Popular Science Monthly/Volume 84/May 1914/The Theory of Relativity and the New Mechanics
|THE THEORY OF RELATIVITY AND THE NEW MECHANICS|
By Professor WILLIAM MARSHALL
HE who elects to write on a mathematical topic is confronted with a choice between two evils. He may decide to handle his subject mathematically, using the conventional mathematical symbols, and whatever facts, formulas and equations the subject may demand—save himself who can! Or he may choose to abandon all mathematical symbols, formulas and equations, and attempt to translate into the vernacular this language which the mathematician speaks so fluently. In the one case there results a finished article which only the elect understand, in the other, only a rather crude and clumsy approximation to the truth. A similar condition exists in all highly specialized branches of learning, but it can safely be said that in no other science must one fare so far, and accumulate so much knowledge on the way, in order to investigate or even understand new problems. And so it is with some trepidation that the attempt is made to discuss in the following pages one of the newest and most important branches of mathematical activity. For the writer has chosen the second evil, and, deprived of his formulas, to borrow a figure of Poincaré's, finds himself a cripple without his crutches.
After this mutually encouraging prologue let us introduce the subject with a definition. What is relativity? By relativity, the theory of relativity, the principle of relativity, the doctrine of relativity, is meant a new conception of the fundamental ideas of mechanics. By the relativity mechanics, or as we may sometimes say, the new mechanics, is meant that body of doctrine which is based on these new conceptions. Now this is a very simple definition and one which would be perfectly comprehensible to everybody, provided the four following points were made clear: first, what are the fundamental concepts of mechanics, second, what are the classical notions about them, third, how are these modified by the new relativity principles, and fourth, how did it come about that we have been forced to change our notions of these fundamental concepts which have not been questioned since the time of Newton? These four questions will now be discussed, though perhaps not in this order. The results reached are, to say the least, amazing, but perhaps our astonishment will not be greater than it was when first we learned, or heard rather, that the earth is round, and that there are persons directly opposite us who do not fall off, and stranger yet, do not realize that they are in any immediate danger of doing so.
In the first place then, how has it come about that our conceptions of the fundamental notions of mechanics have been proved wanting? This crime like many another may safely be laid at the door of the physicists, those restless beings who, with their eternal experimenting, are continually raising disturbing ghosts, and then frantically imploring the aid of the mathematicians in order to exorcise them. Let us briefly consider the experiment which led us into those difficulties from which the principle of relativity alone apparently can extricate us.
Consider a source of sound A at rest (Fig. 1), and surrounded by air, in which sound is propagated, also at rest. Now, as every schoolboy knows, the time taken for sound to go to B is the same as that taken to go to C, if B and C are at the same distance from A. The same is true
|Fig. 1.||Fig. 2.|
also if A, B and C are all moving with uniform velocity in any direction, carrying the air with them. This may be realized by a closed railway car or a boat. But if the points A, B, and C are moving with uniform velocity, and the air is at rest relative to them, or what is the same thing, if they are at rest and the air is moving past them with uniform velocity, the state of affairs is very different. If the three points are moving in the direction indicated by the arrow (Fig. 2), and if the air is at rest, and if a sound wave is sent out from A, then the time required for this sound wave to go from A to C is not the same as that required to go from A to B. Now as sound is propagated in air, so is light in an imaginary medium, the ether. Moreover, this ether is stationary, as many experiments show, and the earth is moving through it, in its path around the sun with a considerable velocity. Therefore we have exactly the same case as before, and it should be very easy to show that the velocity of light in a direction perpendicular to the earth's direction of motion is different from that in a direction which coincides with it. But a famous experiment of Michelson and Morley, carried out with the utmost precision, showed not the slightest difference in these velocities. So fundamental are these two simple experimental facts, that it will be worth while to repeat them in slightly different form. If the three points A, B, C (Fig. 2), are moving to the right with a uniform unknown velocity through still air, and if a sound wave were sent out from A, it would be exceedingly simple to determine the velocity of the point A by a comparison of the time necessary for sound to travel from A to B and from A to C. But now if the same three points move through stationary ether, and if the wave emanating from A is a light wave, there is absolutely no way in which an observer connected with these three points can determine whether he is moving or not. Thus we are, in consequence of the Michelson and Morley experiment, driven to the first fundamental postulate of relativity: The uniform velocity of a body can not be determined by experiments made by observers on the body.
