CHAPTER VII SYSTEMS OF CO-ORDINATES AND DISTANCE
WE HAVE mentioned in a general way the significance of congruence and of spatial distance. It now remains for us to find a means of defining these concepts in a rigorous mathematical form. We remember that the equality or congruence of two spatial distances between two point pairs and was an indeterminate concept, depending essentially on the behaviour of our measuring rods. An alternative presentation of non-Euclideanism (in the case of two-dimensional geometry) was then found to be afforded by assuming that the distance between points could in all cases be determined by measurements with rigid Euclidean rods; but that, whereas in the case of Euclidean geometry all the points should be considered as existing in the same plane, in the case of non-Euclidean geometry it would be as though the points were situated on a suitably curved surface. Thus, in the case of the earth, the non-Euclidean distance between two points, say New York and Paris, would be given by the Euclidean length of a great circle extending between these points, hence by a curved line following the contour of the earth’s surface. On the other hand, the Euclidean distance between these two same points would be given by the Euclidean length of the straight line joining them and passing, of course, through the earth’s interior.
We now propose to investigate the mathematical expression of distance in two dimensions; we will assume that we are discussing it from the standpoint of Euclidean measurements conducted on surfaces. Let us first consider the case where the surface is an unlimited plane. If we wish to define the position of a point of the plane, we must refer it to some system of reference. Three centuries ago Descartes devised a method whereby this result could be accomplished. He considered two families of Euclidean straight lines which we may call horizontals and verticals, respectively. The lines of these two intersecting families are equally spaced (Euclideanly speaking), so that they form a mesh-system or network of equal Euclidean squares. The scientific name for a mesh-system is co-ordinate system, but the appellation “mesh-system” introduced by Eddington has the advantage of giving a more graphic picture of what is involved. The type of mesh-system constituted by horizontals and verticals introduced by Descartes is called a Cartesian co-ordinate system.
If to each vertical and to each horizontal of the mesh-system we assign consecutive whole numbers from zero on indefinitely, we see that the point of the plane which happens to coincide with the intersection of some particular horizontal and some particular vertical is defined by the numbers which represent the two lines, respectively. In this way [Pg 84] every point of intersection is defined by two numbers, and these numbers are called the Cartesian co-ordinates of the point. Of course, by this method we are unable to define the positions of points which do not happen to coincide with the corners of our squares. But there is nothing to prevent us from assuming that between the verticals and horizontals we have mentioned there lie an indefinite number of other similar lines, to which intermediary fractional numbers will be assigned. Henceforth every point of the plane can be regarded as defined by the intersection of some particular horizontal and some particular vertical.
To what extent is it permissible to say that points on the plane have been defined by this method? If we disregard the existence of the co-ordinate system, nothing has been defined, but if we consider the co-ordinate system as given, then every point of the plane can be considered as defined unambiguously. In short, the points are defined not in the abstract, but in relation to the co-ordinate system. There is nothing mysterious about this method of defining the positions of points. Thus, in everyday life, when we agree to meet a friend at the corner of Fifth Avenue and Forty-second Street, we are inadvertently locating our point of meeting in terms of the Cartesian co-ordinate system defined by the avenues and streets. In the present case the co-ordinate system is not strictly Cartesian, since the streets and avenues may not enclose perfectly equal Euclideanly square blocks, but the general principle involved is the same. Needless to say, the definition of our point of meeting would convey no significance were the avenues and streets non-existent. Hence once again we see that it is only relative to the co-ordinate system that points can be defined.
Now, the essential characteristic of the Cartesian procedure is its use of a system of reference represented by separate families of intersecting lines. The fact that the lines we have considered are mutually perpendicular straight lines forming a network of Euclidean squares is of no particular importance. It would be just as feasible, in place of our horizontals and verticals, to select two families of intersecting curves, which we might call the and curves. Of course, our mesh system would now be curvilinear and the spaces enclosed by the meshes would no longer be Euclidean squares, nor even necessarily equal in area. This generalization of Descartes’ method was introduced by Gauss, and for this reason curvilinear mesh-systems are also called Gaussian mesh-systems. As before, every point will be defined by the numbers designating the two curves of either family, that is, by the numbers designating the curve and the curve which intersect at this point; and these two numbers will be called the Gaussian numbers or co-ordinates of the point.
The necessity of generalising Cartesian co-ordinates by introducing Gaussian ones arises from the fact that Cartesian mesh-systems of equal squares can be traced only on a plane and could never be drawn on a [Pg 85] curved surface, like that of a sphere, for example. Hence, were we to ignore the use of Gaussian mesh-systems, it would be impossible for us to localise points on a curved surface, by means of a mesh-system applied on the surface. The nearest approach to a network of squares on the surface of a sphere would be a network of meridians and parallels, and such a network is not one of equal Euclidean squares; it is a curvilinear or Gaussian mesh-system tapering to points at the North and South Poles.
As a matter of fact, we also make use of Gaussian co-ordinates in everyday life. Such is the case when we state that the position of a ship is so many degrees of latitude and so many degrees of longitude; our mesh-system, being one of meridians and parallels, is a Gaussian one, and the latitude and longitude of the ship constitute its Gaussian co-ordinates.
Having determined how the positions of points may be defined on any surface in terms of some mesh-system, let us now see how it is proposed to express the distance between two points, as measured over the surface with rigid Euclidean rods. Here we must proceed with the utmost caution. Let us recall exactly what is involved. Any definite line is one along which remains constant while varies continuously, and inversely any definite line is one along which remains constant while varies continuously. In much the same way, on the earth’s surface a latitude line or parallel is one along which the latitude remains constant and the longitude varies, whereas a longitude line or meridian is one along which the reverse is true.
Fig. II
Suppose, then, we wish to express the distance between two points and , defined by the intersections of a certain line with two successive lines (Fig. II). The co-ordinates of the two points are () and (), respectively, where represents the increase in the value of as we pass from to on the line.
On no account may we say that the distance (which we shall call ) is given by the value , for is nothing but a difference between numbers serving to localise points; it has nothing in common with a distance. In order to dispel any doubts on this score, we may notice that if we compare the lines to parallels and the lines to meridians, the points and will be given by the intersections of two consecutive meridians [Pg 86] with the same parallel. In this case would correspond to the difference in longitude between the two points; and obviously a difference in longitude is no criterion of a distance, since for the same difference in longitude the distance decreases from equator to pole. In short, the distance between and cannot be fully determined by .
Yet, on the other hand, just as the distance between two points lying on the same parallel is affected by their difference in longitude, so now the distance between and must be a function of the co-ordinate difference . A sufficiently general expression of a quantity when defined in terms of another quantity on which it depends is given by where , , are appropriate magnitudes. If, however, we consider points exceedingly close together, becomes exceedingly small in value, and as a result and , etc., are many times smaller still. At the limit, therefore, when the two points are at an infinitesimal distance apart, the higher powers of become so insignificant that they can be neglected in comparison with . We thus obtain , and in order to specify that we are considering infinitesimal distances we replace the symbol by and obtain It is customary to designate by so that, using squares, our formula becomes
[Pg 87]
And now we have to consider an important question. What is ? What does it represent? We do not propose to enter into its full mathematical significance, but we may mention certain of its important characteristics. In our illustration of meridians and parallels, let us consider the various points of intersection defined by the intersections of the successive parallels with two fixed meridians. The difference in longitude between these various point pairs will, of course, be constant and will be given by the same invariable quantity . Inasmuch as the distance between these point pairs varies from pole to equator, we see from that must vary in an appropriate way as we consider various portions of the sphere’s surface. In fact, owing to the constancy of , we must have proportional to . If, then, the meridians were parallel lines instead of tapering together towards the poles, would remain constant. Whence it follows that a knowledge of the way varies from point to point yields us information on the shape of the mesh-system, and inversely the lay of the mesh-system yields us information as to the value of from place to place.
Then again, if, leaving our two points and fixed, we select some new mesh-system, may change in value, and in fact a difference may also appear, since with a change of mesh-system there is no reason why and should still lie on the same curve. Hence we must conclude that even at a fixed point of our surface the value of will be subject to change when we vary our mesh-system.
Precisely the same arguments would apply were we to consider the expression of the distance between two infinitely close points on the same curve. We should then obtain , where is a magnitude generally differing from .
Fig. III
We must now consider the more general case where our two points lie on different and curves (Fig. III). From what precedes we know that the squared distance is given by and that the squared distance is expressed by . But this information is obviously insufficient to tell us what the squared distance will be, since this distance will also be affected by the slant of the lines and . Some new magnitude will obviously have to be introduced in order to specify the value of this slant; and the new magnitude considered will be called and . Under the circumstances, the squared distance of the infinitesimally near points and can be written but as and are found to be always identical we have We may write this formula more concisely by referring to as and to as . In this case our formula becomes where indicates summation and where for and we substitute the values 1 and 2 in all possible ways. This most important mathematical expression was discovered by Gauss; it is destined [Pg 88] to play a part of paramount significance in the physical theory of relativity.
In order to understand the geometrical meaning of , let us consider the special case of a diamond-shaped mesh-system, where the co-ordinate lines make an angle (Fig. IV). Elementary geometry teaches us that
Fig. IV
The lines being always equally spaced, and remain constant throughout the mesh-system; hence we may put them equal to unity and write In this formula we have from which we see that represents the cosine of the angle formed by the and lines, at the point where is calculated.[26] Inasmuch as , we see that where the and lines are perpendicular, vanishes. Hence, whenever we have an expression of , such as from which is absent, we may be certain that the and lines are orthogonal at the point for which has been calculated.
If the and lines remain perpendicular throughout the entire mesh-system, will vanish not only for one particular point, but for all points of the surface.
Let us now consider the particular case of a Cartesian mesh-system of equal squares. In this case, the and lines being orthogonal [Pg 89] and equally spaced, we have and constant, with vanishing. Hence we may replace and by unity, and we obtain a result in complete agreement with the Pythagorean theorem of the square of the hypothenuse.
Fig. V
Suppose, now, that on this same plane we were to trace another type of co-ordinate system, one known as a polar system (Fig. V). It is constituted by symmetrically disposed v lines radiating from a central point . The lines are then given by equally distanced concentric circles having as centre. Under the circumstances the expression for becomes In other words, we have proving once again that the values of the ’s vary with our choice of a co-ordinate system. As before, vanishes for the same reasons as previously stated, namely, because the lines of our mesh-system are orthogonal at their intersections.
Finally, we may consider the mesh-system defined by the parallels and meridians on a sphere, where we may assume that gives a parallel and gives a meridian. Here we have so that
From these various examples the following conclusions may be drawn:
1°. When a definite mesh-system has been traced on a surface and its lines numbered consecutively, every point of the surface is defined unambiguously by its two numbers (its two Gaussian numbers or co-ordinates). These numbers represent, of course, those of the [Pg 90] respective and curves at whose intersection the point stands. A variation in the shape or in the numbering of the and lines entails a change in the co-ordinate numbers of any given point on the surface.
2°. Alongside of these two Gaussian numbers at every point defining the positions of points in the mesh-system, there exist four numbers, at every point, namely , , , . But as and are always identical, we have only three of these magnitudes to consider. The values of these ’s at a point vary when we change our mesh-system. In the majority of cases they also vary from place to place throughout the same mesh-system. If, however, the meshes are always orthogonal at their points of intersection, vanishes at every point of the surface. If the and lines always have the same slant over the surface and are always equally spaced, the three ’s will remain constant throughout; their values being given by 1, , 1. If the mesh-system is Cartesian, hence forms equal squares , of course, vanishes while, as before, and remain constantly equal to unity.
3°. The value of the square of the distance between two infinitesimally distant points is given by where the values of the ’s and of and will vary when the mesh-system is changed or when we consider different regions of the same mesh-system. The only case in which the ’s will remain constant is when the mesh-system is of the uniform type, that is, a network of two families of parallel lines intersecting one another.
It is a remarkable fact that although everything entering into the expression of varies when we change our mesh-system, yet the value of as defining the value of the square of the distance between two infinitesimally distant points remains unchanged. In other words, is a scalar, an invariant. This allows us to place a new interpretation on the numbers; they appear to act as correctives counterbalancing the variations of and . If we compare the variations in the values of and when the mesh-system is changed, to the advance of a squirrel in a drum, we see that the action of the ’s is similar to a backward revolution of the drum, offsetting the squirrel’s advance, so that the squirrel remains motionless in space.[27]
[Pg 91]
Now, up to this point we have been considering the expression of a distance between infinitely close points, but in practice we also wish to establish the length of an extended curve traced on the surface. In this case we proceed from point to point along the curve, computing the successive infinitesimal distances , then summating them, or, as it would be more proper to say, integrating them. We thus obtain . Also we may state that the area of an infinitesimal parallelogram formed by two-line elements and is given by Here again we may calculate any finite area by a process of integration, so we see that the finite geometry of the surface can be studied by concentrating our attention on infinitesimal portions and then extending our results from place to place. In short, the method reduces to an application of the differential calculus to geometrical problems, and for that reason is named differential geometry.
Powerful as this method of differential geometry has proved to be, there are cases in which it cannot be applied. However, as in the problems of physics with which we shall be concerned, difficulties do not arise, we need not dwell on a number of special cases which in the present state of our knowledge are of interest only to the mathematician.[28]
And now we come to the main body of Gauss’ discoveries. We have seen that on a given surface the values of the three ’s at any point, or, more correctly, their variations in value from point to point, are defined by our choice of a mesh-system. But we know that a mesh-system, though in large measure arbitrary, is yet not completely independent of the nature of the underlying surface. For instance, a Cartesian mesh-system of equal squares, or again a diamond-shaped one, both of which hold on a plane, cannot be traced on a sphere. Neither can a network of meridians and parallels which holds on a sphere be traced [Pg 92] on a plane. For this reason the representation of the disposition of oceans and continents is necessarily distorted in some way or other when given on a flat map.
In short, every species of surface possesses an infinite aggregate of possible mesh-systems, but those systems which are applicable to one type of surface are never applicable to surfaces of any other type. Inasmuch as the Cartesian mesh-system and the diamond-shaped variety are the only ones that entail the constancy of the three ’s throughout the surface, and inasmuch as such co-ordinate systems can be traced only on a plane or on surfaces derived therefrom without stretching (cylinder, cone), we see that the constancy of the three ’s is characteristic of Euclideanism. This does not mean, of course, that all co-ordinate systems traced on a plane yield constant values for the three ’s; it simply means that on a plane it is always possible to trace a Cartesian mesh-system, whereas on all other types of surfaces the task is impossible.
All this goes to prove that the curvature of a surface must exert a modifying influence on the -distribution, since when the surface is curved no constant distribution is possible. We must infer, therefore, that the -distribution is governed by two separate influences; first, by the lay of the mesh-system over the surface; secondly, by the intrinsic curvature of the surface from place to place. Gauss realised the importance of separating these two influences and of determining in what measure respectively they affected the -distribution.
Obviously, if it were possible to discover some mathematical expression connecting the ’s at one point with the ’s at neighbouring points, and if this mathematical expression remained invariant in value to a change of mesh-system in spite of the variations of the individual ’s which must accompany the change of mesh-system, we should be in the presence of a magnitude which, transcending our choice of a mesh-system, would refer solely to the shape of the surface itself, i.e., to its curvature at the point considered. But before we investigate the nature of Gauss’ discoveries, certain elementary notions must be recalled.
We know that the curvature of a circle at every point of its circumference is a constant given by , where is the radius of the circle. But if in place of a circle we trace any arbitrary curve on our plane, the curvature will vary from point to point along the curve. The curvature of the curve at given point is then defined by the a certain circle which is tangent to the curve at the point . As a matter of fact there exist an indefinite number of circles of varying radii lying tangent to the curve at ; but among these circles one stands out prominently in that it is, so to speak, more perfectly tangent than all the others. Whereas the tangent circles intersect the curve in two points coinciding at , this privileged circle intersects it in three such points. It is called [Pg 93] the osculating circle (osculare meaning to kiss in Latin). The curvature of the curve at is defined by the curvature of its osculating circle at . Calling the radius of this osculating circle, the curvature of the curve at is thus given by (Fig. VI).
Fig. VI
We must now pass to the curvature of a surface. At a point on the surface where the curvature is to be computed we trace a normal to the surface. Then through this normal we trace a plane, which of course intersects the surface along a plane curve. We assume this normal plane to revolve round the normal as axis and we thus obtain a series of plane curves of intersection defined by the normal plane and surface. Each one of these curves passing through the point possesses a definite curvature at , and this curvature can be computed through the medium of the corresponding osculating circle.
Here a geometrical fact is evidenced. It is found that in the general case there exist two remarkable positions of the intersecting normal plane, perpendicular to one another and therefore sectioning the surface along two curves (1) and (2) orthogonal to each other at . The curvature of one of these curves, say, the curve (1), is less than that of all other curves obtained by revolving the plane round the normal at , while the curvature of the second curve (2) is greater than all others. These two curvatures are called the two principal curvatures of the surface at the point and are designated by and respectively.
We then define the total curvature or the Gaussian curvature of the surface at the point by the product . If our surface is a sphere we have , hence the Gaussian curvature becomes or more simply , where is the radius of our sphere; but in the general case, and are unequal. According to [Pg 94] the nature of the surface the two principal curvatures may be of the same or of opposite signs. When of the same sign, the total curvature is obviously positive; hence the surface is said to manifest positive curvature at the point considered. The sphere and ellipsoid are illustrations of surfaces presenting a positive curvature throughout. When, however, the surface is saddle-shaped, the two principal curvatures are of opposite sign; the total Gaussian curvature is then negative, and we have a surface of negative curvature at the point considered.
