APPENDICES

APPENDIX ITHE 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 ourorigin.Thanks to this choice of an origin, thespatial positionof 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 aspace axis; this will represent the embankment. On this space axis we mark a pointwhich 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 eventsmay 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 atime 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.

i009Fig. IX

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 fromand 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 originand 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 aspoint-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.

i010Fig X

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 instantzero. 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.

i011Fig. XI

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 merelydescribes graphicallythe 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 asand(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 eventsandare 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.

i012Fig. XII

Fig. XII

i013Fig. XIII

Fig. XIII

Next consider two stonesandlying on the embankment. Their world-lines will beand,respectively (Fig. XII), and their distance apart at any timewill be given by the lengthof the horizontal situated at the heightabove(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 embankmentpriorto zero hour; those point-events lying above the linewill represent events that occurredafterzero hour, while the point-events lying onwill give the events that occurredatzero hour.

i014Fig. XIV

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 originat 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,andthe point-event of the observer who remained atwill be,the linebeing, 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 lightis no longer equally inclined to; the world-lineof 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 eventsin 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 tapebetween 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 classicalspace-cum-timeconstituted 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 classicalspace-cum-timehad 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.

i015Fig. XV

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-eventsand(Fig. XV). We wish to find for the distance between the point-eventsandsome mathematical expression which will remain the same for all observers regardless of their motion. Now it is obvious that whereas the time distance betweenand,given by thedifference 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 ofandis,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 ofspace-cum-timeas a four-dimensional continuum of events which we callspace-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 originat a time zero, its world-line was given by a linebisecting the angle(Fig. XIV). On the other hand, for an observer travelling to the right and having a world-line,the light-world-linewould no longer bisect the angle formed by his own world-lineand 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 linewould 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 thatwould bisect the angle.In similar fashion, the observer’s time would have to be measured along his own world-line,now called(Fig. XVI).

i016Fig. XVI

Fig. XVI

Hence we conclude that the space and time directionsandare 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 onwhich 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 throughat a time zero) will be contained between the two light-world-linesand(Fig. XVII).

i017Fig. XVII

Fig. XVII

Any line passing through and not contained between these twolight-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 throughat the time zero form a cone called thelight-conewith its apex at.Straight lines passing throughand contained within the light-cone are possible world-lines, or time directions; those passing throughbut 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.

i018Fig. XVIII

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 transmissionin 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, thespace-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 theEinsteinian intervaldiscovered 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 greatimportance 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 passsimultaneously. 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?”

i019Fig. XIX

Fig. XIX

We know that the point-event of the observer, who does not move from the pointon 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 throughand which has for asymptotes the two light-world-linesand.For instance, the observer whose world-line iswill find himself at the pointwhen his clock marks zero hour plus one second, that is to say, when he has travelled along the embankment away fromduring a time of one secondas 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 fromto 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 withas its centre.

And now let us consider the world-line of a light ray leavingat 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,whereis infinitely distant. Hence we must conclude that since the pointmust be infinitely distant forto have a finite space-time value, any such space-time distance as,whereis 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 anull-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 casegives 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 passmust 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 fromto 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 beand,respectively (Fig. XX). If, now, an observer passesat 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 lengthis shorter than the lengthof 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.

i020Fig. XX

Fig. XX

i021Fig. XXI

Fig. XXI

We might, as a final example, consider the trip to the star (Fig. XXI). Ifdenotes the world-line of the star, andthe world-line of the travelling twin, we see that he will have lived a timeduring his trip. His brother remaining on earth will havefor world-line. Hence, he will have lived a time.As before, thoughappears longer than,it is, in reality, shorter, as can be understood by referring toFig. 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 anabsolute,non-reciprocaldifference 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 isentirely 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 theaxis. 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.

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