The evolution of scientific thought from Newton to Einstein

by A. D'Abro

APPENDICES

APPENDIX I
THE SPACE AND TIME GRAPHS

THE theory of relativity appeals to what is known as the space-time graphical representation of Minkowski, but aside from certain peculiarities which the relativity theory entails, the general method of graphical representation in space and in time was known to classical science. Indeed, the graph traced by a thermometer needle is an illustration of this method. In it we have a graphic description of the variations in height of the mercury as time passes by.

The essence of these space and time graphs is to select a frame of reference, then to represent the successive positions of a body moving through this frame, in terms of its successive space and time co-ordinates. As a simple case let us consider a railroad embankment which will serve as our frame of reference. We shall restrict ourselves to considering the graphical space and time representation of events occurring on the surface of the embankment; not above it or beneath it. In other words, the space we shall be dealing with will for all practical purposes be reduced to one line, hence to one dimension. We then select a fixed point (any one at all) on the embankment and call it our origin

. Thanks to this choice of an origin, the spatial position of any event occurring on the embankment may be specified by a number. Thus the position of an event occurring two units of distance to the right or to the left of the origin will be given by the number +2 or -2, and an event occurring at the origin itself will have zero for its number.

In order to represent these results on a sheet of paper, we shall draw a straight line, say a horizontal, called a space axis; this will represent the embankment. On this space axis we mark a point

which will represent our origin on the embankment. Then, in order to represent on our paper the positions of events occurring at, say, one mile, two miles, three miles, etc., to the right of the origin on the embankment, we mark off points along our space axis at one, two and three units of distance from the point

. Obviously, we cannot manipulate a sheet of paper miles in length; hence we agree to represent a distance of one mile along the embankment by a length of one foot or one inch or one centimetre along the space axis. It matters not what unit we select so long as, once specified, it is maintained consistently throughout. As the reader can understand, the procedure is exactly the same as that followed in the plan of a city.

Thus far, our graph reduces to a space graph of the points situated along the embankment. But we have now to introduce time. Two events may happen at the same point of the embankment, but at widely different times, and our graph in its present form offers us no means of differentiating graphically between the occurrence of the two events. Accordingly, we shall agree to represent such differences in time on our sheet of paper by placing our representative points of the events at varying heights above or below the space axis. If, then, we assume that all points on our space axis represent the space and time positions of events occurring on the embankment at a time zero or at noon, it will follow that all points above or below the space axis will represent events occurring on the embankment either after or before the time zero. This is equivalent to considering a vertical axis called a time axis, along which durations, hence instants, of time will be measured. Of course, just as in the case of distances, we must agree on some unit of length in our graph, in order to represent one second in time. We may choose this unit as we please; we may, for example, represent one second by one foot, or by one inch, along the vertical

. However, for reasons which will become apparent in the theory of relativity, it is advantageous, though by no means imperative, to select the same unit of length in our graph in order to represent both one second in time and 186,000 miles in distance.

Fig. IX

Suppose, for instance, we agree to represent these magnitudes by a length of one inch on our graph; then a point such as

, one inch from

and one inch from

, will represent the space and time position of an event occurring on the embankment 186,000 miles to the right of the origin

and one second after the time zero ([Fig. IX]).

We see, then, that in our space and time graph, a point traced on our sheet of paper represents not merely a position in space along the embankment, but also an instant in time. For this reason such graph-points are known as point-events. Thus a point-event constitutes the graphical representation of an instantaneous event occurring anywhere and at any time along the embankment. The position of the point-event with respect to our space and time axes will then define without ambiguity the spatio-temporal position of the physical event with respect to the embankment and to the time zero, provided the units of measurement in the graph have been specified.

And now let us consider the representation of events that last and are not merely instantaneous. Here let us note that the existence of a body, say a stone on the embankment, constitutes an event, since the position of the stone can be defined in space and in time. But the stone endures: its existence is not merely momentary; hence its permanency is given by a continuous succession of point-events forming a continuous line. This line giving the successive positions of the stone both in space and in time is called a world-line. For the stone to possess a world-line, it is not necessary that it should be in motion along the embankment; it may just as well remain at rest. The sole difference will be that if the stone is at rest, its world-line will be a vertical, whereas if it is in motion along the embankment, its world-line will be slanting, since in this case the spatial position of the stone will vary as time passes by. If the speed of the stone along the embankment is constant, its world-line will be straight, whereas if the motion is uneven or accelerated, the world-line will be more or less curved. Assuming the motion to be uniform, the greater the speed of the body, the more will its world-line slant away from the vertical and tend to become horizontal.

Fig X

Of course, as can easily be understood, the slants of the world-lines in the graph (aside from exceptional ones such as that of a body at rest) will be influenced by our choice of units. With the particular units we have chosen, the world-lines of bodies moving along the embankment at a speed of 186,000 miles a second will possess a slant of 45° with respect to both space and time axes. In other words, the world-lines of such bodies, hence also of rays of light, will be inclined equally to our space and time axes. Thus any straight line inclined in this way, either to the right or to the left ([Fig. X]), will represent the world-line of a body moving with respect to the embankment with the speed of light, either to the right or to the left. The reason we selected our units of space and time as we did, was precisely in order to confer this symmetrical position of the world-lines of light, on account of the important rôle which the velocity of light plays in the theory of relativity. We may consider still another case, that of a body moving with infinite speed along the embankment (assuming, of course, that the existence of such a motion is physically possible). The body will obviously be everywhere along the embankment at the same instant of time; hence its world-line will be a horizontal. Thus the space axis is itself the world-line of a body moving along the embankment with infinite speed at the instant zero. Conversely, the time axis is the world-line of a body remaining motionless at the origin. Again, we may say that the space axis represents the totality of events occurring on the embankment at the instant zero, hence simultaneous with one another and with the instant zero. Likewise, any horizontal represents the totality of events occurring on the embankment at some given time, the height of this horizontal above or below the space axis defining the time.

Thus far, we have been considering happenings with reference to an observer at rest on the embankment, and everything we have said applies in an identical way to classical science and to relativity. It will only be when observers in relative motion are considered that differences in our graphical representation will arise.

Fig. XI

Let us first consider the case of classical science. And here it is important to realise that our graph is nothing but a description: it merely describes graphically the relationships of duration, distance and motion which we wish to represent. Hence it is for us to discover through the medium of experiment what these relationships are going to be. Only after this preliminary work has been done can we represent these relationships graphically. Now classical science, both as the result of crude experience and, later, of more refined measurements, held to the view that duration and distance were absolutes, in that their magnitudes would never be modified by our circumstances of motion. Accordingly, regardless of whether we were at rest on the embankment or in motion, the duration separating two events or the distance between two fixed points on the embankment was assumed to remain the same. Interpreted graphically, this meant that the space and time axes would never have to be changed in our graph, regardless of the motion of the embankment observer whose measurements we were seeking to represent.

Consider two point-events, such as

and

([Fig. XI]). As referred to the embankment, these two point-events represent two instantaneous events occurring on the embankment at a definite distance apart in space and in time. Suppose, now an observer leaves the origin at a time zero and moves along the embankment to the right. His world-line will be given by some such line as

. Since, regardless of his motion, these two events

and

are to manifest exactly the same time separation as before, we must assume that the moving observer must adhere to measurements computed along the same time axis

.

Fig. XII

Fig. XIII

Next consider two stones

and

lying on the embankment. Their world-lines will be

and

, respectively ([Fig. XII]), and their distance apart at any time

will be given by the length

of the horizontal situated at the height

above

(the world-lines being vertical, this distance can of course never vary with time). Now, since the distance of the two stones must remain the same for the moving observer, he also must measure this distance along a horizontal, hence along the space axis

. In other words, the space axis and the time axis are unique, or absolute.

It follows, of course, that since the space and time axes remain unchanged regardless of our motion along the embankment, all horizontals will represent events occurring simultaneously not only for an observer at rest on the embankment, but for all observers. We thus get the absolute nature of simultaneity.

Then again, since all observers measure time along the same direction, they will all recognise one same absolute distinction between past and future, regardless of their position and motion along the embankment. Thus, point-events below

([Fig. XIII]), will represent events that occurred on the embankment prior to zero hour; those point-events lying above the line

will represent events that occurred after zero hour, while the point-events lying on

will give the events that occurred at zero hour.

