Space Time and Gravitation/Chapter 12

Space Time and Gravitation: An outline of the general relativity theory  (1920)
Arthur Eddington
On the Nature of Things

Cambridge University Press, pages 180–201

CHAPTER XII

ON THE NATURE OF THINGS

Hippolyta. This is the silliest stuff that ever I heard.
Theseus. The best in this kind are but shadows; and the worst are no worse, if imagination amend them.

The constructive results of the theory of relativity are based on two principles which have been enunciated—the restricted principle of relativity, and the principle of equivalence. These may be summed up in the statement that uniform motion and fields of force are purely relative. In their more formal enunciations they are experimental generalisations, which can be admitted or denied; if admitted, all the observational results obtained by us can be deduced mathematically without any reference to the views of space, time, or force, described in this book. In many respects this is the most attractive aspect of Einstein's work; it deduces a great number of remarkable phenomena solely from two general principles, aided by a mathematical calculus of great power; and it leaves aside as irrelevant all questions of mechanism. But this mode of development of the theory cannot be described in a non-technical book.

To avoid mathematical analysis we have had to resort to geometrical illustrations, which run parallel with the mathematical development and enable its processes to be understood to some extent. The question arises, are these merely illustrations of the mathematical argument, or illustrations of the actual processes of nature. No doubt the safest course is to avoid the thorny questions raised by the latter suggestion, and to say that it is quite sufficient that the illustrations should correctly replace the mathematical argument. But I think that this would give a misleading view of what the theory of relativity has accomplished in science.

The physicist, so long as he thinks as a physicist, has a definite belief in a real world outside him. For instance, he believes that atoms and molecules really exist; they are not mere inventions that enable him to grasp certain laws of chemical combination. That suggestion might have sufficed in the early days of the atomic theory; but now the existence of atoms as entities in the real world of physics is fully demonstrated. This confident assertion is not inconsistent with philosophic doubts as to the meaning of ultimate reality.

When therefore we are asked whether the four-dimensional world may not be regarded merely as an illustration of mathematical processes, we must bear in mind that our questioner has probably an ulterior motive. He has already a belief in a real world of three Euclidean dimensions, and he hopes to be allowed to continue in this belief undisturbed. In that case our answer must be definite; the real three-dimensional world is obsolete, and must be replaced by the four-dimensional space-time with non-Euclidean properties. In this book we have sometimes employed illustrations which certainly do not correspond to any physical reality—imaginary time, and an unperceived fifth dimension. But the four-dimensional world is no mere illustration; it is the real world of physics, arrived at in the recognised way by which physics has always (rightly or wrongly) sought for reality.

I hold a certain object before me, and see an outline of the figure of Britannia; another observer on the other side sees a picture of a monarch; a third observer sees only a thin rectangle. Am I to say that the figure of Britannia is the real object; and that the crude impressions of the other observers must be corrected to make allowance for their positions? All the appearances can be accounted for if we are all looking at a three-dimensional object—a penny—and no reasonable person can doubt that the penny is the corresponding physical reality. Similarly, an observer on the earth sees and measures an oblong block; an observer on another star contemplating the same object finds it to be a cube. Shall we say that the oblong block is the real thing, and that the other observer must correct his measures to make allowance for his motion? All the appearances are accounted for if the real object is four-dimensional, and the observers are merely measuring different three-dimensional appearances or sections; and it seems impossible to doubt that this is the true explanation. He who doubts the reality of the four-dimensional world (for logical, as distinct from experimental, reasons) can only be compared to a man who doubts the reality of the penny, and prefers to regard one of its innumerable appearances as the real object.

Physical reality is the synthesis of all possible physical aspects of nature. An illustration may be taken from the phenomena of radiant-energy, or light. In a very large number of phenomena the light coming from an atom appears to be a series of spreading waves, extending so as to be capable of filling the largest telescope yet made. In many other phenomena the light coming from an atom appears to remain a minute bundle of energy, all of which can enter and blow up a single atom. There may be some illusion in these experimental deductions; but if not, it must be admitted that the physical reality corresponding to light must be some synthesis comprehending both these appearances. How to make this synthesis has hitherto baffled conception. But the lesson is that a vast number of appearances may be combined into one consistent whole—perhaps all appearances that are directly perceived by terrestrial observers—and yet the result may still be only an appearance. Reality is only obtained when all conceivable points of view have been combined.

That is why it has been necessary to give up the reality of the everyday world of three dimensions. Until recently it comprised all the possible appearances that had been considered. But now it has been discovered that there are new points of view with new appearances; and the reality must contain them all. It is by bringing in all these new points of view that we have been able to learn the nature of the real world of physics.