Consider now one of the fundamental concepts of mechanics, time. Physicists have not attempted to define it, admitting the impossibility of a definition, but still insisting that this impossibility was not owing to our lack of knowledge, but was due to the fact that there are no simpler concepts in terms of which time can be defined. As Newton says:
Absolute and real time flows on equably, having no relation in itself or in its nature to any external object.
Let us examine this statement, which embodies fairly our notion of time, in the light of the first fundamental principle of relativity just laid down. Suppose A and B (Fig. 3) are two observers, some distance apart, and they wish to set their clocks together. At a given instant agreed upon before hand, A sends out a signal, by wireless if you wish, and B sets his clock at this instant. But obviously the signal has taken some time to pass from A to B, so B's clock is slow. But this seems easy to correct; B sends a signal and A receives, and they take the mean of the correction. But says the first principle of relativity, both A and B are moving through the ether with a velocity which neither knows, and which neither can know, and therefore the time taken for the signal to pass from A to B is not the same as that taken to pass from B to A. Therefore the clocks are not together, and never can be, and when A's clock indicates half-past two, B's does not indicate this instant, and worse yet, there is absolutely no way of determining what time it does indicate. Time then is purely a local affair. The well-known phrase, "at the same instant" has no meaning for A and B, unless a definition be laid down giving it a meaning. The "now" of A may be the "past" or "future" of B. To state the case in still other words, two events can no more happen simultaneously at two different places, than can two bodies occupy the same position.
But doubtless the reader is anxious to say, this matter of adjusting the clocks together can still be settled. Let there be two clocks having the same rate at a point A, and let them be set together. Then let one of them be carried to the point B, can not they then be said to be together? Let us examine this relative motion of one clock with respect to another, in the light of the first principle of relativity. Let there be two observers as before with identical clocks, and for simplicity, suppose A is at rest and B moving on the line BX (Fig. 4). Suppose further BX parallel to AY. Let now A send out a light signal which is reflected on the line BX and returns to A. The signal has then traveled twice the distance between the lines in a certain time. B then repeats the same experiment, for, as far as he knows, he is at rest, and A moving in the opposite direction. The signal traverses twice the distance between the lines, and B's clock must record the same interval of time as A's did. But now suppose B's experiment is visible to A. He sees the signal leave B, traverse the distance between the lines, and return, but not to the point B, but to the point to which B has moved in consequence of his velocity. That is, A sees the experiment as in Fig. 5, where the position of B' depends on B's velocity with respect to A. The state of affairs is to A then simply this: A signal with a certain known velocity has traversed the distance ABA while his (A's) clock has registered a certain time interval. The same signal, moving with the same velocity, has traversed the greater distance BOB' while B's clock registers exactly the same time interval. The only conclusion is that to A, B's clock appears to be running slow as we say, and its rate will depend on the relative velocity of A and B. Thus we are led to a second conclusion regarding time in the relativity mechanics. To an observer on one body the time unit of another body moving relative to the first body varies with this relative velocity. This last conclusion regarding time is certainly staggering, for it takes away from us what we have long regarded as its most distinguishing characteristic, namely, its steady, inexorable, onward flow, which recognizes neither place nor position nor movement nor anything else. But now in the new mechanics it appears only as a relative notion, just as velocity is. There is no more reason why two beings should be living at the same rate, to coin an expression, than that two railroad trains should be running at the same speed. It is no longer a figure of speech to say that a thousand years are but as yesterday when it is past, but thousand years and yesterday are actually the same time interval provided the bodies on which these two times are measured have a sufficiently high relative velocity.
It is to be noted that in the above discussion, use was made of the fact that the light signal sent out by B appeared to A to have the same velocity as one sent out by A himself. This stated in general terms, the velocity of light in free space appears the same to all observers, regardless of the motion of the source of light or of the observer, is the second fundamental postulate of relativity. It is an assumption pure and simple, reasonable on account of the analogy between sound and light, and does not contradict any known facts.