Fig. XIV

As a final illustration ([Fig. XIV]), let us consider the case of an observer rushing after a light wave, both observer and light disturbance having left the origin

at the time zero. The world-line of the light ray will be represented by

, and that of the observer by

, less slanting than

, at least if we consider the case of an observer moving along the embankment with a speed inferior to that of light. At a definite instant, say at one second after zero hour, the point-event defining the observer’s position will be

, that is to say, a point on his world-line at a height of one inch above

. The point-event of the light wave at the same instant will be

, also at one inch above

, and the point-event of the observer who remained at

will be

, the line

being, of course, horizontal. Hence the distance of the light wave from the moving observer, at that precise; instant, will be

, which is less than

, its distance from the stationary observer. It follows that the light ray is moving with decreased speed with respect to the moving observer. We might have foreseen this; directly since the world-line of light

is no longer equally inclined to; the world-line

of the moving observer (hence to the successive positions of his body) and to his space line

.

We have now mentioned the chief characteristics of the space and time: graph of classical science, and we should find that it reduced to a mere geometric representation of the classical Galilean space and time transformations of classical science, namely,

Thus all the problems we have discussed could be solved either geometrically or analytically.

It is a matter of common knowledge that this space and time graph we have discussed was rarely mentioned in classical science; and such appellations as point-events and world-lines were unknown prior to Minkowski’s discoveries. It may seem strange, therefore, that the theory of relativity should have made so extensive a use of a graphical method of representation. The reason is that in classical science the graph reduced to a mere geometrical superposition of space and time which was convenient in certain cases, but which had no profound significance. By reason of the separateness of space and time exemplified by the absoluteness of the space and time axes, there existed no amalgamation, no unity between space and time. They did not form one four-dimensional continuum of events in any deep sense. Let us endeavour to understand the meaning of these statements.

In classical science, space was regarded as a three-dimensional continuum of points, because it was possible to localise the position of a point in our frame by referring to its three co-ordinates. Suppose, then, that we wished to measure the distance between two points in space. All we should have to do would be to stretch a tape between the two points, and the length of the tape would define the distance between them.

And now let us consider the space and time of classical science. Here, again, we might claim that space and time constituted a four-dimensional continuum of events; for we could always localise the occurrence of an instantaneous event by measuring three spatial co-ordinates in our frame of reference, and by computing the instant at which the event occurred. But suppose we wished to measure the distance between two events occurring, say, one in New York on Monday, and the other in Washington on Tuesday. Obviously the problem would be meaningless. We might say that the distance in space between the two events was so many miles, and their distance in time so many hours; but we should be unable to measure the distance in space-time itself directly, whereas, with the aid of the tape, in our previous example, we were able to measure immediately the spatial distance between the two points.

We see, then, that whereas both classical space and classical space-cum-time constituted a three-dimensional and a four-dimensional continuum, respectively, yet there was a vast difference between these two continua. The first one was a metrical continuum, whereas classical space-cum-time had no four-dimensional metrics.

We may present these arguments in a slightly different way, as follows: We say that a necessary condition for a continuum of points to form a metrical continuum possessing a definite geometry is that a definite unambiguous expression, invariant in magnitude, be found for the distance between any two points. Now this condition is certainly not realised in our classical space and time graph.

Fig. XV

Consider, as before, the embankment, an observer at rest at

(whose world-line is then

), an observer in motion (whose world-line is

), and finally two point-events

and

([Fig. XV]). We wish to find for the distance between the point-events

and

some mathematical expression which will remain the same for all observers regardless of their motion. Now it is obvious that whereas the time distance between

and

, given by the difference in height of these two point-events above

, remains the same for all observers, this is no longer true of their space distance. Thus, for the observer at rest, the space distance of

and

is

, whereas, for the moving observer, it is

, i.e., a different magnitude. Owing, then, to the fixedness of the time separation and the variableness of the spatial one, it becomes impossible to construct an invariant mathematical relation capable of expressing a distance. This is what is meant when we say that the space and time of classical science could not be regarded as a four-dimensional metrical continuum of events. Space and time, when considered jointly, reduced to the juxtaposition of a continuum of points in space and of a continuum of instants in time; for space alone and time alone constituted separate metrical continua.

To Minkowski belongs the honour of having established the fusion between the two. Now and only now can we speak of the space-time distance, or Einsteinian interval, between the two events—say, one occurring in New York on Monday, and the other in Washington on Tuesday. Now and only now, thanks to ultra-precise experiment and to the genius of Einstein and Minkowski, is there any advantage in speaking of space-cum-time as a four-dimensional continuum of events which we call space-time. Prior to these achievements, the concept of space-time was as artificial as that of an

-dimensional continuum of space, time, pressure, temperature, colour, etc.

We shall now investigate in what measure the graphical representation of classical science will have to be modified in order to harmonise with the empirical facts revealed by ultra-refined experiment. There is no need to modify our understanding of point-events and world-lines; these will remain undisturbed.

The bifurcation between the two graphs arises when we consider the principle of the invariant velocity of light. We saw that if a ray of light was sent along the embankment to the right, leaving the origin

at a time zero, its world-line was given by a line

bisecting the angle

([Fig. XIV]). On the other hand, for an observer travelling to the right and having a world-line

, the light-world-line

would no longer bisect the angle formed by his own world-line

and the space axis

. The physical significance of this fact was that the velocity of light for the moving observer would be less than for the embankment observer. Now this result contradicted the principle of the invariant velocity of light. If, therefore, we wished to conceive of a graph capable of yielding results in conformity with the principle, we should have to assume that for all observers moving along the embankment, the line

would bisect the angle formed by their respective world-lines and space axes. It followed that the space axis of the moving observer could no longer be Ox but a new line

, such that

would bisect the angle

. In similar fashion, the observer’s time would have to be measured along his own world-line

, now called

([Fig. XVI]).

Fig. XVI

Hence we conclude that the space and time directions

and

are no longer absolute; every observer will have to measure time along his world-line and space along a line orthogonal to this world-line.[164] It follows that there exist an indefinite number of time directions given by the world-lines of the various observers, and a correspondingly indefinite number of space directions.

A first consequence of this novelty is that simultaneity can no longer be absolute. For whereas, in the classical graph, all events on the same horizontal or space direction were simultaneous for all observers, we now realise that with this variation in the space directions, or lines of simultaneous occurrence, the absoluteness of simultaneity must vanish. For instance, all the point-events lying on

which are therefore simultaneous with the time zero for the moving observer, appear to be unfolding themselves in succession for the stationary observer.

Also it follows that as, according to relativity, no observer can travel faster than light, all the permissible world-lines of the observers (passing through

at a time zero) will be contained between the two light-world-lines

and

([Fig. XVII]).

Fig. XVII

Any line passing through and not contained between these two

light-lines can never be a world-line, hence can never be a time direction; it will be a space direction of some possible observer. We may extend these results to two space dimensions. In this case our world-lines of light rays passing through

at the time zero form a cone called the light-cone with its apex at

. Straight lines passing through

and contained within the light-cone are possible world-lines, or time directions; those passing through

but lying outside the cone constitute possible space directions for appropriate observers, or again possible loci of simultaneous events. Of course each point-event taken as point of departure gives rise to a light-cone, so that in a general way we may say that all lines parallel to lines contained within any given light-cone and passing through its summit constitute possible world-lines or time directions. It is thus apparent that any light-cone defines limiting directions in space-time. All lines whose slant is less than that of the generators of the cone are possible time directions, while all those whose slant is greater are possible space directions. We thus realise the importance of the light-cone in defining the particularities of structure of space-time. In the case of four-dimensional space-time the cone becomes a three-dimensional surface, which it is not easy to visualise, but this, of course, need not trouble us when we reason analytically.

And here a further point must be mentioned. Since no disturbance can reach us with a speed greater than that of light, the world-line of the disturbance can never lie outside the light-cone at whose apex we momentarily stand. In other words, the events of which we may become conscious (otherwise than visually) at a given instant will always be represented by point-events situated within the cone below us, in the direction of the past. Events which we perceive visually at an instant will always lie on the cone’s surface.

Fig. XVIII

Thus, if a star suddenly becomes visible in the sky, the point-event of that star when it burst into prominence is situated on the surface of our instantaneous light-cone somewhere in the past ([Fig. XVIII]).