Let us briefly recapitulate the steps of our synthesis. We found one step already accomplished. The immediate perception of the world with one eye is a two-dimensional appearance. But we have two eyes, and these combine the appearances of the world as seen from two positions; in some mysterious way the brain makes the synthesis by suggesting solid relief, and we obtain the familiar appearance of a three-dimensional world. This suffices for all possible positions of the observer within the parts of space hitherto explored. The next step was to combine the appearances for all possible states of uniform motion of the observer. The result was to add another dimension to the world, making it four-dimensional. Next the synthesis was extended to include all possible variable motions of the observer. The process of adding dimensions stopped, but the world became non-Euclidean; a new geometry called Riemannian geometry was adopted. Finally the points of view of observers varying in size in any way were added; and the result was to replace the Riemannian geometry by a still more general geometry described in the last chapter.

The search for physical reality is not necessarily utilitarian, but it has been by no means profitless. As the geometry became more complex, the physics became simpler; until finally it almost appears that the physics has been absorbed into the geometry. We did not consciously set out to construct a geometrical theory of the world; we were seeking physical reality by approved methods, and this is what has happened.

Is the point now reached the ultimate goal? Have the points of view of all conceivable observers now been absorbed? We do not assert that they have. But it seems as though a definite task has been rounded off, and a natural halting-place reached. So far as we know, the different possible impersonal points of view have been exhausted—those for which the observer can be regarded as a mechanical automaton, and can be replaced by scientific measuring-appliances. A variety of more personal points of view may indeed be needed for an ultimate reality; but they can scarcely be incorporated in a real world of physics. There is thus justification for stopping at this point but not for stopping earlier.

It may be asked whether it is necessary to take into account all conceivable observers, many of whom, we suspect, have no existence. Is not the real world that which comprehends the appearances to all real observers? Whether or not it is a tenable hypothesis that that which no one observes does not exist, science uncompromisingly rejects it. If we deny the rights of extra-terrestrial observers, we must take the side of the Inquisition against Galileo. And if extra-terrestrial observers are admitted, the other observers, whose results are here combined, cannot be excluded.

Our inquiry into the nature of things is subject to certain limitations which it is important to realise. The best comparison I can offer is with a future antiquarian investigation, which may be dated about the year 5000 A.D. An interesting find has been made relating to a vanished civilisation which flourished about the twentieth century, namely a volume containing a large number of games of chess, written out in the obscure symbolism usually adopted for that purpose. The antiquarians, to whom the game was hitherto unknown, manage to discover certain uniformities; and by long research they at last succeed in establishing beyond doubt the nature of the moves and rules of the game. But it is obvious that no amount of study of the volume will reveal the true nature either of the participants in the game—the chessmen—or the field of the game—the chess board. With regard to the former, all that is possible is to give arbitrary names distinguishing the chessmen according to their properties; but with regard to the chess-board something more can be stated. The material of the board is unknown, so too are the shapes of the meshes—whether squares or diamonds; but it is ascertainable that the different points of the field are connected with one another by relations of two-dimensional order, and a large number of hypothetical types of chess-board satisfying these relations of order can be constructed. In spite of these gaps in their knowledge, our antiquarians may fairly claim that they thoroughly understand the game of chess.

The application of this analogy is as follows. The recorded games are our physical experiments. The rules of the game, ascertained by study of them, are the laws of physics. The hypothetical chess-board of 64 squares is the space and time of some particular observer or player; whilst the more general relations of two-fold order, are the absolute relations of order in space-time which we have been studying. The chessmen are the entities of physics—electrons, particles, or point-events; and the range of movement may perhaps be compared to the fields of relation radiating from them—electric and gravitational fields, or intervals. By no amount of study of the experiments can the absolute nature or appearance of these participants be deduced; nor is this knowledge relevant, for without it we may yet learn "the game" in all its intricacy. Our knowledge of the nature of things must be like the antiquarians' knowledge of the nature of chessmen, viz. their nature as pawns and pieces in the game, not as carved shapes of wood. In the latter aspect they may have relations and significance transcending anything dreamt of in physics.

It is believed that the familiar things of experience are very complex; and the scientific method is to analyse them into simpler elements. Theories and laws of behaviour of these simpler constituents are studied; and from these it becomes possible to predict and explain phenomena. It seems a natural procedure to explain the complex in terms of the simple, but it carries with it the necessity of explaining the familiar in terms of the unfamiliar.

There are thus two reasons why the ultimate constituents of the real world must be of an unfamiliar nature. Firstly, all familiar objects are of a much too complex character. Secondly, familiar objects belong not to the real world of physics, but to a much earlier stage in the synthesis of appearances. The ultimate elements in a theory of the world must be of a nature impossible to define in terms recognisable to the mind.

