CHAPTER XIXVARIOUS POSSIBLE WORLDS
WE have seen that the great distinction between Einstein’s theory and classical science arises from the value to be attributed to the invariant velocity of the world. In the belief of classical science, this velocity was infinite, and we were thus led to a world of separate space and time. According to relativity, the value of the invariant velocity is finite and is given by,the constant which enters into Maxwell’s equations; this constant being illustrated physically by the velocity of lightin vacuo.
From a purely mathematical standpoint, however, if we disregard the relations of reality, we may consider other purely formal possibilities. Suppose that the invariant velocity, though finite, were some numberdiffering from Maxwell’s constant.Obviously,being a maximum velocity, would have to be greater than or equal to the highest velocity known to physicists, hence would have to be equal to or greater than,velocity of light. Let us therefore suppose thatis greater than.As in Einstein’s theory, we should still be faced with a world of four-dimensional space-time, but this world would differ from that of relativity, owing to the discrepancy existing between the invariant velocityand Maxwell’s constant.This would indicate that the equations of electromagnetics would no longer maintain the same form when we changed Galilean frames. As a result, the erstwhile negative experiments in electromagnetics should now yield positive results, and absolute velocity through space would be detected. The theory of relativity would have to be abandoned.
But we may consider still another case. Suppose that the invariant velocity of the world were some imaginary quantity,wherestands, as before, for,andfor some arbitrary finite number. The square of this imaginary velocity being a negative number, the square ofwould now assume the formThuswould be given by a sum of four squares instead of by a sum of three squares and a difference. This would prove that the space-time continuum was Euclidean, having its four dimensions of the same category, instead of semi-Euclidean, as in Einstein’s theory, with an imaginary dimension for time. Time would now fail to be differentiated from space. It is difficult to realise how a world of this sort would manifest itself to us. The only reason we mention this case is in order to show with greater clarity that the classical belief in the separateness of time and space was equivalent to the classical assumption that the invariant velocity was infinite, and not a finite magnitude, whether real or imaginary.
A point that appears to have misled a number of persons relates to the meaning to be given to a four-dimensional space and time continuum. In all the cases mentioned in this chapter, it would be permissible to speak of the world as a four-dimensional continuum of events. But with the separate space and time of classical science the statement was artificial, owing to the absoluteness of time, which stood out by itself as distinct from space; and so the four-dimensional aspect of the world was never stressed. It is different with space-time. For now this aspect can no longer be disregarded, since it becomes impossible to divorce space from time in any absolute way holding for all observers. Thus we see that the four-dimensional nature of the world is a fact which has been disclosed by the theory of relativity. Prior to Einstein’s discoveries, any reference to space and time as one continuum would have been an unjustified extension of the accepted meaning of words.
Now let us return to the various types of worlds we have discussed in the preceding paragraphs. It is possible to give an alternative presentation of the results mentioned. Thus, we start from the mathematical expression of:whererepresents the invariant velocity without any specification as to its value. (It may be infinite, as in classical science, or equal to Maxwell’s constant,as in Einstein’s theory, or even equal to some imaginary number, as in the case last discussed.) Then it can be shown that by dividing this equation byand equating the result to zero, we obtain an equation which corresponds to relations holding in three-dimensional space. This equation is given bywhere the’s are the components of this velocity along the three mutually perpendicular directions of space. When a mathematical expression of this kind is given as invariant or absolute, the theory of groups enables us to determine the nature of the geometrical rules according to which the variables entering into the expression may be added together. In the present case these variables represent velocities, so that we are on the way to discover the rules governing the addition of velocities. From this equation the following results are obtained.
1°. If,the invariant velocity, is infinite, as in classical science, it is found that the composition of velocities must be Euclidean; so we should be able to combine velocities graphically by the well-known rule of the parallelogram or of the Euclidean triangle. In this case, of course, velocities which lie in the same direction can be added up and subtracted like numbers.
2°. If,the invariant velocity, is no longer infinite but finite, and if it is a real number, as in Einstein’s theory, the rules according to which velocities combine are found to be those of Lobatchewskian geometry. If, therefore, in Einstein’s theory, we wish to compound velocities, we must operate no longer on a Euclidean but on a Lobatchewskian triangle. We might, for example, trace our triangles on the surface of a pseudosphere, since the geometry of this surface is, as we know, Lobatchewskian. In this case velocities lying in the same direction will no longer add up like the numbers of arithmetic.
