CHAPTER II
RELATIVITY
The views of time and space, which I have to set forth, have their foundation in experimental physics. Therein is their strength. Their tendency is revolutionary. From henceforth space in itself and time in itself sink to mere shadows, and only a kind of union of the two preserves an independent existence.
There are two parties to every observation—the observed and the observer.
What we see depends not only on the object looked at, but on our own circumstances—position, motion, or more personal idiosyncracies. Sometimes by instinctive habit, sometimes by design, we attempt to eliminate our own share in the observation, and so form a general picture of the world outside us, which shall be common to all observers. A small speck on the horizon of the sea is interpreted as a giant steamer. From the window of our railway carriage we see a cow glide past at fifty miles an hour, and remark that the creature is enjoying a rest. We see the starry heavens revolve round the earth, but decide that it is really the earth that is revolving, and so picture the state of the universe in a way which would be acceptable to an astronomer on any other planet.
The first step in throwing our knowledge into a common stock must be the elimination of the various individual standpoints and the reduction to some specified standard observer. The picture of the world so obtained is none the less relative. We have not eliminated the observer's share; we have only fixed it definitely.
To obtain a conception of the world from the point of view of no one in particular is a much more difficult task. The position of the observer can be eliminated; we are able to grasp the conception of a chair as an object in nature—looked at all round, and not from any particular angle or distance. We can think of it without mentally assigning ourselves some position with respect to it. This is a remarkable faculty, which has evidently been greatly assisted by the perception of solid relief with our two eyes. But the motion of the observer is not eliminated so simply. We had thought that it was accomplished; but the discovery in the last chapter that observers with different motions use different space- and time-reckoning shows that the matter is more complicated than was supposed. It may well require a complete change in our apparatus of description, because all the familiar terms of physics refer primarily to the relations of the world to an observer in some specified circumstances.
Whether we are able to go still further and obtain a knowledge of the world, which not merely does not particularise the observer, but does not postulate an observer at all; whether if such knowledge could be obtained, it would convey any intelligible meaning; and whether it could be of any conceivable interest to anybody if it could be understood—these questions need not detain us now. The answers are not necessarily negative, but they lie outside the normal scope of physics.
The circumstances of an observer which affect his observations are his position, motion and gauge of magnitude. More personal idiosyncracies disappear if, instead of relying on his crude senses, he employs scientific measuring apparatus. But scientific apparatus has position, motion and size, so that these are still involved in the results of any observation. There is no essential distinction between scientific measures and the measures of the senses. In either case our acquaintance with the external world comes to us through material channels; the observer's body can be regarded as part of his laboratory equipment, and, so far as we know, it obeys the same laws. We therefore group together perceptions and scientific measures, and in speaking of "a particular observer" we include all his measuring appliances.
Position, motion, magnitude-scale—these factors have a profound influence on the aspect of the world to us. Can we form a picture of the world which shall be a synthesis of what is seen by observers in all sorts of positions, having all sorts of velocities, and all sorts of sizes. As already stated we have accomplished the synthesis of positions. We have two eyes, which have dinned into our minds from babyhood that the world has to be looked at from more than one position. Our brains have so far responded as to give us the idea of solid relief, which enables us to appreciate the three-dimensional world in a vivid way that would be scarcely possible if we were only acquainted with strictly two-dimensional pictures. We not merely deduce the three-dimensional world; we see it. But we have no such aid in synthesising different motions. Perhaps if we had been endowed with two eyes moving with different velocities our brains would have developed the necessary faculty; we should have perceived a kind of relief in a fourth dimension so as to combine into one picture the aspect of things seen with different motions. Finally, if we had had two eyes of different sizes, we might have evolved a faculty for combining the points of view of the mammoth and the microbe.
It will be seen that we are not fully equipped by our senses for forming an impersonal picture of the world. And it is because the deficiency is manifest that we do not hesitate to advocate a conception of the world which transcends the images familiar to the senses. Such a world can perhaps be grasped, but not pictured by the brain. It would be unreasonable to limit our thought of nature to what can be comprised in sense-pictures. As Lodge has said, our senses were developed by the struggle for existence, not for the purpose of philosophising on the world.