The fact that he has to deal with entities of unknown nature presents no difficulty to the mathematician. As the mathematician in the Prologue explained, he is never so happy as when he does not know what he is talking about. But we ourselves cannot take any interest in the chain of reasoning he is producing, unless we can give it some meaning—a meaning, which we find by experiment, it will bear. We have to be in a position to make a sort of running comment on his work. At first his symbols bring no picture of anything before our eyes, and we watch in silence. Presently we can say "Now he is talking about a particle of matter"…"Now he is talking about another particle"…"Now he is saying where they will be at a certain time of day"…"Now he says that they will be in the same spot at a certain time." We watch to see.—"Yes. The two particles have collided. For once he is speaking about something familiar, and speaking the truth, although, of course, he does not know it." Evidently his chain of symbols can be interpreted as describing what occurs in the world; we need not, and do not, form any idea of the meaning of each individual symbol; it is only certain elaborate combinations of them that we recognise.

Thus, although the elementary concepts of the theory are of undefined nature, at some later stage we must link the derivative concepts to the familiar objects of experience.

We shall now collect the results arrived at in the previous chapters by successive steps, and set the theory out in more logical order. The extension in Chapter xi will not be considered here, partly because it would increase the difficulty of grasping the main ideas, partly because it is less certainly established.

In the relativity theory of nature the most elementary concept is the point-event. In ordinary language a point-event is an instant of time at a point of space; but this is only one aspect of the point-event, and it must not be taken as a definition. Time and space—the familiar terms—are derived concepts to be introduced much later in our theory. The first simple concepts are necessarily undefinable, and their nature is beyond human understanding. The aggregate of all the point-events is called the world. It is postulated that the world is four-dimensional, which means that a particular point-event has to be specified by the values of four variables or coordinates, though there is entire freedom as to the way in which these four identifying numbers are to be assigned.

The meaning of the statement that the world is four-dimensional is not so clear as it appears at first. An aggregate of a large number of things has in itself no particular number of dimensions. Consider, for example, the words on this page. To a casual glance they form a two-dimensional distribution; but they were written in the hope that the reader would regard them as a one-dimensional distribution. In order to define the number of dimensions we have to postulate some ordering relation; and the result depends entirely on what this ordering relation is—whether the words are ordered according to sense or to position on the page. Thus the statement that the world is four-dimensional contains an implicit reference to some ordering relation. This relation appears to be the interval, though I am not sure whether that alone suffices without some relation corresponding to proximity. It must be remembered that if the interval ${\displaystyle s}$ between two events is small, the events are not necessarily near together in the ordinary sense.

Between any two neighbouring point-events there exists a certain relation known as the interval between them. The relation is a quantitative one which can be measured on a definite scale of numerical values[1]. But the term "interval" is not to be taken as a guide to the real nature of the relation, which is altogether beyond our conception. Its geometrical properties, which we have dwelt on so often in the previous chapters, can only represent one aspect of the relation. It may have other aspects associated with features of the world outside the scope of physics. But in physics we are concerned not with the nature of the relation but with the number assigned to express its intensity; and this suggests a graphical representation, leading to a geometrical theory of the world of physics.

What we have here called the world might perhaps have been legitimately called the aether; at least it is the universal substratum of things which the relativity theory gives us in place of the aether.

We have seen that the number expressing the intensity of the interval-relation can be measured practically with scales and clocks. Now, I think it is improbable that our coarse measures can really get hold of the individual intervals of point-events; our measures are not sufficiently microscopic for that. The interval which has appeared in our analysis must be a macroscopic value; and the potentials and kinds of space deduced from it are averaged properties of regions, perhaps small in comparison even with the electron, but containing vast numbers of the primitive intervals. We shall therefore pass at once to the consideration of the macroscopic interval; but we shall not forestall later results by assuming that it is measurable with a scale and clock. That property must be introduced in its logical order.

Consider a small portion of the world. It consists of a large (possibly infinite) number of point-events between every two of which an interval exists. If we are given the intervals between a point ${\displaystyle A}$ and a sufficient number of other points, and also between ${\displaystyle B}$ and the same points, can we calculate what will be the interval between ${\displaystyle A}$ and ${\displaystyle B}$? In ordinary geometry this would be possible; but, since in the present case we know nothing of the relation signified by the word interval, it is impossible to predict any law a priori. But we have found in our previous work that there is such a rule, expressed by the formula ${\displaystyle ds^{2}=g_{11}\,{dx_{1}}^{2}+g_{22}\,{dx_{2}}^{2}+\ldots +2g_{12}\,dx_{1}\,dx_{2}+\ldots }$ This means that, having assigned our identification numbers (${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$, ${\displaystyle x_{3}}$, ${\displaystyle x_{4}}$) to the point-events, we have only to measure ten different intervals to enable us to determine the ten coefficients, ${\displaystyle g_{11}}$, etc., which in a small region may be considered to be constants; then all other intervals in this region can be predicted from the formula. For any other region we must make fresh measures, and determine the coefficients for a new formula.

