CHAPTER XVIPRACTICAL CONGRUENCE IN RELATIVITY
IN a preceding chapter we mentioned certain of the most important aspects of the problem of physical space. We saw that the concept of spatial equality or congruence was deemed to have arisen from the facts of experience. Certain objects appeared to maintain the same visual aspect wherever displaced, provided we modified our own positions as observers in an appropriate way. But such fundamental recognitions were too vague to be of any use to science; hence congruent bodies were defined as those which, when maintained at constant temperature and pressure, coincided when placed side by side.
Congruence, as thus defined, involved physical measurements with material bodies; and, as Poincaré remarked, all we could ever discover in this way would reduce to the laws of configuration of solid bodies, space itself transcending our experiments, since the bodies might behave one way or another in the same space. Poincaré’s attitude drives us to complete agnosticism so far as the geometry of space is concerned. If physical measurements are denied us, there is no means of solving the problem of space, for we have noa priorimeans of deciding that the structure of space is this or that. Logical arguments are of no avail, for they do not lead us to any definite solution, only to a variety of possibilities. We are thus thrown back on Poincaré’s main contention,i.e., “space is amorphous.” In it we can define congruence in any way we please (theoretical congruence), although for reasons of practical convenience it is necessary to be guided by the properties of so-called rigid bodies. Thus we obtain a physical definition of practical congruence, which permits us to determine the geometry that for all practical purposes is to be called the geometry of real physical space.
When we consider Poincaré’s arguments, there is very little to be said against them. Nevertheless, the physicist must concentrate his attention on the space with which he can cope, with the one that can be explored by physical methods. This attitude defended by Einstein (which was also the attitude of Gauss and Riemann) leads us to the space whose geometry is defined by the paths of light rays, by the laws of nature when expressed in their simplest form, and again by the numerical results of measurement obtained with material rods maintained under constant conditions of pressure and temperature. It is with this space that we shall be concerned.
But before proceeding, let us make it quite plain that according to the present view the office of our rods is to reveal the pre-existing structure of space much as a thermometer reveals the pre-existing differences of temperature throughout the house. On no account, therefore, may it be claimed that the rods are assumed to create space any more so than the thermometer creates temperature or the weather glass creates the weather. The rods permit us to explore space because they adjust themselves or mould themselves to its structure, or metrical field. The origin of this structure is still very mysterious. For Riemann it was created by matter; but by matter he did not mean the matter of our rod: he meant the total aggregate matter of the entire universe. Einstein’s cylindrical universe confirms Riemann’s views. At all events, regardless of the mysterious problem of the ultimate origin of the metrical field, regardless of whether this structure of space existper seas a characteristic of physical space or whether it be generated by the masses of the cosmos, in either case it pre-exists to our local exploration with rods, whose masses are far too insignificant to modify it to any perceptible degree.
Also let us recall that when we discuss space, we must specify it by referring it to a frame of reference, in particular to a Galilean frame (one in which no forces of inertia are experienced). We then proceed to consider the laws of disposition of our congruent bodies in this frame, and the nature of the geometrical results obtained will define the geometry of space. As we know, Euclidean geometry is the outcome (at least to a first approximation).
Now classical science had never thought it necessary to stipulate that our rods should stand at rest in our frame, but in view of the disclosures of relativity we see that it is essential to specify this condition. For two rods whose extremities would coincide when at relative rest would cease to coincide when in relative motion, so that a square traced on the floor of our frame would turn out to be an elongated rectangle when measured by a rod in relative motion. To be sure, similar results would have been expected even in classical science, had it been assumed that the moving rod was compressed or distorted owing to its motion through the ether. In a case of this sort, the modified readings of the rod would not have involved space and its geometry, any more than when by heating or crushing a rod we alter its readings. The blame for the discrepancy would have been placed solely on the disturbing conditions affecting the rod.
Under Lorentz’s theory this interpretation could be entertained, since the FitzGerald contraction would have been explained in terms of the pressure generated by the rod’s motion through the stagnant ether. But under the theory of relativity this interpretation can no longer be defended. For when relative velocity exists betweenand(not relative acceleration), there is no sense in enquiring whether it isorwhich is in motion. This, indeed, constitutes the essence of the special principle of relativity. Hence, when we consider the moving rod, we can ascribe no absolute significance to its motion. Whatever motion exists between frame and rod might with equal justification be attributed to the motion of the frame gliding beneath a motionless rod, there being no absolute term of comparison like the stagnant Lorentzian ether to decide the issue. And so we cannot claim that the so-called moving rod has suffered a physical change by reason of its motion, a change rendering it unfit for measurement purposes.