And now let us return to Gauss’ discoveries. We saw that the distribution of the ’s over the surface was affected both by our choice of a mesh-system traced on the surface and by the intrinsic nature or curvature of the surface from point to point. Then we also mentioned that if it were possible to discover some mathematical expression connecting the ’s at a point with the ’s at neighbouring points, and that if this mathematical expression remained invariant in value to a change of mesh-system in spite of the variations of the individual ’s which accompany the change of mesh-system, we should be in the presence of a magnitude which, transcending our choice of mesh-system, would refer solely to the shape of the surface itself, i.e., to its curvature at the point considered. This important mathematical invariant, built up with the ’s, was discovered by Gauss; it is generally designated by the letter and is referred to as the scalar or invariant of curvature at the point . Gauss then proved that this scalar of curvature was none other than minus twice the total curvature defined previously. Hence we may write: [Pg 95] Aside from a constant factor, these two curvatures are thus the same, so that we shall often refer to as the Gaussian curvature, even though the appellation is not strictly accurate.[29]
Before proceeding farther, we must recall that the method we have followed of investigating the geometry of surfaces and using Euclidean rigid rods for the purpose of conducting measurements over the surface, leads to the same geometrical results as would be obtained by an exploration of a two-dimensional space (of a plane, for example) by means of non-Euclidean rods. If, therefore, we conducted measurements with non-Euclidean measurements over a plane, we should of course obtain a non-Euclidean geometry; and the Gaussian curvature of the plane would no longer vanish, as it would were we to make use of the Euclidean measuring rods.[30]
In terms of our non-Euclidean measurements the same plane would be curved. We see, therefore, that the non-vanishing of the Gaussian curvature does not necessarily represent curvature in the usual visualising sense. It represents more truly a relationship between the surface and the behaviour of our measuring rods; in other words, it represents non-Euclideanism, and the word “curvature” is apt to be misleading. In a general way, therefore, we may state that the type of geometry of our two-dimensional space from place to place is defined by the value of the Gaussian curvature from point to point, hence by the law of -distribution throughout the space; and that when the space is Euclidean, the -distribution is always such that the Gaussian curvature vanishes at all points, regardless of the particular mesh-system selected.
Now all these discoveries of Gauss relating to a two-dimensional space were extended by Riemann to spaces of any number of dimensions. Riemann found that for spaces of more than two dimensions the results became very much more complex. In the case of a space of dimensions we of course require Gaussian co-ordinates to define the position of a point in our mesh-system, which now becomes -dimensional. As before, we can conceive of Cartesian and Gaussian mesh-systems, [Pg 96] the former being a generalisation to dimensions of our network of Euclidean squares. As for the invariant mathematical expression of the square of a distance between two points, it now contains a greater number of terms. In the case of three-dimensional space we have no longer three separate quantities at every point; this number is increased to six. In the case of a four-dimensional space it is increased to ten, and in the case of an -dimensional space to .
Owing to the importance of a four-dimensional extension in the theory of relativity, we shall write out the expression of the square of the distance for a four-dimensional space. If we call , , , the four differences between the four Gaussian co-ordinates of our two infinitely close points and and designate the ten numbers at every point by , , etc., we have or, more concisely, where we give to and all whole values from 1 to 4, permuting them in all possible ways. Accordingly, in what follows, we shall refer to the ’s as the ’s.
Proceeding by the same method as we did in the case of two dimensions, we find that there now exist several invariant types of relations between the ’s at every definite point and their variations in value at all points near this one. The values of these expressions remain unchanged when we alter our mesh-system; hence, as in the case of the Gaussian curvature for two-dimensional space, they refer to the geometry of the space itself and not to our choice of mesh-system. One of these invariant expressions is the generalisation of the Gaussian curvature extended to four dimensions. The others refer to new types of curvatures which appear only in spaces of more than two dimensions. It is impossible to represent these curvatures with a two-dimensional surface because they then become identified with the ordinary Gaussian curvature.[31] For this reason it was only when curved spaces of more than two dimensions were studied that these new forms of curvature were brought into prominence.
[Pg 97]
From what has been said we may anticipate marked complications when we wish to study the geometry of a space of more than two dimensions. Thus, we saw that in a two-dimensional space the Gaussian curvature fully defines the geometry of the space. For instance, if the Gaussian curvature vanishes throughout, the surface is Euclidean or at least flat.[32] In the same way, if the Gaussian curvature is an invariable positive or negative number throughout, the surface is one of constant curvature, either positive, as with a sphere, or negative, as with a pseudosphere.
But when we consider spaces of more than two dimensions, a knowledge of the generalised Gaussian curvature throughout the space is no longer sufficient to fix its geometry. While this curvature may vanish or present the same non-vanishing value throughout, we cannot infer therefrom that the space is necessarily flat or of constant curvature. Thus, whereas the vanishing of the generalised Gaussian curvature is a necessary condition for the space to be flat, it is by no means sufficient. There is still room for a large measure of indeterminateness in the actual geometry of the space.
For the geometry of a space of more than two dimensions to be determined fully, we must state the values of the components of the tensor (mentioned in the note); there being twenty such components at every point for a four-dimensional space. We may note that, contrary to the Gaussian curvature , is not invariant. It splits up into components each one of which may vary in value when the mesh-system is changed. Yet, in spite of this variability, still defines the curvatures of the space, hence still refers to something that is intrinsic and irrelevant to our choice of mesh-system. Such is one of the characteristics of a tensor. This particular tensor of twenty components (in a four-dimensional space) is known as the Riemann-Christoffel tensor.[33] Curiously enough, it was discovered by Riemann, not when investigating the geometry of space, but when considering a problem in heat.
Only when every one of these twenty components of the tensor vanishes at every point is it possible to assert that the four-dimensional space is Euclidean, or at least flat. Thus, whereas for two dimensions Euclideanism was ensured when one condition was satisfied at every point, namely, the vanishing of the Gaussian curvature, , on the other hand, when we step up to four dimensions, twenty conditions are needed, and these are given by the vanishing at every point of the twenty expressions which in their aggregate constitute the Riemann-Christoffel tensor .
We may also mention that in addition to the Riemann-Christoffel tensor , Einstein has made use of the other tensor (previously discovered by Ricci), which in four-dimensional space is defined by ten separate relations between the values of the [Pg 98] ’s at a point and their values at neighbouring points. The vanishing of this other tensor at every point, that is to say, the vanishing of the ten new component expressions at every point, is insufficient in itself to ensure the Euclideanism of the space; although, of course, this vanishing imposes certain restrictions on the space’s non-Euclideanism. These restrictions are less severe than those defined by the vanishing of the twenty components of the Riemann-Christoffel tensor, which ensures perfect flatness; on the other hand, they are more stringent than those imposed by a mere vanishing of the Gaussian curvature .[34]
Turning to the physical significance of the numbers, we may note that it is these magnitudes which enter into the expression of a distance, hence which serve to define congruence. If we are dealing with a purely amorphous continuum in which no particular definition of congruence or distance is pressed upon us by nature, there can exist no particular distribution of numbers inherent in the continuum. But if, on the other hand, as in the case of space, or, better still, of Einstein’s four-dimensional space-time, a precise definition of practical congruence is imposed upon us by nature—if, in other words, our continuum, in place of being amorphous, possesses a definite metrics or structure—there will exist a -distribution inherent in the continuum, and this distribution will define its metrical field.
The same arguments would apply to the structural tensors of curvature of the continuum, namely, to , to and to . If the continuum is amorphous, these structural tensors, together with the ’s that compose them, are meaningless until we have decided upon some theoretical type of congruence of geometry; and even then they represent characteristics of structure that are as conventional as the geometry we have selected. But if, on the other hand, a definite metrics exists in the continuum, then these structural tensors represent curvatures of the continuum, which we may regard as physically existent and no longer as purely conventional.
[Pg 99]
CHAPTER VIII THE MEANING OF THE WORD RELATIVITY
ALL our previous discussions and explanations have been made in order to prepare the ground for Einstein's theory of relativity. However, before we proceed to discuss Einstein’s theory proper, it may be useful to reach a more precise understanding of the scientific meaning of the words “relativity” and “absoluteness.”
In traditional philosophy an absolute reality is the primary, self-existing reality, the substance, the true being or the metaphysical reality, whereas a relative reality is a secondary, dependent reality owing its form of being to a relation of the genuinely real.[35]
Now, so far as scientific philosophy is concerned, absolute reality is a myth. All we can ever become cognisant of in nature reduces in the final analysis to structure, that is, to relationships. Hence, if we adopt this very general understanding of the word “relativity,” we may anticipate even at this stage that Einstein’s theory can add nothing to our belief in the fundamental relativity of all knowledge. Obviously, then, when physicists speak of absolutes and relatives, they are referring to categories of a nature differing from those of philosophers.
Consider, for instance, the length of an object, or again the comparison of two lengths situated in different parts of space. We have seen, when discussing mathematical space, that according to our measuring standards, which were conventional, these lengths might be equal or unequal; and it was this aspect of relativity which expressed the fundamental mathematical relativity of length. But we also saw that although in theory the definition of congruence was conventional, in practice, if we wished to define congruence so as not to enter into conflict with our sense perceptions, a definite type of congruence (practical congruence) was imposed upon us by nature. We saw further that bodies which were thereby defined as congruent constituted what are known as Euclidean solids, and that these were assumed to remain absolute in length, regardless of the observer’s motion.
Under these circumstances, when a physicist such as Einstein, in contradistinction to a mathematician, asserts that length is relative, he is referring to the actual length measured out by a so-called rigid body; and he wishes to imply that this length will turn out to be indeterminate and to depend essentially on our conditions of observation. We see, therefore, that the relativity of the physicist is not inevitable, as is that of the mathematician. It is not an a priori type of relativity expressing a trivial relativity to [Pg 100] standard. It is essentially an empirical type of relativity, always subject to proof or disproof by precise empirical tests. The two following illustrations may be helpful:
Consider a telegraph pole rising vertically from the ground. The visual angle under which this pole will appear to us is essentially relative, since it varies in value with the distance of the observer from the pole. Of course, this visual angle is fundamentally relative, since its value depends on the system of geometry we may adopt, but we are no longer discussing this mathematical aspect of relativity. We are assuming that our system of measurements has been fixed according to the requirements of practical congruence, i.e., that our measurements are performed with ordinary rigid rods and hence are Euclidean. In this event the visual angle under which we perceive a given pole is perfectly determinate when our distance from the pole is specified. But, on the other hand, it is relative, inasmuch as it is not immanent in the pole; it varies with our relative distance and thereby reduces to a mere relationship between the pole and our distance therefrom.
In contradistinction to the relativity of the visual angle, classical science considered that the length of the pole itself was absolute, being irrelevant to our relative position as observers. Once again, if we argue as mathematicians, we may claim that this length in turn is relative, since it depends on our measuring conventions. But if we assume that our measuring conventions have been fixed by the requirements of practical congruence, this type of relativity is excluded. Such being the case, classical science believed that the length of the pole was an absolute, inasmuch as its numerical value was so many feet or so many metres regardless of our position and regardless of our motion. It is this last assumption that is contested by Einstein. The absolute magnitude is not the length of the pole; for this, analogously to visual angle, is relative. The absolute magnitude is rather a something which transcends space and time, and relates to the absolute world of space-time.
A second illustration may be given by the concepts of mass and weight. As a result of Newton’s discoveries, the weight of a body was realised to be relative, since its value varied with the proximity and distribution of matter. On the earth’s surface the value was so many tons, on the surface of the moon it was considerably reduced, and in interstellar space it vanished completely. A body of itself had no weight. Weight was not immanent in the body; it reduced to the mere expression of a physical relationship between the position of the body and that of surrounding matter.
In contradistinction to the relativity of weight, classical science assumed that mass was absolute. The mass of a body, wherever situated and in whatever circumstances observed, was supposed to be an invariant remaining always the same. For this reason, mass, in contrast to weight, was assumed to be immanent in matter. It was not the result of a physical relationship, but was a reflection of the very existence of matter.
True, when Bucherer’s experiments had shown that the mass of an electron appeared to increase with its velocity, mass unquestionably [Pg 101] appeared to be relative. But it was assumed that this apparent increase in the mass of the electron was to be ascribed not to any variation in the mass of the electron itself, but to a deformation of its electromagnetic field brought about by motion through the ether. It was as though the ether was dragged along by the electron in its motion, just as a ship moving through the ocean carries along a more or less considerable volume of surrounding water according to the speed with which it is moving. In other words, there had been no real change in the mass of the electron itself, and this belief of classical science was expressed by the principle of the conservation of mass.
The preceding examples teach us two things: First of all, it is not sufficient to state that a concept is a relative; we must complete our statement by stipulating with respect to what surrounding conditions this concept happens to be relative: whether it be, e.g., to the distance of the observer, to his motion, or to the distribution of matter. Secondly, we see that the only experimental means we have of distinguishing a relative quantity from an absolute is by varying surrounding conditions and by modifying the nature of the motion and the position of the observer. Were the value of the quantity to vary under these conditions, we should be assured that it was a relative. But obviously it may be impossible for us to vary all surrounding conditions; we cannot vary, for instance, the star distribution. Inasmuch as in this universe the whole influences the part and the part influences the whole, it would appear that our differentiations between absolutes and relatives might reduce to a mere expression of our limitations.
This view is undoubtedly correct in theory, but it must be remembered that physics can proceed only by successive approximations. For example, experiment proves that the action of a certain phenomenon on another decreases with the distance. The physicist is therefore perfectly entitled to assume that if the disturbing cause is far enough removed its influence can be neglected. In this way, by a process of progressive elimination, he is enabled to conceive phenomena to proceed in complete isolation from all foreign disturbing influences. Henceforth it will be these ideally isolated phenomena which he will subject to the tests of relativity by submitting them in succession to those foreign influences over which he can exercise some control. Under these conditions the question of the absoluteness or the relativity of a given concept or magnitude acquires a definite physical meaning.
In classical science it was always assumed that a distance in space, a duration in time and a simultaneity between distant events were absolute concepts. Regardless of the relative motion or position of the observer, regardless of the distribution of matter, these concepts remained unchanged. Einstein succeeded in proving that these opinions must be erroneous, since they were incompatible with certain refined optical experiments. He showed that the relative motion of the observer could not help but exert a modifying influence. Henceforth, distance, duration and simultaneity become relatives expressing relationships [Pg 102] between the magnitudes measured and the relative motion of the observer. Just as the visual angle under which an object appeared to an observer was in no wise immanent in the object itself, since it varied with the observer’s position, so now length, duration and simultaneity are in no wise immanent in the real world, since they vary with the observer’s motion.
Once more we must draw attention to the fact that the relativity we are discussing is essentially physical; it is posterior to our definitions of practical congruence. It follows that this relativity of distance and duration must correspond to variations which would be seen and felt. These variations are not due to mere arbitrary changes which we may introduce into our mathematical systems of measurement. They are not merely conceptual; they are perceptual.
The fact is that the systems of measurement are imposed upon us by nature, and by that we mean that they translate what we actually see and feel and not that which we might see and feel if the world happened to be such as some nebulous mathematician might posit it to be. In a similar way, Einstein in the special theory proved that mass, which classical science had considered absolute, was in reality relative, since it depended on the relative motion of the observer. The general theory of relativity went still farther, proving that mass was relative in yet another sense. It established that not only the relative motion of the observer, but also the distribution of surrounding matter, would influence the mass of a given body. In this way Mach’s premonitions were vindicated. Following the relativity of mass, a number of other concepts, such as temperature, pressure, electric and magnetic forces, and gravitational force, were in turn proved to be relative, depending on the motion of the observer.
Now, although the epithet relative in its widest sense implies relative to surrounding conditions, we shall find that in Einstein’s theory it is employed most generally with reference to the observer’s motion. A magnitude which is relative will, therefore, mean one whose value depends on the relative motion of the observer; a magnitude which is absolute will imply one whose value remains unaffected by this motion. The major aim of the theory will be to separate those magnitudes which are relative from those which are absolute. The absolute magnitudes will then be representative of the absolute, common world of which the various observers will obtain but private perspectives. This absolute world will be the world of space-time standing in contrast to the relative world of space and time.
Among the absolutes which the theory has discovered, we may mention electric charge, entropy, the velocity of light in vacuo (in the special theory, at least) and the length of what is known as the Einsteinian interval between any two points in four-dimensional space-time; this latter type of absolute holding throughout the entire theory, both special and general.
[Pg 103]
CHAPTER IX THE PRINCIPLES OF RELATIVITY
MUCH of the difficulty which philosophers and laymen experience in understanding Einstein’s theory arises from a confusion between the different meanings that may be attributed to the concepts “relativity of space and motion.” For the student who already possesses some knowledge of classical science, confusions of this type, of course, are not to be feared. But in view of the general misconceptions on the subject we will mention briefly the various principles, elucidating them further as we come across them in the course of this book. We must name here:
1. The primordial mathematical relativity of space and time.
2. The kinematical or visual principle of relativity.
3. The dynamical or classical Galilean and Newtonian principle of relativity.
4. Einstein’s special principle of relativity.
5. Einstein’s general principle of relativity.
6. The radical Mach-Einstein principle of relativity.
Let us consider these various principles in their order. The mathematical type of relativity is the one we have already had occasion to mention when discussing mathematical space and time. It implies that a distance in space, even after the space has been referred to the observer’s frame, has a purely relative magnitude; so that two distances in space may be congruent or unequal according to our measuring conventions. Even after we have decided upon our measuring conventions, the magnitude of a spatial distance can only be expressed by the magnitude of our measuring rod; and if during the night all lengths were to contract in the same way, no difference could be detected when we awoke on the following day. Similar conclusions would apply to time.
Now, this mathematical aspect of the relativity of space and time would appear to be in conflict with everyday experience; for if during the night all things were to move twice as fast, there is not the slightest doubt that certain very apparent physical changes would be manifest. For instance, a rapidly rotating flywheel might burst under the tremendous strain of centrifugal stresses. However, it must be remembered that the mathematician is discussing pure amorphous mathematical space; and even though Weyl’s theory throws a new light on the relativity of magnitude by referring it to the radius of the universe as a whole, the entire question is still somewhat obscure. We need lose no more time over these purely mathematical conceptions, but shall concern ourselves with the principles of relativity which [Pg 104] relate to the real universe of physics, to real space, together with its metrical field.
The kinematical or visual principle of relativity, which is at least as old as the Greeks, states that a body can be considered in motion only when referred to some other body. For instance, if we consider the particular case of the earth and sun, we would conclude, according to whether we referred motion to the earth or sun, either that the sun was rotating round the earth every twenty-four hours, or else that the earth was rotating on its axis.
The two methods of presentation would be equivalent. Again, it would be impossible for us to state whether a body was approaching us faster and faster with some accelerated motion or whether it was we who were travelling with accelerated motion towards the body, visual appearances being the same in either case. Obviously this kinematical or visual principle would connote the complete relativity of all motion and rest, hence the complete relativity of space.
But a closer study of the dynamics of material systems proves the visual principle to be untenable. It was found that the behaviour of material systems, and thus the results of mechanical experiments, were influenced by absolute acceleration and rotation through space, although they remained totally unaffected by absolute velocity.