I think it is unlikely that the individual interval-relations of point-events follow any such definite rule. A microscopic examination would probably show them as quite arbitrary, the relations of so-called intermediate points being not necessarily intermediate. Perhaps even the primitive interval is not quantitative, but simply 1 for certain pairs of point-events and 0 for others. The formula given is just an average summary which suffices for our coarse methods of investigation, and holds true only statistically. Just as statistical averages of one community may differ from those of another, so may this statistical formula for one region of the world differ from that of another. This is the starting point of the infinite variety of nature.

Perhaps an example may make this clearer. Compare the point-events to persons, and the intervals to the degree of acquaintance between them. There is no means of forecasting the degree of acquaintance between ${\displaystyle A}$ and ${\displaystyle B}$ from a knowledge of the familiarity of both with ${\displaystyle C}$, ${\displaystyle D}$, ${\displaystyle E}$, etc. But a statistician may compute in any community a kind of average rule. In most cases if ${\displaystyle A}$ and ${\displaystyle B}$ both know ${\displaystyle C}$, it slightly increases the probability of their knowing one another. A community in which this correlation was very high would be described as cliquish. There may be differences among communities in this respect, corresponding to their degree of cliquishness; and so the statistical laws may be the means of expressing intrinsic differences in communities.

Now comes the difficulty which is by this time familiar to us. The ten ${\displaystyle g}$'s are concerned, not only with intrinsic properties of the world, but with our arbitrary system of identification-numbers for the point-events; or, as we have previously expressed it, they describe not only the kind of space-time, but the nature of the arbitrary mesh-system that is used. Mathematics shows the way of steering through this difficulty by fixing attention on expressions called tensors, of which ${\displaystyle B_{\mu \nu \sigma }^{\rho }}$ and ${\displaystyle G_{\mu \nu }}$ are examples.

A tensor does not express explicitly the measure of an intrinsic quality of the world, for some kind of mesh-system is essential to the idea of measurement of a property, except in certain very special cases where the property is expressed by a single number termed an invariant, e.g. the interval, or the total curvature. But to state that a tensor vanishes, or that it is equal to another tensor in the same region, is a statement of intrinsic property, quite independent of the mesh-system chosen. Thus by keeping entirely to tensors, we contrive that there shall be behind our formulae an undercurrent of information having reference to the intrinsic state of the world.

In this way we have found two absolute formulae, which appear to be fully confirmed by observation, namely

 in empty space, ⁠${\displaystyle G_{\mu \nu }=0}$, in space containing matter, ⁠${\displaystyle G_{\mu \nu }=K_{\mu \nu }}$,

where ${\displaystyle K_{\mu \nu }}$ contains only physical quantities which are perfectly familiar to us, viz. the density and state of motion of the matter in the region.

I think the usual view of these equations would be that the first expresses some law existing in the world, so that the point-events by natural necessity tend to arrange their relations in conformity with this equation. But when matter intrudes it causes a disturbance or strain of the natural linkages; and a rearrangement takes place to the extent indicated by the second equation.

But let us examine more closely what the equation ${\displaystyle G_{\mu \nu }=0}$ tells us. We have been giving the mathematician a free hand with his indefinable intervals and point-events. He has arrived at the quantity ${\displaystyle G_{\mu \nu }}$; but as yet this means to us—absolutely nothing. The pure mathematician left to himself never "deviates into sense." His work can never relate to the familiar things around us, unless we boldly lay hold of some of his symbols and give them an intelligible meaning—tentatively at first, and then definitely as we find that they satisfy all experimental knowledge. We have decided that in empty space ${\displaystyle G_{\mu \nu }}$ vanishes. Here is our opportunity. In default of any other suggestion as to what the vanishing of ${\displaystyle G_{\mu \nu }}$ might mean, let us say that the vanishing of ${\displaystyle G_{\mu \nu }}$ means emptiness; so that ${\displaystyle G_{\mu \nu }}$, if it does not vanish, is a condition of the world which distinguishes space said to be occupied from space said to be empty. Hitherto ${\displaystyle G_{\mu \nu }}$ was merely a formal outline to be filled with some undefined contents; we are as far as ever from being able to explain what those contents are; but we have now given a recognisable meaning to the completed picture, so that we shall know it when we come across it in the familiar world of experience.

The two equations are accordingly merely definitions—definitions of the way in which certain states of the world (described in terms of the indefinables) impress themselves on our perceptions. When we perceive that a certain region of the world is empty, that is merely the mode in which our senses recognise that it is curved no higher than the first degree. When we perceive that a region contains matter we are recognising the intrinsic curvature of the world; and when we believe we are measuring the mass and momentum of the matter (relative to some axes of reference) we are measuring certain components of world-curvature (referred to those axes). The statistical averages of something unknown, which have been used to describe the state of the world, vary from point to point; and it is out of these that the mind has constructed the familiar notions of matter and emptiness.