Now there is a second fundamental concept of mechanics, very much resembling time in that we are unable to define it, namely, space. Instead of being one-dimensional, as is time, it is three-dimensional, which is not an essential difference. From the days of Newton and Galileo, physicists have agreed that space like time is everywhere the same, and that it too is independent of any motion or external object. To fix the ideas, consider any one of the units in measuring length, the yard, for example. To be sure, the bar of wood or iron, which in length more or less nearly represents this yard, may vary, as every one knows, in its dimensions, on account of varying temperature or pressure or humidity, or what not, but the yard itself, this unit of linear space which we have arbitrarily chosen, according to all our preconceived notions, neither depends on place nor position, nor motion, nor any other thinkable thing. But let us follow through another imaginary experiment in the light of the two fundamental postulates of relativity. Consider again our two observers A and B (Fig. 6), each furnished with a clock and a yardstick, A at rest, B moving in the direction indicated by the arrow. Suppose A sends out a light signal and adjusts a mirror at C say, so that a ray of light goes from A to C and returns in say one second. A then measures the distance AC with his yardstick and finds a certain number. Then B, supposing that he himself is at rest and A in motion, sends out a light signal and adjusts a mirror at D so that a ray travels the distance BD and back again in one second of his time. B then measures the distance BD with his yardstick, and since the velocity of light is the same in any system, B comes out with the same number of units of length in BD as A found in AC But A watching B's experiment sees two remarkable facts: first, that the light has not traversed the distance BDB at all, but the greater distance BD'B' (Fig. 7), where D' and B' are the points, respectively, to which D and B have moved in consequence of the motion; second, since B's clock is running slow, the time taken for light to traverse this too great distance is itself too great. Now if too great a distance is traversed in too great a time, then the velocity will remain the same provided the factor which multiplies the distance is the same as that which multiplies the time. But unfortunately, or fortunately, a very little mathematics shows that this multiplier is not the same. A sees too short a distance being traversed by light in a second of time, and therefore B's yardstick is too short, and by an amount depending on the relative velocity of A and B. Thus we are led to the astonishing general conclusion of the relativity theory with reference to length: If two bodies are moving relative to each other, then to an observer on the one, the unit of length of the other, measured in the direction of this relative velocity, appears to be shortened by an amount depending on this relative velocity. This shortening must not be looked upon as due to the resistance of any medium, but, as Minkowski puts it, must be regarded as purely a gift of the gods, a necessary accompaniment of the condition of motion. The same objection might be raised here as in the case of the time unit. Perhaps the length of the yardstick appears to change, but does the real length change? But the answer is, there is no way of determining the real length, or more exactly, the words real length have no meaning. Neither A nor B can determine whether he is in motion or at rest absolutely, and if B compares his measure with another one traveling with him, he learns nothing, and if he compares it with one in motion relative to him, he finds the two of different length, just as A did.
This startling fact, that a railway train as it whizzes past us is shorter than the same train at rest, is at first a trifle disturbing, but how much of our amazement is due to our experience, or lack of it. A certain African king, on beholding white men for the first time, reasoned that as all men were black, these beings, being white, could not be men. Are we any more logical when we say that since in our experience no yardsticks have varied appreciably on account of their velocity, hence it is absurd to admit the possibility of such a thing.
Perhaps it might be well at this point to give some idea of the size of these apparent changes in the length of the time unit and the space unit, although the magnitude is a matter of secondary importance. The whole history of physics is a record of continual striving after more exact measurements, and a fitting of theory to meet new corrections, however small. So it need not occasion surprise to learn that these differences are exceedingly minute; the amazing thing, and the thing of scientific interest, is that they exist at all. If we consider the velocity of the earth in its orbit, which is about 19 miles per second, the shortening of the earth's diameter due to this velocity as seen by an observer at rest relative to the earth would be approximately a couple of inches only. Similarly for the relative motion of the earth and the sun, the shortening of the time unit would be approximately one second in five years. Even if this were the highest relative velocity known, the results would still be of importance, but the earth is by no means the most rapid in its movement of the heavenly bodies, while the velocity of the radium discharge is some thousand times the velocity of the most rapidly moving planet.