The Newtonian or Galilean or classical or dynamical principle of relativity expresses this elusiveness of absolute velocity or Galilean motion through space so far as mechanical experiments are concerned, while it stresses by contrast the physical significance of absolute acceleration and rotation. As is well known, it was this absoluteness or physical significance of acceleration and rotation which compelled Newton to recognise that space could not be relative. That the Newtonian principle of relativity will considerably restrict the scope of the visual principle can be gathered easily from the following example:
If a body were moving with respect to our frame of reference with a relative velocity, but with no relative acceleration, we should be justified according to the Newtonian principle in maintaining that it would be impossible to decide by mechanical experiments whether it was we who were moving towards the body or the body that was moving towards us. To this extent the visual principle is satisfied. But, on the other hand, if a relative acceleration of rotation existed as between ourselves and the body, mechanical experiments would enable us to ascertain what fraction of the relative acceleration or rotation was due to our own absolute acceleration or rotation in empty space, hence also what fraction was due to the body’s motion.
We now come to Einstein’s special principle of relativity. It is an exact replica of the Newtonian principle, upholding the relativity of velocity and the absoluteness of acceleration.[36] The sole difference between the two principles is that in Einstein’s, the [Pg 105] elusiveness of absolute velocity through space is no longer restricted to mechanical experiments and to material systems; electromagnetic and optical experiments are now placed on exactly the same footing.
While we are on the subject of the various principles of relativity, we may discuss two further types which play an important part in that extension of Einstein’s special theory known as the general theory of relativity. Here we are introduced to the General Principle of Relativity. This principle states that the mathematical expressions of the laws of nature must maintain the same form regardless of our choice of a frame of reference, be it Galilean (i.e., unaccelerated through empty space), accelerated, or even squirming like an octopus, while our clocks situated throughout the frame may beat at the most capricious rates. The classical principle of relativity conceded this invariance of natural laws in the case of Galilean frames, and synchronised clocks alone, and then only in the case of the laws of mechanics. The special principle recognised that this invariance held true even for the electromagnetic laws, but, as before, only for Galilean frames. The general principle of relativity, by extending the invariance of the laws of nature to all types of motions of the frame of reference, marks the starting point for the possible relativisation of acceleration, which had heretofore stood out as aloof and as distinctly absolute.
The radical Mach-Einstein principle of relativity is the result of a natural desire to bring about the complete relativisation of all manner of motion, rotationary and accelerated, as well as uniform. This is achieved by ascribing all the dynamical effects which accompany the acceleration and rotation of material and electrodynamic systems, to motion with respect to the material universe as a whole. According to this principle, which is still highly speculative, there can exist no observable difference between the rotation of a body with respect to the universe of stars and the rotation of the stars round the body; exactly the same dynamical effects of centrifugal force would be set up in either case, so that no trace of absolute motion through empty space would be left. We see that Mach’s principle constitutes an attempt to vindicate the kinematical principle in spite of the difficulties of a dynamical nature which had been the cause of its rejection. It was in part with a view to satisfying Mach’s principle that Einstein elaborated the hypothesis of the cylindrical universe. The principle would fall if astronomical observation should prove this form for the universe to be unacceptable.
[Pg 106]
CHAPTER X CLASSICAL MECHANICS AND THE NEWTONIAN PRINCIPLE OF RELATIVITY
WE mentioned, when discussing amorphous mathematical space, that position and hence motion could have no meaning other than that of expressing conditions with respect to some particular frame of reference. In short, the visual principle of relativity would have to be adhered to.
We will accordingly select some arbitrary system of reference, which we will assume to be defined by three mutually perpendicular planes meeting at a common point . The instantaneous position of a body will henceforth be defined when we have measured with rigid rods its distances from the three planes of our frame; and the only type of position and of motion of the body which we can define in an unambiguous way will be its position and motion as referred to our frame.
The measurement of the velocity of the body with respect to our frame necessitates the addition of a clock. We will assume that this clock beats out congruent intervals of time, where congruence is, of course, defined by the requirements of practical congruence for time, that is, by the beatings of some isolated periodic mechanism.
Now, it is perfectly apparent that unless referred to some particular frame, the path followed by a body and its motion along this path can have no determinate meaning. Thus, if in a uniformly moving train we throw a ball into the air and catch it again in our hands, then, as referred to the train, the ball will have followed an up-and-down motion along the same vertical; on the other hand, as referred to the embankment, the ball will have described a parabola. Again, if a stone is allowed to fall from a great height, its motion with respect to the earth will be accelerated, whereas, with respect to a frame of reference falling together with the stone, it will have remained at rest.
It would thus appear that we might credit to the path followed by a body any arbitrary shape we wished; all we should have to do would be to select some appropriate frame of reference. The same indeterminateness would, of course, apply to the motion of the body along its path. Thus the shape of a path in space (rectilinear or curved) or the species of a motion (uniform or accelerated) could have no absolute significance. From the standpoint of mathematical space these would all be essentially relative to the reference frame we had selected.
But it is obvious that this attitude into which our understanding of mathematical space has forced us presupposes that all the frames of reference we might select were on exactly the same footing, that is, were indistinguishable from one another. Galileo and Newton noticed that such was not the case. In certain frames we were pulled hither [Pg 107] and thither by strange forces unsymmetrically distributed, and objects placed on the floor did not remain where they had been put. Space around us appeared to be in a chaotic condition, as, for example, on a rotating disk or in a train rounding a curve. In other frames no such forces were apparent (excluding the force of gravitation), and space appeared stagnant and quiet, the same everywhere. These latter frames are called Galilean or Inertial.
Now, a Galilean frame did not manifest itself as unique. So far as the most delicate mechanical experiments could detect, there appeared to exist an indefinite number of such frames, all moving in respect to one another with various constant speeds along straight lines, without suffering any relative rotation during their motion. Mechanical experiments conducted in any one of these Galilean frames showed that free bodies followed straight lines with constant speeds (Newton’s law of inertia), and when referred to these frames, mechanical phenomena, including the planetary motions, were susceptible of being formulated by very simple laws. As viewed from Galilean frames, the other frames, the non-Galilean ones (those filled with strange forces), were found to be moving with accelerated or rotationary motions.
Mechanical experiments conducted in the non-Galilean frames proved that the disposition of the strange unsymmetrical so-called inertial forces bore a close relationship with the apparent motions of these frames as viewed from the Galilean ones. It was possible, therefore, for an experimenter situated in one of these strange frames to anticipate, without looking outside, exactly how his frame would appear to be moving when viewed by a Galilean observer (i.e., an observer in a Galilean frame), at least so far as acceleration and rotation were concerned. Thus, in a non-Galilean frame moving with respect to a Galilean frame along a straight line with some definite acceleration, the forces of inertia would be disposed in a parallel way, pulling in the direction opposite to that of the relative acceleration. In a non-Galilean frame rotating with respect to a Galilean observer, the disposition of the inertial forces would be much more complicated. There would be the centrifugal force pulling outwards and the so-called Coriolis forces pulling sideways.
As viewed from all these non-Galilean frames, the laws of mechanics would appear hopelessly complex. Free bodies would no longer appear to follow straight lines with constant speeds, but would describe capricious curves with varying velocities. Newton’s Law of Gravitation would no longer account for the planetary motions as observed from these frames, and a study of mechanics would present tremendous difficulties. In short, all the simple laws observed from Galilean frames would have to be compounded with the effects of these strange inertial forces when we wished to formulate laws controlling phenomena as viewed from the non-Galilean frames.
No surprise need be felt, therefore, when it is said that classical science selected the Galilean frames as standard frames. All the laws of mechanics and of physics were referred to them, and this, indeed, [Pg 108] was the essence of Copernicus’ discovery. Incidentally, we see that we are in possession of another definition of a Galilean frame; for we may say that a Galilean frame is one with respect to which the laws of classical mechanics (law of inertia, Newton’s law, etc.) hold true.
We may also mention that the very possibility of there being definite laws of mechanics, such as the law of inertia, is contingent upon the existence of certain fundamental differences between the various frames. If all frames were identical, as the complete relativity of motion would demand, the law of inertia would express nothing at all. In every case we might make the motion of a body appear anything we wished by selecting our frame suitably; and if we had no means of distinguishing one frame from another the straight path of a body through empty space would convey no physical meaning.
We may now give still another definition of a Galilean frame. It follows from the law of inertia, according to which free bodies, when viewed from Galilean frames, will describe straight lines with constant speeds, that the stars, being presumably free bodies, will obey this law and, owing to their remoteness, appear to suffer very slight displacements even in the course of a century This fact led to a more easily obtainable definition of a Galilean or inertial frame, namely, a frame with respect to which the stars would appear fixed. This definition is not meant to supplant the one given previously of a frame in which no forces of inertia are experienced, for theoretically both definitions are equivalent. Its sole advantage is that it leads to a more accurate empirical determination of a Galilean frame and shows us immediately that our earth does not constitute such a frame, since the stars appear to circle round the Pole star. As viewed from a true Galilean frame, our earth would therefore be rotating round the polar axis every twenty-four hours.
Had we sought to discern the non-Galilean nature of the earth frame by endeavouring to detect traces of inertial, centrifugal and Coriolis forces, the task would have been more difficult owing to the minuteness of these forces. Nevertheless, the forces have been detected, and they account for the protuberance of the equator, the clockwise and anti-clockwise motions of cyclones, the rotation of the plane of Foucault’s pendulum, the directions of trade winds and of ocean currents, etc.
Now, thus far we have not attempted to draw any general conclusions from the existence of these various frames. We have merely stated that they undeniably exist, since they have been revealed by experiment, and that it is simpler to refer the laws of physics, and particularly those of mechanics, to Galilean or inertial frames. When it comes to interpreting the deeper significance of all these facts, various alternatives present themselves, and in this chapter we will discuss Newton’s solution at some length.
The problem to determine is, “Why do different types of frames exist when our understanding of amorphous mathematical space would seem [Pg 109] to preclude any such differences?” Here we may make one of two assumptions. Either we may assume that the differences in conditions which exist in the various frames (differences which are exemplified by varied distributions in the lay of the inertial forces, or again in their total absence) are intrinsic, or else we may assume that they are due to foreign influences. Newton adopted the first alternative, Mach and Neumann the second.
If we follow Newton we must assume that the peculiarities existing in our frames are not to be attributed to causes existing beyond them. We may therefore assume that the inertial forces would be exactly the same were the balance of the universe to be annihilated, so that to all intents and purposes we might consider our frames isolated from all external influences. We assume, then, that the various distributions of the inertial forces manifesting themselves in our various frames must be attributed to the conditions of rest and of motion of these frames, not with respect to the remainder of the universe, which would be as good as annihilated, but with respect to those limited portions of the ether or of space in which the frames are situated.
Newton thus objectivises space and, owing to the symmetrical conditions which endure in Galilean frames, assumes these latter to be at rest in space, whereas the non-Galilean ones are assumed to be accelerated or rotating. If, then, we remember that a strict correspondence exists between the centrifugal and Coriolis distribution of forces in a non-Galilean frame, and its rate of rotation as viewed from a Galilean one, we may assume that the space in which a Galilean frame is at rest is also that in which a non-Galilean frame is accelerated or rotating. There is, then, but one space, an absolute space with respect to which motion and rest have a physical significance. Galilean frames are those which are non-accelerated; non-Galilean ones are those which are accelerated or rotating in this absolute space. This was Newton’s solution, and provided we accept his postulate of isolation there appears to be no way of escaping it.
Under the circumstances Newton’s law of inertia, according to which a free body follows a straight course with constant speed in relation to a Galilean frame, acquires a more determinate significance. The law now implies that this motion is being described in absolute space—in a space, therefore, that possesses a definite structure and which is thereby able to guide the body along its straight course. Had space been entirely amorphous and relative, as we assumed originally, there would have been nothing to guide the body; and even had the latter possessed enough intelligence to guide itself, there would have been nothing in amorphous space for it to direct itself by.
A number of philosophers attacked the Newtonian hypothesis of absolute motion and absolute space. For instance, some of them remarked that this so-called absolute space might well be assimilated to a bubble of ether floating and rotating in a more embracing space, this other in yet another, ad infinitum; so that Newton’s absolute rotation [Pg 110] would be meaningless. Needless to say, arguments of this sort are totally irrelevant to Newton’s stand and to science generally.
Classical science, by absolute space, meant one with respect to which motion, or at any rate accelerated motion and rotation, appeared to manifest themselves dynamically; and experiment proved conclusively that such a space existed. There was no use, and in fact no sense, in wondering whether this absolute space should be conceived of as rotating in some other space; for even if a hypothesis of this sort were ever to be vindicated, the fact remained that rotation appeared to be absolute in a certain space regardless of whether or not this space was the super-space par excellence. This was all that was meant by the absoluteness of rotation so far as science was concerned. In short, as scientists clearly recognised, there was only one way to refute Newton’s absolute space and motion, and that was to deny his postulate of isolation. Such, indeed, was the course adopted by Mach.
If we follow Mach and Neumann and assume that the various internal dynamical conditions which distinguish the various frames are due to influences of an extrinsic nature, we shall no longer be able to subscribe to Newton’s conclusions; for we shall no longer be able to assume that were the entire outside universe to be annihilated, the internal conditions in our frames would endure as before. Henceforth, when ascribing the appearance of centrifugal and Coriolis forces to the rotation of the body, we should have to imply that this rotation was relative to a something in the universe that was not the empty space in which the frame was situated. The natural conclusion was that this rotation was relative to that other form of existence, namely, to matter. Neumann suggested a mysterious body called the “Alpha Body,” to whose dynamical influence he ascribed the rise of inertial forces; but, as Poincaré remarked, “Of this mysterious Alpha Body all we can ever know is its name.” A hypothesis such as Neumann’s, which is incapable of verification, reduces to a mere metaphysical speculation of no scientific value whatsoever.
Mach adopted a much more plausible alternative. He assumed that the rotation of the frame which appeared to accompany centrifugal and Coriolis forces was a rotation with respect to the remainder of the physical universe, hence to the totality of the matter of the universe. As the vastest agglomeration of matter in the universe is given by the totality of the star masses, we are led to Mach’s conclusions, namely, that the forces of inertia arise from a relative rotation or acceleration of the frame with respect to the universe of stars. Were the stars, then, to be annihilated, centrifugal force and all forces of inertia would vanish, a rotating mass of water would not splash over the sides of a bucket, and so forth. It is, of course, scarcely necessary to mention that the actual visibility of the stars or our mere ability to imagine their existence has nothing to do with the problem; it is solely the causal influences produced by their masses and relative acceleration which are relevant.
As a matter of fact, there is no very essential difference between the solutions of Mach and Newton. Even if we follow Mach, unless we are [Pg 111] to believe in action at a distance, we must assume that the stellar influences can reach us only as a result of a propagation from place to place through space. We may perfectly well assume that as a result of this influence space receives a definite structure, so that in fine the physical reality of rotation may still be ascribed to a rotation in space. There is a difference, however; for with Newton this structure of space is intrinsic to space itself and has nothing to do with the contingent presence of matter generally, whereas if we follow Mach this structure of space must be conceived of as extrinsic and caused by the matter of the universe. In the absence of matter, the structure of space would automatically disappear and the various frames become indistinguishable.
As we can see, Mach’s views lead to a partial vindication of the visual or kinematical principle of relativity. But the vindication is only partial, since in all cases the relativity of acceleration and rotation exists only as between the totality of the stars and a body, and not between any two bodies taken arbitrarily, as in the visual principle.
Against Mach’s solution there is the argument that the stars are too remote to produce any such palpable effects as the bursting of a rotating flywheel or the splashing of a revolving mass of water over the sides of the bucket. But, still more important, it appeared impossible for classical science to account for these dynamical actions attributed to the stars. Nevertheless, with the tremendous advance in our understanding of nature which the general theory of relativity has given us, Mach’s ideas become highly probable, though not completely vindicated at the present time. Astronomical observations conducted on the globular clusters may eventually solve the problem.
Let us now return to Newton’s solution of the problem, since it constitutes the embodiment of classical mechanics. As we have seen, Newton, by positing his postulate of isolation, rendered inevitable a belief in some real absolute medium, space or the ether, from which rotation and acceleration would derive a real meaning. But here we must recall that mechanical experiments, though clearly differentiating a Galilean frame from a non-Galilean frame, were totally incapable of differentiating one Galilean frame from another. It appeared impossible to discover, therefore, which particular Galilean frame was to be considered at absolute rest in stagnant absolute space and which ones were to be regarded as moving through space with uniform translationary motions. In other words, when we confined ourselves to mechanical experiments, velocity through space was relative, the only velocity susceptible of mechanical detection being relative velocity, that is, velocity of one frame with respect to another. Acceleration and rotation, on the other hand, were absolute, and could be detected and measured mechanically without our having to appeal to a Galilean frame taken as term of comparison.
Were we to appeal to a Galilean frame in order to check up our computations, it would be a matter of complete indifference which particular Galilean frame we might select. As referred to any one of these Galilean frames, a body would manifest exactly the same acceleration and rate of rotation, whereas its velocity would, of [Pg 112] course, vary with the Galilean frame to which it was referred. This is what is meant by the absolute character of acceleration in contradistinction to the relative character of velocity.
Of course, even an accelerated body could be made to appear at rest or to be moving with uniform motion if we selected a suitable frame. But the frame would have to be non-Galilean or accelerated in order to ensure this result, and we have seen that non-Galilean frames can never (at this stage, at least) be regarded as being at rest in space. It follows that the appearance of a body’s motion relatively to any of these non-Galilean frames could never be considered representative of the real motion of the body in space.
The Principle of Relativity of Classical Science merely summarises these discoveries by stating that it is impossible for an observer situated in a Galilean frame to ascertain by any mechanical experiment whether he is at rest in space or in a state of uniform translationary motion. A simple illustration will clarify the meaning of these statements.
Consider a train moving with constant speed along a straight course; obviously there is no trace of acceleration in the train’s motion, since no forces of inertia are felt in its interior. In common practice, we should speak of the train as having a definite speed of so many miles per second. But if we reflect we shall see that this constant speed is computed with respect to the earth’s surface, considered as defining a Galilean frame of reference. Were the earth suddenly to become invisible, we should have no physical means of determining the exact speed of the train through space. But suppose, now, that the train were slowing down, speeding up or rounding a curve. In this case, even if the earth were invisible, the train would still possess at each instant a definite acceleration and this acceleration would still be perceptible, and would be measured by the forces of inertia pulling and pushing the passengers about in the train. We see, then, that acceleration has an existence per se and that, in contrast to velocity, it is determinate even in the absence of a frame of reference. Now, it follows from the classical principle of relativity that, absolute velocity being meaningless (at least so far as mechanics is concerned), the only type of uniform translationary motion which can be detected by mechanical means reduces to relative uniform translationary motion of matter with respect to matter; that is, motion with respect to something observable. The magnitude of this relative motion depends essentially on the Galilean frame we may select.