The law of gravitation is not a law in the sense that it restricts the possible behaviour of the substratum of the world; it is merely the definition of a vacuum. We need not regard matter as a foreign entity causing a disturbance in the gravitational field; the disturbance is matter. In the same way we do not regard light as an intruder in the electromagnetic field, causing the electromagnetic force to oscillate along its path; the oscillations constitute the light. Nor is heat a fluid causing agitation of the molecules of a body; the agitation is heat.

This view, that matter is a symptom and not a cause, seems so natural that it is surprising that it should be obscured in the usual presentation of the theory. The reason is that the connection of mathematical analysis with the things of experience is usually made, not by determining what matter is, but by what certain combinations of matter do. Hence the interval is at once identified with something familiar to experience, namely the thing that a scale and a clock measure. However advantageous that may be for the sake of bringing the theory into touch with experiment at the outset, we can scarcely hope to build up a theory of the nature of things if we take a scale and clock as the simplest unanalysable concepts. The result of this logical inversion is that by the time the equation ${\displaystyle G_{\mu \nu }=K_{\mu \nu }}$ is encountered, both sides of the equation are well-defined quantities. Their necessary identity is overlooked, and the equation is regarded as a new law of nature. This is the fault of introducing the scale and clock prematurely. For our part we prefer first to define what matter is in terms of the elementary concepts of the theory; then we can introduce any kind of scientific apparatus; and finally determine what property of the world that apparatus will measure.

Matter defined in this way obeys all the laws of mechanics, including conservation of energy and momentum. Proceeding with a similar development of Weyl's more general theory of the combined gravitational and electrical fields, we should find that it has the familiar electrical and optical properties. It is purely gratuitous to suppose that there is anything else present, controlling but not to be identified with the relations of the fourteen potentials (${\displaystyle g}$'s and ${\displaystyle k}$'s).

There is only one further requirement that can be demanded from matter. Our brains are constituted of matter, and they feel and think—or at least feeling and thinking are closely associated with motions or changes of the matter of the brain. It would be difficult to say that any hypothesis as to the nature of matter makes this process less or more easily understood; and a brain constituted out of differential coefficients of ${\displaystyle g}$'s can scarcely be said to be less adapted to the purposes of thought than one made, say, out of tiny billiard balls! But I think we may even go a little beyond this negative justification. The primary interval relation is of an undefined nature, and the ${\displaystyle g}$'s contain this undefinable element. The expression ${\displaystyle G_{\mu \nu }}$ is therefore of defined form, but of undefined content. By its form alone it is fitted to account for all the physical properties of matter; and physical investigation can never penetrate beneath the form. The matter of the brain in its physical aspects is merely the form; but the reality of the brain includes the content. We cannot expect the form to explain the activities of the content, any more than we can expect the number 4 to explain the activities of the Big Four at Versailles.

Some of these views of matter were anticipated with marvellous foresight by W. K. Clifford forty years ago. Whilst other English physicists were distracted by vortex-atoms and other will-o'-the-wisps, Clifford was convinced that matter and the motion of matter were aspects of space-curvature and nothing more. And he was no less convinced that these geometrical notions were only partial aspects of the relations of what he calls "elements of feeling."—"The reality corresponding to our perception of the motion of matter is an element of the complex thing we call feeling. What we might perceive as a plexus of nerve-disturbances is really in itself a feeling; and the succession of feelings which constitutes a man's consciousness is the reality which produces in our minds the perception of the motions of his brain. These elements of feeling have relations of nextness or contiguity in space, which are exemplified by the sight-perceptions of contiguous points; and relations of succession in time which are exemplified by all perceptions. Out of these two relations the future theorist has to build up the world as best he may. Two things may perhaps help him. There are many lines of mathematical thought which indicate that distance or quantity may come to be expressed in terms of position in the wide sense of the analysis situs. And the theory of space-curvature hints at a possibility of describing matter and motion in terms of extension only." (Fortnightly Review, 1875.)