In a similar way, the only events we can affect must lie in or on the upper part of the cone. An event which occurs outside the cone can never affect us or be affected by us now. It may, however, affect us ultimately, when, as a result of our light-cone’s rising with us along our world-line, it finally touches the cone’s surface. The event will then become visible, provided it be luminous. Once in the interior of the cone, the event may be perceived in other ways, by sound, etc., but no longer as a result of light transmission in vacuo. We may also state that when two point-events lie on the same time direction, their order of succession will remain the same for all observers; hence, in this case, it will be possible to conceive of a causal relation as existing between them. But if the two events lie on the same space direction, they may be simultaneous, or subsequent and antecedent, or antecedent and subsequent, according to our motion. Events represented by such point-events can never be considered as manifesting any causal relationship one with the other. We see, then, that inasmuch as it is the light-cone which differentiates space from time directions, and events which may be causally connected from those which may not, the irreversibility of time and the problem of causality are linked with the existence of the light-cone.

In a general way, we may say that whereas the classical graph showed an absolute past, present and future for all observers, in the new graph this statement must be modified as follows: If we consider all the observers situated at a definite point at some definite time, that is to say, all observers whose spatio-temporal position is given by the same common point-event, then, regardless of their relative motions, the same instantaneous light-cone holds for all. The apex of the cone is given by the common point-event, and all point-events situated within or on the surface of the cone stand in the instantaneous past or future for all the observers. There exist, therefore, an absolute past and an absolute future even in the theory of relativity. On the other hand, all point-events situated outside the cone will be neither past nor future in any absolute sense, for they may be in the past, the present or the future, according to the motion of the observer.

We may also mention that, thanks to the indefiniteness of the space and time directions, our graph is now dealing with a veritable continuum, the space-time continuum, which cannot be separated in any unique way into space and time.

We should also find that the space-time distance between any two point-events,

and

, for which no absolute expression existed in the classical graph, would now assume an absolute value, the same for all observers. This distance is the Einsteinian interval discovered by Minkowski, which permits us to associate a definite geometry with the four-dimensional continuum of events. The geometry is not Euclidean, but semi-Euclidean. When account is taken on Minkowski’s discoveries, it is seen that a space-time distance, when taken along a line which can be traced inside a light-cone, hence on a world line, is an imaginary magnitude; whereas, when taken along a line lying outside any light-cone, the distance becomes real. However, no great importance need be attributed to time being imaginary and space being real; for we could just as well have conceived of space as imaginary and time as real. All that it is important to note is that there exists a mathematical difference between the various directions in space-time, those which lie inside the cone and are time-like, and those which lie without and are space-like. The former alone can be followed by physical disturbances and material bodies.

It may be instructive to look into this strange geometry of space-time a little more closely. We shall restrict ourselves to two dimensions, that is to say, to one space dimension and to the time dimension. In other words, our space-time graph will refer to the measurements of observers moving along the embankment.

Suppose, then, that a number of observers, moving with various uniform speeds to the right or to the left, pass the observer on the embankment at

, at the time zero. These observers carry ordinary clocks in their hands, and these all mark zero hour as the observers pass

simultaneously. The question we wish to decide is as follows:

“Where will the point-events of the various observers be situated on the graph when their respective clocks mark one second past zero hour?”

Fig. XIX

We know that the point-event of the observer, who does not move from the point

on the embankment, will be situated on the vertical

, at a distance

= one second or one centimetre on our graph ([Fig. XIX]). Calculation then shows that the point-events of all the other observers will be situated on the equilateral hyperbola which passes through

and which has for asymptotes the two light-world-lines

and

. For instance, the observer whose world-line is

will find himself at the point

when his clock marks zero hour plus one second, that is to say, when he has travelled along the embankment away from

during a time of one second as measured by his clock. In classical science, of course, the various observers’ point-events would have been situated at the intersection of their world-lines with the horizontal

, no longer with the hyperbola

. It is not that the relativistic clocks differ from the classical ones; we are always considering our ordinary chronometers. The novelty is solely due to the fact that, thanks to ultra-precise experiment, our understanding of the behaviour of clocks and rods is more accurate than it used to be.

From all this, we may infer that the space-time distances from

to any of the points on the hyperbola represent congruent space-time distances in the space-time geometry. We may also note that, were space-time Euclidean instead of semi-Euclidean, the locus of points defined by the hyperbola would be given by a circle with

as its centre.

And now let us consider the world-line of a light ray leaving

at a time zero. We see that this world-line (

or

) does not intersect the hyperbola, or, if we prefer, intersects it at infinity. It follows that the space-time distances

,

, etc., are all equal to

, where

is infinitely distant. Hence we must conclude that since the point

must be infinitely distant for

to have a finite space-time value, any such space-time distance as

, where

is any point on

, must be nil. In other words, from the standpoint of space-time geometry, the point-events on the same world-line of light are all at a zero distance apart. Accordingly, a world-line of light, i.e., any line inclined at 45°, is called a null-line. Lines of this sort were well known to geometricians long before Einstein’s theory, so we need not suppose that a null-line is one of the queer conceptions we owe to relativity. All Minkowski has done has been to give us a physical interpretation of a null-line. It would be illustrated by the world-line of an observer moving with the speed of light. For this observer the events of his life would present no temporal separation; though, of course, another observer would realise that these various events were separated in time.

Next, let us examine the measurements of spatial lengths. Suppose that the various observers, when they pass

, carry poles which they hold as lances parallel to the embankment. If these poles are of equal length when placed side by side at rest, the point-events of their further extremities, at the instant the observers pass

, will lie on a hyperbola

. In this case

gives the length of the rod at rest on the embankment. Of course, the positions of the extremities of the rods at the instant the observers pass

must be computed according to the simultaneity determinations of the respective observers. If the rods are 186,000 miles in length,

= one centimetre, and the two hyperbolas will be geometrically alike. We may also infer that the space-time distances from

to any of the points on the second hyperbola are all equal to one another. In fact, we may repeat for the second hyperbola, or space hyperbola, the same arguments we made when discussing the time one.

We are now in a position to understand how the FitzGerald contraction arises. Consider, for instance, a rod,

, lying on the embankment. The world-lines of its two extremities will be

and

, respectively ([Fig. XX]). If, now, an observer passes

at time zero and moves to the right with velocity

, his world-line will be

; but then his space direction will be

. For him, therefore, the length of the rod lying on the embankment is no longer

, as for the embankment observer, but

. The graph shows us that this length

is shorter than the length

of the rod the observer is carrying with him. And as these two rods, when placed side by side at rest, are equal we see that as measured by the observer the rod, on the embankment past which he is moving, will have suffered a contraction.

Fig. XX

Fig. XXI

We might, as a final example, consider the trip to the star ([Fig. XXI]). If

denotes the world-line of the star, and

the world-line of the travelling twin, we see that he will have lived a time

during his trip. His brother remaining on earth will have

for world-line. Hence, he will have lived a time

. As before, though

appears longer than

, it is, in reality, shorter, as can be understood by referring to [Fig. XIX]. Incidentally, we see how the absolute nature of acceleration causes the traveller’s world-line to bend; and it is this absolute bend in the world-line which differentiates the life-histories of the two twins and which is responsible for an absolute, non-reciprocal difference in their respective aging.

Let us also state that just as was the case with the classical graph which represented geometrically the classical Galilean transformations, so now the Minkowski graph merely translates the Lorentz-Einstein transformations.

And here a matter of some importance must be noted. We remember that before proceeding to draw our graph, we were compelled to settle on a co-ordination of units of measurement for space and time. We then chose the same length on our sheet of paper to represent a duration of one second and a distance of 186,000 miles. As a result of this choice the world-lines of light rays became diagonals. But suppose we had selected other units. Then, obviously, the slant of the world-lines, hence the angle of the light-cone, would have been modified and the entire appearance of the graph changed. Inasmuch as our choice of units is entirely arbitrary, we might be led to believe that the graph could not depict reality. But this opinion would be unfounded. While it is true that owing to the arbitrariness of our units, the graph cannot aspire to represent absolute shape, yet it does express certain definite relationships which a change of units could not disturb. In fact we might conceive the graph to be distorted by stretching, but still the relationships would endure; and relationships are all that science (or, we might even say, the human mind) can ever aspire to approach.

On the other hand, this question of units allows us to give a graphical solution of a point which is of great philosophical interest. Here we are living in a world which, theoretically at least, is vastly different from the world of separate space and time, and yet it is only thanks to ultra-refined experiment and to the genius of Einstein and Minkowski that we have finally realised it to be a four-dimensional continuum of events. How is it that ordinary perception is so blind to facts?

In order to understand this point, we must mention that though our choice of units for co-ordinating space and time measurements is arbitrary, since there is no rational connection between the magnitude of a distance in space and that of a duration in time, yet our daily activities suggest a common standard of comparison. The fact is that the distances which we ourselves and other material bodies cover in one second over the earth’s surface are always comprised within certain narrow limits. This leads us to couple one second and one yard, rather than one second and 186,000 miles.

