracing relativistic mechanics must take its place, so now relativistic mechanics ceases to be applicable when minute regions of atomic dimensions are contemplated. ”

“ It is not quite true to say that all the physical quantities are relative to frames of space. We can construct new physical quantities by multiplying, dividing, etc.; thus we multiply mass and velocity to give momentum, divide energy by time to give horse-power. We [Pg 23] can set ourselves the mathematical problem of constructing in this way quantities which shall be invariant, that is to say, shall have the same measure whatever frame of space may be used. One or two of these invariants turn out to be quantities already recognised in pre-relativity physics ”

“ Also the physical laws, which are differential equations defining as functions of and certain constants, are in general altered by a change of the system of reference. Physical science has, however, been so built up that there are certain transformations which leave the form of the equations unaltered. In the classical, or Newtonian, mechanics these are all orthogonal transformations of the three coordinates combined with an arbitrary uniform velocity of these axes and an arbitrary change of the units of length and time. ”

“ The metrical, gravitational and electromagnetic fields are all included. We build the quantities enumerated above; and they obey the great laws of field-physics in virtue of the way in which they have been built. That is the special feature; the field laws—conservation of energy, mass, momentum and of electric [Pg 237] charge, the law of gravitation, Maxwell’s equations—are not controlling laws. [38] They are truisms. Not truisms when approached in the way the mind looks out on the world, but truisms when we encounter them in a building up of the world from a basal structure. ”

“ In order to remedy this indeterminateness, it is usual to specify that the body on which the force is acting is one of unit mass; it is then called a test-body. Under these circumstances, we can explore a field of force by placing our test-body in successive regions of space and determining the magnitude and direction of the force which is acting on the body. When we have mapped out the magnitude and direction of the force for every point of space, we are in a position to state that we have determined the lay of the field of force.
In physics we are often concerned with fields of force. ”

“ In classical science, laws were always tensor equations in separate space and time; but the relativity theory, by compelling us to substitute space-time for space, narrows down our choice of laws in a remarkable manner. Thus, in classical science, the law of gravitation might have been practically anything ”

“ It must indeed be said, it is not alone experimental physics that has given birth to them; mathematical physics has well contributed. It is the experimenters who have seen radium throw out energy, but it is the theorists who have put in evidence all the difficulties raised by the propagation of light across a medium in motion ”

“ The special theory of relativity can, therefore, be regarded only as a stepping-stone to a more general principle, which orders gravitational phenomena satisfactorily into the principles of mechanics. ”

“ Merely because in mathematics—in the case of , for example—the laws of Euclidean geometry were accepted from the start, whereas in physics preliminary information of this sort is denied us. And it is the same with all the magnitudes and constants of physics. It is impossible to predetermine by logical means the value of the invariant velocity, or of the gas constant, or of the size of the earth or of an electron. In every case, we must have recourse to physical measurements of one sort or another, for the constants of physics present no character of rational necessity. ”

“ Energy theorem and momentum theorems could be generally deduced from the fundamental equations of the dynamics of the electron. To derive the second axiom of Newton, we confined ourselves to quasi-stationary translational motion. There, the concept of mass experienced an extension; the electromagnetic mass is not a scalar, but a tensor with symmetry of an ellipsoid of revolution; both masses, longitudinal and transverse, are functions of velocity. ”

“ Hence, if we are to be consistent, we must regard Keplerian motion as an appearance, not as a reality. And here we come in contact with the general problem of the dynamics of the electron, which in the historical sense is responsible for the introduction of the principle of relativity. In this wider problem there is a profound modification of the notion of mass, in addition to the transformation which we have admitted in the time and the spatial coordinates. ”

“ Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. ”

“ The laws of electrodynamics require that such a force be generated, and yet no force can be observed; so we must assume that the mysterious compensations that conceal our motion through the ether are of so perfectly general a nature that they affect mechanics as well as electromagnetics and optics. ”

“ Therein the task is treated to determine the preferred form of the fundamental equations of mechanics, which take the place of the usual Newtonian equations of motion of a free mass point:
when the relativity principle should have general validity. ”