In addition to space and time there is a third fundamental concept of mechanics, though the physicists have not yet settled to the satisfaction of everybody whether it is force or mass. But in any case, the one taken as the fundamental motion, mass say, is, in the classical mechanics, independent of the velocity. Mass is usually defined in physics as the quantity of matter in a body, which means simply that there 13 associated with every body a certain indestructible something, apart from its size and shape, independent of its position or motion with respect to the observer, or with respect to other masses. But in the relativity mechanics this primary concept fares no better than the other ones, space and time. Without going into the details of the argument by means of which the new results are obtained, and this argument, and the experiment underlying it, are by no means simple, it may suffice to say that the mass of a body must also be looked upon as depending on the velocity of the body. This result would seem at first glance to introduce an unnecessary and almost impossible complication in all the considerations of mechanics, but as a matter of fact exactly the opposite is true. It has been known for some time, that electrons moving with the great velocity of the electric discharge, have suffered an apparent increase of mass or inertia due to this velocity, so that physicists for some time have been accustomed to speak of material mass and electromagnetic mass. But now in the light of the principles of relativity, this distinction between material mass and electromagnetic mass is lost, and a great gain in generality is made. All masses depend on velocity and it is only because the velocity of the electric discharge approaches that of light, that the change in mass becomes striking. This perhaps may be looked upon as one of the most important of the consequences of the theory of relativity in that it subjects electromagnetic phenomena to those laws which underlie the motions of ordinary bodies.
In consequence of this revision of our notions of space, time and mass, there result changes in the derived concepts of mechanics, and in the relations between them. In fact the whole subject of mechanics has had to be rewritten on this new basis, and a large part of the work of those interested in the relativity theory has been the building up of the mathematics of the new subject. Some of the conclusions, however, can be understood without much mathematics. For example, we can no longer speak of a particle moving in space, nor can we speak of an event as occurring at a certain time. Space and time are not independent things, so that when the position of a point is mentioned, there must also be given the instant at which it occupied this position. The details of this idea, as first worked out by Minkowski, may be briefly stated. With every point in space there is associated a certain instant of time, or to drop into the language of mathematics for a moment, a point is determined by four coordinates, three in space and one in time. We still use the words space and time out of respect for the memory of these departed ideas, but a new term including them both is actually in use. Such a combination, i. e., a certain something with its four coordinates, is called by Minkowski a world point. If this world point takes a new position, it has four new coordinates, and as it moves it traces out in what Minkowski calls the world, a world-line. Such a world-line gives us then a sort of picture of the eternal life history of any point, and the so-called laws of nature can be nothing else than statements of the relations between these world-lines. Some of the logical consequences of this world-postulate of Minkowski appear to the untrained mind as bordering on the fantastic. For example, the apparatus for measuring in the Minkowskian world is an extraordinarily long rod carrying a length scale and a time scale, with their zeros in coincidence, together with a clock mechanism which moves a hand, not around a circle as in the ordinary clock, but along the scale graduated in hours, minutes and seconds.
Some of the conclusions of the relativity mechanics with reference to velocity are worth noting. In the classical mechanics we were accustomed to reason in the following way: Consider a body with a certain mass at rest. If it be given certain impulse, as we say, it takes on a certain velocity. The same impulse again applied doubles this velocity, and so on, so that the velocity can be increased indefinitely, and can be made greater than any assigned quantity. But in the relativity mechanics, a certain impulse produces a certain velocity, to be sure; this impulse applied again does not double the velocity; a third equal impulse increases the velocity but by a still less amount, and so on, the upper limit of the velocity which can be given to a body being the velocity of light itself. This statement is not without its parallel in another branch of physics. There is in heat what we call the absolute zero, a value of the temperature which according to the present theory is the lower limit of the temperature as a body is indefinitely cooled. No velocity then greater than the velocity of light is admitted in the relativity mechanics, which fact carries with it the necessity for a revision of our notion of gravitational action, which has been looked upon as instantaneous.
In consequence of the change in our ideas of velocity, there results a change in one of the most widely employed laws of velocity, namely the parallelogram law. Briefly stated, in the relativity mechanics, the composition of velocities by means of the parallelogram law is no longer allowable. This follows evidently from the fact that there is an upper limit for the velocity of a material body, and if the parallelogram law were to hold, it would be easy to imagine two velocities which would combine into a velocity greater than that of light. This failure of the parallelogram law to hold is to the mathematician a very disturbing conclusion, more heretical perhaps than the new doctrines regarding space and time.
