CHAPTER XVIIITHE DISCOVERY OF SPACE-TIME
THE discovery of this invariantwhose value we shall designate by,marks a date of immense importance in the history of natural philosophy. The fact is that with Einstein’s discoveries such familiar absolutes as lengths, durations and simultaneities were all found to squirm and vary in magnitude when we passed from one Galilean system to another; that is, when we changed the constant magnitude of the relative velocity existing between ourselves as observers and the events observed. On the other hand, here at last was an invariant magnitude,representing the square of the spatial distance covered by a body in any Galilean frame, minustimes the square of the duration required for this performance (the duration being measured, of course, by the standard of time of the same frame). It mattered not whether we were situated in this frame or in that one; in every case, ifhad a definite value when referred to one frame, it still maintained the same value when referred to any other frame.
Obviously, we were in the presence of something which, contrary to a distance in space or a duration in time, transcended the idiosyncrasies of our variable points of view. This was the first inkling we had in Einstein’s theory of the existence of a common absolute world underlying the relativity of physical space and time.
Minkowski immediately recognised in the mathematical form of this invariant the expression of the square of a distance in a four-dimensional continuum. This distance was termed theEinsteinian interval, or, more simply, theinterval. The invariance of all such distances implied the absolute character of the metric relations of this four-dimensional continuum, regardless of our motion, and thereby implied the absolute nature of the continuum itself. The continuum was neither space nor time, but it pertained to both, since a distance between two of its points could be split up into space and time distances in various ways, just as a distance in ordinary space can be split up into length, breadth and height, also in various ways. For these reasons it was calledSpace-Time, and the interval thus became synonymous with the distance between two points in space-time.
And here certain aspects of the problem must be noted. In the first place, it may appear strange that measurements with clocks can be co-ordinated with measurements with rods or scales. This difficulty, however, need not arrest us; for althoughis a time which can only be measured with a clock, yet,being the product of a velocity by a time, is a spatial length since it represents the spatial distance covered by light in the time.For this reason we may consider our four-dimensional continuum to possess the qualifications of an extensional space.
Our next problem is to determine the nature of the geometry of this mysterious continuum. The mere fact that it has been possible to write the expression of this square of the interval with squared magnitudes, such as,,etc., without having recourse to terms such as,,etc., would of itself imply that the continuum was flat and free from all trace of non-Euclideanism or curvature. Furthermore, it implies that the mesh-system to which we refer this distance is one of straight lines, and is not curvilinear (seeChapter VII).
The only flat continuum we have studied so far is the Euclidean one, but in the present case we see that though flat, our continuum is not strictly Euclidean owing to the minus sign preceding;or, again, owing to the fact that,instead of,as in a four-dimensional Euclidean space. Yet, because of the constancy in the values of the’s throughout the mesh-system, hence because of the flatness or absence of non-Euclideanism or curvature of the continuum, the analogy with a Euclidean continuum is very great, and for this reason it is called semi-Euclidean.[64]
We say, therefore, that space-time is a four-dimensional semi-Euclidean continuum, and that it is differentiated from a truly Euclidean four-dimensional one solely because it has one imaginary dimension and three positive ones in place of four positive ones. In this continuum, time represents the so-called imaginary dimension, and the three dimensions of space represent the three positive ones; though it must be remembered that all we wish to imply by this statement is that there exists a difference between the dimensions of space and that of time. We should be equally justified in calling time a positive dimension, and the three dimensions of space imaginary dimensions.
Now we have also seen that the mathematical form of the expression which gives the square of the distance depends not only on the geometry of the continuum, but also on the system of co-ordinates we may have adopted. In the present case the form of the mathematical expression tells us that our co-ordinate systems or mesh-systems are necessarily Cartesian, that is, are constituted by mesh-systems of four-dimensional cubes. Hence we see that all our different Galilean mesh-systems, when taken as frames of reference, correspond to differently orientated Cartesian mesh-systems in four-dimensional space-time.[65]
In short, space-time must be regarded as the fundamental continuum of the universe. Of itself it is neither space nor time. Any Galilean observer splits it up into four directions by means of his particular Cartesian mesh-system; one of these directions corresponds to his time measurements, while the three others correspond to his space measurements.
Henceforth an instantaneous event occurring at a certain point of our Galilean system and at a definite instant of our time will be represented by one definite point in space-time, a point in four-dimensional space-time being called aPoint-Event. Space-time itself appears then as a continuum of point-events, just as ordinary space was considered to be a continuum of points, and time a continuum of instants. Of course point-events, like points and instants, are mere abstractions; for a point without extension and an instant without duration are mathematical fictions introduced for purposes of definiteness; they are mere abstractions from experience.
Now just as an instantaneous event is represented by a point-event, so a prolonged event, such as a body having a prolonged existence and occupying therefore the same point or successive points in the space of our frame at successive instants of time, will be represented by a continuous line called aWorld-Line. If this line is straight, the body is at rest in our Galilean system, or else is animated with some constant motion along a straight line. If the world-line is curved, the motion of the body is accelerated or non-rectilinear in our Galilean system; hence it is accelerated or curvilinear in space (owing to the absolute character of acceleration). In this way all phenomena reducible to motions of particles are represented by absolute drawings in space-time.