If, now, with these more homely units we were to set up Minkowski’s graph, we should find that it was virtually identical with the classical one. The world-lines of the light rays would appear to coincide with the space axis

, and it would need a graph thousands of miles in length to detect their deviations from this line. A light-cone would cover the entire graph; hence the permissible space directions lying outside the cone would appear to be limited to the

axis. To all intents and purposes, there would be but one permissible space direction entailing the absoluteness of simultaneity and of time. We see, then, that it is by our immediate needs rather than by cosmic conditions, or, again, because slow velocities predominate around us, contrasted with which the velocity of light appears infinite, that we have been misled into believing in a world of separate space and time.

APPENDIX II
THE CURVATURES OF SPACE-TIME

IT may be of interest to explain the significance of Einstein’s gravitational equations more fully. We will restrict our attention to the equations of the general theory in the case of an infinite universe. These equations may be written:

Here it must be noticed that

and

are letters for which the four numbers representing the four dimensions of space-time must be substituted in pairs. There exist, therefore, a number of gravitational equations—sixteen in all. When we discard those which are mere repetitions of the others, hence which yield us no new information, we find that this number reduces to ten.

Now we have already a general idea of the significance of these equations. The left-hand side gives us the curvatures of space-time from point to point as measured in our mesh-system, and the right-hand side gives us the various aspects of matter, energy and momentum situated at the same points where the curvature is being calculated. If we agree to select the nearest possible approach to a Galilean mesh-system, and for reasons of simplicity agree to treat it as a Galilean system, we are able to write out the components of the energy-tensor of matter in this mesh-system, and as a result we obtain:

This gives us all the equations, since the six we have omitted to write out, namely,

are mere repetitions of

We have thus written out the ten equations. In them,

represents the density of the matter at the point considered;

,

,

, its component velocities at the point considered, and

,

,

, etc., the strains and stresses existing in its interior. If we are dealing with non-coherent matter, we may ignore these strains and stresses.

We see that the curvatures of space-time are connected with the various conditions of the matter. If we heat it, we increase its energy, its molecules vibrate faster, and the curvatures of space-time, hence the field of gravitation, are affected correspondingly. We may also note that whereas the density of mass of the matter at a point, namely,

, affects the curvature

, the momentum components of the matter

,

,

affect the curvatures

,

,

, and its vis viva components

,

,

affect the curvatures

;

;

. Whereas one type of curvature is produced by mass, another is produced by momentum, and still another by energy.

Of course, these mathematical equations merely express relationships; and it is impossible to deduce from them alone whether it is mass, momentum and vis viva which produce the respective space-time curvatures, or whether it is these space-time curvatures which arise in some mysterious way and are interpreted by our senses as mass, momentum and vis viva. Eddington, as we know, prefers the second attitude, whereas the majority of thinkers prefer the first.

It is to be noted that if we adopt Eddington’s views, a velocity (say,

), being equivalent to

, appears as a ratio of two of the space-time curvatures; hence the co-presence of two special types of space-time curvature would reveal itself to our consciousness as a velocity. Inasmuch as a velocity implies the passage of time, it would seem as though this mysterious passage might be connected in some way with certain of the space-time curvatures. It would be but a step to assume that our entire perceptual world might eventually be reduced to these curvatures. We do not insist on this aspect of the question, first, because we are not certain that they correspond to Eddington’s views, and, secondly, because the theory of relativity has, thus far at least, been unable to interpret electromagnetic phenomena in terms of space-time curvatures.

But there is an interesting idea that Eddington suggests in his book, “Space, Time and Gravitation.” He argues that we are now in a position to understand why it is that velocity is always relative, that is to say, is meaningless otherwise than in relationship to matter. The fact is that the curvatures of space-time, such as

, or

, connote

and

respectively, and not simply

. The presence of these curvatures connotes, therefore, matter together with velocity, and no curvature taken by itself defines bare velocity without matter.

Acceleration, on the other hand, is connected with the metrical

’s of space-time; and these, of course, subsist regardless of the presence or absence of matter, since they represent aspects of the structure of absolute space-time itself. We thus understand how it comes that acceleration manifests itself as absolute. Mach’s views lead us to ascribe the existence of these

’s of acceleration not to space-time itself, but to the totality of the matter of the universe. Under this aspect the

-distribution acquires the likeness of a species of ether conditioned by matter, which in turn conditions the behaviour of bodies in space and the evolution of phenomena in time. It should be emphasised, however, that rash philosophical conclusions on these and kindred subjects should be avoided. In the present state of our knowledge, even the most competent of scientists are far from having succeeded in solving all these arduous problems.

APPENDIX III
THE GRAVITATIONAL EQUATIONS

FROM a purely mathematical point of view, without any regard to physical applications, we may always select a co-ordinate system, or mesh-system, in a continuum regardless of whether the continuum be Euclidean or non-Euclidean. As referred to this co-ordinate system, the position of every point of the continuum is defined by its co-ordinates

, ...

. There are as many of these as the continuum has dimensions.

In addition to these co-ordinates of every point, there exist certain structural tensors (the

’s), their number being

at every point, where

represents the dimensionality of the continuum. The values both of the

’s and the

’s at a point will depend on the co-ordinate system selected; but the

’s are remarkable in that they depend not solely on the co-ordinate system, but also on the structure (curved or flat) of the continuum which they define. Once, then, a particular co-ordinate system is selected in a continuum whose structure is known from place to place, the values of the

’s in this co-ordinate system can be determined at all points of the continuum. We then find that the square of the distance between two infinitely close points of co-ordinates

and

(hence whose co-ordinates differ by

), is given by

where

represents summation for all values attributable to

and

. Note that

and

can receive all whole values from 1 to

, where

is the dimensionality of the continuum.

It is a remarkable fact that when, and only when, the continuum is flat, this expression for

can break up into a sum or difference of squares. When it breaks up into a sum of squares, the continuum is called Euclidean; when into sums and differences of squares, it is called semi-Euclidean, and is said to have positive and imaginary dimensions; but in all such cases it remains flat. (Space-time is an illustration of this latter species of continuum.)

And now suppose we wish to discover the equations of geodesics, that is to say, of the straightest of lines compatible with the structure of the continuum. We obtain a geodesic between two points

and

by expressing the fact that the line stretching between

and

is a maximum or a minimum. Mathematically, this means that by summating the value of

, expressed above, along the line joining the two points

and

, this value must be greater or less than that of all other lines stretching between these two points. The calculus of variations enables us to solve the problem. As a result, we obtain the following equations, in which the co-ordinates (

,

, ...) of each and every point of the geodesic, as referred to our co-ordinate system, are given by

These equations are written for short:

where

,

,

must be given all whole values from 1 to

;

being the dimensionality of the continuum.

It follows that, having selected a co-ordinate system, all we have to do is to determine the values of the

’s in this co-ordinate system, and then substitute these values in the expression of

and of the geodesics; and the entire geometry of the continuum is thereby established.

Thus far we have been reasoning as though the continuum possessed a structure; but in a purely amorphous continuum no such structure exists, and the mathematician must postulate it. When, however, we are dealing with physical space, we find that the laws of dispositions of material rods, as also the laws of moving bodies (law of inertia), suggest that space is three-dimensional and Euclidean. This implies that it is possible to obtain a particular co-ordinate system, called a Cartesian one, in which the

’s will have constant values equal to unity. Einstein’s and Minkowski’s great achievement was to prove that this conception of the world was incompatible with the verdict of the negative experiments. Instead it appeared necessary to assume that the fundamental continuum was four-dimensional, possessing one imaginary dimension, which was none other than time. In this four-dimensional space-time continuum, experiment proved that

had the following form when a Galilean frame of reference was resorted to:

We have reversed the sign of

, but this is of no particular importance. This formula proves that the

’s have the following values when computed in a Cartesian space-time co-ordinate system:

all other

’s vanishing. Now, while leaving the time axis unaltered, we might also have split up space, no longer with parallel planes, but with concentric spheres and with lines radiating from their centre. We then obtain a polar co-ordinate system. In this case, neglecting one of the spatial dimensions for purposes of simplicity, and calling

the distance from the centre, and

an angle, we get

so that

all other

’s vanishing. It may be noted that this change in the value of the

’s is due solely to our change of co-ordinate system; it is in no wise attributable to any change in the structure of space-time. This remains flat, as before.