Let us compare two well-known books, which might be described as elementary treatises on relativity, Alice in Wonderland and Gulliver's Travels. Alice was continually changing size, sometimes growing, sometimes on the point of vanishing altogether. Gulliver remained the same size, but on one occasion he encountered a race of men of minute size with everything in proportion, and on another voyage a land where everything was gigantic. It does not require much reflection to see that both authors are describing the same phenomenon—a relative change of scale of observer and observed. Lewis Carroll took what is probably the ordinary scientific view, that the observer had changed, rather than that a simultaneous change had occurred to all her surroundings. But it would never have appeared like that to Alice; she could not have "stepped outside and looked at herself," picturing herself as a giant filling the room. She would have said that the room had unaccountably shrunk. Dean Swift took the truer view of the human mind when he made Gulliver attribute his own changes to the things around him; it never occurred to Gulliver that his own size had altered; and, if he had thought of the explanation, he could scarcely have accustomed himself to that way of thinking. But both points of view are legitimate. The size of a thing can only be imagined as relative to something else; and there is no means of assigning the change to one end of the relation rather than the other.
We have seen in the theory of the Michelson-Morley experiment that, according to current physical views, our standard of size—the rigid measuring-rod—must change according to the circumstances of its motion; and the aviator's adventures illustrated a similar change in the standard of duration of time. Certain rather puzzling irregularities have been discovered in the apparent motions of the Sun, Mercury, Venus and the Moon; but there is a strong family resemblance between these, which leads us to believe that the real phenomenon is a failure of the time-keeping of our standard clock, the Earth. Instances could be multiplied where a change of the observer or his standards produces or conceals changes in the world around him.
The object of the relativity theory, however, is not to attempt the hopeless task of apportioning responsibility between the observer and the external world, but to emphasise that in our ordinary description and in our scientific description of natural phenomena the two factors are indissolubly united. All the familiar terms of physics—length, duration of time, motion, force, mass, energy, and so on—refer primarily to this relative knowledge of the world; and it remains to be seen whether any of them can be retained in a description of the world which is not relative to a particular observer.
Our first task is a description of the world independent of the motion of the observer. The question of the elimination of his gauge of magnitude belongs to a later development of the theory discussed in Chapter xi. Let us draw a square on a sheet of paper, making the sides equal, to the best of our knowledge. We have seen that an aviator flying at 161,000 miles a second in the direction , would judge that the sides , had contracted to half their length, so that for him the figure would be an oblong. If it were turned through a right angle and would expand and the other two sides contract—in his judgment. For us, the lengths of and are equal; for him, one length is twice the other. Clearly length cannot be a property inherent in our drawing; it needs the specification of some observer.
We have seen further that duration of time also requires that an observer should be specified. The stationary observer and the aviator disagreed as to whose cigar lasted the longer time.
Thus length and duration are not things inherent in the external world; they are relations of things in the external world to some specified observer. If we grasp this all the mystery disappears from the phenomena described in Chapter i. When the rod in the Michelson-Morley experiment is turned through a right angle it contracts; that naturally gives the impression that something has happened to the rod itself. Nothing whatever has happened to the rod—the object in the external world. Its length has altered, but length is not an intrinsic property of the rod, since it is quite indeterminate until some observer is specified. Turning the rod through a right angle has altered the relation to the observer (implied in the discussion of the experiment); but the rod itself, or the relation of a molecule at one end to a molecule at the other, is unchanged. Measurement of length and duration is a comparison with partitions of space and time drawn by the observer concerned, with the help of apparatus which shares his motion. Nature is not concerned with these partitions; it has, as we shall see later, a geometry of its own which is of a different type.
Current physics has hitherto assumed that all observers are not to be regarded as on the same footing, and that there is some absolute observer whose judgments of length and duration are to be treated with respect, because nature pays attention to his space-time partitions. He is supposed to be at rest in the aether, and the aether materialises his space-partitions so that they have a real significance in the external world. This is sheer hypothesis, and we shall find it is unsupported by any facts. Evidently our proper course is to pursue our investigations, and call in this hypothetical observer only if we find there is something which he can help to explain.