Relative uniform translationary motion can, of course, be detected visually, but it can be detected also in a number of other ways. Thus the relative motion of a razor blade on a strop causes heat due to friction. The relative displacement of a knife between the two poles of an electromagnet produces induced currents which manifest themselves by rendering it difficult to cut through the lines of force of the field. The relative motion of an incandescent atom produces a shift in the [Pg 113] spectral lines (Doppler effect), and it is thanks to this shift that astronomers are able to determine the radial velocities of the stars with respect to the earth. In other words, though absolute velocity can never be detected by mechanical experiments and is therefore meaningless in mechanics, yet relative velocity of matter with respect to matter can be detected in a number of ways; so that this relative type of velocity is the only one that has any physical significance, or at least any mechanical significance.
We have now to consider a number of difficulties that arise from Newton’s conception of absolute space. In the first place, although Newton had postulated absolute space and motion, the fact remained that of all the absolute motions, absolute acceleration and rotation were the only ones that had ever been detected, at least by mechanical means. It seemed strange, if absolute velocity were to have a meaning, as indeed it must if space were absolute, that it should be so obstinate in refusing to reveal itself. We could not hold that our inability to discover which one of the Galilean frames was truly at rest was due to the crudeness of our mechanical experiments; for the entire structure of Newtonian mechanics would have come crashing down had any such suggestion been proved correct.
The reason for this statement is easily understood. The fundamental law of Newtonian mechanics states that the force acting on a body is equal to the mass of the body (an invariant) multiplied by the acceleration. Now, the acceleration of a body as computed in any Galilean frame is always the same; for the various Galilean frames differ only in their velocity through absolute space, and a velocity added to or subtracted from an acceleration can never modify an acceleration. The mass of the body being an invariant according to classical mechanics, the force must also measure out the same in all Galilean frames. In other words, it was essential to classical science that the fundamental law of mechanics should remain the same in all Galilean frames, and similar conclusions would have to apply to all the mechanical laws. But then no mechanical experiment, however precise, executed in one Galilean frame or another, could ever yield varying results; so that absolute velocity could never be known. In short, if Newtonian science were to stand, the failure of mechanical experiments to detect the absolute velocity of our Galilean frame through space could never be attributed to their lack of refinement, since any positive results would have overthrown the entire structure of Newtonian dynamics.
Here, then, was an absolute velocity which was a reality since space was absolute, but which on no account could ever manifest itself!
It was, of course, possible to suggest that space had a dual structure, partly absolute and partly relative; absolute for rotation and acceleration, but relative for velocity. But this intrinsic duality in the nature of space and motion was hard to accept; it seemed as though space and motion should be one thing or the other: either entirely absolute or else entirely relative. Moreover, it was extremely difficult to conceive of a duality in the structure of space, so the [Pg 114] entire situation was most mysterious.
There was, however, a possible way out of the difficulty, and that was to assert that absolute velocity was real, truly enough, and that it might perfectly well be detected provided we appealed to other types of physical experiment—to electrodynamic and optical experiments, for instance, and not solely to mechanical ones. But we shall see that precise experiment has since proved that these expectations were also doomed to disappointment; hence it appeared perfectly clear that no manner of experiment, mechanical, optical or electrodynamic, would ever reveal this elusive absolute velocity.
Even had optical and electromagnetic experiments succeeded in detecting absolute velocity, everything would not have been plain sailing. It would then always have been possible to assert that the absolute velocity detected in this way was not absolute velocity through space, but velocity through the ether floating in space. But we may point out that this last objection carries very little weight. For reasons which we shall explain in the next chapter, it does not appear legitimate to dissociate the ether from space. Had optical experiments detected velocity through the ether, we should have had to assume that they had also disclosed the long-sought-for absolute velocity through space; and the Newtonian belief in absolute space would have been considerably strengthened.
However, it is unnecessary to dwell on what might have happened, seeing that optical and electromagnetic experiments have failed to reveal the slightest trace of absolute velocity. Accordingly we are led to Einstein’s special principle of relativity, which states that not only mechanically, but in every way, absolute velocity through space must escape empirical detection. Under the circumstances, unless we wish to retain in science an absolute velocity which can never be detected and which is therefore any one’s guess, there is no alternative but to state that space together with its ether has a dual structure: relative for velocity, absolute for acceleration. Einstein appears to have thrown us back on the extremely difficult task of conceiving of this dual nature for space. But the solution of this particular problem was forthcoming as soon as Minkowski discovered space-time. With four-dimensional space-time the reason for the duality becomes apparent; for the absoluteness of space-time (as will be explained in a later chapter) accounts for the relativity of velocity and the absoluteness of acceleration.
Thus in Einstein’s theory, in its original form at least, velocity remains relative and acceleration remains absolute, as in classical science,[37] the only difference being that the relativity of velocity is now more thorough, applying as it does to all manner of experiments and not merely to mechanical ones. It is only when we consider the more speculative part of the theory, that pertaining to the form of the universe, that the complete relativisation of all motion in the Machian sense occurs. Space-time is then found to be no longer an absolute [Pg 115] existing per se but to be conditioned entirely by the matter of the universe. But, as we have already had occasion to mention, this part of the theory is still highly speculative, and in order to avoid unnecessary confusion, we would strongly advise the reader to eliminate it entirely from his mind until such time as he has acquired a proper understanding of the special and of the first part of the general theory. Accordingly we will ignore it for the present; and we may state that in Einstein’s theory motion possesses a dual nature, just as was the case in classical science.
[Pg 116]
CHAPTER XI THE ETHER
CLASSICAL science assumed that, those manifestations we called electricity, magnetism, and light were nothing but strains, compressions and wavelike motions in an imponderable medium, the stagnant ether, floating in space. This belief was not altogether blind guesswork. In view of the stresses and strains that clearly appeared to surround electrified bodies, electric currents and magnets, and in view of the well-established wave nature of light propagation, some medium had to be postulated in order to give these stresses, strains and waves physical significance. Of this mysterious ether itself, practically nothing was known; but from what scant information could be gathered, it appeared to possess strange and contradictory properties.
Its resistance to motion was practically nil—far less than that of a gas like the atmosphere; since the frictional effects due to the atmosphere are sufficient to cause shooting stars to become incandescent, while no such effect was observed in those regions of ether-filled space extending beyond the limits of the atmosphere. Besides, if motion through the ether generated friction, mechanical phenomena should be influenced by the velocity of our Galilean frame through space and hence through the ether; but so far as experiment was able to decide, no such influences had ever been detected. (This was indeed the essence of the Newtonian principle of relativity.) It appeared, therefore, that the ether must be assimilable to some ultra-rarefied gas. On the other hand, the transverse vibrations of light waves were compatible only with the existence of a rigid medium; but between the properties of rigidity and of extreme rarefaction there was an obvious contradiction. Yet, however baffling and artificial the ether appeared to be, it was extremely hard to banish it from science owing to the great mass of experimental evidence that appeared to lend support to its existence.
At any rate, once the existence of a semi-material ether was accepted, it was natural to suppose that the motion of our optical and electrical apparatus through its substance should exert a perceptible influence on the results of experiment. Just as the velocity of the wind adds itself to the velocity of sound waves propagated through the atmosphere, so was it logical to anticipate, in view of our understanding of light waves and of electrical phenomena, that an ether wind should carry the light waves and ether strains along with it. Optical and electrical experiments conducted with sufficient refinement should therefore be susceptible of detecting the velocity of our planet through the ether; so that in contradistinction to the relativity of Galilean motion for [Pg 117] mechanical phenomena with which the ether was not concerned, Galilean motion would be anything but relative were electromagnetic or optical experiments to be attempted.
Now this duality which was thus assumed to exist in the nature of Galilean motion or velocity, making it relative for mechanical phenomena and absolute for electrodynamic ones, was perfectly acceptable, provided it was permissible to draw a sharp distinction between mechanics and electrodynamics, between space and the ether. But the entire trend of recent research had been to show that this distinction was unjustified. In the first place, matter seemed to be constituted by electrons and protons, hence to be reducible to phenomena of an electromagnetic nature. Then again, electromagnetic phenomena were known to develop mechanical forces. Where, then, was the essential difference between mechanical and electromagnetic phenomena? Under the circumstances how could we account for this anticipated duality in the palpable effects caused by a velocity through the ether, relative or physically meaningless for mechanical experiments, and capable of producing physical effects in electromagnetic ones?
It appeared as though mechanical and electromagnetic phenomena should be susceptible of being placed on the same footing. We should then have to assume that mechanical phenomena together with electromagnetic ones would be affected by velocity through the ether, hence through space, or else to recognise that both electrodynamic and mechanical phenomena could never be influenced by this type of motion. If we should accept the first hypothesis, the Newtonian or mechanical principle of relativity would be overthrown, and with it the entire structure of Newtonian mechanics; and if we should accept the second hypothesis, our understanding of electromagnetic processes and of the ether would have to be modified profoundly. In either case, whether or not electromagnetic and optical experiments succeeded in detecting absolute Galilean motion, science seemed to be faced with a difficult situation.
Classical physicists held to the view that optical and electromagnetic experiments would surely reveal absolute Galilean motion, and they were not perturbed by the fact that as a result Newtonian mechanics might have to be abandoned. Indeed, doubts as to the validity of Newtonian mechanics already presented themselves when Kaufmann and Bucherer showed that electrons in very rapid motion appeared to increase in mass—a phenomenon which was of course incompatible with the Newtonian belief in the invariance of mass, unless some hypothesis ad hoc was to be invoked. It is true that Michelson’s very refined test attempted in 1887 failed to detect any influence of absolute Galilean motion in a certain optical experiment; but it was always assumed that an explanation for this failure would be forthcoming sooner or later, and that other experiments would be successful where Michelson’s had failed. When, therefore, Einstein suggested as a principle of nature that, regardless of whether or not the ether was a necessary hypothesis, Galilean motion through the ether or through space would [Pg 118] never be detected by any type of experiment, whether optical or electromagnetic or mechanical, the majority of physicists paid little heed to the new doctrine.
It must not be thought that this hostility of classical physicists to the Einstein theory in its initial stages was due solely to their belief in the absolute and objective nature which it appeared inevitable to credit to the ether. It was due also to other reasons which we shall now proceed to investigate.
From a mathematical point of view, a physical phenomenon reduces to the systematic variation in intensity and in orientation from place to place and from time to time of a number of magnitudes. A physical phenomenon as referred to some arbitrary frame of reference can therefore be expressed by mathematical equations in which the space and time co-ordinates with respect to the frame figure as variables. Now a fundamental problem of theoretical physics is to determine how the equation of a phenomenon referred to a frame , which is in motion with respect to a frame , can be obtained from its equation referred to the frame , by purely mathematical means without any further appeal to experiment. The solution of this problem would enable us to anticipate the change in appearance of a phenomenon when observed, first from the frame , then from the frame . Obviously some kind of mathematical operation would have to be performed on the equation as referred to . This operation is called a mathematical transformation; and the equation operated upon is said to have been transformed. In the type of problem we are considering, where one frame of reference is followed by another, the transformation will bear on the space and time variables or co-ordinates present in the equation; and transformations of this character are termed space and time transformations. The entire problem reduces, therefore, to the discovery of those space and time transformations which hold in this world of ours. These transformations cannot be guessed at a priori; and a study of the behaviour of physical phenomena is a prerequisite condition for their determination. Inasmuch as all the experiments known to classical science appeared to corroborate the impression that a distance in space and a duration in time were absolutes, remaining unaltered when we changed our frame of reference, a certain definite species of space and time transformations followed as a matter of course.
Consider, for example, a train moving with constant speed along an embankment. If an event occurs in the moving train at any instant of time, say at two o’clock, this event will obviously have occurred at precisely the same instant (two o’clock) when referred to the embankment, since time is assumed to be absolute, the same for all. In other words, time and duration for the observer in the moving train are identical in all respects with time and duration for the observer on the embankment. This fact is expressed by , where is the time for the train, and the time for the embankment.
Now suppose that a ball is rolling along the moving train with a speed [Pg 119] , and suppose that owing to the train’s motion the train observer is carried past the embankment observer at the instant of time when the clocks mark noon. If the ball leaves the train observer at this precise instant of time, its distance from the train observer at a time (i.e. seconds after the noon instant) will be (since and are identical). But then its distance from the embankment observer will be the sum of the distance between the two observers, and of this distance we have just expressed (distances being absolute). It will be, therefore, (where is the velocity of the train along the embankment).[38] This last deduction is obvious provided we assume that a distance is an absolute concept which would turn out the same whether measured by one observer or another, and such was precisely the assumption that classical science always made. In short, knowing the point where an event takes place at a given instant as referred to the train we can deduce immediately the point where the same event takes place when referred to the embankment. All we have to do is to replace by , and the trick is done.
These space and time transformations permitting us to pass from one Galilean frame to another were the celebrated Galilean transformations of classical science, and they were, as we have seen, the necessary consequences of our belief in absolute duration and distance. They were given by: or again by Applying them to particular problems, as, e.g., to the change in the shape of the trajectory of a falling body when viewed successively from a moving train and from the embankment, or to the change in the colour of a monochromatic source of light, or in the pitch of a musical note, the anticipations of the transformations were always found to be verified, to the order of precision of our experiments. Corroborations of this sort, among others, were considered to prove the correctness of the transformations, hence also of our traditional belief in the absoluteness of duration and distance.
Now if, instead of restricting ourselves to any particular phenomenon, we consider all phenomena of the same type, such as all mechanical or all electrodynamic phenomena, we know that these phenomena in their aggregate satisfy the requirements of certain general laws or equations: the laws of mechanics or the laws of electrodynamics, as the case may be. To say, therefore, that Galilean motion or velocity is relative so far as a certain type of phenomenon is concerned means that the general laws governing such types of phenomena remain unchanged [Pg 120] in form when we pass from one Galilean frame to another. Expressed mathematically, this means that when the space and time transformations are applied to the general laws, these laws or equations are transformed into equations possessing exactly the same mathematical appearance; they are then said to have been transformed into themselves.
When the transformations were applied to the general laws of mechanics, these laws suffered no change in form; and this fact constituted the mathematical expression of the Newtonian or mechanical principle of the relativity of Galilean motion. But when the transformations were applied to the equations or laws of electrodynamics, a distinct change in form was noted; the laws or equations lost their simple appearance. This suggested that the electrodynamical laws as accepted by classical science held only in their simple form for a privileged Galilean frame, which for reasons of symmetry was naturally assumed to be one at rest in the stagnant ether. But then, the laws changing in form according to the Galilean frame to which they were referred, it followed as a necessary consequence that electromagnetic experiments should pursue a different course and yield different results according to the frame in which the observer and his instruments were placed. It should then be possible to discover the magnitude of the velocity of the frame through the ether by means of electromagnetic or optical experiments, so that once again, independently of any particular ether hypothesis, mathematical reasoning as well as commonplace reasoning confirmed the physical reality of velocity through the ether.
Of course these mathematical anticipations were dependent on a twofold hypothesis: first, that the classical transformation-formulæ based on absolute distance and duration were accurate; and, secondly, that the equations or laws of electrodynamics were correct. To question the first assumption appeared unjustified, and to question the validity of the mathematical expression of the laws led to conflict with experiment. True, we could modify the laws of electrodynamics (by suppressing those mathematical terms which were the cause of their variability), rendering them thereby invariant under the Galilean transformations, but these terms turned out to be those responsible for the phenomenon of electromagnetic induction. To suppress them would have been equivalent to denying the existence of induction, hence of wireless, radio, dynamos and light propagation. Obviously, a solution of this kind could not be accepted, since dynamos, etc., were known to exist.
In view of all these facts, not the slightest doubt was entertained by scientists that experiments sufficiently precise in nature would yield us the velocity of our instruments, hence of our planet, through the stagnant ether.
We must now pass on to a rapid survey of those so-called negative experiments which were primarily responsible for all the trouble. Let us recall once more the point at issue. We wish to ascertain to what extent the velocity of our Galilean frame through the ether is capable [Pg 121] of modifying the results of our electromagnetic and optical experiments.
First in order among these tests we must mention the well-known phenomenon of astronomical aberration, according to which the direction of incidence of a ray of starlight varies annually as the earth circumnavigates the sun. The obvious analogy of this phenomenon is the slanting direction followed on the window pane of a moving train by raindrops which to an observer stationed on the embankment would appear to be falling vertically. This analogy might tempt us to assume that inasmuch as the slant of the raindrops on the pane permits us to deduce the speed of our train through the stagnant atmosphere, so now the slant of the ray of starlight resulting from the earth’s motion should yield us our velocity through the stagnant ether. But the conclusion would be faulty; for if we reflect we shall see that we were able to deduce our motion through the atmosphere, solely because we had reason to assume that had our train stood motionless in the atmosphere the raindrops would have appeared to fall vertically. But in the case of astronomical aberration we have no means of basing our deductions on any such preliminary information. We cannot tell from what directions the rays of starlight would appear to issue were we at rest in the ether; for we ignore the true positions of the stars. And so all that astronomical aberration can yield us is the annual variation in our velocity through the ether, hence our velocity relative to the sun.[39] In short, astronomical aberration does not enable us to detect our absolute speed through the ether, and we must rely on other means.
In all the experiments we shall have occasion to mention, the velocity through the ether which we shall seek to determine will be that of our earth; for, attached as we are to the earth’s surface, this will be the speed we shall best be in a position to determine. Unfortunately, the earth does not constitute a truly Galilean frame on account of its acceleration or rotation round its axis. However, this acceleration of the points on the earth’s surface is very small and the centrifugal force to which it gives rise is scarcely perceptible. Calculation shows that, so far as optical and electromagnetic experiments are concerned, we may regard a frame attached to the earth as constituting a Galilean frame, possessing velocity but no acceleration, at least as a first approximation. Nevertheless this statement is not altogether correct. Experiments can be devised both in mechanics and in optics which would reveal the earth’s acceleration or rotation. Of course this is only natural, since acceleration, in contradistinction to velocity through space, is known to be absolute.
A proper understanding of Einstein’s theory requires that this point be fully understood, so we must be excused for repeating once more that all we are concerned with at present is to reach a decision on the [Pg 122] relativity or the absoluteness of velocity or Galilean motion through the ether. In the case of the earth there is no danger of our confusing any possible effects of acceleration with those due to velocity because, owing to the comparative slowness of the earth’s rotationary motion, any effects that might arise from acceleration would in any case be far feebler than those arising from velocity through the ether.