Another striking consequence of the relativity theory is that the hypothesis of an ether can now be abandoned. As is well known, there have been two theories advanced in order to explain the phenomena connected with light, the emission theory which asserts that light effect is due to the impinging of particles actually sent out by the source of light, and the wave theory which assumes that the sensation we call light is due to a wave in a hypothetical universal medium, the ether. Needless to say this latter theory is the only one which recently has received any support. And now the relativists assert that the logical thing to do is to abandon the hypothesis of an ether. For they reason that not only has it been impossible to demonstrate the existence of an ether, but we have now arrived at the point where we can safely say that at no time in the future will any one be able to prove its existence. And yet the abandoning of the ether hypothesis places one in a very embarrassing position logically, as the three following statements would indicate:
1. The Michelson and Morley experiment was only possible on the basis of an ether hypothesis.
2. From this experiment, follow the essential principles of the relativity theory.
3. The relativity theory now denies the existence of the ether. Whether there is anything more in this state of affairs than mere filial ingratitude is no question for a mathematician.
It should perhaps be pointed out somewhat more explicitly that these changes in the units of time, space and mass, and in those units depending on them, are changes which are ordinarily looked upon as psychological and not physical. If we imagine that A has a clock and that about him move any number of observers, B, C, D,. . ., in different directions and with different velocities, each one of these observers sees A′s clock running at a different rate. Now the actual physical state of A′s clock, if there is such a state, is not affected by what each observer thinks of it; but the difficulty is that there is no way for any one except A to get at the actual state of A′s clock. We are then driven to one of the two alternatives: Either we must give up all notion of time at all, for bodies in relative motion, or we must define it in such a way as will free it of this ambiguity, and this is exactly what the relativity mechanics attempts to do.
Any discussion of the theory of relativity would be hardly satisfactory without a brief survey of the history of the development of the subject. As has been stated, for many years the ether theory of light has found general acceptance, and up to about twenty-five years ago practically all of the known phenomena of light, electricity and magnetism were explained on the basis of this theory. This hypothetical ether was stationary, surrounded and permeated all objects, did not, however, offer any resistance to the motion of ponderable matter. There came then, in 1887, into this fairly satisfactory state of affairs, the famous Michelson and Morley experiment. This experiment was directly undertaken to discover, if possible, the so-called ether drift.
In this experiment, the apparatus was the most perfect that the skill of man could devise, and the operator was perhaps one of the most skilful observers in the world, but in spite of all this no result was obtained. Physicists were then driven to seek some theory which would explain this experiment, but with varying success. It was proposed that the ether was carried along with the earth, but a host of experiments show this untenable. It was suggested that the velocity of light depends on the velocity of the source of light, but here again there were too many experiments to the contrary. Michelson himself offered no theory, though he suggested that the negative result could be accounted for by supposing that the apparatus underwent a shortening in the direction of the velocity and due to the velocity, just enough to compensate for the difference in path. This idea was later, in 1892, developed by Lorentz, a Dutch physicist, and under the name of the Lorentz-shortening hypothesis has had a dignified following. The Michelson and Morley experiment, together with certain others undertaken for the same purpose, remained for a number of years as an unexplained fact—a contradiction to ascertained well-established and orderly physical theory. Then there appeared in 1905, in the Annalen der Physik, a modest article by A. Einstein, of Bern, Switzerland, entitled, "Concerning the Electrodynamics of Moving Bodies," In this article Einstein, in a very unassuming way, and yet in all confidence, boldly attacked the problem and showed that the astonishing results concerning space and time which we have just considered, all follow very naturally from very simple assumptions. Naturally a large part of his paper was devoted to the mathematical side—to the deduction of the equations of transformation which express mathematically the relation between two systems moving relative to each other. It may safely be said that this article laid the foundation of the relativity theory.