The equation ${\displaystyle G_{\mu \nu }=K_{\mu \nu }}$ is a kind of dictionary explaining what the different components of world-curvature mean in terms ordinarily used in mechanics. If we write it in the slightly modified, but equivalent, form ${\displaystyle G_{\mu \nu }-{\tfrac {1}{2}}g_{\mu \nu }G=-8\pi \,T_{\mu \nu }}$ we have the following scheme of interpretation ${\displaystyle {\begin{matrix}T_{11}{\text{,}}&T_{12}{\text{,}}&T_{13}{\text{,}}&T_{14}&=p_{11}+\rho u^{2}{\text{,}}&p_{12}+\rho uv{\text{,}}&p_{13}+\rho uw{\text{,}}&-\rho u{\text{,}}\\&T_{22}{\text{,}}&T_{23}{\text{,}}&T_{24}&&p_{22}+\rho v^{2}{\text{,}}&p_{23}+\rho vw{\text{,}}&-\rho v{\text{,}}\\&&T_{33}{\text{,}}&T_{34}&&&p_{33}+\rho w^{2}{\text{,}}&-\rho w{\text{,}}\\&&&T_{44}&&&&\rho {\text{.}}\end{matrix}}}$ Here we are using the partitions of space and time adopted in ordinary mechanics; ${\displaystyle \rho }$ is the density of the matter, ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$ its component velocities, and ${\displaystyle p_{11}}$, ${\displaystyle p_{12}}$, ${\displaystyle \ldots p_{33}}$, the components of the internal stresses which are believed to be analysable into molecular movements.

Now the question arises, is it legitimate to make identifications on such a wholesale scale? Having identified ${\displaystyle T_{44}}$ as density, can we go on to identify another quantity ${\displaystyle T_{34}}$ as density multiplied by velocity? It is as though we identified one "thing" as air, and a quite different "thing" as wind. Yes, it is legitimate, because we have not hitherto explained what is to be the counterpart of velocity in our scheme of the world; and this is the way we choose to introduce it. All identifications are at this stage provisional, being subject to subsequent test by observation.

A definition of the velocity of matter in some such terms as "wind divided by air" does not correspond to the way in which motion primarily manifests itself in our experience. Motion is generally recognised by the disappearance of a particle at one point of space and the appearance of an apparently identical particle at a neighbouring point. This manifestation of motion can be deduced mathematically from the identifying definition here adopted. Remembering that in physical theory it is necessary to proceed from the simple to the complex, which is often opposed to the instinctive desire to proceed from the familiar to the unfamiliar, this inversion of the order in which the manifestations of motion appear need occasion no surprise. Permanent identity of particles of matter (without which the ordinary notion of velocity fails) is a very familiar idea, but it appears to be a very complex feature of the world.

A simple instance may be given where the familiar kinematical conception of motion is insufficient. Suppose a perfectly homogeneous continuous ring is rotating like a wheel, what meaning can we attach to its motion? The kinematical conception of motion implies change—disappearance at one point and reappearance at another point—but no change is detectable. The state at any one moment is the same as at a previous moment, and the matter occupying one position now is indistinguishable from the matter in the same position a moment ago. At the most it can only differ in a mysterious non-physical quality—that of identity; but if, as most physicists are willing to believe, matter is some state in the aether, what can we mean by saying that two states are exactly alike, but are not identical? Is the hotness of the room equal to, but not identical with, its hotness yesterday? Considered kinematically, the rotation of the ring appears to have no meaning; yet the revolving ring differs mechanically from a stationary ring. For example, it has gyrostatic properties. The fact that in nature a ring has atomic and not continuous structure is scarcely relevant. A conception of motion which affords a distinction between a rotating and non-rotating continuous ring must be possible; otherwise this would amount to an a priori proof that matter is atomic. According to the conception now proposed, velocity of matter is as much a static quality as density. Generally velocity is accompanied by changes in the physical state of the world, which afford the usual means of recognising its existence; but the foregoing illustration shows that these symptoms do not always occur.

This definition of velocity enables us to understand why velocity except in reference to matter is meaningless, whereas acceleration and rotation have a meaning. The philosophical argument, that velocity through space is meaningless, ceases to apply as soon as we admit any kind of structure or aether in empty regions; consequently the problem is by no means so simple as is often supposed. But our definition of velocity is dynamical, not kinematical. Velocity is the ratio of certain components of ${\displaystyle T_{\mu \nu }}$, and only exists when ${\displaystyle T_{44}}$ is not zero. Thus matter (or electromagnetic energy) is the only thing that can have a velocity relative to the frame of reference. The velocity of the world-structure or aether, where the ${\displaystyle T_{\mu \nu }}$ vanish, is always of the indeterminate form 0 ÷ 0. On the other hand acceleration and rotation are defined by means of the ${\displaystyle g_{\mu \nu }}$ and exist wherever these exist[2]; so that the acceleration and rotation of the world-structure or aether relative to the frame of reference are determinate. Notice that acceleration is not defined as change of velocity; it is an independent entity, much simpler and more universal than velocity. It is from a comparison of these two entities that we ultimately obtain the definition of time.