3°. In a similar way we should find that if the invariant velocity,while finite, happened to be an imaginary number, the composition of velocities would follow the rules of Riemann’s geometry. In this case we saw that four-dimensional space-time would be Euclidean and not semi-Euclidean.[66]We merely mention this third type of universe for motives of symmetry. Obviously it does not correspond to the world we live in, hence we will not refer to it in future, confining ourselves to the classical world and to that of relativity.
When we are called upon to decide which of the two alternatives is correct (space-time or separate space and time), which one corresponds to reality,a priorispeculations are futile, since both solutions are conceivable and satisfy the facts of crude observation. Our only recourse is then to appeal to experiment, and by experiment we mean observations that are more reliable than our crude perceptions unaided by ultra-precise instruments. It is only thanks to such experiments that we may succeed in ascertaining whether or not a finite invariant velocity is demanded by the world-structure. Should our experiments point to the existence of a finite invariant velocity, our problem would be settled in favour of a world of space-time as against one of separate space and time.
Now, experiment suggests that the propagation of lightin vacuofurnishes us with a concrete example of this critical velocity. It follows that a measure of the speed of light propagation will yield us the precise numerical value of the critical velocity. For this reason, the propagation of light assumes a position of vast theoretical importance. Needless to say, this importance of light is due solely to its physical behaviourin vacuo, solely to the fact that it moves with the invariant speed,notto the fact that because of some accidental circumstance it happens to be visible to the human eye. For light passing through glass or water is also visible, but its speed, being reduced, is no longer invariant. Hence light propagation through matter can no longer serve to define the invariant velocity directly. Besides we can get away from light as a visible propagation by considering an invisible electromagnetic propagation in its stead.
If we wish, we can do better still and obviate electromagnetic waves entirely. For when account is taken of the negative experiments in electrodynamics, we realise that the invariant velocity is none other than the constantwhich enters into Maxwell’s equations, whence any experiment capable of measuringwould automatically yield us the invariant velocity. In particular, we might measure the ratio of an electric charge when computed in terms of electrostatic, then of electromagnetic units; this again would yield us the value of Maxwell’s constant,hence of the critical invariant velocity. We might also appeal to the necessary consequences of the existence of a critical velocity, for instance, to the peculiar Lobatchewskian law which it would entail for the additions of velocities. We should then check these consequences by experiment, discover whether they were justified by facts, and deduce therefrom the existence and value of the invariant velocity.
Fizeau’s experiment permits this verification and proves that the composition of velocities is Lobatchewskian, not Euclidean, and corresponds to the existence of a critical invariant velocity of 186,000 miles per second. It is true that in Fizeau’s experiment we are again dealing with the propagation of light, though no longerin vacuo, where its speed is invariant, but through water. It is precisely because its speed through water is reduced that the velocity of the water will affect the velocity of the light, thereby permitting us to test out the relativistic law of the composition of velocities. But once again the reason why light propagations enter into the majority of these tests is due solely to the fact that optical experiments permit of greater accuracy. Apart from this, there is no need to limit ourselves to experiments dealing with the propagation of light, whetherin vacuoor through material bodies such as water. Any speed, be it the speed of sound or of a bullet shot from a passing train, should enable us to detect the same law of Lobatchewskian composition and thus allow us to demonstrate the existence of space-time.
At all events, one point should be clear by now. Whatever method we follow, we can never get away from performing physical measurements of one kind or another. It is therefore meaningless to speak of defining the invariant critical velocity without reference to a measurement of the speed of lightin vacuo, or more generally, without reference to physical measurements bearing on the phenomena which the existence of a finite invariant speed in the universe would necessarily entail. From this it follows that without physical measurements of one kind or another we have no means of establishing whether the world is one of separate space and time or one of space-time.