And yet, if we retain all our rods as equally valid, whether in motion or at rest, we obtain conflicting results which are such as to deprive the geometry of space of all significance. When we analyse this anomalous situation we find that the following facts stand out clearly: Rods remain unmodified whether in motion or at rest, and yet their length varies. The truth is that length is not a definite characteristic of a body; it appears as a shadow of a something else projected into the space of our frame. When the rod is at rest in our frame, its length, or the shadow, presents a maximum; it is as though the something and its shadow coincided. But when relative motion is present, the something becomes tilted—tilted along a fourth dimension out of the space of our frame; and its shadow is shortened in proportion. Yet this rod which is moving in our frame is at rest in some other frame, and in this other frame, therefore, the something again coincides with its shadow. And so we must assume that the various Galilean frames and the spaces they serve to define are variously tilted, rotated in a fourth dimension with respect to one another.
There is then no longer one all-embracing Euclidean space which may be defined with respect to one Galilean frame or another. In its place we must conceive of an indefinite member of Euclidean spaces variously tilted, yet fused together.[61]To each separate Galilean frame one of these spaces will correspond, so that when we change our motion we are also changing our space.
The situation is analogous to that which arises when we consider various verticals drawn to the earth’s surface.
John, in London, will assert that his vertical is truly vertical and that Peter’s at New York is slanting, and Peter will return the compliment. Owing to these discrepancies, we cannot regard the two verticals as defining one same direction, though they both may be truly vertical for John and Peter respectively. Rather must we say that there exist an indefinite number of different vertical directions, no one of which is more truly vertical in any absolute sense than any other. Now replace the concept of vertical by that of three-dimensional space, and we have a picture of the spaces of relativity.
Very similar conclusions apply to time. Thus, consider two observersandin their respectively moving frames. Each observer holds a watch in his hand, and these watches, when at rest with respect to each other, beat the same time; they define congruent stretches of time. The classical idea of practical time-congruence, founded on approximate observations, led us to maintain that these conditions would still endure regardless of the positions of the clocks in space and regardless of their relative motions.
But it now appears that the time-stretches defined by the ticking of a clock at rest in the frame of referencewould appear to an observerin relative motion to be longer than they would appear to.The result is quite general and applies to all processes. Thus, the duration of the revolution of a top will be possessed of varying values according to the relative motion of the observer. Arguments similar to those we elaborated in the case of space go to show that time is no longer unique. A multiplicity of different time directions, variously tilted, must coexist, each one referring to some particular frame of reference, hence to some particular observer. As a result, the simultaneity of two events occurring at different points of space can no longer be absolute, since simultaneity depends essentially on the time which we select, and therefore on our frame of reference.
Just as each frame had its own space, so also would it now have its own time and its own definition of simultaneity. We may summarise these various discoveries by stating that there can exist no universal definition of practical congruence either for space or for time; there can exist, therefore, no geometry for a universal space or for a universal time.
The philosophical consequence of these new points of view is to deprive universal space and time of the objective significance with which they were formerly credited. Henceforth, in the words of Minkowski, “space and time, by themselves, sink to the position of mere shadows.”
Thus, consider a yardstick. If this same yardstick is measured by various observers passing it with different velocities, it will be possessed of different lengths. The space this rod occupies can then be credited with no particular magnitude, since this magnitude itself depends as much on the point of view of the observer as on the characteristics of the rod. This is what is meant by the relativity of distance. We must not confuse this discovery of Einstein’s with the fundamental relativity of all distance in conceptual space. For while classical science recognised that there was no such thing as absolute distance in conceptual space, yet it believed that in the real space of the physicist the distance between two given points in a frame of reference could be defined unambiguously for all observers, by means of rigid rods. It is this last opinion which is shattered by Einstein’s discoveries.
It is the same for duration when we remember that the duration of any phenomenon will be possessed of varying values according to the relative motion of the observer; and it is this fact which is expressed by the statement that time or duration is relative.