We have been leading up from the older physics to the new outlook of relativity, and the reader may feel some doubt as to whether the strange phenomena of contraction and time-retardation, that were described in the last chapter, are to be taken seriously, or are part of a reductio ad absurdum argument. The answer is that we believe that the phenomena do occur as described; only the description (like that of all observed phenomena) concerns the relations of the external world to some observer, and not the external world itself. The startling character of the phenomena arises from the natural but fallacious inference that they involve intrinsic changes in the objects themselves.
We have been considering chiefly the observer's end of the observation; we must now turn to the other end—the thing observed. Although length and duration have no exact counterparts in the external world, it is clear that there is a certain ordering of things and events outside us which we must now find more appropriate terms to describe. The order of events is a four-fold order; we can arrange them as right-and-left, backwards-and-forwards, up-and-down, sooner-and-later. An individual may at first consider these as four independent orders, but he will soon attempt to combine some of them. It is recognised at once that there is no essential distinction between right-and-left and backwards-and-forwards. The observer has merely to turn through a right angle and the two are interchanged. If he turns through a smaller angle, he has first to combine them, and then to redivide them in a different way. Clearly it would be a nuisance to continually combine and redivide; so we get accustomed to the thought of leaving them combined in a two-fold or two-dimensional order. The amalgamation of up-and-down is less simple. There are obvious reasons for considering this dimension of the world as fundamentally distinct from the other two. Yet it would have been a great stumbling-block to science if the mind had refused to combine space into a three-dimensional whole. The combination has not concealed the real distinction of horizontal and vertical, but has enabled us to understand more clearly its nature—for what phenomena it is relevant, and for what irrelevant. We can understand how an observer in another country redivides the combination into a different vertical and horizontal. We must now go further and amalgamate the fourth order, sooner-and-later. This is still harder for the mind. It does not imply that there is no distinction between space and time; but it gives a fresh unbiassed start by which to determine what the nature of the distinction is.
The idea of putting together space and time, so that time is regarded as a fourth dimension, is not new. But until recently it was regarded as merely a picturesque way of looking at things without any deep significance. We can put together time and temperature in a thermometer chart, or pressure and volume on an indicator-diagram. It is quite non-committal. But our theory is going to lead much further than that. We can lay two dimensional surfaces—sheets of paper—on one another till we build up a three-dimensional block; but there is a difference between a block which is a pile of sheets and a solid block of paper. The solid block is the true analogy for the four-dimensional combination of space-time; it does not separate naturally into a particular set of three-dimensional spaces piled in time-order. It can be redivided into such a pile; but it can be redivided in any direction we please.
Just as the observer by changing his orientation makes a new division of the two-dimensional plane into right-and-left, backwards-and-forwards—just as the observer by changing his longitude makes a new division of three-dimensional space into vertical and horizontal—so the observer by changing his motion makes a new division of the four-dimensional order into time and space.
This will be justified in detail later; it indicates that observers with different motions will have different time and space-reckoning—a conclusion we have already reached from another point of view.
Although different observers separate the four orders differently, they all agree that the order of events is four-fold; and it appears that this undivided four-fold order is the same for all observers. We therefore believe that it is inherent in the external world; it is in fact the synthesis, which we have been seeking, of the appearances seen by observers having all sorts of positions and all sorts of (uniform) motions. It is therefore to be regarded as a conception of the real world not relative to any particularly circumstanced observer.
The term "real world" is used in the ordinary sense of physics, without any intention of prejudging philosophical questions as to reality. It has the same degree of reality as was formerly attributed to the three-dimensional world of scientific theory or everyday conception, which by the advance of knowledge it replaces. As I have already indicated, it is merely the accident that we are not furnished with a pair of eyes in rapid relative motion, which has allowed our brains to neglect to develop a faculty for visualising this four-dimensional world as directly as we visualise its three-dimensional section.