“ Chemistry here comes to the aid of mechanics, and we may say boldly that that matter, which is identical with ours, is subject, like it, to the laws of gravitation. Certainly, when Newton decomposed a beam of white light, and laid the first basis of the theory of the spectrum, he had not the slightest suspicion that his law of gravitation would, at a later period, find in it wings to carry it into regions where all measurement ceases and all calculation is powerless. ”

“ If our ideas possess a certain degree of artificiality, this is also true of others which have long since been adopted into mechanics. The apparent change in rate of a moving clock, and the apparent change in length and mass of a moving body, are completely analogous to that apparent change in energy of a body in motion, which we have long been accustomed to call its kinetic energy. ”

“ First, there is no more matter, since the positive electrons no longer have real mass, or at least no constant real mass. The present principles of our mechanics, founded upon the constancy of mass, must therefore be modified. Again, an electromagnetic explanation must be sought of all the known forces, in particular of gravitation, or at least the law of gravitation must be so modified that this force is altered by velocity in the same way as the electromagnetic forces. ”

“ We have shown how observers stationed on systems in motion relative to one another have been able to preserve their fundamental principles of mechanics only by adopting certain novel conclusions. These conclusions are self-consistent; in the one case where they have been tested they are in accord with experiment; and they enable us to save all the fundamental physical concepts which have been found useful in the past. We have, however, considered primarily only systems which are initially in uniform relative motion. ”

“ This ideal has often played an effective part in popular lectures, but in the workshop of the serious inquirer it has discharged scarcely the least function. What has really been achieved in mechanical physics is either the elucidation of physical processes by more [Pg 160] familiar mechanical analogies, (for example, the theories of light and of electricity,) or the exact quantitative ascertainment of the connexion of mechanical processes with other physical processes, for example, the results of thermodynamics. ”

“ If, on the other hand, the nature of gravity were essentially not electrodynamic, it would be possible for some deviation from the accepted laws of dynamics, owing to a motion of translation through space, to be revealed by direct astronomical observations. ”

“ Are we then entitled to assert that every particle in the universe attracts every other particle with a force which is proportional to the product of their masses, and which varies inversely as the square of their distance? We have, indeed, but a slender basis of fact on which to rest a proposition so universal. Let us attempt to enunciate the law of gravitation so as to commit ourselves to no assertion not absolutely proved. ”

“ I may here point out that by reforming mechanics, where instead of the Newtonian Relativity-Postulate with we assume a relativity-postulate with a finite c, then the axiomatic construction of Mechanics appears to gain considerably in perfection. ”

“ Since the time enters to the second power, the product of mass and acceleration is smaller by the ratio than it would be if the system were at rest. And we conclude, therefore, that the unit of force, or the dyne, in a direction transverse to the line of motion is smaller in a moving system than in one at rest by this same ratio. ”

“ We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. [Pg 143] It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example. ”

“ To make my position more definite, let us consider one of the methods by which the equations of motion of an electron are obtained in the usual electromagnetic theory. The principle is adopted that at each instant the integral over the electron of the total force on each element must be zero. Now before this principle can be used to write down the equations of motion we must know the design of the electron, and we must know the way in which the motions of the different elements are co-ordinated. ”

“ A force which belongs to individual atoms, is independent of chemical and physical conditions, and cannot be altered or destroyed by any known means, must be closely related to the fundamental nature of the atoms. One of the most essential parts in our concept of matter is mass, and the force of gravitation of an atom is proportional to its mass. Mass and gravitation stand, therefore, either as co-effects of the same cause or as cause and effect. ”

“ The mathematical argument which shows that the energy of the electron in its orbit only depends upon proceeds on Newtonian principles; more particularly, it treats the mass of the electron as constant. But in the modern theory of relativity, the mass of a body is increased by rapid motion. This increase is not noticeable for ordinary velocities, but becomes very great as we approach the velocity of light, which is a limit that no material body can quite reach. ”

“ They concern the adaptation of the principles of mathematical physics to the form of the relativity principle which is here maintained. Einstein’s method of using the theory of tensors is adopted, but the application is worked out on different lines and from different assumptions. ”

“ It consists, broadly speaking, in showing that, although the particles, points, and instants with which physics operates are not themselves given in experience, and are very likely not actually existing things, yet, out of the materials provided in sensation, it is possible to make logical constructions having the mathematical properties which physics assigns to particles, points, and instants. ”