Thus, neither space nor time by itself can exist in the real phenomenal world which the physicist explores. Space and time appear as mere modes of perception, mere relations, varying and changing according to the conditions of relative motion existing between the observer and the observed. With the disappearance of the objectivity of space and time regarded as absolute and universal forms of our perception, the objectivity of the entire physical world seemed to sink into a mist. If, then, any kind of objectivity was to be retained, it was necessary to effect a synthesis of all the individual points of view of all the different observers, to mould into one sole representation the multiplicity of individual spaces and the multiplicity of individual durations.
Could this result be accomplished, the relativity of practical congruence for both space and time would give way to some universal definition of practical congruence holding for all observers regardless of their relative motion, and connoting, therefore, the existence of some absolute continuum no longer a mere shadow. In place of the individual point of view varying with the relative motions of the observers, we should obtain an impersonal and hence common objective understanding of nature.
The achievement of this supreme synthesis of the points of view of all observers was accomplished by Minkowski in 1908. He succeeded in obtaining an invariant definition of congruence by combining any given observer’s definitions of practical congruence for space and for time, and by showing that this combination possessed an invariant value holding equally for all other observers. Of course, the type of congruence obtained was one neither of space nor of time. It was a combination of both. But its existence showed us that, transcending space and time, there existed an impersonal and fundamental four-dimensional metrical continuum which Minkowski called space-time. The definition of congruence obtained by him for this mysterious continuum proved it to be Euclidean (more properly, semi-Euclidean).
The elusiveness of space and time, or rather the ambiguity of these concepts, depending as they do on the observer who measures and senses them, is replaced by the common objectivity of this fundamental continuum, which is the same for all and transcends the particular conditions of motion of the observer. This continuum, though of itself neither space nor time, yet pertains to both in that the observer is able to carve it up into that particular space and that particular time which are characteristic of the frame of reference in which he is stationed, and which constitute the space and time in which he lives and experiments.
From now on, the real extension of the world has a fourfold order, and the geometry of relativity is necessarily four-dimensional. What we call the space of our Galilean frame at an instant, is but a cross-section of this four-dimensional space-time world; and what we call the time of our frame lies along a perpendicular to this section. As we pass from one Galilean frame to another, we change our space and we change our time. Our new instantaneous space, though still a three-dimensional cross-section of the four-dimensional space-time world, is tilted with respect to our former cross-section; and the same tilting ensues for the perpendicular direction called time. In fact, the situation presents a striking analogy with the various directions of the verticals to the earth’s surface. They, also, manifest different directions because the various two-dimensional portions of the earth’s surface are variously tilted in three-dimensional space. The great change in our understanding of the spatio-temporal background, the great difference between space-time and the separate space and time of classical science, resides, therefore, in this tilting process which accompanies a change of relative motion.
In order to make these points clearer, let us understand that when we speak of the instantaneous spaces of two observers in relative motion as being tilted one with respect to the other, hence as lying outside of each other (except for their common two-dimensional cross-section), we do not mean that the space of one is some incomprehensible entity in the opinion of the other. The tilted instantaneous spaces will enter into our perception, but not as spaces alone: they will appear as a succession of two-dimensional surfaces moving with uniform speed.
All these spaces are equally justified. No one of them stands out more prominently than any other on the ground of symmetry with respect to the four-dimensional world. Only when we specify the frame of the observer will one particular space and one particular duration be allotted. Hence we may say that practical congruence exists for space and for time, as in classical science, provided the Galilean observer effects his measurements with rods and periodic mechanisms, such as clocks, which do not move about in his Galilean frame while performing measurements. There is thus a perfectly definite physical meaning in stating that the distances between two point-pairs at rest in the frame are congruent or unequal; but we must specify that this distance is measured according to the standards of the frame; and the same applies to time-stretches. Likewise, as in classical science, the Galilean observer will discover upon measurement with his congruent rods that his space is Euclidean. It is only when the rods and clocks are in relative motion or when, while at rest in our frame, the frame happens to be accelerated or submitted to gravitational action, that they cease to measure congruent stretches. It is only, therefore, when we reason in an impersonal way, without specifying any particular frame, only when we reason from the standpoint of all possible observers, whatever be their motion, that space and time fade away into shadows; and it is only then that we are compelled, whether we like it or not, to reason in terms of the common objective world of relativity, that is, in terms of four-dimensional space-time.