It is now easy to see that length and duration must be the components of a single entity in the four-dimensional world of space-time. Just as we resolve a structure into plan and elevation, so we resolve extension in the four-dimensional world into length and duration. The structure has a size and shape independent of our choice of vertical. Similarly with things in space-time. Whereas length and duration are relative, the single "extension" of which they are components has an absolute significance in nature, independent of the particular decomposition into space and time separately adopted by the observer.
Consider two events; for example, the stroke of one o'clock and the stroke of two o'clock by Big Ben. These occupy two points in space-time, and there is a definite separation between them. An observer at Westminster considers that they occur at the same place, and that they are separated by an hour in time; thus he resolves their four-dimensional separation into zero distance in space and one hour distance in time. An observer on the sun considers that they do not occur at the same place; they are separated by about 70,000 miles, that being the distance travelled by the earth in its orbital motion with respect to the sun. It is clear that he is not resolving in quite the same directions as the terrestrial observer, since he finds the space-component to be 70,000 miles instead of zero. But if he alters one component he must necessarily alter the other; so he will make the time-component differ slightly from an hour. By analogy with resolution into components in three-dimensions, we should expect him to make it less than an hour—having, as it were, borrowed from time to make space; but as a matter of fact he makes it longer. This is because space-time has a different geometry, which will be described later. Our present point is that there is but one separation of two events in four dimensions, which can be resolved in any number of ways into the components length and duration.
We see further how motion must be purely relative. Take two events and in the history of one particle. We can choose any direction as the time-direction; let us choose it along . Then and are separated only in time and not in space, so the particle is at rest. If we choose a slightly inclined time-direction, the separation will have a component in space; the two events then do not occur at the same place, that is to say, the particle has moved. The negation of absolute motion is thus associated with the possibility of choosing the time-direction in any way we please. What determines the separation of space and time for any particular observer can now be seen. Let the observer place himself so that he is, to the best of his knowledge, at rest. If he is a normal human being, he will seat himself in an arm-chair; if he is an astronomer, he will place himself on the sun or at the centre of the stellar universe. Then all the events happening directly to him will in his opinion occur at the same place. Their separation will have no space-component, and they will accordingly be ranged solely in the time-direction. This chain of events, marking his track through the four-dimensional world, will be his time-direction. Each observer bases his separation of space and time on his own track through the world.
Since any separation of space and time is admissible, it is possible for the astronomer to base his space and time on the track of a solar observer instead of that of a terrestrial observer; but it must be remembered that in practice the space and time of the solar observer have to be inferred indirectly from those of the terrestrial observer; and, if the corrections are made according to the crude methods hitherto employed, they may be inferred wrongly (if extreme accuracy is needed).
The most formidable objection to this relativist view of the world is the aether difficulty. We have seen that uniform motion through the aether cannot be detected by experiment, and therefore it is entirely in accordance with experiment that such motion should have no counterpart in the four-dimensional world. Nevertheless, it would almost seem that such motion must logically exist, if the aether exists; and, even at the expense of formal simplicity, it ought to be exhibited in any theory which pretends to give a complete account of what is going on in nature. If a substantial aether analogous to a material ocean exists, it must rigidify, as it were, a definite space; and whether the observer or whether nature pays any attention to that space or not, a fundamental separation of space and time must be there. Some would cut the knot by denying the aether altogether. We do not consider that desirable, or, so far as we can see, possible; but we do deny that the aether need have such properties as to separate space and time in the way supposed. It seems an abuse of language to speak of a division existing, when nothing has ever been found to pay any attention to the division.
Mathematicians of the nineteenth century devoted much time to theories of elastic solid and other material aethers. Waves of light were supposed to be actual oscillations of this substance; it was thought to have the familiar properties of rigidity and density; it was sometimes even assigned a place in the table of the elements. The real death-blow to this materialistic conception of the aether was given when attempts were made to explain matter as some state in the aether. For if matter is vortex-motion or beknottedness in aether, the aether cannot be matter—some state in itself. If any property of matter comes to be regarded as a thing to be explained by a theory of its structure, clearly that property need not be attributed to the aether. If physics evolves a theory of matter which explains some property, it stultifies itself when it postulates that the same property exists unexplained in the primitive basis of matter.