Einstein's article created no great stir at the time, but within a couple of years his theory was claiming the attention of a number of prominent mathematicians and physicists. Minkowski, a German mathematician of the first rank, just at this time turning his attention to mathematical physics, came out in 1909 with his famous world postulate, which has been briefly described. It is interesting to note that within a year translations of Minkowski's article appeared in English, French and Italian, and that extensions of his theories have occupied the attention of a number of Germany's most famous mathematicians. Next Poincaré, perhaps the most brilliant mathematician of the last quarter century, stamped the relativity theory with the unofficial approval of French science, and Lorentz, of Holland, one of the most famous in a land of famous physicists, aided materially to the development of the subject. Thus we find within five years of the appearance of Einstein's article, a fairly consistent body of doctrine developed, and accepted to a surprising degree by many of the prominent mathematical physicists of the foremost scientific nations. No sooner was the theory in a fairly satisfactory condition, than the attempt was made to verify some of the hypotheses by direct experiment. Naturally the difficulties in the way of such experimental verification were very great—insurmountable in fact for many experiments, since no two observers could move relative to each other with a velocity approaching that of light. But the change in mass of a moving electron could be measured, and a qualitative experiment by Kaufmann, and a quantitative one by Bucherer gave results which were in good agreement with the theoretical equations. It was the hope of the astronomers that the new theory would account for the long-outstanding disagreement between the calculated and the observed motion of Mercury's perihelion, but while the relativity mechanics gave a correction in the right direction, it was not sufficient. To bring this very brief historical sketch down to the present time, it will perhaps be sufficient to state that this theory is at present claiming the attention of a large number of prominent mathematicians and physicists. The details are being worked out, the postulates are being subjected to careful mathematical investigation, and every opportunity is being taken to substantiate experimentally those portions of the theory which admit of experimental verification. Practically all of the work which has been done is scattered through research journals in some six languages, bo that it is not very accessible. Some idea of the number of articles published may be obtained from the fact that a certain incomplete bibliography contains the names of some fifty-odd articles, all devoted to some phase of this subject—varying all the way from the soundest mathematical treatment, at the one end of the scale, to the most absurd philosophical discussion at the other. And these fifty or more articles include only those in three languages, only those which an ordinary mathematician and physicist could read without too great an expenditure of time and energy, and with few exceptions, only those which could be found in a rather meager scientific library.
In spite of the fact that the relativity theory rests on a firm basis of experiment, and upon logical deductions from such experiments, and notwithstanding also that this theory is remarkably self-consistent, and is in fact the only theory which at present seems to agree with all the facts, nevertheless it perhaps goes without saying that it has not been universally accepted. Some objections to the theory have been advanced by men of good standing in the world of physics, and a fair and impartial presentation of the subject would of necessity include a brief statement of these objections. I shall not attempt to answer these objections. Those who have adopted the relativity theory seem in no wise concerned with the arguments put forward against it. In fact, if there is one thing which impresses the reader of the articles on relativity, it is the calm assurance of the advocates of this theory that they are right. Naturally the theory and its consequences have been criticized by a host of persons of small scientific training, but it will not be necessary to mention these arguments. They are the sort of objections which no doubt Galileo had to meet and answer in his famous controversy with the Inquisition. Fortunately for the cause of science, however, the authority back of these arguments is not what it was in Galileo's time, for it is not at all certain just how many of those who have enthusiastically embraced relativity would go to prison in defence of the dogma that one man's now is another man's past, or would allow themselves to be led to the stake rather than deny the doctrine that the length of a yardstick depends upon whether one happens to be measuring north and south with it, or east and west.
In general it may be said that the chief objection to the relativity theory is that it is too artificial. The end and aim of the science of physics is to describe the phenomena which occur in nature, in the simplest manner which is consistent with completeness, and the objectors to the relativity theory urge that this theory and especially its consequences, are not simple and intelligible to the average intellect. Consider, for example, the theory which explains the behavior of a gas by means of solid elastic spheres. This theory may be clumsy, but it is readily understood, rests upon an analogy with things which can be seen and felt, in other words is built up of elements essentially simple. But the objectors to the relativity theory say that it is based on ideas of time and space which are not now and which never can be intelligible to the human mind. They claim that the universe has a real existence quite apart from what any one thinks about it, and that this real universe, through the human senses, impresses upon the normal mind certain simple notions which can not be changed at will. Minkowski's famous world-postulate practically assumes a four-dimensional space in which all phenomena occur, and this say the objectors, on account of the construction of the human mind, can never be intelligible to any one in spite of its mathematical simplicity. They insist that the words space and time, as names for two distinct concepts, are not only convenient, but necessary. Nor can any description of phenomena in terms of a time which is a function of the velocity of the body on which the time is measured ever be satisfactory, simply because the human mind can not now nor can it ever appreciate the existence of such a time. To sum up, then, this model of the universe which the relativists have constructed in order to explain the universe, can never satisfactorily do this, for the reason that it can never be intelligible to everybody. It is a mathematical theory and can not be satisfactory to those lacking the mathematician's sixth sense.