Summarising, we may say that a veritable ether hurricane must be blowing over the earth, varying both in intensity and in direction at the different points of the earth’s surface and also at the various times of the day and year. It is this ether hurricane or ether drift that we must proceed to detect. Now all experiments, whether optical or electrical, obstinately refused to detect the slightest trace of this ether drift; so that one of two things was obvious. Either there was no drift; or else, if drift there was, it was completely concealed by compensating phenomena of which as yet nothing was known. Stokes adopted the first alternative and suggested that the earth in its motion round the sun might possibly drag the ether along with it, in its immediate vicinity at least. It would follow that there could never be any drift on the earth’s surface; and all the negative experiments which had failed to detect the drift would be explained. The obvious objection to Stokes’ hypothesis was the existence of astronomical aberration, which seemed to imply that the earth was moving through a stagnant ether. Stokes, however, proved mathematically that aberration was no argument against his hypothesis; that this phenomenon would still be in order, provided the earth’s motion did not set up whirlpools in the convected ether.
In addition to a number of other difficulties to be explained shortly, we may mention that for the motion of the ether to be such as to satisfy Stokes’ requirements, hence the observed phenomenon of astronomical aberration, calculation proved that the ether could not be incompressible. In fact, Planck showed that on the earth’s surface the ether would have to be 60,000 times denser, and thus correspondingly more compressed than in distant regions; and it would seem strange indeed that this considerable increase in the density of the ether should fail to manifest itself when we compared the results of optical experiments performed on the summit of a high mountain and at sea level. Moreover, light vibrations being transversal, the hypothesis of a rigid ether appeared to be inevitable, so long as we wished to conceive of light as due to vibrations in the ether. Here again the Stokes-Planck theory was difficult to accept.
Lodge endeavoured to subject Stokes’ hypothesis to a direct experimental test. He argued that if the earth were capable of dragging the ether along with it in its motion, the same should be true of other moving bodies. But when he investigated this anticipated drag effect in the case of rapidly revolving disks of steel, he found it entirely absent.
Even if we ignore these contradictions and assume that the terrestrial atmosphere is able to drag the ether along bodily, a new series of difficulties awaits us. A total drag of this sort would be incompatible with a general phenomenon predicted by Fresnel and [Pg 123] verified by Fizeau according to which dielectrics in motion (electrical non-conductors), such as water, glass and air, drag the ether along only in a partial way; the percentage of drag increasing with the refractivity of the dielectric. Inasmuch as the refractivity of the atmosphere is comparatively feeble it would be impossible to credit the total drag of the ether to the atmosphere.
A quite recent experiment of Michelson (not to be confused with the celebrated experiment of Michelson and Morley) also renders Stokes’ hypothesis extremely unlikely. The essence of this particular experiment is to send two light rays round the earth in opposite directions. Its result was to prove that the ray travelling eastward required a longer time to complete its circuit than the one travelling westward. Such would not be the case were the earth to drag the ether around with it in its rotation on its axis. True, this experiment did not prove that the ether was not dragged along by the earth’s motion along its orbit, but, on the other hand, it did prove that the earth’s rotationary motion produced no such drag. This, of course, rendered extremely improbable the hypothesis that the earth’s motion along its orbit should be any more successful in producing an ether drag. The aforementioned experiment of Michelson is an apt illustration of the type of experiment which enables us to detect by optical means the earth’s rotation or acceleration through the ether, and is therefore irrelevant to the problem we are here investigating, namely, that of its velocity through the ether. The only reason we mentioned the experiment was because of its bearing on Stokes’ hypothesis.
Stokes’ hypothesis was thus discarded, and the most satisfactory explanation of all negative experiments that had been suggested was given by Fresnel. It must be realised that up to this stage all the experiments attempted were of an optical nature, and related to the transmission of light rays through dielectrics (glass, water, etc.). Such experiments had been performed by Foucault, Airy, and Arago. Fresnel succeeded in proving by mathematical analysis that if dielectrics carried the ether along in their interior in a partial manner, increasing in a definite degree with their refractive power, all the negative results attained up to his day would be explained, provided they were not pursued to too high an order of accuracy. This required percentage of drag of the ether by dielectrics was called Fresnel’s convection coefficient.
Fresnel’s hypothesis was subjected to a direct experimental test. Fizeau discovered, exactly as had been anticipated by Fresnel, that when a ray of light was propagated through running water, the total speed of the light ray with respect to the tube was equal to its speed through the water, when the water was stagnant, plus only a fraction of the water’s velocity in the tube; and that this fraction was precisely equal to the convection coefficient predicted by Fresnel.
In view of this remarkable confirmation, Fresnel’s hypothesis was considered fairly established, so that the situation confronting science was the following: The ether was stagnant in that the earth moved through it without disturbing it in any way. Velocity through [Pg 124] the ether had therefore a definite physical significance, and a real ether hurricane must be blowing over the earth’s surface in various directions according to the time of day. Experiments had failed to detect it, not because it did not exist, but on account of this compensating effect of the partial drag of the ether by dielectrics in motion. Such a failure, however, could only be regarded as temporary; for Fresnel’s hypothesis specifically showed that this compensating effect was but partial, and that experiments of a still more refined order ought certainly to detect the ether drift on the earth’s surface, revealing thereby the earth’s absolute velocity.
Obviously Fresnel’s hypothesis would fall if more refined experiments should also fail to detect the ether hurricane. Furthermore, even to the order of precision contemplated by Fresnel, if in our experiments we should appeal to phenomena in which dielectrics played no part, his showed that this compensating effect was but partial, and that experiment of Michelson and Morley was of this type. Not only was it far more precise than the ones hitherto attempted, but in addition refraction played no part in it. Yet, in spite of all, the Michelson and Morley experiment again gave a negative result.
The next great advance in our understanding of the problem was due to Lorentz. He, however, like all his predecessors, dared not take the great step, but still endeavoured to explain the negative results of experiment while holding to the view of a real stagnant ether. As might have been expected, Lorentz was compelled to appeal to new compensating influences. But before going farther it appears indispensable to devote a few pages to the general subject of electrodynamics; for as we proceed, we shall see that theoretical considerations are destined to play a part of ever-increasing importance.
[Pg 125]
CHAPTER XII THE EQUATIONS OF ELECTROMAGNETICS AND LORENTZ’S THEORY
WHEN amber is rubbed, it develops the peculiar property of attracting small bodies, such as bits of paper, particles of dust and the like. This phenomenon, which was known to the Greeks, was described by saying that the amber had become electrified (electron meaning amber in Greek). To-day, however, we should say that it had received an electric charge. Soon it was found that electrified bodies did not always attract one another, but that in many instances they appeared to exert a repulsive action. For this reason it was assumed that there existed two different species of electricity, the positive and the negative; and the laws governing the phenomena involved were compressed into the statement that like charges repelled whereas opposite charges attracted. Similar conditions were found to endure between magnetic poles, so that the existence of two different types of magnetism was also assumed. But magnetism and electricity remained entirely distinct; no reciprocal action appeared to exist between an electric charge and a magnetic pole.
We may illustrate these phenomena of attraction and repulsion in a more concrete way by assuming that invisible fields of electric and magnetic forces surround electrified bodies and magnetic poles. To these fields we may ascribe the mechanical actions of attraction and repulsion detected by experiment; the electric fields act on electrified bodies, and the magnetic ones on magnetic poles. It was not until the early years of the nineteenth century that Coulomb submitted the mutual attractions and repulsions of charged bodies to a quantitative test. He found them to be expressed by a law (Coulomb’s law) which was of the inverse-square variety; the same as Newton’s law of gravitation, except that the electric charges took the place of masses, and that the actions could be either attractive or repulsive.
But with the invention of the electric cell by Volta a new manifestation of electricity presented itself for study and experiment, in the form of the electric current. Oersted discovered that an electric current was able to deflect a magnetized needle placed in its vicinity. This was of great significance, as showing that electricity and magnetism were closely allied phenomena. The nature of their relationship became better understood when, by placing iron filings round a wire, it was shown that an electric current was surrounded by a magnetic field. The deviation of the magnet in Oersted’s experiment was accordingly ascribed to the existence of the magnetic field surrounding the current. In the following year Ampère gave an exact quantitative formulation of the laws involved.
Now the fact that an electric current generated a magnetic field rendered it legitimate to suspect that conversely a varying magnetic field should generate an electric current. If this were the case, [Pg 126] a magnet displaced near a closed wire should generate or induce an electric current in the wire. This most important phenomenon, known as electromagnetic induction, was discovered by Henry and Faraday, and as a result the connection between magnetism and electricity became still more pronounced. Our present-day dynamos and generators are nothing but machines constructed with a view to utilising this phenomenon of induction to generate electric currents on a commercial basis.
Faraday was the first scientist to realise the enormous importance of the electromagnetic field. He saw in it a reality of a new category differing from matter. It was capable of transmitting effects from place to place, and was not to be likened to a mere mathematical fiction such as the gravitational field was then assumed to be. In his opinion, the phenomena of electricity and magnetism should be approached via the field rather than via the charged bodies and currents. In other words, according to Faraday, when a current was flowing along a wire, the most important aspect of the phenomenon lay not in the current itself but in the fields of electric and magnetic force distributed throughout space in the current’s vicinity. It is this elevation of the field to a position of pre-eminence that is often called the pure physics of the field. Faraday was not a mathematician and was unable to co-ordinate the phenomena he foresaw in a mathematical way, and derive the full benefit from his ideas. Before dying, however, he entrusted this task to his colleague Maxwell; and one of the most astounding theories of science, eclipsed only in recent years by Einstein’s theory of relativity, was the outcome.
In order to appreciate the nature of Maxwell’s contributions, let us recall how matters stood in his day. If we confine-our attention to regions of space where no electric currents, no charged bodies or magnets are present, so that solely the electric and magnetic fields need be considered, the fundamental law of electromagnetism is Faraday’s law of induction. This law states that a variable magnetic field generates an electric field.
Maxwell, however, considered that this law, standing alone, lacked symmetry; so he formulated the hypothesis that conversely a variable electric field should generate a magnetic one, and proceeded to construct his theory with this idea in view.[40] In the particular case of free space in which only fields but no charges or currents are present, Maxwell’s equations of electromagnetics, termed field-equations (since they describe the state of the electromagnetic field), can be written: [Pg 127] where represents electric field intensity, magnetic field intensity, denotes time, and a most important constant about which we shall have much to say later. As for “div” and “curl,” they represent mathematical operations of a peculiar sort. At any rate, the reader need not be alarmed at these equations, for it is not necessary that we understand them. Suffice it to say that (2) represents Faraday’s law of induction, while (4) constitutes the hypothetical law postulated by Maxwell in order to ensure symmetry. Prior to Maxwell’s investigations the fourth equation would have been written “”; hence it is the adjunction of the term which constitutes the originality of the theory.
From his field equations, Maxwell by purely mathematical means was able to deduce the two additional ones: where represents a particular species of mathematical operation. In ordinary language, these last two equations connote that varying electric and magnetic intensities will be propagated through the ether in wave form with a velocity , where is a magnitude yet to be determined. This discovery removed all possibility of action at a distance, since the field perturbations now appeared to be propagated from place to place with a finite velocity. It was, of course, of interest to determine the precise value of . The most obvious means would have been to create electromagnetic disturbances and then measure their speed of propagation directly. Physicists, however, were unable, in Maxwell’s day, to devise a means of performing such delicate experiments. But there existed another way of determining the value of . Maxwell remarked that it would be given by the ratio of the magnitude of any electric charge, measured successively in terms of electrostatic units (based on electricity) and then of electromagnetic units (based on magnetism).[41] Precise measurements conducted on electrified bodies and magnets then proved that the value of this ratio was about 186,000 miles per second; whence it became necessary to assume that periodic perturbations in the strains and stresses of the field would be propagated in the form of waves moving through the ether with this particular speed. But this velocity was precisely that of light waves propagated through the luminiferous ether.
[Pg 128]
The conclusion was obvious. Unless we were to assume that this extraordinary coincidence in the values of these two characteristic velocities, that of electromagnetic induction and that of light waves, was due to blind chance, there was no other alternative but to recognise that what we commonly called a ray of light was nothing else than a series of oscillations in the electromagnetic field, propagated from point to point. Electromagnetic waves and luminous waves were thus all one. Henceforth the two ethers, the electrical ether and the luminiferous ether, were seen to be the same continuum; electricity, magnetism and optics were merged into one single science.
Now, although experimenters had attempted by various means to submit Maxwell’s views to a test, the technical difficulties were so great that no success had been achieved. It appeared clearly from Maxwell’s equations that no appreciable effects could be anticipated unless was very great; and this meant that the electric intensity would have to vary with extreme rapidity. The simplest means of obtaining a result of this kind would be to produce an oscillating field of electric intensity in which the oscillations were extraordinarily rapid, several billions a second. But no mechanical contrivance could yield such results, and apart from mechanical contrivances no other methods suggested themselves. For these reasons a number of years passed before Maxwell’s views could be put to the test.
In 1885 Helmholtz directed the attention of his pupil, Hertz, to the problem. Hertz was one of the most remarkable experimenters of the nineteenth century; he succeeded at last in vanquishing the technical difficulties and in generating by purely electrical means an oscillating electric field of extremely high frequency. Electromagnetic waves of sufficient intensity were thus produced; and after having been sidetracked for a time by a secondary phenomenon whose nature was elucidated by Poincaré, Hertz verified the fact that the waves advanced with the speed of light and indeed possessed all the essential properties of light waves other than those of visibility to the human eye. Thus, as a result of Hertz’s experiments, the foundations were laid for the commercial use of wireless and radio; but, more important still, Maxwell’s electromagnetic theory of light establishing the intimate connection between electricity and optics had at last been vindicated.
[Pg 129]
Just as Hertz had proved that electromagnetic waves were of the same nature as light waves, so now, conversely, Lebedew succeeded in showing that light waves were of the same nature as electromagnetic ones. It is, indeed, a necessary consequence of Maxwell’s theory that electromagnetic waves should exert a definite pressure on bodies upon which they impinge, and this pressure was verified and measured by Lebedew and again by Nichols and Hull in the case of light waves falling on matter.[42]
Yet it must be realised that marvellous as were Maxwell’s equations, so far as the field phenomena in empty space were concerned, they led to conflict with experiment in the presence of matter. The conflict was notably apparent when the phenomenon of dispersion was considered; and by dispersion we mean the splitting up of white light by a prism into a rainbow of colors. Maxwell’s equations proved that on entering a transparent dielectric such as glass, the electromagnetic vibrations which constitute light would be slowed down; and this slowing down, as had long been known, would account for refraction. But dispersion demands that the angle of refraction, hence the slowing down, should depend on the frequency, hence on the colour of the light vibrations; and this is what Maxwell’s theory failed to account for. His equations showed that the slowing down would be the same for all frequencies. A similar difficulty was encountered when the phenomenon of the magnetic rotation of the plane of polarisation was considered. Faraday had discovered the existence of this phenomenon, and any theory of electromagnetics would have to account for it. Nevertheless, it will be seen that all these difficulties which beset Maxwell’s theory arose solely when experiments involving matter were taken into consideration. It appeared, therefore, that the fault resided not in the field equations themselves, but in the too crude assumptions that had been made with respect to the constitution of matter.
Furthermore, Maxwell had considered more especially the case of matter at rest in the ether, and it was necessary to investigate the general case of matter in motion. This extension of Maxwell’s theory, known as the electrodynamics of moving bodies, was attempted by Hertz, who assumed that matter in motion through the ether would drag the ether along with it in its interior—a hypothesis not so radical, but somewhat in line with that of Stokes. So long as conductors were contemplated, Hertz’s hypothesis seemed to be in accord with experiment; but when we came to consider dielectrics in motion, Hertz’s anticipations were contradicted. Fizeau’s experiment in particular, proving that the velocity of running water did not compound with the velocity of the light in stagnant water, was incompatible with Hertz’s hypothesis of a total drag. Also, in an experiment of a totally different nature, dealing with the motion of a dielectric through a magnetic field, Wilson proved the existence of the same partial drag, given once again by Fresnel’s convection coefficient. It appeared, therefore, that Fresnel’s convection coefficient was a fact which would have to be fitted into any theoretical scheme of the electrodynamics of moving dielectrics.
Now, Fresnel’s partial drag hypothesis, though seemingly confirmed by experiment, had yet to be accounted for theoretically, and this appeared impossible so long as we remained in such utter ignorance of the ultimate constitution of dielectrics and of matter in general. At this stage Lorentz started his theoretical investigations which were to lead him to a refinement of Maxwell’s equations, extending their application to cases where matter was present, either at rest or in motion through the ether. As we have mentioned, Maxwell’s equations [Pg 130] had proved themselves incapable of accounting for dispersion. It appeared necessary to conceive of some structure for dielectrics which would act selectively, imposing different degrees of retardation on light waves of different frequencies. Lorentz achieved this result by assuming that electricity was atomic and that matter was constituted by more or less complicated groupings of these electric atoms or electrons.
In the case of dielectrics or non-conductors such as glass, the electrons were assumed to be tied down to their atoms by elastic forces. When disturbed they would vibrate with some characteristic frequency depending on the magnitude of the elastic forces which held them in place. If, then, a wave of light were transmitted through the transparent dielectric, the electrons would be set into forced vibration and the retarding effect these vibrations would impose on the incident light would depend on the relative frequencies of the incident light and of the natural frequencies of the electrons. In this way it was possible to obtain a mathematical interpretation of dispersion.
Further phenomena were accounted for by taking into consideration the frictional resistances that would interfere with rapid vibrations of the electrons. When these frictional resistances were weak, oscillatory disturbances, such as rays of light, could be propagated through the dielectric, which was then termed transparent (glass). When these frictional forces were considerable, the light ray was unable to set the electrons into vibration; its energy was consumed in the attempt, and as a result it could not proceed; the dielectric was then opaque (ebonite, sulphur).
In the case of conductors such as metals, the electrons were assumed to be very loosely held to their atoms so that the slightest difference of potential would tear them away and cause them to rush in the same direction, thereby producing an electric current. It was precisely because electrons in conductors were not tied down to fixed positions by elastic forces that they were incapable of vibrating; and so conductors were necessarily opaque to electromagnetic vibrations or to light. Conversely, it was because the electrons were all tied down to fixed positions in dielectrics, that they could not rush along in one direction. As a result dielectrics were opaque to currents, and hence were non-conductors. According to these views of Lorentz, an electric current passing through matter was nothing but a rush of electrons. If this were the case, we should expect the motion of a charged body to generate the magnetic field of a current. Rowland had established the existence of this effect many years previously, so that Lorentz’s theory conformed to facts in this respect. This same effect is illustrated in a more vivid way by the magnetic field which surrounds cathode rays. These are constituted by streams of electrons travelling with enormous velocities through a partial vacuum; and for this reason the cathode rays have much in common with a current passing through a metal wire.
