Chapter V: Section V: that all attempts of this nature led to a negative result

Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galileian transformation for changing over from one body of reference to another. Now assuming that the Maxwell-Lorentz equations hold for a reference-body _K_, we then find that they do not hold for a reference-body _K′_ moving uniformly with respect to _K_, if we assume that the relations of the Galileian transformation exist between the co-ordinates of _K_ and _K′_. It thus appears that, of all Galileian co-ordinate systems, one (_K_) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding _K_ as at rest with respect to a hypothetical æther of space. On the other hand, all coordinate systems _K′_ moving relatively to _K_ were to be regarded as in motion with respect to the æther. To this motion of _K′_ against the æther (“æther-drift” relative to _K′_) were attributed the more complicated laws which were supposed to hold relative to _K′_. Strictly speaking, such an æther-drift ought also to be assumed relative to the earth, and for a long time the efforts of physicists were devoted to attempts to detect the existence of an æther-drift at the earth’s surface.

In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. A ray of light requires a perfectly definite time _T_ to pass from one mirror to the other and back again, if the whole system be at rest with respect to the æther. It is found by calculation, however, that a slightly different time _T′_ is required for this process, if the body, together with the mirrors, be moving relatively to the æther. And yet another point: it is shown by calculation that for a given velocity _v_ with reference to the æther, this time _T′_ is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have been clearly detectable. But the experiment gave a negative result—a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the æther produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section XII shows that also from the standpoint of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a “specially favoured” (unique) co-ordinate system to occasion the introduction of the æther-idea, and hence there can be no æther-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory, without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it _is_ shortened for a co-ordinate system which is at rest relatively to the sun.

XVII.

MINKOWSKI’S FOUR-DIMENSIONAL SPACE

The non-mathematician is seized by a mysterious shuddering when he hears of “four-dimensional” things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world in which we live is a four-dimensional space-time continuum.

Space is a three-dimensional continuum. By this we mean that it is possible to describe the position of a point (at rest) by means of three numbers (co-ordinates) _x, y, z_, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by co-ordinates such as _x1, y1, z1_, which may be as near as we choose to the respective values of the co-ordinates _x, y, z_, of the first point. In virtue of the latter property we speak of a “continuum,” and owing to the fact that there are three co-ordinates we speak of it as being “three-dimensional.”

Similarly, the world of physical phenomena which was briefly called “world” by Minkowski is naturally four dimensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates _x, y, z_, and a time co-ordinate, the time value _t_. The “world” is in this sense also a continuum; for to every event there are as many “neighbouring” events (realised or at least thinkable) as we care to choose, the co-ordinates _x1, y1, z1, t1_ of which differ by an indefinitely small amount from those of the event _x, y, z, t_ originally considered. That we have not been accustomed to regard the world in this sense as a four-dimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent rôle, as compared with the space coordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, _i.e._ it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the Galileian transformation (_t′_ = _t_).

The four-dimensional mode of consideration of the “world” is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transformation:

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Moreover, according to this equation the time difference Δ_t′_ of two events with respect to _K′_ does not in general vanish, even when the time difference Δ_t_ of the same events with reference to _K_ vanishes. Pure “space-distance” of two events with respect to _K_ results in “time-distance ” of the same events with respect to _K_. But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.[14] In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude

image031

proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure.

[14] Cf. the somewhat more detailed discussion in Appendix II.

These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes. Minkowski’s work is doubtless difficult of access to anyone inexperienced in mathematics, but since it is not necessary to have a very exact grasp of this work in order to understand the fundamental ideas of either the special or the general theory of relativity, I shall leave it here at present, and revert to it only towards the end of Part II.