In theory even crude observations should have led us to a decision on this score, for crude observations differ from those of the physicist only in the matter of precision. Theoretically, therefore, prior to Einstein we should have known of the incorrectness of the Euclidean composition of velocities and hence known of space-time. The reason for our failure to make the momentous discovery is easily understood when we realise that owing to the very great magnitude of the invariant finite velocity,the Einsteinian rule can scarcely be differentiated from the classical rule unless very great velocities are involved. From this it is seen that the reason why the effects anticipated by Einstein’s theory are so difficult to detect is because of the very high value of the invariant velocity.Had this finite invariant velocity happened to be small, say a mile a second, the classical conception of a world of separate space and time would have been abandoned years ago; for the curious effects predicted by Einstein would have become perceptible without effort. Again, were we to conceive of this invariant velocity as becoming greater and greater, the distinguishing features of the Einsteinian universe would gradually fade away, until finally, if the invariant velocity became infinite, space-time would give way to the separate space and time of classical science, and all the peculiarities of Einstein’s universe would vanish.
A direct study of the Lorentz-Einstein transformations would lead to the same conclusions; for if in these transformations we makeinfinite, we find that they degenerate into the classical Galilean ones. And here a rather delicate point must be noted. The constant—invariant velocity of the universe—has been identified in Einstein’s theory with the velocityof lightin vacuo. This identification was inevitable, as we explained when discussing the equations of electromagnetics. But while it is correct to state that Einstein’s theory would degenerate into classical science were,the invariant velocity of the universe, infinite instead of finite, it would be wrong to infer therefrom that were the velocity of light infinite we should be faced with the world such as classical science thought it to be. As we know, in classical science the velocity of light wasnotinfinite; it was only the invariant velocity that was infinite; and these two velocities had nothing in common.
If we wished to conceive of a world in which both the velocity of light and the invariant velocity were infinite, we should have to makeinfinite in the equations of electromagnetics, since in these equationsrepresents the velocity of light. But then all the terms in these equations in whichentered as denominator would disappear. Since these terms are those responsible for induction, in a world in which the velocity of light was infinite, no such phenomenon as induction could exist. There would be no dynamos, no radio; in fact, there would be no light rays, since light is itself a phenomenon of electromagnetic induction. Obviously such a world would not be the one in which we live.
Summarising, we must say that in classical science the velocity of lightis finite and that the invariant velocity is infinite. In relativity,,the velocity of light, is finite, as before, but is identical with the invariant velocity, which is therefore also finite. We also understand the deep-seated reason for the curious complicated form of the relativistic space and time transformations when we pass from one Galilean frame to another (the Lorentz-Einstein transformations) as contrasted with the classical transformations. We realise now that this more complicated form is due to the fact that the real world is one of space-time, and not of separate space and time, as was formerly believed.
Before closing this chapter, there is a point which we feel it advisable to discuss more fully. It deals with purely formal definitions in physics, and in particular with that of the critical velocity. We have mentioned that definitions which profess to dispense with the use of the rod, clock and physical measurements in general, are recognised as impossible. Einstein has repeatedly stressed this fact.
Possibly these points will be best understood if we discuss an illustration of a purely formal definition given by Dr. Whitehead. Thus, he writes:
“Experiment shows that this critical maximum velocity is a near approximation to the velocity of lightin vacuo, but its definition in no way depends upon any reference to light.”[67]
Now any physicist would readily agree with this statement were it Dr. Whitehead’s desire merely to obviate an appeal to light propagation in the definition of the critical velocity of relativity. For, as we have seen, the critical or maximum velocity can be defined and has been defined by other physical means. Indeed, so far as we know, the existence of the critical velocity is not contingent on that of light. But Whitehead’s idea, it would seem from his writings, is to eliminate physical measurements entirely from the basic definitions of physics; whence his definition, or at least the nearest approach to a definition we have been able to find in his writings:
“The physical meaning ofis also well known; namely, any velocity which in any time-system is of magnitudeis of the same magnitude in every other time-system. No assumption of the existence of a velocity with this property or of the electromagnetic invariance has entered into the deduction of the kinematical equations of the hyperbolic type.”
Note that in this definition Dr. Whitehead has failed to identifywith either Maxwell’s constant or any other physical magnitude. Thusis some unspecified magnitude; and we may express Dr. Whitehead’s definition in more familiar language as follows:
“The critical or maximum velocity is a velocity of undetermined value which may or may not be demanded by the world-structure. If, however, it should exist, it will be given by the same invariant in every Galilean frame, whatever the value of this invariant may be.”