Now we know that in a gravitational field space-time is no longer flat; it becomes curved, and its law of curvature outside matter is given by

. But

is what is known as a differential equation in the

’s. By this we mean that it contains in its expression the differences in the values of the

’s at neighbouring points. What we wish to discover, however, are the precise values of the

’s, and not their mere differences in value. For this purpose the differential equation must be manipulated in a certain way, or, as mathematicians would say, integrated. Only after the integration has been accomplished can solutions be obtained. Now all differential equations have a character in common in that they yield classes of solutions. In order to single out that particular solution which answers to the problem we may be considering, it will be necessary to impose certain conditions upon it. In the present case, the differential equations of the gravitational field are of the second order, which means that two separate conditions will have to be imposed.

Schwarzschild succeeded in integrating these equations for the particular case of the radially symmetrical field of gravitation produced by a mass-particle. The flatness of space-time at infinity, or, in other words, the degeneration of

to

at infinity, yielded a first condition; and the second condition was obtained by introducing into the solution the numerical value of the mass

exciting the gravitational field. In this way the solution became determinate. Schwarzschild then found that for a polar co-ordinate system the expression of

assumed the following form:

or, in other words,

all other

’s vanishing. We can see that when

, the mass of the particle, vanishes, or when

becomes infinite, we obtain the values of flat space-time; this is as it should be.

Now that we are in possession of these values of the

’s, all we have to do is to introduce them into the equations of the geodesics, and automatically we shall obtain the equations of free motion of particles moving in our system of co-ordinates under the influence of a central mass

. It is these equations which yield us the double bending of light and the motion of the planet Mercury.

We may note that if we neglect to consider the influence due to the variations in value of

as

increases, that is, if we assign to

the value of -1, which it had in flat space-time, we obtain approximately Newton’s law. It is then the deviation of

from the value -1 which is the distinguishing feature of Einstein’s law of gravitation. But it is obvious that

was not adjusted for the mere satisfaction of accounting for Mercury’s motion, since its value follows mathematically from the integration of

.

We may also add that the gravitational equations

are not linear. In ordinary language, this means that the resultant field produced by two masses cannot be obtained by superposing the two separate fields; additional cross-terms must also be taken into consideration. In the case of weak fields, however, these cross-terms may be neglected, so that in practice a superposition of separate fields is possible. Nevertheless, owing to this non-linearity of the gravitational equations, the problem of two bodies (when neither of the masses may be neglected) presents insurmountable mathematical difficulties. And so it happens that the problem of double stars, so easy to solve in Newtonian science, has as yet failed to receive a solution in the case of Einstein’s law of gravitation.

One last point is worth noting. We have seen that there were ten gravitational equations, and that there were ten

’s whose values were to be determined. Owing to the fact that there were as many equations as unknowns, it might be thought that one single set of values for the ten

’s would be obtained. But inasmuch as the values of the

’s deduced from

must necessarily differ as we choose one mesh-system or another, it would appear as though some definite mesh-system were imposed on us—a fact incompatible with the arbitrariness of mesh-system demanded by the general theory of relativity. According to relativity, there should be, therefore, a fourfold arbitrariness in the values of the

’s of space-time corresponding to the four arbitrarily chosen meshes of our space-time mesh-system. Mathematically, this would imply that four of our gravitational equations should be superfluous, hence linked to one another by relations of identity. Now we know that four such identical relations actually exist. Physically, they express the conservation principles of momentum and energy. We are thus led to the remarkable conclusion that the relativity of motion entails the existence of permanent entities in the world. In Eddington’s words:

“The argument is so general that we can even assert that corresponding to any absolute property of a volume of a world of four dimensions (in this case, curvature), there must be four relative properties which are conserved. This might be made the starting-point of a general inquiry into the necessary qualities of a permanent perceptual world, i.e., a world whose substance is conserved.”[165]

APPENDIX IV
SPACE, GRAVITATION AND SPACE-TIME

A NUMBER of philosophers have expressed the view that the special theory is too paradoxical to be accepted, but that the general theory meets with their approval. Statements of this sort are indefensible; they arise from too meagre an understanding of mechanics and geometry. Nevertheless, it may be of interest to investigate the problem more fully.

The special theory is but a particular limiting case of the general theory, one in which space-time remains flat instead of manifesting curvature. Now it is space-time which is the source of all the paradoxes of feeling (trip to the star; relativity of simultaneity, etc.). And since this new continuum is as essential to the general as to the special theory, we may be quite certain that all the paradoxes of the special theory will subsist in the generalised case, aggravated, however, by an additional element of complication produced by curvature. In short, the rejection of space-time would entail the downfall of both the general and special theories. In much the same way, if there were no omelet there might still be eggs, but if there had been no eggs in the first place there could certainly be no omelet. Indeed, we may summarise Einstein’s work by saying that in the special theory space-time was discovered, and that in the general theory its possibilities were investigated mathematically.

However, we may consider the problem under a different aspect. The main achievement of the general theory has been to account for gravitation in a purely geometrical way in terms of the curvatures of the fundamental continuum (space-time in the present case). But we may examine whether similar developments might not have been pursued had we been content to abide by the classical picture of a separate space and time. Gravitation would then be accounted for in terms of space-curvature alone, and space-time would be obviated together with the paradoxes of feeling it entails. However, we shall see that there are a number of reasons which would render success along these lines impossible. But it is not by a mere line of philosophical talk that the situation can be made clear; hence a short digression seems necessary.

When Galileo and Newton initiated their epochal discoveries in mechanics, they assumed that space was three-dimensional and Euclidean. But when we state that space is Euclidean, what is it we wish to imply? We mean that if ideally accurate measurements could be performed, the numerical results of pure Euclidean geometry would be obtained. If, therefore, on performing measurements with material rods, slight discrepancies are noted, we shall assume that owing to contingent influences of one sort or another, our rods are not true to standard.

Now it is obvious that by proclaiming space to be Euclidean a priori and then laying the blame on our physical measurements if our a priori assumption is not borne out in practice, we are placing ourselves in a position which is impregnable but useless. We are professing to know everything before we start, and hence are depriving ourselves of the aid of experiment in our study of nature.

It might therefore appear strange to find Newton, the great empiricist, following an a priori procedure which he was never weary of combating in science. But here we must remember that in Newton’s day Euclidean space was the only type known; hence it was regarded as axiomatic, not alone by Kant, but, more important still, by the most competent mathematicians of the seventeenth and eighteenth centuries. In view of this supposed inevitableness of Euclidean geometry, it was assumed that if the geometry of space was to convey any meaning at all, it could not help but be Euclidean. And so the danger of an a priori attitude towards the geometry of space failed to impress itself.

But the entire situation changed when non-Euclidean geometry was discovered. For since it was now established that various types of spaces could exist, it was impossible to anticipate a priori which one of these possible types would be realised in nature. Euclidean geometry was thus deprived of its position of inevitableness. Henceforth, our sole means of ascertaining the geometry of space was to appeal to physical measurements; and the geometry of space would be that of light rays and of material bodies maintained under constant conditions of temperature and pressure.

It is, however, permissible to question whether such physical measurements could ever teach us anything about the geometry of space; it might be claimed that they merely yielded information as to the behaviour of light rays and material bodies. This was Poincaré’s attitude. But we have seen that if the aid of physical measurements is denied us, the geometry of space escapes us completely; and there is nothing further to argue about. In view of the various types that are rationally possible, any species we might select would be in the nature of an arbitrary definition, hence would be purely conventional. It would be as useless to try and prove that our definition was incorrect as it would be to state that the system of numbers should be decimal, or binary, or duodecimal; convenience alone could guide us in such cases.

But when we analyse what is implied by the word “convenience,” we see that in the final analysis it is often based on the results of physical measurements. Thus, were all physical measurements to yield non-Euclidean results, were all the laws of nature to be expressed more simply in terms of the same non-Euclidean geometry, it would certainly be more convenient to attribute a non-Euclidean structure to space. For the same reason it is more convenient to attribute three dimensions to space, since this hypothesis permits us to account very simply for our inability to get out of a closed room without opening the door or window. The reason the word “convenient” is still adhered to in this case is because it might still be possible to account for the observed facts in terms of, say, a four-dimensional space, provided we were willing to vary our fundamental ordering relation and with it our understanding of sameness.