In perfectly empty space no electrons were assumed to be present; and the propagation of electromagnetic disturbances through space was credited entirely to the oscillations of the field which stood out [Pg 131] as a manifestation of energy differing radically from the substantial electrons.
In contradistinction to Hertz, Lorentz assumed that the ether was never disturbed by the motion of matter; and just as in Maxwell’s theory, a luminous source in motion through the ether would never communicate its velocity to the light waves it emitted. With the constitution of matter postulated in this way, Lorentz set out to extend Maxwell’s theory of electromagnetics in the free ether to cases where matter was present, either at rest or in motion. As already mentioned, he succeeded in giving a theory of dispersion, one of the phenomena which Maxwell had been unable to explain. Furthermore, such highly complicated phenomena as abnormal dispersion, absorption, metallic reflection, selective reflection, body colour and many others were accounted for by the theory with great precision.
The apparent partial drag of the ether verified in Fizeau’s experiment and also in Wilson’s (Fresnel’s convection coefficient) was proved to be a necessary consequence of the theory. It had nothing to do with a partial dragging of the ether, since the ether was in no wise disturbed by the motion of the matter; the apparent drag was accounted for solely in terms of the electronic constitution of the moving matter. In this way Fresnel’s convection coefficient received the rational explanation which it had thus far lacked. The most refined experiments confirmed Lorentz’s theory in a number of minute details; and when the electron was finally isolated and studied, all doubts appeared to be removed, at least as to the general correctness of the doctrine.
Theories, however, are especially convincing, not so much when they account for what has been observed but has not been explained, as when they allow us to foretell phenomena which no one has anticipated. Triumphs of this sort were soon forthcoming in the case of Lorentz’s theory. For instance, the theory required that light waves be produced by electrons vibrating in the interior of the atom. Inasmuch as the presence of a magnetic or an electric field affects the motion of electrified bodies, the vibrations of the electrons in the luminous atom should be modified by the presence of a strong magnetic field. Calculation then showed Lorentz that a magnetic field would cause an atom normally emitting a monochromatic light to emit two or three separate lights, according to the relative inclination of the magnetic field to the line of sight of the observer. This totally unexpected effect was soon verified by Zeeman, one of Lorentz’s colleagues, and is now known as the Zeeman effect. It has since been observed in the light emitted by sunspots, proving that very intense magnetic fields must there be present.
It is quite unnecessary to dwell on a number of further verifications of Lorentz’s theory. All that we have wished to show is that the empirical discoveries that lend weight to Lorentz’s electronic theory are so numerous that it can certainly not be cast aside lightly. Although no achievement in theoretical physics can ever claim to be permanent, yet in view of the wonderful accuracy of Lorentz’s previsions it would require some reason of a truly imperative nature [Pg 132] for us to feel justified in tampering with it in any essential detail.
Lorentz’s electronic theory is also perfectly consistent with the negative experiments we have heretofore mentioned, since it explains the existence of the Fresnel convection coefficient; and we have seen that this coefficient in turn accounts for the negative results of experiments conducted to the first order of approximation in dielectric media moving through the ether. But Lorentz’s theory as it now stands suggests that experiments conducted to a higher order of precision, to the second order, for example, would cease to yield negative results; so that with experiments of this sort our motion through the ether would be detectible.[43]
Now, the Michelson and Morley experiment is precisely of this type. It is a second-order experiment. It is assumed that the reader is sufficiently familiar with it to render its detailed description unnecessary. Stated briefly, the experiment proves that if waves of light leave the centre of a sphere simultaneously, they will return in perfect unison to the centre of the sphere after having been reflected against the sphere’s inner surface, no matter in what direction through the ether, or with what constant velocity, the sphere attached to the earth may be moving. An explanation was suggested by both FitzGerald and Lorentz, and was to the effect that the sphere had contracted in a definite degree in the direction of its motion through the ether. The sphere accordingly became an ellipsoid, and Michelson’s negative result was explained.
More generally, any body, a yardstick, for example, when lying in the direction of the earth’s motion through the ether, would be contracted and become shorter than when placed perpendicularly to the ether hurricane. However, we should never have any means of detecting this contraction, for all things, the human body included, would participate with it, whence observationally, at least, nothing would be changed.
This Lorentz-FitzGerald contraction hypothesis was a hypothesis formulated ad hoc for the sole purpose of explaining the null result of Michelson’s experiment; and like all such hypotheses, it could be accepted only provided it was justified by a wide variety of experiments. At the time this hypothesis was suggested, scientists were busying themselves, with the study of the electron when in motion through the ether. Abraham assumed that an electron moving at high speed through the ether would remain spherical and rigid; and [Pg 133] he worked out mathematically the path a rigid electron would pursue when acted upon by electric and magnetic forces. Lorentz, holding to his view of the FitzGerald contraction, attacked the same problem, assuming the electron to contract more and more as its speed through the ether increased. Experimenters were called upon to decide between the hypotheses of the rigid and the contractile electron. Accordingly, highly refined experiments were conducted on the electrons emitted by radium ( rays). Kaufmann, by photographing the paths followed by the electrons, came to the conclusion that Abraham’s rigid electron corresponded to reality. But, still more precise experiments conducted by Bucherer reversed the verdict, and little doubt remained that the electron suffered the FitzGerald contraction in the precise degree demanded by Lorentz.
Inasmuch as matter was assumed to be electronically constituted, these experiments of Bucherer lent weight to the FitzGerald contraction of matter, hence to the Lorentz-FitzGerald explanation of Michelson’s null result. This contraction being seemingly established, attempts were made by experimenters to detect it in bodies which were attached to the earth. Could this result be achieved, we should have a means of ascertaining the earth’s velocity through the ether; and it was confidently expected that this time success would attend us.
Now, as we have mentioned, if in our frame of reference, on the earth’s surface, for instance, all things, including our measuring rods and our own bodies, contracted in exactly the same way owing to the motion of our planet through the ether, it would, of course, be idle for us to expect this contraction to be observable by direct measurement. But although not observable in a direct way, it was argued that this contraction of a body should modify its optical and electrical properties so that a study of the variations in these properties when the body was placed successively in different directions should yield us the earth’s motion through the ether. In particular the contraction of a solid transparent body, owing to its motion through the ether, should give rise to a phenomenon of double refraction, much as would follow from an ordinary physical compression of the body. The experiment of Rayleigh and of Brace was based on this idea, but the result was a complete disappointment. The FitzGerald contraction gave rise to no such effect, and a negative result was registered. The experiment of Trouton and Rankine aimed to detect a variation in the electrical conductivity of a strip of metal as a result of the FitzGerald contraction. Once more a negative result was the outcome. These experiments having failed, still another was suggested and carried out by Trouton and Noble. The essence of this experiment was based on the well-known phenomenon verified by Rowland, that an electrified body in motion produces all the effects of an electric current. If, therefore, two electrified bodies were placed side by side, the motion of the earth through the ether should produce two parallel electric currents; and since parallel electric currents attract one another, a very sensitive torsion balance should be able to detect this attraction. Again, the result of this experiment [Pg 134] was negative; the velocity of the earth through the ether evaded us completely.
Now, it should be noticed that the last two experiments pertain purely to electricity and electrodynamics; there is nothing of an optical nature about them. Hence, it would appear that all electromagnetic phenomena behave in exactly the same way, regardless of the velocity of the ether hurricane. In other words, they, too, yield negative results. But there is yet another aspect to these experiments.
If we regard the FitzGerald contraction as physically real—and we must do so in order to explain Michelson’s negative result—we must recognise that it is certainly compensated in a miraculous way; since no experiments, whether optical or electrical, appear to be capable of revealing it. Furthermore, in the last experiment, that of Trouton and Noble, we are not dealing primarily with the FitzGerald contraction; we are here in the presence of an experiment of electrodynamics, and we are endeavouring to detect a purely mechanical force arising as a result of the motion of our electrified bodies through the ether. The laws of electrodynamics require that such a force be generated, and yet no force can be observed; so we must assume that the mysterious compensations that conceal our motion through the ether are of so perfectly general a nature that they affect mechanics as well as electromagnetics and optics.
True, none of the experiments we had been able to execute exceeded the second order in precision; and there was nothing in Lorentz’s theory, even when the FitzGerald contraction was taken into consideration, to suggest that all experiments, whatever their nature and however precise they might be, would fail to detect our velocity through the ether. A very simple hypothetical experiment will make this point clear. Suppose, for instance, that the earth were rushing through the ether with a speed approximating that of light; and suppose that we suddenly stepped in front of a mirror. All that the FitzGerald contraction hypothesis would ensure would be that the time required by the light rays to leave our body and be reflected back to our eyes should be independent of the orientation of the mirror with respect to the earth’s motion. But the actual duration which the light rays would require, hence the time it would take for our image to appear to us, would depend essentially on the earth’s speed through the ether; so that theoretically, at least, a measure of duration should reveal this mysterious speed.
However, in view of the fact that the highly precise experiments that had been performed were incapable of detecting this velocity through the ether, it appeared legitimate to extrapolate, to assume that something deeper was at stake, and that in all probability this velocity would remain undetected, however far we might extend the precision of our electrodynamic or optical experiments, and however much we might vary their nature. Lorentz accepted what appeared to him as the inevitable, and asserted that the time had come to recognise that nature seemed to have entered into some giant conspiracy to defraud us of a knowledge of our velocity through the ether. Accordingly, he laid down his celebrated principle of [Pg 135] correlation, according to which adjustments were so regulated in nature that the velocity of our planet through the ether could never be detected, however precise our experiments.
Lorentz applied his mathematical talents to the discovery of the necessary adjustments which would have to exist in nature for this correlation to be satisfied completely so far as electrodynamics and optics were concerned. In order to understand the significance of his investigations, we must recall what we have said of the laws of nature and of the space and time transformations.
We remember that Newtonian relativity expressed the fact that mechanical phenomena, hence the laws of mechanics regulating these mechanical phenomena, remained unchanged when we passed from one Galilean frame to another. The significance of this statement is as follows:
To begin with, when we wish to fix the position of a body in our frame, it is also necessary to specify the instant of time at which the body occupies this position. We thus have four co-ordinates to take into consideration: three for space, , , ; and one for time, . If, now, we change our frame of reference, taking our stand in some other Galilean system, which is in motion with respect to the first, these co-ordinates will change in value or will be transformed. For instance, if a body remains at rest at the meeting point of the three axes of our first frame, its co-ordinates will always be , , in this frame for all values of the time . But computed with respect to our second frame, the new co-ordinates represented by , , will vary with time, since with respect to this second frame the body is moving—i.e., is adopting successive positions in time.
The classical Galilean transformations had for their object to tell us exactly how these co-ordinates would vary when we passed from one Galilean frame to another. As it was always assumed by classical science that time was absolute, the same for all systems, the only co-ordinates susceptible of variation as we changed frames were the space co-ordinates , , . As for , it had the same value in all frames, and this was expressed by writing . Now the laws or equations which govern phenomena contain in their expression, among other quantities, these co-ordinates, , , , . Hence, in the general case it is only to be expected that the laws will change in form when we refer them first to one Galilean frame, then to another; or again, when we replace , , , by , , , . When we say that the laws suffer no change in form or are transformed into themselves we mean that this substitution of the new co-ordinates for the old does not modify the form of the equations or laws.
We can now understand what mathematical condition will have to be satisfied for Lorentz’s principle of correlation to hold. The principle asserts that all things are so adjusted in nature that absolute velocity through the ether must ever elude us. In other words, all electrodynamic experiments will yield negative results. But this obviously makes it necessary that the laws of electrodynamics which govern all electrodynamic phenomena, hence all electrodynamic and all optical experiments, should remain unchanged in form when we pass from [Pg 136] one Galilean system to another. In mathematical language, the laws of electrodynamics will remain invariant when we submit them to the space and time transformations.
Now we have said that when submitted to the classical space and time transformations, the invariance of the equations of electrodynamics does, not hold. It appeared then that, contrary to the negative results of experiment and contrary to Lorentz’s principle of correlation, velocity through the ether should be detectible by electromagnetic experiments. Science was thus placed in an exceedingly difficult position; the equations of electrodynamics had been so strongly confirmed by extremely accurate experiment that we could scarcely question their accuracy. There remained, but one alternative, namely, to suspect the accuracy of the classical transformations.
This was indeed the course followed by Lorentz. He succeeded in establishing the transformations which would be in harmony with the invariance of the electrodynamic equations, and he found these to differ perceptibly from the classical ones; although reducing to the latter in the case of low velocities. The new transformations constitute the celebrated Lorentz transformations.[44]
[Pg 137]
Translated into ordinary language, these transformations expressed the existence of two separate phenomena: first, the FitzGerald contraction of bodies moving through the ether, a phenomenon with which we are already acquainted; and, secondly, a new phenomenon consisting in the slowing down (with increase of velocity through the ether) of the rate of time-flow as applying to electromagnetic processes. Thus, contrary to the classical transformations, where time was absolute (), we see that in the Lorentz transformations this absoluteness of time begins to be questioned. For psychological reasons, however, imbued as he was with the spirit of classical science, Lorentz was unable to realise the importance of his discovery; and he never, succeeded in ridding himself of his belief in the absoluteness of time. For him, as also for Larmor, who contributed to these discoveries, this new species of variable duration, depending as it did on the motion through the ether of the Galilean frame, was not real time. It was a species of “local time”—a distortion of real time—and it was not assumed that this local time corresponded in any way to the time which the observer in the moving frame would live and sense.
And so Lorentz assumed that these new transformations applied only to purely electromagnetic quantities, and no reference was made to their being applicable to mechanical phenomena as well. Though, as a result of these transformations, the velocity of light proved to have always the same invariant value through all Galilean frames when measured by the observer in the frame, no suspicion was cast on the classical formula for the composition of the velocities of material bodies; and this in spite of the fact that these two circumstances were mutually incompatible.
It appeared then as if Lorentz still considered the classical transformation-formulæ to be the real ones, the new transformations being merely of limited application and referring solely to the electromagnetic quantities. When, therefore, in such experiments as that of Trouton and Noble, mechanical and purely electromagnetic effects were indissolubly connected, a large measure of obscurity was involved. Lorentz, however, succeeded in extricating himself from his difficulties and in accounting for all the negative experiments; but he was compelled to appeal to additional hypotheses, such as a modification in those elastic forces which cause the electrons to vibrate in transparent bodies (Rayleigh and Brace experiments). In a similar way it was possible to explain the negative result of the Trouton and Noble experiment. Nevertheless, we can understand the numerous difficulties into which Lorentz’s theory was leading us. We knew very little about the constitution of matter, and here, in Lorentz’s theory, we were compelled to account for negative experiments by taking this unknown constitution of matter into consideration.
Taking Lorentz’s theory as it stands, one cannot help but recognise that this accumulation of hypotheses postulated ad hoc makes it painfully artificial. According to the theory, the ether must be regarded as stagnant and the earth as moving through it with some definite velocity. This velocity must, however, remain eternally unknowable to us. Its elusiveness arises from the slowing down of time and the FitzGerald contraction, in the case of the simplest electromagnetic experiments; in the case of experiments dealing with dielectrics and conductors, we have to appeal to still more complicated adjustments pertaining to the constitution of matter. But although forever indiscernible, our velocity through the ether was a reality; it was there, and had it not been for all these compensating effects, it would have been observed. All the forces of the cosmos appeared to have conspired to deprive us of a knowledge we were so anxious to obtain, namely, the velocity of our planet through the ether.
The existence of the FitzGerald contraction, holding in exactly the same degree for all bodies, was an instance of this marvellous conspiracy of nature. Even if we admitted that all matter were electronically constituted, and that the constituent electrons became flattened as a result of their motion through the ether, it was still impossible to conceive of an identical flattening for all bodies, whether rigid or soft, unless we assumed that an appropriate adjustment had also taken place in the other factors entering into the constitution of matter (elastic forces, etc.). The whole theory [Pg 138] was unsatisfactory in the extreme. If Nature was blind, by what marvellous coincidence had all things been so adjusted as to conceal a velocity through the ether? And if Nature was wise, she had surely other subjects to attend to, more worthy of her consideration, and would scarcely be interested in hampering our feeble attempts to philosophise. In Lorentz’s theory, Nature, when we read into her system all these extraordinary adjustments ad hoc, is made to appear mischievous; it was exceedingly difficult to reconcile one’s self to finding such human traits in the universal plan.
Possibly more space than was needed has been devoted to a discussion of these difficulties of pre-Einsteinian physics, but the purpose of these lengthy explanations has been to show that science was confronted with an exceedingly difficult situation. Before proceeding to a study of Einstein’s solution, we must mention a tentative hypothesis put forward by the Swiss physicist Ritz.
Ritz remarked that Michelson’s experiment could be explained by assuming that, contrary to the requirements of the electromagnetic theory of light, waves of light were shot out from the luminous source just as bullets are shot out from a gun. With this species of ballistic theory, the speed of the light ray with respect to the observer would be given by the constant speed of the ray with respect to its source, compounded with the relative speed of the observer and source. Now, there is not the slightest doubt that Ritz’s hypothesis affords an explanation of Michelson’s negative result, but, on the other hand, it leads us into serious difficulties when we wish to interpret a number of other experiments, Bucherer’s, for example. This can be understood when we realize that the Ritz hypothesis compels us to deny the existence of the FitzGerald contraction, since were the contraction to subsist in addition to Ritz’s assumption, Michelson’s experiment would no longer yield a negative result. But, as we have said, the FitzGerald contraction permits a very simple explanation of the moving electron’s increase in mass in Bucherer’s experiment. This difficulty would not be insuperable, but still it would constitute an argument adverse to the acceptance of Ritz’s idea. Other experiments which it would be extremely difficult to account for are those connected with the Fresnel convection coefficient, e.g., Fizeau’s experiment.
Furthermore, with Ritz’s conception of the velocity of a light wave through the ether being compounded with the velocity of the source, the frequency of light issuing from a source, moving towards us through the ether, would be increased, whereas the wave length would remain unchanged. Now, it so happens that the angle of refraction of a monochromatic light ray passing through a prism depends on the frequency of the incident light, whereas the dispersion caused by a grating depends on the wave length. It should, therefore, be possible to determine whether frequency and wave length truly behaved like independent variables as demanded by Ritz’s hypothesis. Experiments conducted by Stark in 1910 on canal rays disproved Ritz’s anticipations. Subsequently other experiments led to the same results.