This finally resolves the difficulty encountered in Chapter x—the apparent difference in the Principle of Relativity as applied to uniform and non-uniform motion. Fundamentally velocity and acceleration are both static qualities of a region of the world (referred to some mesh-system). Acceleration is a comparatively simple quality present wherever there is geodesic structure, that is to say everywhere. Velocity is a highly complex quality existing only where the structure is itself more than ordinarily complicated, viz. in matter. Both these qualities commonly give physical manifestations, to which the terms acceleration and velocity are more particularly applied; but it is by examining their more fundamental meaning that we can understand the universality of the one and the localisation of the other.

It has been shown that there are four identical relations between the ten qualities of a piece of matter here identified, which depend solely on the way the ${\displaystyle G_{\mu \nu }}$ were by definition constructed out of simpler elements. These four relations state that, provided the mesh-system is drawn in one of a certain number of ways, mass (or energy) and momentum will be conserved. The conservation of mass is of great importance; matter will be permanent, and for every particle disappearing at any point a corresponding mass will appear at a neighbouring point; the change consists in the displacement of matter, not its creation or destruction. This gives matter the right to be regarded, not as a mere assemblage of symbols, but as the substance of a permanent world. But the permanent world so found demands the partitioning of space-time in one of a certain number of ways, viz. those discussed in Chapter iii[3]; from these a particular space and time are selected, because the observer wishes to consider himself, or some arbitrary body, at rest. This gives the space and time used for ordinary descriptions of experience. In this way we are able to introduce perceptual space and time into the four-dimensional world, as derived concepts depending on our desire that the new-found matter should be permanent.

I think it is now possible to discern something of the reason why the world must of necessity be as we have described it. When the eye surveys the tossing waters of the ocean, the eddying particles of water leave little impression; it is the waves that strike the attention, because they have a certain degree of permanence. The motion particularly noticed is the motion of the wave-form, which is not a motion of the water at all. So the mind surveying the world of point-events looks for the permanent things. The simpler relations, the intervals and potentials, are transient, and are not the stuff out of which mind can build a habitation for itself. But the thing that has been identified with matter is permanent, and because of its permanence it must be for mind the substance of the world. Practically no other choice was possible.

It must be recognised that the conservation of mass is not exactly equivalent to the permanence of matter. If a loaf of bread suddenly transforms into a cabbage, our surprise is not diminished by the fact that there may have been no change of weight. It is not very easy to define this extra element of permanence required, because we accept as quite natural apparently similar transformations—an egg into an omelette, or radium into lead. But at least it seems clear that some degree of permanence of one quality, mass, would be the primary property looked for in matter, and this gives sufficient reason for the particular choice.

We see now that the choice of a permanent substance for the world of perception necessarily carries with it the law of gravitation, all the laws of mechanics, and the introduction of the ordinary space and time of experience. Our whole theory has really been a discussion of the most general way in which permanent substance can be built up out of relations; and it is the mind which, by insisting on regarding only the things that are permanent, has actually imposed these laws on an indifferent world. Nature has had very little to do with the matter; she had to provide a basis—point-events; but practically anything would do for that purpose if the relations were of a reasonable degree of complexity. The relativity theory of physics reduces everything to relations; that is to say, it is structure, not material, which counts. The structure cannot be built up with out material; but the nature of the material is of no importance. We may quote a passage from Bertrand Russell's Introduction to Mathematical Philosophy.

"There has been a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realised. For example it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world…. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes words and baffles description, but which for that very reason is irrelevant to science."

This is how our theory now stands.—We have a world of point-events with their primary interval-relations. Out of these an unlimited number of more complicated relations and qualities can be built up mathematically, describing various features of the state of the world. These exist in nature in the same sense as an unlimited number of walks exist on an open moor. But the existence is, as it were, latent unless someone gives a significance to the walk by following it; and in the same way the existence of any one of these qualities of the world only acquires significance above its fellows, if a mind singles it out for recognition. Mind filters out matter from the meaningless jumble of qualities, as the prism filters out the colours of the rainbow from the chaotic pulsations of white light. Mind exalts the permanent and ignores the transitory; and it appears from the mathematical study of relations that the only way in which mind can achieve her object is by picking out one particular quality as the permanent substance of the perceptual world, partitioning a perceptual time and space for it to be permanent in, and, as a necessary consequence of this Hobson's choice, the laws of gravitation and mechanics and geometry have to be obeyed. Is it too much to say that mind's search for permanence has created the world of physics? So that the world we perceive around us could scarcely have been other than it is[4]?

The last sentence possibly goes too far, but it illustrates the direction in which these views are tending. With Weyl's more general theory of interval-relations, the laws of electrodynamics appear in like manner to depend merely on the identification of another permanent thing—electric charge. In this case the identification is due, not to the rudimentary instinct of the savage or the animal, but the more developed reasoning-power of the scientist. But the conclusion is that the whole of those laws of nature which have been woven into a unified—scheme mechanics, gravitation, electrodynamics and optics—have their origin, not in any special mechanism of nature, but in the workings of the mind.