Moreover the aether has ceased to take any very active part in physical theory and has, as it were, gone into reserve. A modern writer on electromagnetic theory will generally start with the postulate of an aether pervading all space; he will then explain that at any point in it there is an electromagnetic vector whose intensity can be measured; henceforth his sole dealings are with this vector, and probably nothing more will be heard of the aether itself. In a vague way it is supposed that this vector represents some condition of the aether, and we need not dispute that without some such background the vector would scarcely be intelligible—but the aether is now only a background and not an active participant in the theory.
There is accordingly no reason to transfer to this vague background of aether the properties of a material ocean. Its properties must be determined by experiment, not by analogy. In particular there is no reason to suppose that it can partition out space in a definite way, as a material ocean would do. We have seen in the Prologue that natural geometry depends on laws of matter; therefore it need not apply to the aether. Permanent identity of particles is a property of matter, which Lord Kelvin sought to explain in his vortex-ring hypothesis. This abandoned hypothesis at least teaches us that permanence should not be regarded as axiomatic, but may be the result of elaborate constitution. There need not be anything corresponding to permanent identity in the constituent portions of the aether; we cannot lay our finger at one spot and say "this piece of aether was a few seconds ago over there." Without any continuity of identity of the aether motion through the aether becomes meaningless; and it seems likely that this is the true reason why no experiment ever reveals it.
This modern theory of the relativity of all uniform motion is essentially a return to the original Newtonian view, temporarily disturbed by the introduction of aether problems; for in Newton's dynamics uniform motion of the whole system has not—and no one would expect it to—have any effect. But there are considerable difficulties in the limitation to uniform motion. Newton himself seems to have appreciated the difficulty; but the experimental evidence appeared to him to be against any extension of the principle. Accordingly Newton's laws of mechanics are not of the general type in which it is unnecessary to particularise the observer; they hold only for observers with a special kind of motion which is described as "unaccelerated." The only definition of this epithet that can be given is that an "unaccelerated" observer is one for whom Newton's laws of motion hold. On this theory, the phenomena are not indifferent to an acceleration or non-uniform motion of the whole system. Yet an absolute non-uniform motion through space is just as impossible to imagine as an absolute uniform motion. The partial relativity of phenomena makes the difficulty all the greater. If we deny a fundamental medium with continuous identity of its parts, motion uniform or non-uniform should have no significance; if we admit such a medium, motion uniform or non-uniform should be detectable; but it is much more difficult to devise a plan of the world according to which uniform motion has no significance and non-uniform motion is significant.
It is through experiment that we have been led back to the principle of relativity for uniform motion. In seeking some kind of extension of this principle to accelerated motion, we are led by the feeling that, having got so far, it is difficult and arbitrary to stop at this point. We now try to conceive a system of nature for which all kinds of motion of the observer are indifferent. It will be a completion of our synthesis of what is perceived by observers having all kinds of motions with respect to one another, removing the restriction to uniform motion. The experimental tests must follow after the consequences of this generalisation have been deduced.
The task of formulating such a theory long appeared impossible. It was pointed out by Newton that, whereas there is no criterion for detecting whether a body is at rest or in uniform motion, it is easy to detect whether it is in rotation. For example the bulge of the earth s equator is a sign that the earth is rotating, since a plastic body at rest would be spherical.
This problem of rotation affords a hint as to the cause of the incomplete relativity of Newtonian mechanics. The laws of motion are formulated with respect to an unaccelerated observer, and do not apply to a frame of reference rotating with the earth. Yet mathematicians frequently do use a rotating frame. Some modification of the laws is then necessary; and the modification is made by introducing a centrifugal force—not regarded as a real force like gravitation, but as a mathematical fiction employed to correct for the improper choice of a frame of reference. The bulge of the earth's equator may be attributed indifferently to the earth's rotation or to the outward pull of the centrifugal force introduced when the earth is regarded as non-rotating.