We may summarise these statements by saying that “reality” for the scientist reduces to the simplest co-ordination of experimental facts and that in view of the various possible types of spaces suggested by non-Euclidean geometry, physical measurements constitute our only means of determining which is the simplest, hence which is the real solution. This explains why Gauss (who as far back as 1804 had mastered in secret the essentials of non-Euclidean geometry) endeavoured to establish the geometry of space by means of light-ray triangulations. But in Newton’s day the fusion of geometry and physics was not contemplated, since Euclidean geometry was thought to constitute the only possible type. Furthermore, the greater the care in performing measurements, the nearer did results approximate to those of pure Euclidean geometry; and so it appeared reasonable enough to assume that space was Euclidean.

We have deemed it necessary to recall these fundamental notions in view of their importance in the discussions which are to follow. But for the present we shall revert to the problem as it stood in the days of Newton, when non-Euclidean geometry was unknown and space was assumed to be Euclidean. In particular, we shall examine briefly the foundations of rational mechanics.

In all problems of motion it is essential to specify the frame of reference in which the motions of bodies are to be computed since the velocities and accelerations that are measurable vary with our choice of this frame. Following Copernicus’ suggestion, scientists defined the standard frame of reference as one in which the stars appeared to be fixed, or at least non-rotating. A frame of this sort is termed inertial or Galilean.

In addition to the choice of a frame, the measurement of a motion entails measurements of both space and time. The Euclideanism of space gave significance to the equality of two spatial distances measured in our Galilean frame. To a high degree of approximation, equal spatial distances were assumed to be defined by the displacement of the same material rod maintained under standard conditions. As for measurements in time, they were assumed to be given by clocks regulated ultimately by the earth’s rotation.

The operations of measurement being thus defined physically, both Galileo and Newton considered that on the strength of the empirical evidence, it was permissible to assert that a free body in motion, unsolicited by forces, would pursue a straight course with constant speed. This statement constitutes the law of inertia. Nevertheless, the empirical methods whence Galileo and Newton had derived the law of inertia were extremely crude. Not only were more or less inaccurate physical measurements involved, but, worse still, perfectly free bodies could never be contemplated, since the earth’s gravitational effect and frictional influences could never be eliminated. It follows that the law of inertia could lay claim to no direct empirical justification.

But, on the other hand, if mechanics were to be developed along mathematical lines, it was essential that certain rigorous premises be accepted. Newton could not content himself with the statement that free bodies pursued more or less straight courses with more or less constant speeds. Hence the necessity of elevating the law of inertia to the position of a principle. Thus the principle, though originally adduced as a generalisation from experience, now assumed the position of the definition of the motion of a free body. When, therefore, a body appeared to deviate from the dictates of the principle, it was agreed that the body could not be free; just as, when rods did not yield perfectly Euclidean results, it was assumed that they could not be perfectly rigid (since, in classical science, the Euclideanism of space had been accepted as a principle).

The principle of inertia having been posited in this way, we see that it yields us an ideal definition of time, for, by definition, a perfectly free body describes equal Euclidean distances in rigorously equal intervals of time. Thanks to the principle of inertia, Newton thus gave an accurate (though ideal) definition of time.

In addition to this first principle, it was necessary to consider two others, also obtained as generalisations from experience; namely, the principle of force and acceleration, and the principle of action and reaction. These three basic principles constitute the foundations of rational mechanics.

Henceforth mechanics, originally an empirical science, becomes rationalised and is placed on a mathematical basis. It can now be developed without further appeal to experience by purely mathematical methods.[166] For this reason it is called rational mechanics. If, however, we consider the mechanics of the solar system, further empirical information must be obtained. The observational data were furnished by Tycho Brahe and Kepler. From this information Newton deduced the law of gravitation. When, therefore, the mutual force acting between bodies is of the inverse-square variety, we obtain a special branch of mathematical mechanics, called celestial mechanics.

From what has been said, we realise that the solutions of these problems of rational and celestial mechanics are of a purely mathematical nature, involving technical difficulties of their own. And so we need not be surprised to find the names of pure mathematicians on every page of the treatises consecrated to these arduous sciences. Thus, in rational mechanics we find the great names of Newton, Euler, d’Alembert, Lagrange and Hamilton, whereas in celestial mechanics we come upon the equally great names of Laplace, Jacobi and Poincaré. We have stressed these points in order to show how mechanics, originally, an empirical science, has since become mathematised.

Now, theoretically at least, it is of no interest to the pure mathematician to know whether these mathematical speculations correspond in any way to the workings of the real world; and regardless of whether Newton’s law is correct or at fault, the solution of the problem of the n bodies acting under Newton’s law of the inverse square would still present considerable interest. But, on the other hand, it would be a grave mistake to assume that these abstruse mathematical problems constitute mere intellectual pastimes, cross-word puzzles of a higher sort. Rational and celestial mechanics claim to correspond to the real workings of the world. The only way to verify the justice of this claim will be continually to check the anticipations of theory by the results of observation.

If this checking up proves harmony to prevail between theory and observation, we may assume that our fundamental principles are correct. If not, three courses are open: Either we may recognise that our fundamental principles are at fault, and attempt to replace them by others; or else we may retain our fundamental principles, and adjust theory to observation by introducing suitable disturbing influences or hypotheses ad hoc; finally, we may assume that nature is irrational, hence is not amenable to mathematical investigation.

It is impossible to give any golden rule in this respect. The hypothesis of Neptune was an illustration of the second procedure, where disturbing influences were introduced while the basic principles were maintained. The Einstein theory is an example of the first procedure, where the fundamental principles have been abandoned. As for the third procedure, it would be adopted only as a last resort, since it would be equivalent to throwing up the sponge. Nevertheless, it is far from certain that we may not have to resign ourselves to this attitude eventually, when our observations deal with phenomena on a microscopic scale.

Reverting to Newtonian science, we may say that until quite recently the fundamental principles of rational mechanics appeared to be in no danger. The most accurate astronomical observations yielded results in full conformity with theoretical previsions, when account was taken of a minimum of disturbing influences and of the inevitable inaccuracy in our knowledge of the planetary masses. We understand, therefore, whence arises the justification for the law of inertia, hence, incidentally, for the classical definitions of time and space.

It was not because stones thrown on the ice appeared to follow straight courses with constant speeds that the law of inertia was deemed to be established, for stones eventually come to a stop. It was because the numerous consequences of the law appeared to be verified indirectly to a high degree of accuracy. Hence, although it is undoubtedly correct to say that the fundamental principles of mechanics were nothing but hazardous generalisations from exceedingly crude observations, their justification appeared to have been established a posteriori, at least until quite recently.

From the principles of mechanics it is possible to deduce a certain very general principle. We refer to the principle of Least Action, alternative forms of which are given by Hamilton’s principle, and by Gauss’ principle of minimum effort. Any one of these general principles comprises all the principles of mechanics in a highly condensed form. Least action enables us to anticipate the following result: Free bodies unsolicited by forces invariably follow geodesics with constant speeds. Even when only partially free, as, for instance, when constrained to move over a fixed surface (without friction), the bodies will invariably follow the geodesics of the surface, also with constant speed.

Here a word of explanation may be necessary. The geodesics of three-dimensional Euclidean space are the familiar Euclidean straight lines; and our previous statement is but a presentation of the law of inertia. But when we consider a curved surface, the straight lines of space cannot lie on its surface. The geodesics of the curved surface are then no longer straight lines in the Euclidean sense, but they are the nearest approach to such lines, i.e., the least curved of all lines traceable on the curved surface. In the case of the sphere, geodesics are great circles. All surfaces have their geodesics more or less capriciously distorted in shape according to the variation in curvature from place to place of the surface. Again, as in the case of three-dimensional space, a geodesic is the shortest line between two points of the surface (when the distance is computed along the surface).[167]

An inextensible thread stretched between two points over a curved surface lies along a geodesic. Of course, in practice, it would be difficult to obtain adherence of the thread to the surface, were the surface concave. But this difficulty can be remedied if we consider a second surface fitting over the original one. The thread would then be stretched in the interstice between the two surfaces; and a body moving over the surface would be given by a small ball rolling without friction in the interstice between the two surfaces. In this way the ball would always remain in contact with the surface.

We may say, then, that in the absence of forces—for instance, in the interior of a falling elevator—the ball, if started with some initial speed, would describe that geodesic of the surface which lay tangentially to its initial velocity. In addition, the ball’s speed along the geodesic would remain constant (in the absence of friction).

And so, were we to discover one of the geodesics of the surface by stretching a string, and were we to mark the lay of this geodesic with a line of ink, we should find that the ball, when constrained to move without friction over the surface, would follow this line provided its initial velocity were directed tangentially to it. To all intents and purposes, geodesics act, therefore, as grooves guiding the progress of free bodies; and the free bodies follow these grooves with whatever initial speed we may have given them. All these considerations apply solely when no external forces are acting on the moving body. The presence of a force tears the body away from the geodesic it would normally follow, and also accelerates its speed.