[Pg 139]
Besides these objections to Ritz’s views, it must be realised that Michelson’s experiment was only one of many. Here it cannot be emphasised too strongly that the gravity of the situation arose from the fact that as a result of a large number of different experiments (many of which we have not mentioned), the entire structure of classical electrodynamics seemed to be crumbling on every side. Ritz’s hypothesis consists in shoving one individual brick back into place. Lorentz, thanks to his mathematical knowledge, realised that something much deeper was at the root of all the trouble and that more radical means would have to be adopted. Accordingly, he proceeded to tackle the very foundations of science, namely, the space and time transformations which had endured for centuries. But Lorentz only proceeded in a half-hearted way and did not have the courage to push his discoveries to their logical conclusion, contenting himself with patching up rather than reconstructing. Einstein, as we shall see, took the bull by the horns, abandoned the classical structure, and proceeded to build up an entirely new edifice, superbly coherent and free from all artificial support and scaffoldings. Thus it was that even before Ritz’s hypothesis was disproved by numerous direct investigations (experiments of Majorana and of Sagnac, and astronomical observations on double stars by de Sitter), scientists had ceased to give it serious consideration.
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PART II THE SPECIAL THEORY OF RELATIVITY
[Pg 143]
CHAPTER XIII EINSTEIN’S SPECIAL THEORY OF RELATIVITY
HOWEVER unsatisfactory, for the numerous reasons already discussed, Lorentz’s theory might appear, it constituted for some years the only possible explanation of the negative results of all electromagnetic experiments performed on the earth’s surface. But in 1905 Einstein published a paper on the “Electrodynamics of Moving Bodies,” a document destined to prove one of the most epochal events in the history of human thought. In this paper, all the cobwebs of electrodynamics were swept aside.
The essence of Einstein’s position was that the difficulties which had beset the study of electrodynamics (we have mentioned only a small number of them) had arisen from our retention of the stagnant ether as a fundamental frame of reference; but that in view of the anomalies this attitude had brought about, the time had come to submit it to a critical analysis. After all, we knew little or nothing about the ether; why then start by stating it to be an absolute medium floating in space and hampering thereby all future progress? Why state that velocity through the ether must have a physical significance, then under the evidence of the negative experiments proceed to postulate complicated hypotheses in order to explain away the absoluteness we had ourselves introduced? Would it not be simpler to adopt a more cautious attitude, deriving knowledge from experiment rather than trying to reconcile experiment with a series of a priori beliefs which, for all we knew, might be totally erroneous?
If with Einstein we follow this line of argument, we must assume that the large number of negative experiments prove conclusively that velocity through the stagnant ether or through space is physically meaningless, not only from a purely mechanical point of view, as was the belief of classical science, but from every point of view. If we accept this relativity of the ether and space for Galilean motion as a fundamental fact of nature, holding for all manner of experiments, our past difficulties are seen to be of our own making. We had endeavoured to discover that which was meaningless, and then blamed nature for tricking us and hiding it from us. Instead of saying that velocity through the ether appeared meaningless because it was meaningless, we had complicated matters by saying that velocity appeared meaningless but of course was not meaningless.
Quite apart from the negative experiments, Einstein lays special stress on another type of phenomenon. Thus, whether we displace a magnet before a closed circuit or the closed circuit before the magnet, the [Pg 144] current induced in the wire is exactly the same in either case, so far as experiment can detect. That which appears relevant is the relative motion between magnet and circuit; the respective absolute velocities of magnet and circuit through the ether, which are of course different in both cases seem to be totally irrelevant.
In view of the importance of the question, let us mention another example. We remember that when discussing electrical phenomena in the first pages of the preceding chapter, we mentioned that an electrified body was surrounded by an electric field of force, whereas a current was surrounded by both an electric and a magnetic field. Subsequent experiments performed by Rowland confirmed the view that a charged body in motion generated the same electric and magnetic fields as a current flowing along a wire, so that we had to agree that a charged body in motion developed a magnetic field by the sole virtue of its motion.
Now here again we are in a quandary to understand what is to be meant by motion. Classical science assumed that motion in this case meant motion with respect to the stagnant ether, and that an electric current was constituted by the rushing of electrons through the ether. Numerous difficulties beset this view. Owing to the earth’s motion through the ether, every charged body on the earth’s surface would constitute an electric current; so that around every electrically charged body a magnetic field should be present. Yet experiment failed to detect this magnetic field. Was it due to the crudeness of our experiments? This solution was scarcely possible; for if we displaced our charged body before a magnetised needle (Rowland’s experiment), the needle was deflected, proving that experiments were perfectly able to detect the magnetic field when it was truly produced. It appeared as though the type of velocity that constituted a current was not velocity through the ether at all, but velocity relative to the recording instrument or, more generally, to the observer. Once again, the only type of velocity which appeared to have any significance in nature was relative velocity, and never velocity through the stagnant ether or absolute space.
Hence, Einstein postulated his special principle of relativity, according to which Galilean motion through the ether or space is meaningless. This principle is, as we see, merely an extension of the Newtonian mechanical principle to the case of the ether, or in other words to electromagnetic and optical experiments.
What is called the special theory of relativity concerns the rational consequences that must follow from the special principle. These consequences may be anticipated as follows: If Galilean motion through the ether is meaningless, the laws of electrodynamics must remain invariant in form when we change our Galilean system of reference. But this implies that the space and time transformations which the co-ordinates of all points undergo, when we change systems, must be given by the Lorentz transformations.
In Einstein’s theory, however, the significance of the transformations is entirely different. It will be remembered that in ordinary language these transformations implied the FitzGerald contraction [Pg 145] and the slowing down of time. In Lorentz’s theory these modifications arose as a result of a real velocity through the ether; and though it was impossible for the observer in the moving frame to detect their presence, yet in spite of all they were physically real. In Einstein’s theory, on the other hand, velocity through the ether being meaningless, the modifications arise only as a result of the relative motion of one frame with respect to the other, and are thus due to conditions of observation and not to any intrinsic change. For instance, if in any two Galilean frames originally at relative rest, two identical clocks and two identical cubes are placed, and if then the two frames are set in relative motion, either observer would discover as a result of his measurements that the other man’s clock had slowed down and that his cube had become flattened in the direction of the relative motion. Nothing would have happened to the cubes and clocks; only the conditions of observation would have been changed.
Expressed mathematically, this means that the variable which enters into the Lorentz transformations now stands for the relative velocity of one frame with respect to the other, and no longer for velocity through the ether. In this measure the transformations of Einstein’s theory, though identical in appearance with those of Lorentz’s, are far different in meaning. To avoid confusion, the appellation Einstein-Lorentz or Lorentz-Einstein transformations is accordingly often made use of.[45]
There is still a further difference, a most important one, distinguishing the implications of the transformations in the two theories. In Lorentz’s theory there was a privileged observer situated in the frame at rest in the ether. In Einstein’s theory there is no such privileged observer; for since a privileged state of rest in the ether is meaningless, all observers in Galilean frames are on the same footing. Hence, Lorentz’s distinction between the true time of the frame at rest and the distorted, local or electromagnetic times of the frames in motion through the ether loses its significance. Henceforth, all these so-called local times given by the Lorentz transformations constitute the true times lived, sensed and experienced by the respective observers in their respective Galilean frames.
There being no longer any reason to subordinate one frame to another, the descriptions of all phenomena in terms of space and time are on the same footing in all Galilean frames; and there is no longer any reason to follow Lorentz in limiting the validity of the transformations to the sole case of electromagnetic phenomena. Henceforth, these transformations appear of universal application, holding for mechanics, for the composition of velocities, and indeed [Pg 146] for the entire universe of physical phenomena. This belief is not dependent on the fact that, matter being constituted electronically, all mechanical experiments must be regarded as electrodynamic experiments viewed microscopically. Rather is it due to the fact that these changes in space and time are absolutely general. They arise from the relative motion of the observer and not from the peculiar microscopic constitution of the phenomena observed.
The method of presentation of Einstein’s theory we have followed up to this stage is not the one usually adhered to, but it has appeared preferable to proceed as we have done in order to bring into prominence certain important features which are obscured in the customary presentation (these features will be better understood later). We will now fall in with the usual procedure.
As Maxwell proved many years ago, a necessary consequence of the laws of electromagnetics was that light waves in vacuo in the stagnant ether should travel with a velocity of 186,000 miles per second. This was the celebrated law of light propagation of classical science. Now we have seen that the laws of electromagnetics (and with them their immediate consequence, the law of light propagation) held only for privileged observers, those situated in any frame at rest in the stagnant ether. As referred to all other Galilean observers situated in frames moving through the ether, an application of the classical transformations showed that the equations of electromagnetics, and with them the velocity of light propagation, would be modified. But since the essence, of the relativity of Galilean motion is to deprive any particular Galilean observer of his privileged position, the laws of electromagnetics must now maintain the same form for all Galilean observers. Hence the law of light propagation, which is a mathematical consequence of these electromagnetic laws, must likewise hold in exactly the same way for all Galilean observers. Whence the new result: “Light waves must travel with the same invariant speed of 186,000 miles through any Galilean frame when this speed is measured by the observer located in the frame.” It is this statement which Einstein has called the principle or the postulate of the invariant velocity of light.
If we combine this principle with the relativity of the ether for Galilean motion and endeavour to construct the transformations which will be compatible with these two principles, we are again led to the Lorentz-Einstein transformations.[46] This should not surprise us since, as we have seen, the principle of the invariant velocity of light is but a direct consequence of the invariance of the electrodynamical equations; so that the transformations ensuring invariance will be the same in either case.
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In short, we see that without appealing directly to the equations of electrodynamics, we can deduce the Lorentz-Einstein transformations merely by taking one of their consequences into consideration, namely, the invariant velocity of light, and combining it with the relativity of Galilean motion. From a mathematical point of view, this procedure is the simpler. Hence, Einstein posits as his fundamental assumptions:
1. The relativity of space or of the ether for Galilean motion, or, more simply, the relativity of Galilean motion without particular reference to the ether or to space.
2. The postulate or principle of the invariant velocity of light.
This procedure followed by Einstein offers a number of advantages. In the first place, as we have said, the mathematical discovery of the Lorentz-Einstein transformations is considerably simplified. Secondly, as the classical law of light propagation, though a consequence of the laws of electromagnetics, insusceptible of being tested by direct experiment without any reference to the laws of electromagnetics, there is no necessity for dragging in these highly complex laws. In fact, even had the laws of electromagnetics been unknown, the relativity of velocity and the classical law of light propagation are all that would have been required to construct the theory. Hence, from the standpoint of mathematical elegance, the procedure is certainly one of extreme simplicity. Furthermore, inasmuch as in Einstein’s theory the Lorentz-Einstein transformations are of general application and do not concern only electromagnetic phenomena, it is preferable to avoid conferring on them an exclusively electromagnetic aspect. This aim is achieved by deducing them from a general law of propagation, such as that of light.
On the other hand, this method of presentation, by appealing to the propagation of light, is likely to confuse the beginner, who is apt to assume that Einstein postulated the invariant velocity of light as a hypothesis ad hoc for the sole purpose of accounting for Michelson’s negative experiment. In this way the entire theory is supposed to hinge on Michelson’s experiment, and the critic assumes that could Michelson’s experiment be explained in some other way, Einstein’s theory would be obviated. This assumption appears all the more natural to the critic as Michelson’s experiment is, nine times out of ten, the only negative experiment he is acquainted with. The result is that he assumes Einstein’s theory to be nothing but a wild guess grafted on one of those highly delicate experiments where the chances of error are always great. As a matter of fact, by reasoning in this way, the critic loses sight of the entire raison d’être of the theory. It is safe to say that even had Michelson’s experiment never been performed, Einstein’s theory would have been forthcoming just the same (though, of course, had Michelson’s experiment given a positive result, enabling us to measure our velocity through the ether, the theory of relativity would have been untenable).
This explains why, in presenting the theory, we started by showing how it arose as a necessary consequence of the irrelevance of absolute velocity in all electromagnetic experiments, hence sprang from the invariance of the equations of electrodynamics, which expresses [Pg 148] mathematically the aggregate of all the negative results. If the reader has grasped the significance of these Lorentz-Einstein transformations, we may proceed to examine certain of their particular consequences, and to show how, quite apart from the negative experiments, the theory of relativity has cleared up a number of obscure points in our understanding of electromagnetics.
For instance, we mentioned that a charged body at rest was surrounded by an electric field of force, whereas the same body in motion (or an electric current) was surrounded both by an electric and a magnetic field. Ampère had given a formula describing the distribution and intensity of the electromagnetic field surrounding an electric current of any given intensity. But this formula was purely empirical, and it was felt that we should have been able to anticipate the existence of this electromagnetic field and derive an exact expression of its disposition and magnitude by purely rational methods. These hopes were disappointed; for there appeared to be no rational connection between the field developed by an electrified body when at rest and its field when set in motion.
Contrast our ignorance on this score with our ability to understand the nature of other problems attendant on relative motion. For example, a monochromatic source of light at rest appears yellow, whereas, when moving towards us with a sufficiently high speed, it appears greenish. But there is nothing mysterious about this change in colour; we could have foreseen it, knowing what we do of the wave nature of light. The situation was much more obscure in the case of electromagnetics.
Einstein’s theory solves the entire problem. Consider the simple case of a charge in uniform motion. We have said that owing to its motion it develops a magnetic field (just like a current), in addition to its electric field. But in Einstein’s theory velocity through the ether is meaningless; hence there is no such thing as a charge in uniform motion in any absolute sense. If we were to rush after the electrified body and accompany it in its motion, it would cease, according to Einstein, to be a moving charge and would become a charge at rest in our frame; but then its erstwhile magnetic field would necessarily have vanished, the electric field alone remaining. We must assume, therefore, that the appearance of the magnetic field arises only because we have changed our Galilean frame of reference by passing from one fixed with respect to the charge to one in relative motion.
Now, the Lorentz-Einstein transformations tell us how our space and time measurements change under these circumstances; and by applying the transformations to the equations of electrodynamics we are able to ascertain how an electric field which is purely electric when viewed from one frame should appear to us when viewed from another Galilean frame. Proceeding in this way, we do in fact find that the electric field of the electrified body at rest will appear as an electromagnetic field when the body is set in relative motion. Likewise, the precise mathematical formula for the new field is obtained, proving [Pg 149] incidentally that our classical empirical formula was only approximate. Thus a rational justification for the appearance of a magnetic field round an electrified charge in motion is finally obtained, and this field is seen to be a direct consequence of the variations in those fundamental space and time forms of perception—variations which are expressed by the Lorentz-Einstein transformations.
In exactly the same way a magnetic pole in motion develops an electric field around it in addition to its magnetic field when at rest, and it is this phenomenon that gives rise to the electromagnetic induction on which we base our generators and dynamos. Here again the Lorentz-Einstein transformations throw complete light on the subject.[47]
In a general way the transformations show us that a field which appears to be exclusively electric or exclusively magnetic in one Galilean frame will appear to be electromagnetic in another frame, thereby compelling us to recognise the relativity of electric and magnetic forces. We shall see in due course that this relativity of force is general.
Let us consider another type of phenomenon which is also explained by the transformations. We refer to the Fresnel convection coefficient illustrated by Fizeau’s experiment. Experiment proves that the ray of light progressing through the running water is not carried along bodily by the water; it lags behind a little. Lorentz who, though he had discovered the transformations, did not recognise their general validity, sought to explain this phenomenon by postulating a suitable electronic constitution for the water. But Einstein by merely applying the transformations, without having to appeal to our highly problematic knowledge of the precise electronic constitution of matter, found that they anticipated Fizeau’s result with marvellous precision; for according to the transformations, velocities do not add up like numbers, as they did in classical science. It follows that the velocity of the light with respect to the tube must necessarily be less than the velocity of the light through the water plus the velocity of the water in the tube.[48]
Consider again the FitzGerald contraction. Here Lorentz thought it permissible to apply the transformations; but owing to the slight difference in their significance in his theory, he concluded that a body in motion was really contracted owing to its real motion through the ether. Although the observer carried along with the body could not detect the contraction, yet it was physically real and would be observed by the observer at rest in the ether. A similar [Pg 150] interpretation would have to be placed on the slowing down of phenomena. In Lorentz’s theory the difficulty consisted in accounting for an identical contraction manifesting itself in exactly the same way for all bodies, soft or hard. Lorentz again appeals to the electronic and atomic constitution of matter and has to take into consideration elastic forces.
With Einstein the explanation is simple. The contraction is due solely to a modification in our space and time measurements due to relative motion, and is completely irrelevant to the hardness or softness of the body, whose atomic or electronic structure need not be taken into consideration at all. In much the same way an object appears magnified under the microscope, and this magnification is independent of the body’s nature.
In short, the modifications are due to variations (as a result of relative motion) in our space and time measurements and perceptions, and in every case they are irrelevant to the microscopic constitution of matter. We see, then, that so far as all these curious modifications are concerned, Einstein s theory does not require any particular knowledge of the microscopic constitution and hidden mechanisms that are assumed to underlie matter. Herein resides one of the principal advantages of Einstein’s theory over that of Lorentz; for we know very little about the mysterious nature of electricity and matter, and were all progress to be arrested until such knowledge was forthcoming, we might have to wait many a day without result.
One of the chief difficulties attendant on an atomistic theory such as Lorentz’s was due precisely to this interpretation of observable effects in terms of invisible ones about which practically nothing could be known and where hypotheses ad hoc had to be invoked at each stage. As for the mathematical difficulties, they, of course, grew in proportion to the complexity of the hidden mechanisms which we had to postulate. Einstein’s theory is thus a return to universal principles induced from experiment, and in this respect is analogous to the physics of the general principles which appeared in the course of the nineteenth century. There again such comprehensive universal principles as those of entropy, of least action, of the conservation of energy, mass and momentum, took the place of the hidden mechanisms whereby the atomists had endeavoured to account for observed phenomena.
These periodic swings in the scientific viewpoint have always been necessitated by the continuous advance of knowledge. Both attitudes are fruitful and have yielded important results. In the present case Lorentz had progressed about as far as it was possible to go in his speculations on hidden mechanisms, and a theory such as Einstein’s was the necessary antidote to the increasing difficulties which were hampering further developments.