A second serious objection urged against the relativity theory is that it has practically abandoned the hypothesis of an ether, without furnishing a satisfactory substitute for this hypothesis. As has been previously stated, the very experiment which the relativity theory seeks to explain depends on interference phenomena which are only satisfactorily accounted for on the hypothesis of an ether. Then too, there are in electromagnetism certain equations of fundamental importance, known as the Maxwell equations, and it is perhaps just as important that the relativity theory retain these equations, as it is that it explain the Michelson and Morley experiment. But the electro-magnetic equations were deduced on the hypothesis of an ether, and can be explained, or at least have been explained only on the hypothesis that there is some such medium in which the electric and magnetic forces exist. So, say the objectors to the relativity theory, the relativists are in the same illogical (or worse) position that they occupy with reference to the Michelson and Morley experiment, in that they deny the existence of the medium which made possible the Maxwell equations, which equations the relativity theory must retain at any cost. Professor Magie, of Princeton, who states with great clearness the principal objections to the theory, waxes fairly indignant on this point, and compares the relativists to Baron Munchausen, who lengthened a rope which he needed to escape from prison, by cutting off a piece from the upper end and splicing it on the lower. The objectors to the relativity theory point out that there have been advocated only two theories which have explained with any success the propagation of light and other phenomena connected with light, and that of these two, only the ether theory has survived. To abandon it at this time would mean the giving up of a theory which lies at the foundation of all the great advances which have been made in the field of speculative physics.
It remains finally to ask and perhaps also to answer the question, whither will all this discussion of relativity lead us, and what is the chief end and aim and hope of those interested in the relativity theory. The answer will depend upon the point of view. To the mathematician the whole theory presents a consistent mathematical structure, based on certain assumed or demonstrated fundamental postulates. As a finished piece of mathematical investigation, it is, and of necessity must remain, of theoretical interest, even though it be finally abandoned by the physicists. The theory has been particularly pleasing to the mathematician in that it is a generalization of the Newtonian mechanics, and includes this latter as a special case. Many of the important formulas of the relativity mechanics, which contain the constant denoting the velocity of light become, on putting this velocity equal to infinity, the ordinary formulas of the Newtonian mechanics. Generality is to the mathematician what the philosopher's stone was to the alchemist, and just as the search for the one laid the foundation of modern chemistry, so is the striving after the other responsible for many of the advances in mathematics.
On the other hand, those physicists who have advocated the theory of relativity see in it a further advance in the long attempt to rightly explain the universe. The whole history of physics, is, to use a somewhat doubtful figure of speech, strewn with the wrecks of discarded theories. One does not have to go back to the middle ages to find amusing reading in the description of these theories, which were seriously entertained and discarded only with the greatest reluctance. But all the arguments of the wise, and all the sophistries of the foolish, could not prevent the abandoning of a theory, if a few stubborn facts were not in agreement with it. Of all the theories worked out by man's ingenuity, no one has seemed more sure of immortality than the one we know as the Newtonian mechanics. But the moment a single fact appears which this system fails to explain, then to the physicist with a conscience this theory is only a makeshift until a better one is devised. Now this better one may not be the relativity mechanics—its opponents are insisting rather loudly that it is not. But in any case, the entire discussion has had one result pleasing alike to the friends and foes of relativity. It has forced upon us a fresh study of the fundamental ideas of physical theory, and will give us without doubt, a more satisfactory foundation for the superstructure which grows more and more elaborate.
It can well happen that scientists, some generations hence, will read of the. relativity mechanics with the same amused tolerance which marks our attitude towards, for example, Newton's theory of fits of easy transmission and reflection in his theory of the propagation of light. But whatever theory may be current at that future time, it will owe much to the fact that in the early years of the twentieth century, this same relativity theory was so insistent and plausible, that mathematicians and physicists in sheer desperation were forced either to accept it, or to construct a new theory which shunned its objectionable features. Whether the relativity theory then is to serve as a pattern for the ultimate hypothesis of the universe or whether its end is to illustrate what is to be avoided in the construction of such a hypothesis, is perhaps after all not the important question.