We now come to Newton’s law of gravitation. It was a remarkable fact that although, according to the law of inertia, the motions of free bodies should be directed along the straight geodesics of three-dimensional Euclidean space, yet the courses and motions of the planets round the sun and of projectiles near the earth’s surface deviated widely from these geodesic paths. In the case of projectiles, a cause for this deviation was easy to discover. It was attributed to the disturbing action of the force of gravitation exerting itself in the earth’s neighbourhood. But in the case of the planets the discrepancy was harder to explain. Newton solved the problem by extending the scope of gravitation; namely, by assuming that a gravitational force varying inversely as the square of the distance, and directed towards the sun, was acting on the planets. He was thus able to account for the elliptical orbits of the planets, and also for their motions along these orbits, as given by Kepler’s laws.

Thus, in classical science, we were faced with straight geodesics and with forces emanating from matter and compelling bodies to depart from their normal geodesic courses when moving in the proximity of matter.

But we, who, in contradistinction to Newton, know of non-Euclidean geometry, might be tempted to offer another solution of gravitation. Viewed from the standpoint of non-Euclidean geometry, the motion of the point over the surface (discussed previously) can be interpreted as the free motion of a point in a two-dimensional non-Euclidean space (the space of the surface). Hence we may say that whether space be Euclidean or non-Euclidean, least action demands that free bodies follow its geodesics with constant speed. This holds regardless of the dimensionality of the space, whence we may conclude that were our three-dimensional space non-Euclidean instead of Euclidean, Newton’s law of inertia would have to be revised. It would no longer be correct to state that free bodies described Euclidean straight lines with constant speeds. Instead we should say: “Free bodies pursue the geodesics of space with constant speed.”

Thanks to this generalisation of the law of inertia, we are enabled to extend classical mechanics to spaces of all kinds. Only in the particular case where space is Euclidean does our generalised law of inertia coincide with that given by Newton.

From an observational standpoint, what would be the result of space manifesting itself as non-Euclidean? The geodesics would be curved, as contrasted with Euclidean ones, and free bodies unsolicited by forces would appear to follow curves with constant speeds. Under the circumstances, it would seem to be possible to account for the curved paths of planets and projectiles without having to introduce a disturbing force of gravitation. All we should do would be to assume that space was suitably curved around large masses, such as the sun and earth; and as a result the geodesics, hence the paths of free bodies, would be curved in turn. In particular, planets would now circle round the sun, not because the attraction of the sun compelled them to do so, tearing them away from their straight geodesics, but because the space around the sun being now curved, its geodesics would automatically become curved lines.

Ideas of this sort were advanced tentatively by Lobatchewski, Riemann and Clifford; but, as we shall proceed to explain, a solution along these lines was physically and mathematically impossible so long as we restricted our investigations to the separate space and time of classical science.

A first difficulty would be encountered when we contrasted an exploration of the metrics of space by the method of rod measurements, with that of free bodies following geodesics. We remember that when discussing the free motion of a ball constrained to move on a curved surface, we said that the ball would describe the shortest course between two points on the surface, in other words, a geodesic; that had we stretched a string over the surface between these two same points, the string in turn would have placed itself along the same geodesic (it was in this sense that the geodesic could be considered as defining the shortest line between the two points over the surface). In short, measurements with rods or motions of bodies yielded the same geometry, the same metrics, for the two-dimensional surface. Similar considerations would apply to a three-dimensional space.

But with our tentative theory of gravitation this accord would disappear and we should be faced with an insufferable dualism. Consider, for instance, the case of a stone thrown into the air; it describes a parabola. According to our present views, we should have to assume that this parabola represented a geodesic of three-dimensional space, and therefore the shortest distance between two points on the trajectory. But obviously, were a string to be stretched between the starting point and the point of fall of the stone, it would never coincide with the parabolic trajectory. In the same way, the distance between two points on the earth’s orbit—say, at six months’ interval—is not given by the line of the orbit. Thus the geometry of space established by freely moving bodies would be incompatible with that established by rod measurements. In the first case, intense non-Euclideanism would manifest itself, whereas, in the second case, space would appear to be Euclidean.

Under such conditions, it would be difficult to see what argument we could invoke for maintaining that the curved course of the parabola was a geodesic, hence constituted the shortest spatial distance between two points on the trajectory. In short, we should be faced with a dualism subsisting as between rod measurements and explorations with moving bodies. Inasmuch as it is the constant endeavour of science to obviate dualism wherever possible and to establish unity, we may anticipate that an interpretation of gravitation along the preceding lines would have had no chance of being accepted by scientists. Rather would they have retained Newton’s Euclidean space, which would account perfectly for the results of rod measurements and light-ray triangulations, and then appealed to an additional force of gravitation whose action would be to tear bodies away from the straight geodesics they would normally have followed.

Furthermore, there is still another aspect to the problem. If we assume that moving points describe geodesics in a non-Euclidean space which presents a high degree of non-Euclideanism, we must recognise that our material bodies (which behave Euclideanly) must diverge widely from standard when displaced. If, then, we consider the translational motion of a large non-rotating mass, all its molecules should describe geodesics. But this would be impossible, since, owing to the enormous variations in the shape of the body, as contrasted with the geometry of space, the majority of the molecules would be torn from their natural courses. Enormous forces of reaction would therefore arise, so that the body would be either torn apart, crushed or unable to move at all.

With space-time, however, all these drawbacks are removed, since no dualism arises as between measurements with rods and the courses of free bodies. In Einstein’s theory it is the geodesics of space-time, not those of space, which are followed by free bodies. This curvature of space-time is sufficient to account for the motions of projectiles and planets; and when space-time is split up into space and time, we find that over limited areas the geodesics of space remain Euclidean straight lines (owing to the enormous value of

), as in classical science. For this reason, it is fully in order that in spite of the space-time curvature, measurements with rods and stretched strings should yield Euclidean results.

Thus we see that in Einstein’s theory the metrics of space-time permits us to account both for gravitation and for the results of ordinary measurements with rods. It also enables us to co-ordinate temporal processes as exhibited in the Einstein shift-effect. In short, with the rejection of space-time, the unity of nature revealed by the relativity theory would be lost.

Let us now consider a second objection to the space-curvature hypothesis. The law of motion for falling bodies is not concerned solely with spatial courses. For instance, projectiles describe parabolas, but their motions along these trajectories are just as characteristic as are the shapes of the trajectories themselves. Again, in the case of the planets, we are telling only half the tale when we state that they describe ellipses round the sun, situated at one of the foci. It was not from an incomplete statement of this sort that Newton could ever have obtained his law of the inverse square. Kepler’s second and third laws complete the description of the planetary motions by defining the nature of the motions and velocities of the planets along their elliptical orbits. Only when all three of Kepler’s laws are taken into consideration can Newton’s law of gravitation be deduced without ambiguity. For example, if the planets pursued their elliptical orbits with constant speeds, Newton’s law would be unable to account for their motions.

Turning, then, to our tentative scheme of gravitation in terms of a curved space, we see that there is nothing in it to suggest the peculiar accelerated motions of planets and projectiles, so that we are unable to account for all the effects that Newton ascribed to gravitation. Worse still, with a theory of space-curvature the motions of bodies along their trajectories would be uniform, since geodesics are described with constant velocity. And this would be in utter conflict with the accelerated motions of falling bodies and of the planets.

We might, however, seek an avenue of escape by claiming that the curvature of space would account for the accelerated appearance of these motions, much as the apparent velocity of a body depends on its orientation with respect to our line of vision. To be sure, a curvature of space would entail a visual distortion of this sort, since the light rays emanating from a luminous source would now reach us along curved routes. As a result, we should misjudge the position of the source, much as occurs in the case of refraction. This effect has to be taken into consideration in Einstein’s cylindrical universe, for example. The distances of stars, as deduced from a measurement of their parallaxes, have to be decreased somewhat, owing to the spherical curvature of space. It follows that a star leaving us radially with uniform speed would appear to possess an increasing acceleration as it moved away from us, as though repelled from the point where we happened to be standing.[168]

But it is obvious that if separate space and time are retained, our belief in the accelerated motions of bodies falling in a gravitational field can in no wise be attributed to effects due to optical distortions varying with the relative position of the percipient. These accelerations are objective in that they exist to the same degree for all observers in the objective space and time world of science. To take a simple case, if we allow a stone to fall on our foot, its velocity of impact will vary with the height whence it was released. We cannot say that these variations in velocity are to be ascribed to a mere optical distortion of the light rays reflected from the stone. We know, indeed, that quite aside from any visual image of motion, the pain we experience will tell the tale. Added to all these considerations, it would be quite impossible to account for the precise accelerations of falling bodies and of the planets, even were we to interpret them by means of optical distortions.