We have now to consider certain important consequences of Einstein’s theory which affect our traditional concepts of space and time. The FitzGerald contraction is no longer a real physical contraction, as it [Pg 151] was assumed to be in Lorentz’s theory. It no longer arises as a result of a perfectly concrete motion of a body through a stagnant ether; the stagnant ether with respect to which velocity acquired its significance having been banished by Einstein. The FitzGerald contraction is now solely due to the relative motion existing between the observer and the body observed.
If the observer remains attached to the body, there is no contraction; if the observer is moving with respect to the body, or the body moving with respect to him, the FitzGerald contraction appears. If the observer once more changes his velocity relative to the body, the FitzGerald contraction of the body will likewise change in magnitude. Nothing has happened to the body and yet its length has altered. Obviously, physical length is not what we once thought it to be; it can in no wise be immanent in the body, since a body has no determinate length until the relative motion of the observer has been specified. A length is therefore but the expression of a relationship between the observer and the observed, and the two partners of the relationship must be specified before the length can have any meaning.
In the same way the colour of an opal has no meaning. It is red from here, green from there, blue from elsewhere, and yet the opal has not changed. It is our position with respect to the opal which has changed, and the colour of the opal is indeterminate until such time as we have specified our relative position. In other words, length and the colour of the opal both express relations and not immanent characteristics.
Similar considerations apply to the slowing down of time. Duration, and time are mere relatives, mere expressions of relationships, and have no absolute significance per se. This does not mean that the duration we sense is a myth; for as our consciousness always accompanies our human body wherever we go, we always experience the same flow of time. All that is implied is that this rate of time-flow cannot be credited with any unique significance in nature, and a comparison of time-flows characteristic of various Galilean frames will reveal differences in the rate of flow.
We have mentioned elsewhere a further illustration of relativity, when discussing electric and magnetic forces. We showed how it was that an electric field, as such, was indeterminate; how according to the observer it would present itself as a purely electric or as an electromagnetic field. As all these conditions of observation are on the same footing, there is no sense in distinguishing between apparent fields and real fields, or apparent lengths and real lengths, or apparent durations and real durations. All these concepts, of themselves, are mere phantoms, to which substance can be given only when the conditions of observation are specified.
A certain number of lay philosophers have been confused by this continual reference to the observer in Einstein’s theory, and have assumed that all things occurred in the observer’s mind, the outside common objective world of science playing no part.[49] But this [Pg 152] extreme idealistic interpretation cannot be defended.
The word “observer” is a very loose term and does not necessarily mean a living human being. We might replace the observer’s eyes by a photographic camera, his computation of time-flow by a clock—in fact, all his senses and measurements by recording instruments of a suitable nature, whose readings any man situated in any frame could check later. The results would still be the same.
Now it might be feared that, with this vanishing of the absoluteness of such fundamental forms of perception as duration and distance, the entire objective world of science would sink to a shadow; and without a common objective world, science would be impossible. However, we need have no fear of any such catastrophe; for, as will be explained in later pages, a new common objective world of space-time will take the place of the classical one of space and time. But even at the present stage, without appealing to space-time, we can see that objectivity is not denied us for the following reasons:
When two different Galilean observers measure the same object, or time the duration of the same phenomenon, their computations will differ according to the rules specified in the Lorentz-Einstein transformations. It follows that the observers may infer that they are discussing the same objects, not when their respective measurements agree according to the classical standards of absolute distance and duration, but when they agree with the new standards set by the Lorentz-Einstein transformations. Thus a definite criterion of objectivity is still possible, though its form differs from that of classical science. So in spite of the indefiniteness of the concepts on which classical science was founded (space, time, force, mass), Einstein’s theory does not deprive us of the possibility of conceiving of the existence of a common outside world.
Enough has been said even at this stage to show that Einstein’s theory cannot be considered a mere mathematical dream. The extraordinary difficulties with which classical science was confronted owing to the negative results of the experiments we have mentioned (and to numerous other problems) have disappeared. With them have vanished the miraculous compensations which Lorentz was compelled to invoke in order to explain these negative results. The complete relativity of Galilean motion explains all our troubles. Nature is no longer mischievous, but the ether or space is relative for Galilean motion. We had failed to recognise this relativity which was staring us in the face; in so doing we ourselves were the creators of our difficulties. Yet rather than abandon the classical concepts of space and time, physicists in general refused to follow Einstein. To-day criticisms have subsided (on the part of the great majority of scientists) as experiment after experiment has confirmed Einstein’s previsions. The majority of experiments, however, concerned the consequences of the theory and not [Pg 153] its foundations. The experiments of de Sitter and Majorana have filled this gap by proving that a ray of light always passes the observer with the same invariant speed regardless of the relative velocity of observer and source.
Now one of the most important criticisms that has been directed against Einstein’s theory is that it deprives us of the possibility of conceiving of an objective ether. There is not the slightest doubt that Einstein’s theory compels us to abandon our conception of the stagnant ether of classical science, with reference to which motion could be measured. But this is not quite the same thing as holding that the theory banishes the ether entirely. Nevertheless, upon first inspection, the fact that whatever our Galilean motion may be, experiments conducted in our frame will always yield exactly the same results, would seem to relegate the ether to the realm of ghosts, making it a useless hypothesis. If this were the case we could no longer conceive of electromagnetic fields and light rays as expressing states of the ether, but should have to regard these fields as constituting independent realities of some new category, differing from matter but susceptible of existence in space without the support of a carrier, or without being the mere manifestations of its states. However, in the general theory which we shall discuss in the second part of this book we shall find that the ether is reinstated in the guise of the metrical field of space-time. But as this new ether has only its name in common with the stagnant Lorentzian ether, there does not appear to be much advantage in retaining the older appellation.
If, then, we wish to emphasise the great distinction between the classical view and the relativistic view, we must say: “According to classical science, the speed of light waves, and of electromagnetic waves generally, is constant in all directions with respect to the stagnant ether, hence also with respect to that particular observer who happens to be at rest in the stagnant ether. According to the relativistic point of view, the speed of light waves is constant throughout empty space when measured by any Galilean observer; that is, by any observer in any non-accelerated frame” (such frames being recognised by the absence of all centrifugal or inertial pushes and pulls).
For the present we are in no position to predict what would happen if the observer, instead of being posted in a Galilean frame, were situated in a non-Galilean, i.e., accelerated or rotating, frame. All the statements we have made up to this point concern only Galilean observers. Such are the restrictions which the special principle and the special theory of relativity impose upon us. It is most important to understand this fact, as many of the criticisms levelled at Einstein’s theory are due to a failure to grasp the point. As the theory now stands, acceleration and rotation remain absolute and are therefore excluded from the special principle of relativity, which refers solely to motions in space that are relative. These are Galilean motions.
Perhaps a definite illustration will make these points clearer. Consider, for example, Michelson’s latest experiment (not the celebrated one), or, again, consider Sagnac’s experiment. The essence [Pg 154] of both these experiments is to show that a ray of light travelling round the earth in the direction of the earth’s rotation requires a longer time to return to its starting point than would be the case for a ray travelling in the opposite direction. Obviously the velocity of the light waves with reference to the earth is not the same in all directions, so that we are able to detect the rotation of the earth on which we stand. The critic then infers that Einstein’s principle of relativity is upset by experiment. But the critic fails to realise that the motion that has been detected is a rotation, hence an acceleration, and that Einstein’s special principle confines itself to denying any significance to absolute velocities, that is, to motions which are not accelerated. Had this not been the case, Einstein’s principle would have been untenable since it is a fact of common knowledge that a large number of experiments (Foucault’s pendulum, the gyroscope, etc.) are capable of revealing the earth’s rotation. It is absolute velocity and not acceleration that experiment has ever obstinately refused to reveal.
A very similar criticism consists in stating that inasmuch as the speed of light with reference to the earth’s surface is greater from east to west than from west to east, the postulate of the invariant velocity of light is refuted. But once again the same error is involved. The postulate states that the velocity of light is invariant through any Galilean frame when measured by the observer in the frame. The postulate ceases to be true in an accelerated frame, and the rotating earth does constitute an accelerated frame. True, in the negative experiments, the earth was treated as a Galilean frame; but it was always stated that this attitude was only approximately correct, and that experiments could be devised to detect the earth’s acceleration. However, the effects due to the earth’s absolute velocity (which was then assumed to exist) would have been so much more pronounced than those due to acceleration that as a first approximation we could afford to neglect these weaker effects of acceleration.
There is still another point on which we must insist, as it has given rise to a number of criticisms. The Newtonian principle of relativity stated that the velocity of matter through empty space was meaningless or relative. Einstein’s special theory of relativity, on the other hand, compels us to consider the velocity of light as an absolute. How, then, can Einstein’s special principle be a mere extension to electrodynamics or to the ether of the Newtonian principle?
To answer this question we must realise that although velocity was a relative in Newtonian science, yet there did exist one definite velocity which was assumed to be absolute. This was the infinite velocity. It was assumed that a velocity that was infinite or instantaneous for one observer would remain infinite or instantaneous for all other observers. So far, therefore, as velocity is concerned, the sole difference between Einstein and Newton is that with Einstein the absolute or invariant velocity is no longer infinite. Though very great (186,000 miles a second), it is now finite. It is this [Pg 155] difference between the invariant velocities of Newton and Einstein which is responsible for all the major differences between classical and Einsteinian science, as will be explained more fully in a following chapter. In particular, it is this finiteness of the invariant velocity which precludes us from attaching any absolute value to shape and distance.
Viewing the question in another way, we should notice that as a matter of fact the Newtonian principle of relativity merely states that velocity of matter with respect to matter alone has physical significance. Velocity of matter through empty space is physically meaningless. It is the same in Einstein’s special principle. Einstein’s theory proves (see next chapter) that molar matter can never move with the absolute speed of light. We are therefore perfectly justified in saying that the velocity of matter remains essentially relative, since it can never attain that critical velocity (i.e. that of light) which is absolute.
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CHAPTER XIV RELATIVISTIC MECHANICS
WE saw in the preceding chapter that the time and space variables which entered into the Lorentz-Einstein transformations should be considered of absolutely general validity. They were assumed to represent the duration actually sensed and lived by the observer and the spatial measurements actually determined by his rigid rods in his Galilean frame. It was no longer a case, as in Lorentz’s theory, of supposing that these space and time variables applied only in certain electromagnetic cases and not in others. They were now of absolutely general application, and in particular should hold for all mechanical phenomena.
But this realisation brought us face to face with a peculiar difficulty. Einstein’s fundamental postulate was that Galilean motion was meaningless, or at least could never be detected by experiment—whether of a mechanical or electromagnetic nature. Now the complete relativity of Galilean motion is expressed mathematically by the invariance of form of the laws of nature for transformations corresponding to that type of motion. The classical laws of mechanics satisfied this requirement in classical science, since they remained invariant when submitted to the classical transformations. But if we accept Einstein’s views, these laws must remain invariant no longer to the classical transformations but to the Lorentz-Einstein ones.
The classical laws of mechanics did not fulfil this new condition of invariance. From this it would appear that the relativity of Galilean motion which Einstein had succeeded in establishing for electrodynamic and optical experiments could be maintained only at the price of abandoning this same relativity for mechanical ones. What had been gained on one side had been lost on the other. Inasmuch as the laws of mechanics would certainly have to remain invariant to the Lorentz-Einstein transformations if absolute velocity were to remain meaningless, and inasmuch as the classical laws did not satisfy this condition, there was only one way out of the difficulty. It was necessary to assume that the classical laws were incorrect.
Accordingly Einstein was compelled so to modify the classical laws of mechanics as to ensure their invariance to the Lorentz-Einstein transformations. This modification was then found to entail the relativity of mass. By this we mean that the mass of a given body could no longer be regarded as an invariant, independent of the body’s relative motion. On the contrary, it was now found to increase with the relative velocity of the body, becoming infinite when the velocity of light was reached. In the same way the relativity of force was [Pg 157] established, and classical Newtonian mechanics, with its invariance of mass and force, was proved to be incorrect, and acceptable only as a first approximation for slow velocities.
It also followed that no material body could ever move with a speed greater than or even equal to the speed of light, since it would require a force of infinite magnitude to give it this critical velocity. A number of paradoxes associated with the theory are due to the critic’s initial assumption that the observer is moving with the speed of light. But all these paradoxes may be dismissed at the outset, since no observer could ever move with such a speed.
When the mathematical expression of the mass of a body in relative motion was considered, it was found to be decomposable into several parts. First of all we had the mass of the body as at rest (this was the mass which classical science assumed to be invariant, i.e., independent of the body’s motion). To this was added the vis viva, or energy of motion of the body with respect to the observer, as well as a number of other terms depending on the relative velocity. These last terms were, however, of very minor importance unless the relative motion of the body was considerable. Thus the mass of a body in relative motion was increased over its mass at rest by its energy of motion or vis viva, so that the distinction between mass and energy became obscured.
Since part of the mass of the body when in motion was due to its vis viva, it appeared probable that the entire mass of the body, whether at rest or in motion, was due to its energy of one kind or another. Now matter even at rest was known to contain energy, as exhibited by radioactive phenomena and by the thermal effects produced by chemical combinations. So it was natural to assume that the mass of the body at rest was due to its contained or bound energy, which would be released were the body to disintegrate into nothingness. The formula connecting this energy of constitution of the body with its mass was obtained from the mathematical expression of the mass, and it was found that in C.G.S. units the contained energy was equal to the mass multiplied by the square of the velocity of light. From this formula it is easy to realise the tremendous liberation of energy and the dire consequences that would result from the complete disintegration of even a small parcel of matter.
Another consequence of these discoveries was that any variation in the internal energy of a body maintained at rest, such as would result from compressing it, heating it or electrifying it, would be accompanied by an increase in its mass at rest.[50] This discovery enabled us to account for a curious discrepancy in the comparative masses of [Pg 158] the helium and of the hydrogen atom, a mystery which had never been accounted for by classical science.
To illustrate: The nucleus of the hydrogen atom is composed of one proton; and the nucleus of the helium atom is composed of four protons, held together by two electrons. The protons alone having a mass of any importance, it seemed logical to assume that the mass of the helium atom would be four times that of the hydrogen atom. Accurate measurements proved, however, that there was a slight deficiency, and that the mass of the helium atom was less than four times that of the hydrogen atom. Classical science had recognised that the closer the electrons and protons were packed together in the nucleus, the smaller would be the electric energy of the atom, but it appeared impossible to account for the deficiency in mass. The theory of relativity by its identification of mass and energy gave an obvious solution to the problem. The deficiency in mass of the helium atom was to be ascribed to the loss of energy radiated under the form of heat and light when the electrons and protons were being packed together. At the same time it became possible to calculate with precision from the loss of mass the precise amount of energy that had been liberated. It followed that in regions of the cosmos where the helium atoms were in process of formation owing to the grouping together of the protons and electrons, enormous amounts of heat and light would be radiated continually. This consequence of the theory enabled astrophysicists to account for the enormous temperature of the sun, maintained through thousands of millions of years in spite of incessant radiation in a frigid space. Helium atoms are in constant process of formation in the sun, and in this fact resides the source of its apparently inexhaustible supply of energy.
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Through this constant radiation of energy, the sun and stars would be gradually losing in mass; for light was of course a form of energy. As such, light should possess mass and momentum and exert a pressure over bodies on which it impinged. This pressure, indeed, was not unknown, since it resulted from Maxwell’s equations and had been verified experimentally by Lebedew and by Nichols and Hull. As a matter of fact it was to this pressure of light that the apparent repulsion of comets’ tails away from the sun was supposed to be due. Nevertheless, though not a new discovery, the fact that Einstein’s theory entailed momentum for light constituted one of the many indirect confirmations of the theory.
An objection which has often been raised by non-mathematical students is the following: They argue that since energy moving with the velocity of light must possess an infinite mass, we should be crushed under the tremendous impact of light rays falling on our bodies. In an elementary book of this sort we cannot go into mathematical details, but we may say that the argument is fallacious. It is only ponderable matter constituted by bound energy which would become infinite in mass when moving with the velocity of light. Light rays represent another type of energy, known as free or loose energy, and calculation shows that the momentum produced by this form of energy moving with the velocity of light is not infinite, but exceedingly minute.
Of course the theory of relativity is not one of pure mathematics. Einstein was not solely interested in juggling with equations and laws to make them fit into his scheme. The final verdict of the correctness of a theory of mathematical physics must always rest with the experimenter, and it remained to be seen whether the Einsteinian equations of mechanics corresponded more accurately to facts than did the classical ones. We may state that the most precise experiments have proved the correctness of the Einsteinian laws of mechanics and that Bucherer’s experiment proving the increase in mass of an electron in rapid motion is a case in point.[51]
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It should be clear by now that very important differences distinguish the theory of Einstein from that of Lorentz. Lorentz also had deduced from his theory that the mass of an electron should increase and grow infinite when its speed neared that of light; but the speed in question was the speed of the electron through the stagnant ether; whereas in Einstein’s theory it is merely the speed with respect to the observer. According to Lorentz, the increase in mass of the moving electron was due to its deformation or FitzGerald contraction. The contraction modified the lay of the electromagnetic field round the electron; and it was from this modification that the increase in mass observed by Bucherer was assumed to arise. In Einstein’s theory, however, the increase in mass is absolutely general and need not be ascribed to the electromagnetic field of the electron in motion. An ordinary unelectrified lump of matter like a grain of sand would have increased in mass in exactly the same proportion; and no knowledge of the microscopic constitution of matter is necessary in order to predict these effects, which result directly from the space and time transformations themselves.
Furthermore, the fact that this increase in mass of matter in motion is now due to relative motion and not to motion through the stagnant ether, as in Lorentz’s theory, changes the entire outlook considerably. According to Lorentz, the electron really increased in mass, since its motion through the ether remained a reality. According to Einstein, the electron increases in mass only in so far as it is in relative motion with respect to the observer. Were the observer to be attached to the flying electron no increase in mass would exist; it would be the electron left behind which would now appear to have suffered the increase. Thus mass follows distance, duration and electromagnetic field in being a relative having no definite magnitude of itself and being essentially dependent on the conditions of observation.
Owing to the general validity of the Lorentz-Einstein transformations, it becomes permissible to apply them to all manner of phenomena. In this way it was found that temperature, pressure and many other physical magnitudes turned out to be relatives. On the other hand, entropy, electric charge and the velocity of light in vacuo were absolutes transcending the observer’s motion. Later on we shall see that a number of other entities are found to be absolutes, the most important of which is that abstract mathematical quantity called the Einsteinian interval, which plays so important a part in the fabric of the new objective world of science, the world of four-dimensional space-time.
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