According to the Galilean system of reference we may adopt, we divide space-time into space and time in one way or in another. Hence the resolution of the drawings of phenomena into space and time components is effected in various ways. We thus obtain variations in the spatio-temporal appearances of phenomena. Notwithstanding these variations, dependent on our relative motion, the absolute space-time drawings themselves, with their intersections, remain unchanged, and a direct study of the absolute world would consist in a study of these absolute drawings without reference to their varying spatio-temporal appearances.
It should be noted that inasmuch as space-time appears as an absolute continuum, existing independently of our measurements, there is no reason to limit ourselves to Cartesian mesh-systems when we wish to explore it. The mesh-system is, as we know, a mere mathematical device facilitating our investigations; and from a purely mathematical point of view we may always with equal justification split up a given continuum with one type of mesh-system or another. However, it must be remembered that in the case of space-time, when we wish to express the results of our measurements in terms of the space and time proper to our frame of reference, the mesh-system we must adopt is no longer arbitrary.
All Galilean observers, as we have seen, must split up space-time with their particular Cartesian mesh-systems. Certain curvilinear mesh-systems will correspond with more or less precision to the partitioning of space-time by accelerated or rotating observers. But in the most general case, arbitrary curvilinear mesh-systems do not correspond to the space and time partitions of any possible observer, though this fact does not detract from their utility.
So far as the essential characteristics of space-time itself are concerned, these Gaussian mesh-systems are just as legitimate as the Cartesian ones. As a matter of fact, when space-time becomes curved, owing to the presence of gravitation, Cartesian co-ordinates cannot be realised, being incompatible with the curvature of the continuum.
Yet, regardless of the mesh-system we use, certain general characteristics of our space-time drawings remain absolute. Thus, we have seen that the interval between two space-time points has a value which is invariant to a change of mesh-system. While it is true that a world-line which measures out as straight from a Cartesian system may appear curved when referred to a curvilinear or Gaussian one, yet, on the other hand, intersections or non-intersections of world-lines are absolute and are in no wise affected by our choice of a mesh-system. It is for this reason that phenomena such as coincidences are absolute, in contrast to simultaneities of events at spatially separated points. These are relative.
Thus, if two billiard balls kiss in the observation of one man, they will continue to kiss in the observation of all other men, regardless of the relative motion of these men. The kissing of the balls constitutes a coincidence, an intersection of the world-lines of the two balls; hence it is an absolute. On the other hand, if two billiard balls hit different cushions simultaneously in the observation of one man, they will not in general hit the cushions simultaneously in the observation of a man in motion with respect to the first. The reason is that our second observer will have adopted a new mesh-system, oriented differently from the first; and in this new mesh-system the two space-time point-events represented by the instantaneous impacts of the two balls against the two cushions will no longer lie necessarily at the same distance along the new time direction.
Owing to the common use of curvilinear mesh-systems in the general theory, we must recall that in a curvilinear mesh-system the square of the interval adopts the more complicated formexpressed more concisely by
The’s are no longer constants, as when we hadand,with all other’s vanishing; they now have variable values from place to place, and as for,,,these no longer necessarily represent space and time differences; they are mere mathematical Gaussian number differences. In spite of all this, the numerical value of,the square of the interval, remains the same regardless of the mesh-system selected.
Psychological duration, or aging, can also be represented in space-time. By psychological duration we mean the flow of time which the observer experiences; or, more precisely still, the duration that would be marked out by a clock held in his hand. Psychological duration is measured by the length of the space-time world-line followed by the observer. When we keep this point in mind the paradox of the travelling twin to be discussed in a later chapter loses much of its paradoxical appearance.
We can also understand how it comes that the discovery of space-time permits us to account for the duality in the nature of motion, relative when translationary and uniform, absolute when accelerated or rotationary. As long as we were endeavouring to account for this duality by attributing an appropriate structure to three-dimensional space, we were met by a peculiar difficulty: If we assumed space to have a flat structure, that is, one of planes and straight lines, it would be possible to understand why a free body, when set in motion with any definite velocity, should be guided along one of these straight lines or grooves in space. It would be following its natural course and there would be complete symmetry all around it; everything would run smoothly and no forces would arise. Then, if we attempted to tear the body away from its straight course, compelling it to follow a curve, we should be forcing it to violate the laws of the space-structure, tearing it away from the space-groove which it would normally follow. It was not impossible to assume that the groove would react and that antagonistic forces would arise. Thus the fact that curved or rotationary motion was accompanied by forces of inertia, hence manifested itself as absolute, did not constitute a difficulty of an insuperable order. But a real difficulty arose when we considered the other species of accelerated motion,i.e., variable motion along a straight line. In this case, the paths of the bodies being straight (hence in harmony with the flat structure of space), no violation of the laws of this spatial structure was evidenced. And yet forces of inertia were again present. It was difficult to interpret their appearance in terms of the space-structure.
When, however, we substitute four-dimensional space-time for separate space and time, the difficulty vanishes. For now we notice thatallaccelerated motions, regardless of whether they be rectilinear or curvilinear, are represented by curved world-lines in space-time. All these accelerated motions violate, therefore, the flat space-time structure; and for this reason forces of inertia will always be generated by them. On the other hand, when we consider Galilean or uniform or translational motions, we see that, as in the case of three-dimensional space, they will be represented by straight lines. These motions will then stand in perfect harmony with the flat space-time structure, and no forces of inertia will arise.
In this way, thanks to space-time, one of the outstanding difficulties which confronted classical science, namely, the dual nature of space and of motion, is accounted for.