And so we see that the best that could be hoped from a curvature of space alone would be an explanation of the precise paths of falling bodies, but not of their motions along these paths. Hence we should not have succeeded in interpreting gravitation, since, once more, the motion along the paths is as essential as the spatial shape of the paths. Newton’s law, on the other hand, accounts both for paths and for motions.

But we may consider another means of escape. We might assume that the spatial geodesics which free bodies follow act not only as grooves, but more like arteries which propel the blood along by their expansions and contractions. We might then assume that these pulsations of the geodesics would give the planets and projectiles those precise variations in motion which are observed. Needless to say, the hypothesis would be absurd and useless, and, so far as the writer is aware, has fortunately never been suggested. At all events, even with this hypothesis ad hoc, we should have failed to account for gravitation in terms of space-curvature, since we should have been compelled to introduce this additional influence. Furthermore, as in the absence of attracting masses free bodies are not accelerated, we should have to assume that in the absence of attracting masses these pulsations of the geodesics would cease; hence it would be the action of matter which would produce the pulsations. In other words, what we formerly called the force of gravitation would now be called the force of pulsation, and we should be thrown back on the Newtonian law of gravitation under a new name, in spite of our absurd hypothesis ad hoc.

With space-time, again, all these difficulties are obviated, for the geodesics of space-time, in contradistinction to those of space, account both for the paths of bodies and for their precise motions along these paths. This is due to the fact that a space-time geodesic deals with both time and space.

Thus far we have seen that one of the main obstacles to a space-curvature theory of gravitation resided in its inability to account for the precise motions of bodies along their orbits or trajectories. Yet, on the other hand, were it not for the other difficulties mentioned previously in this chapter, it might not have appeared impossible to account for the spatial shapes of the orbits of bodies in a gravitational field. We now propose to show that, quite independently of the reasons already given, even this partial success can never be realised. Here, however, we are compelled to consider a more mathematical aspect of the problem.

First, let us consider ordinary three-dimensional Euclidean space. If we select a point

at random, we know that there exists a triple infinity of straight lines passing through

. But if we impose the restriction that these lines shall all be perpendicular to some arbitrary direction

, the possible straight lines are decreased in number. In their aggregate they now constitute a plane, the plane passing through

and perpendicular to

.

Similar reasonings may be applied to any

-dimensional space, whether Euclidean or non-Euclidean. But if the space is non-Euclidean our straight lines, or geodesics, will be curved from the Euclidean standpoint. Hence we see that the geodesics passing through

and perpendicular at

to the direction

will now form a curved surface, no longer a plane as in Euclidean space. A surface of this sort formed by geodesics is called a geodesic surface; and the one we are considering will be the geodesic surface passing through

and perpendicular to the direction

at

. We see, then, that in a Euclidean space the geodesic surfaces are planes, just as the geodesics are straight lines; whereas in a non-Euclidean space the geodesic surfaces are curved, just as the geodesics are curves.

If, now, we assume that the direction

is pivoted round

, we shall obtain a new geodesic surface for each new orientation of

. In the case of Euclidean space these surfaces will always be planes, but in non-Euclidean space the curvatures of the various surfaces at

may vary with the orientation of

, and also with the position of the point

. Riemann then defined the curvature of the space at the point

in the direction

by the curvature of the geodesic surface passing through

and perpendicular to

at

. Thus we see that whereas a two-dimensional surface has but one curvature at every point, a three-dimensional space has various curvatures at a point, depending on the direction in which the curvature is computed. If for all orientations of

, and for all points

, the curvature of the geodesic surface remains the same, we have a space of constant curvature. When the geodesic surfaces are spheres of the same radius, hence are surfaces of the same positive curvature, we have a spherical space; when pseudospheres, hence surfaces of constant negative curvature, we have a Lobatchewskian space, and when planes, hence surfaces of constant zero curvature, we have a Euclidean space.

It remains to be said that, as Riemann discovered, these curvatures of a space at a point can be described fully only in terms of the Riemann-Christoffel tensor

at each point. For this reason the Riemann-Christoffel tensor is the only one which enables us to define the nature of the space in a complete way.

And now let us revert to the space-curvature theory of gravitation. Inasmuch as the orbits of free bodies do not present a constant curvature, we should be in the presence of a space of variable curvature from place to place. This in itself is no drawback. But here comes the difficulty: Consider a point

above the earth’s surface, and let

be directed along a vertical. Then the geodesic surface perpendicular to

at

contains all the geodesics passing through

and proceeding initially in a horizontal direction parallel to the earth’s surface. Let us consider, in particular, one of these geodesics, say the one proceeding from east to west. There exists one and only one such geodesic passing through

and starting out horizontally from east to west. According to our tentative theory of gravitation, this would imply that there existed but one path which a body could follow when projected horizontally from

in a westerly direction. But we know that this is not true, since by varying the initial velocity of a body we obtain an infinite variety of parabolas; whence we must conclude that it is impossible to identify the spatial courses of bodies shot out from

with the geodesics of a curved three-dimensional space. But we have seen that unless we are to introduce forces, least action requires that free bodies should describe geodesics. Furthermore, geodesics are the only lines that possess any absolute structural significance. If, then, bodies moving in a gravitational field do not follow them, we cannot attribute the courses of these bodies to the intrinsic geometry of the space. Once more we are compelled to appeal to a force of gravitation, just as Newton did, and our entire theory collapses. In order to succeed in an identification of the courses of bodies in a gravitational field with the geodesics of the continuum, a fourth non-spatial dimension is unavoidable. Only thus can we obtain the infinite number of geodesics necessitated by the infinite number of paths that bodies are known to follow. Here again, space-time solves all our difficulties.

Summarising, we may say that the great merit of Einstein’s general theory resides in the fact that his law of space-time curvature,

(practically the only one possible) yields us not only the motions of bodies, hence the law of gravitation, but also the metrical properties of the fundamental continuum. By this we mean that the law of space-time curvature permits us to anticipate the precise results of measurements with rods and the precise rates of temporal evolution of processes situated in various parts of the gravitational field. Thanks to the law of curvature, we know that an atom must beat faster here than there, hence that its light must be modified in colour. This is, of course, the celebrated Einstein shift-effect since observed by astronomers. The importance Einstein attributes to this phenomenon is plainly illustrated when, writing in 1918, before the shift was observed, he tells us: “If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.”

Thus we see that this fusion of geometry and physics, this linking together of mechanics and gravitation with the geometry, or metrics, of space and of time, constitutes one of the most beautiful aspects of the general theory. It emphasises a degree of unity in nature heretofore undreamt of. Were it not for this superb unity, were it not that measurements with rods and clocks were in perfect harmony with the motions of bodies, the theory would be abandoned. At all events, the passage previously quoted expresses Einstein’s personal attitude on the subject.

Finally, we may consider still another drawback of the space-curvature hypothesis. Whatever the law of curvature, it would consist in some three-dimensional tensor law—say,

, restricted to space alone. But a tensor law of this sort, while remaining covariant for any change in the orientation of our spatial frame of reference, would cease to manifest this property when referred to a frame in motion. It is admitted that this in itself is in no sense a decisive argument against the space-curvature attitude, since the covariance of natural laws for all observers, regardless of motion, is not a condition which is by any means imposed on us a priori. Classical science, for instance, did not anticipate any such covariance. Nevertheless, it must be admitted that Einstein’s theory, by permitting us to establish this irrelevance of the laws of nature to the observer’s motion, has achieved a result which, from a philosophical point of view, is compelling. And again this covariance entails space-time.

We have thus presented a number of arguments establishing the impossibility of accounting for gravitation in terms of space-curvature. With the exception of the last argument, it is safe to say that any of the preceding ones would prove the point to the satisfaction of scientists. We might have mentioned a number of other reasons, but those discussed seem sufficient. We see, then, why it was that the ideas of Riemann and Clifford were incapable of realisation in their day, when space-time was as yet unknown. And so Newton’s conception of a force emanating from matter and operating in a three-dimensional Euclidean space was in no danger of being overthrown.