"Give me matter and motion," said Descartes, "and I will construct the universe." The mind reverses this. "Give me a world—a world in which there are relations—and I will construct matter and motion."

Are there then no genuine laws in the external world? Laws inherent in the substratum of events, which break through into the phenomena otherwise regulated by the despotism of the mind? We cannot foretell what the final answer will be; but, at present, we have to admit that there are laws which appear to have their seat in external nature. The most important of these, if not the only law, is a law of atomicity. Why does that quality of the world which distinguishes matter from emptiness exist only in certain lumps called atoms or electrons, all of comparable mass? Whence arises this discontinuity? At present, there seems no ground for believing that discontinuity is a law due to the mind; indeed the mind seems rather to take pains to smooth the discontinuities of nature into continuous perception. We can only suppose that there is something in the nature of things that causes this aggregation into atoms. Probably our analysis into point-events is not final; and if it could be pushed further to reach something still more fundamental, then atomicity and the remaining laws of physics would be seen as identities. This indeed is the only kind of explanation that a physicist could accept as ultimate. But this more ultimate analysis stands on a different plane from that by which the point-events were reached. The world may be so constituted that the laws of atomicity must necessarily hold; but, so far as the mind is concerned, there seems no reason why it should have been constituted in that way. We can conceive a world constituted otherwise. But our argument hitherto has been that, however the world is constituted, the necessary combinations of things can be found which obey the laws of mechanics, gravitation and electrodynamics, and these combinations are ready to play the part of the world of perception for any mind that is tuned to appreciate them; and further, any world of perception of a different character would be rejected by the mind as unsubstantial.

If atomicity depends on laws inherent in nature, it seems at first difficult to understand why it should relate to matter especially; since matter is not of any great account in the analytical scheme, and owes its importance to irrelevant considerations introduced by the mind. It has appeared, however, that atomicity is by no means confined to matter and electricity; the quantum, which plays so great a part in recent physics, is apparently an atom of action. So nature cannot be accused of connivance with mind in singling out matter for special distinction. Action is generally regarded as the most fundamental thing in the real world of physics, although the mind passes it over because of its lack of permanence; and it is vaguely believed that the atomicity of action is the general law, and the appearance of electrons is in some way dependent on this. But the precise formulation of the theory of quanta of action has hitherto baffled physicists.

There is a striking contrast between the triumph of the scientific mind in formulating the great general scheme of natural laws, nowadays summed up in the principle of least action, and its present defeat by the newly discovered but equally general phenomena depending on the laws of atomicity of quanta. It is too early to cry failure in the latter case; but possibly the contrast is significant. It is one thing for the human mind to extract from the phenomena of nature the laws which it has itself put into them; it may be a far harder thing to extract laws over which it has had no control. It is even possible that laws which have not their origin in the mind may be irrational, and we can never succeed in formulating them. This is, however, only a remote possibility; probably if they were really irrational it would not have been possible to make the limited progress that has been achieved. But if the law of quanta do indeed differentiate the actual world from other worlds possible to the mind, we may expect the task of formulating them to be far harder than anything yet accomplished by physics.

The theory of relativity has passed in review the whole subject-matter of physics. It has unified the great laws, which by the precision of their formulation and the exactness of their application have won the proud place in human knowledge which physical science holds to-day. And yet, in regard to the nature of things, this knowledge is only an empty shell—a form of symbols. It is knowledge of structural form, and not knowledge of content. All through the physical world runs that unknown content, which must surely be the stuff of our consciousness. Here is a hint of aspects deep within the world of physics, and yet unattainable by the methods of physics. And, moreover, we have found that where science has progressed the farthest, the mind has but regained from nature that which the mind has put into nature.

We have found a strange foot-print on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last, we have succeeded in reconstructing the creature that made the footprint. And Lo! it is our own.

1. There is also a qualitative distinction into two kinds, ultimately identified as time-like and space-like, which for mathematical treatment are distinguished by real and imaginary numbers.
2. Even in Newtonian mechanics we speak of the "field of acceleration," and think of it as existing even when there is no test body to display the acceleration. In the present theory this field of acceleration is described by the ${\displaystyle g_{\mu \nu }}$. There is no such thing as a "field of velocity" in empty space; but there is in a material ocean.
3. When the kind of space-time is such that a strict partition of this kind is impossible, strict conservation does not exist; but we retain the principle as formally satisfied by attributing energy and momentum to the gravitational field.
4. This summary is intended to indicate the direction in which the views suggested by the relativity theory appear to me to be tending, rather than to be a precise statement of what has been established. I am aware that there are at present many gaps in the argument. Indeed the whole of this part of the discussion should be regarded as suggestive rather than dogmatic.