In this definition no reference is made to any physical phenomena or laws, and this is the aspect which appears to constitute its superiority in Whitehead’s opinion. But the trouble is that if physical measurements or the existence of physical laws established empirically by the use of rod, clock, etc., are to be discarded, the definition turns out to be no definition at all. For what sort of definition is one which fails to apprise us of the magnitude of the velocity it proposes to define? The definition of the velocity of sound, for example, is that of a velocity of so many feet a second under specified conditions of observation. A definition which failed to give us this information would be no definition of the velocity of sound.
Now it might be contended that the precise magnitude of the invariant velocity is a matter of minor importance, and that its property of invariance (mentioned in the definition) is the fundamental point. We may safely say, however, that no physicist could ever subscribe to this view. Our entire outlook on nature depends essentially on the value of this invariant velocity.
If it is finite, and if, in addition, it is equal in value to Maxwell’s constant,hence to the velocity of lightin vacuo, we must conceive of a world of space-time in which the laws of nature will be subject to certain rigid conditions. Also, simultaneity, length and duration will be relative.
If the invariant velocity is a finite magnitude greater than the value of Maxwell’s constant,we shall still have a world of space-time in the extended sense, but the relativity of velocity will not hold and we should be able to determine our absolute velocity through space. The rigid restrictions placed on the laws of nature would no longer be present, since the laws of electromagnetics would certainly be in conflict with those restrictions. The entire structure of Einstein’s theory would be overthrown.
Finally, if this invariant velocity were infinite, we should have the separate space and time of classical science entailing the absoluteness of simultaneity, length and duration. There is nothing in Dr. Whitehead’s mathematical deductions to suggest which of these values the invariant velocity is to have; it might even be an imaginary number.
At all events, we see that the precise magnitude of the invariant velocity is not a matter of any small importance; hence it should inevitably be defined by the definition. Yet this essential piece of information Dr. Whitehead’s definition fails to give, by reason of its being compatible with a whole class of possible invariant velocities ranging from zero to infinity, just as a definition of New York as “a city by the sea” gives us a whole class of possible cities. Definitions of this sort are ambiguous, they are indefinite, and therefore are no definitions at all. The best that can be said of them is that they constitute partial descriptions.[68]
Up to this point we have discussed Whitehead’s definition in the light of a self-supporting mathematical definition that would necessitate no appeal to physical measurements in order to acquire a determinate significance. Such appeared to be his intention. Thus interpreted, the definition is a failure, as we have seen.
It should therefore be obvious that our only means of rendering it determinate must be to establish the existence, then the magnitude of this elusive critical velocity.But this can be done only through the medium of physical measurements. It matters not whether the measurements be accomplished with rods, clocks, light rays, or by electromagnetic means. In every case they will be physical; hence, when finally the value of the invariant velocity is obtained, this magnitude will be fixed, determined or defined by means of physical measurements. Dr. Whitehead’s purely formal definition cannot emancipate us from this restriction. It seems strange, therefore, that he should criticise in Einstein a procedure which he himself is unable to avoid.
At any rate, inasmuch as scientists convinced themselves years ago that in the matter of physical definitions, physical measurements could not be avoided, their aim has always been to select a definition which would define the unknown quantity in terms of magnitudes already known. If this method fails them, they attempt to present their definition in a form which will be of aid to the physicist, directing him towards his goal.
When these requirements are considered, it is seen that Einstein’s definition of the invariant velocity as that of lightin vacuocannot easily be improved upon. To understand this point, we shall contrast the situation of an experimenter to whom Dr. Whitehead’s definition, then Einstein’s, would be given.