Now it is generally assumed that the centrifugal force is something sui generis, which could always be distinguished experimentally from any other natural phenomenon. If then on choosing a frame of reference we find that a centrifugal force is detected, we can at once infer that the frame of reference is a "wrong" one; rotating and non-rotating frames can be distinguished by experiment, and rotation is thus strictly absolute. But this assumes that the observed effects of centrifugal force cannot be produced in any other way than by rotation of the observer's frame of reference. If once it is admitted that centrifugal force may not be completely distinguishable by experiment from another kind of force—gravitation—perceived even by Newton's unaccelerated observer, the argument ceases to apply. We can never determine exactly how much of the observed field of force is centrifugal force and how much is gravitation; and we cannot find experimentally any definite standard that is to be considered absolutely non-rotating.
The question then, whether there exists a distinction between "right" frames of reference and "wrong" frames, turns on whether the use of a "wrong" frame produces effects experimentally distinguishable from any natural effects which can be perceived when a "right" frame is used. If there is no such difference, all frames may be regarded as on the same footing and equally right. In that case we can have a complete relativity of natural phenomena. Since the effect of departing from Newton's standard frame is the introduction of a field of force, this generalised relativity theory must be largely occupied with the nature of fields of force.
The precise meaning of the statement that all frames of reference are on the same footing is rather difficult to grasp. We believe that there are absolute things in the world—not only matter, but certain characteristics in empty space or aether. In the atmosphere a frame of reference which moves with the air is differentiated from other frames moving in a different manner; this is because, besides discharging the normal functions of a frame of reference, the air-frame embodies certain of the absolute properties of the matter existing in the region. Similarly, if in empty space we choose a frame of reference which more or less follows the lines of the absolute structure in the region, the frame will usurp some of the absolute qualities of that structure. What we mean by the equivalence of all frames is that they are not differentiated by any qualities formerly supposed to be intrinsic in the frames themselves—rest, rectangularity, acceleration—independent of the absolute structure of the world that is referred to them. Accordingly the objection to attributing absolute properties to Newton's frame of reference is not that it is impossible for a frame of reference to acquire absolute properties, but that the Newtonian frame has been laid down on the basis of relative knowledge without any attempt to follow the lines of absolute structure.
Force, as known to us observationally, is like the other quantities of physics, a relation. The force, measured with a spring-balance, for example, depends on the acceleration of the observer holding the balance; and the term may, like length and duration, have no exact counterpart in a description of nature independent of the observer. Newton's view assumes that there is such a counterpart, an active cause in nature which is identical with the force perceived by his standard unaccelerated observer. Although any other observer perceives this force with additions of his own, it is implied that the original force in nature and the observer's additions can in some way be separated without ambiguity. There is no experimental foundation for this separation, and the relativity view is that a field of force can, like length and duration, be nothing but a link between nature and the observer. There is, of course, something at the far end of the link, just as we found an extension in four dimensions at the far end of the relations denoted by length and duration. We shall have to study the nature of this unknown whose relation to us appears as force. Meanwhile we shall realise that the alteration of perception of force by non-uniform motion of the observer, as well as the alteration of the perception of length by his uniform motion, is what might be expected from the nature of these quantities as relations solely.
We proceed now to a more detailed study of the four-dimensional world, of the things which occur in it, and of the laws by which they are regulated. It is necessary to dive into this absolute world to seek the truth about nature; but the physicist's object is always to obtain knowledge which can be applied to the relative and familiar aspect of the world. The absolute world is of so different a nature, that the relative world, with which we are acquainted, seems almost like a dream. But if indeed we are dreaming, our concern is with the baseless fabric of our vision. We do not suggest that physicists ought to translate their results into terms of four-dimensional space for the empty satisfaction of working in the realm of reality. It is rather the opposite. They explore the new field and bring back their spoils—a few simple generalisations—to apply them to the practical world of three-dimensions. Some guiding light will be given to the attempts to build a scheme of things entire. For the rest, physics will continue undisturbedly to explore the relative world, and to employ the terms applicable to relative knowledge, but with a fuller appreciation of its relativity.