As we remember, Dr. Whitehead’s definition informs the experimenter that he may recognise the critical speed when his measurements have succeeded in detecting a velocity which is invariant. Failing this, he may also fix its value as soon as his measurements succeed in detecting a trace of non-Euclideanism in the law of composition of velocities. But this is obviously a very roundabout way to help him. The experimenter will now be compelled to fire bullets with various speeds and endeavour to detect traces of non-Euclideanism. If his attempts are unsuccessful, he can draw no conclusions, since it might always be supposed that the invariant velocity was too great to exert perceptible effects on the velocities at his disposal. Even were he to detect what might appear to be non-Euclideanism, how would he know that the effect had not arisen from secondary causes? We remember, in this connection, that neither Fresnel nor Lorentz ever thought it necessary to attribute the result of Fizeau’s experiment to a non-Euclidean composition of velocities.[69]Fresnel ascribed this result to interactions between ether and matter, whereas Lorentz attributed it to the electronic constitution of matter. In short, Dr. Whitehead’s definition places the experimenter in the unenviable position of a man looking for a needle in a haystack without having the consolation of knowing that the needle is really there. From this we see that the definition, when interpreted in a purely formal sense, is ambiguous and tells us nothing, and, when interpreted in a physical sense, is of very little use to the physicist.
And now let us consider Einstein’s definition. First we may note that had the bare invariance of the critical or maximum velocity represented a proper definition in his opinion, he would have given it twenty years ago, seeing that this invariance constitutes the A B C of his theory. Yet, as we know, though he often refers to the maximum velocity as the invariant velocity, and vice versa, he considered it necessary to complete this partial description by a proper definition.
For Einstein, the important point was not so much to furnish a physical illustration of this invariant velocity, which might conceivably never be found to be realised in nature as a physical existent. (In the same way, the infinite velocity of classical science was never given by any physical propagation.) The important point was that he should justify his belief that the world-structure demanded the existence of a critical velocity of this sort; and by its existence we mean that were it ever to be realised, it should prove to be an unsurpassable maximum and an invariant.
We are now in a position to understand the advantage of Einstein’s definition. First, he states that the electromagnetic experiments have established the relativity of velocity. This in turn entails the maintenance of form of all natural laws—in particular, Maxwell’s laws of electromagnetics. Hence it follows that there must exist a critical invariant velocity that is given by the constant c which enters into Maxwell’s equations. Inasmuch as Maxwell had proved thatand the velocity of lightin vacuowere one and the same, the invariant velocity is defined by this velocity of light. Thus, with Einstein’s definition, we know that the world-structure demands the existence of a finite invariant velocity, and not of an infinite one, as was believed by classical science. And, in addition, we know that this velocity is given by Maxwell’s celebrated constant,a constant whose value had been determined with accuracy long before the advent of the relativity theory. In other words, Einstein defines the unknown in terms of the known, not in terms of the ambiguous.
To return to our illustration of the needle in the haystack, not only has Einstein given us the assurance that the needle is there, but he also tells us exactly where to look for it. For this reason his definition, in contradistinction to Dr. Whitehead’s, satisfies the requirements demanded by physical science.
In these pages we have discussed Whitehead’s definition at some length because it illustrates the danger there is in confusing mathematics and physics. Thus, whereas in mathematics we may postulate anything we please (with certain reservations) and then proceed to reach our conclusions deductively, in physics this procedure is impossible. We must take our cue from experiment and formulate our premises accordingly. The result is that whereas in mathematics we are concerned with formal possibilities, in physics we are limited to an analysis of actual facts. And in every case the transition from the possibilities of mathematics to the actualities of physics necessitates the introduction of physical measurements.
In this connection it is most important to understand the difference between a physical and a mathematical definition. Take a number like n in mathematics. We can define it as the ratio of the length of a circumference to its diameter in Euclidean geometry. Without performing physical measurements, we can deduce from this definition, by purely mathematical means, the precise value ofto any order of approximation we please. Thus the definition does not lead to ambiguity, hence is a valid one.
On the other hand, try to give a physical definition by some similar method—say, the definition of the “gram” or of the “dyne.” The purely logical type of definition breaks down, and we are compelled to resort to physical determinations. Accordingly, we define the gram as the mass of a cubic centimetre of distilled water under specified conditions of temperature and pressure.
And why was the purely logical definition possible in mathematics and a failure in physics? 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.
Much of the criticism directed against Einstein’s use of rods, clocks and light propagation arises from a failure to understand thisfundamental point. Purely logical, non-empirical definitions in physics can only lead to ambiguity. As for the theory of relativity, it is one of physics; it does not claim or aspire to be anything else; and a theory of physics which would avoid physical measurements is about as reasonable a conception as a forest without trees.