A PHYSICAL law is clothed in mathematical language by setting up a formula. This comprises, and represents in the form of an equation, all measurements which numerically describe the event in question. We make use of such formulæ, not only in cases in which we have the means of checking the results of our calculations at any moment actually at our disposal, but also when the corresponding measurements cannot really be carried out in practice, but have to be imagined, i.e. only take place in our minds: e.g. when we speak of the distance of the moon from the earth, and express it in metres, as if it were really possible to measure it by applying a metre-rule end to end.
By means of this expedient of analysis we have extended the range of exact scientific research far beyond the limits of measurement actually accessible in practice, both in the matter of immeasurably large, as well as in that of immeasurably small, quantities. Now, when such a formula is used to describe an event, symbols occur in it that stand for those quantities which are, in a certain sense, the ground elements of the measurements, with the help of which we endeavour to grip the event; thus, for example, in the case of all spatial measurements, symbols for the "length" of a rod, the "volume" of a cube, and so forth. In creating these ground elements of spatial elements we had hitherto been led by the idea of a rigid body which was to be freely movable in space without altering any of its dimensional relationships. By the repeated application of a rigid unit measure along the body to be measured we obtained information about its dimensional relationships. This idea of the ideal rigid measuring rod, which is only partially realizable in practice, on account of all sorts of disturbing influences such as the expansion due to heat, represents the fundamental conception of the geometry of measure.
The discovery of suitable mathematical terms, which can be inserted in a formula as symbols for definite physical magnitudes of measurements, such as e.g. length of a rod, volume of a cube, etc., in order to shift the responsibility, as it were, for all further deductions upon analysis, is one of the fundamental problems of theoretical physics and is intimately connected with the two postulates enunciated in § 2.
To realize this fully, we must revert to the foundations of geometry, and analyse them from the point of view adopted by Helmholtz in various essays, and by Riemann in his inaugural dissertation of 1854: "On the hypotheses which lie at the bases of geometry." Riemann points almost prophetically to the path now taken by Einstein.
(a) THE LINE-ELEMENT IN THE THREE-DIMENSIONAL MANIFOLD OF POINTS IN SPACE, EXPRESSED IN A FORM COMPATIBLE WITH THE TWO POSTULATES
Every point in space can be singly and unambiguously defined by the three numbers , , , which may be regarded as the co-ordinates of a rectangular system of co-ordinates, and which distinguish it from all other points; a continuous variation of these three numbers enables us to specify every single point of space in turn. The assemblage of points in space represents, in Riemann's notation, "a multiply extended magnitude" (an -fold manifoldness or manifold) between the single elements (points) of which a continuous transition is possible. We are familiar with diverse continuous manifolds, e.g. the system of colours, of tones and various others. A feature which is common to all of them is that, in order to specify a single element out of the entire manifold (to define a particular point, a particular colour, or a particular tone), a characteristic number of magnitude-determinations, i.e. co-ordinates, is required: this characteristic number is called the dimensions of the respective manifold. Its value is three for space, two for a plane, one for a line. The system of colours is a continuous manifold of the dimension three, corresponding to the three "primary" colours, red, green, and violet, by mixing which in due proportions every colour can be produced.
But the assumption of continuity for the transition from one element to another in the same manifold, and the determination of the dimensions of the latter, does not give us any information about the possibility of comparing limited parts of the same manifold with one another, e.g. about the possibility of comparing two tones with one another or two single colours; i.e. nothing has yet been stated about the metric relations (measure-conditions) of the manifold, about the nature of the scale, according to which measurements can be undertaken within the manifold. In order to be able to do this, we must allow experience to give us the facts from which to establish the metric (measure-) laws which hold for each particular manifold (space-points, colours, tones) under various physical conditions; these metric laws will be different according to the set of empirical facts chosen for this purpose.[3]
In the case of the manifold of space-points, experience has taught us that finite rigid point-systems can be freely moved in space without altering their form or dimensions; the conception of "congruence" which has been derived from this fact, has become a vital factor for a measure-determination.[4] It sets us the problem of building up a mathematical expression from the numbers , , , and , , , which are assigned to two definite points in space, and which we may imagine as the end-points of a rigid measuring rod, such that this expression may be regarded as a measure of the distance between them, that is, as an expression for the length of the rod, and may be introduced as such into the formulae expressing physical laws.
The equations of physical laws, which—in order to fulfil the conditions of continuity—must be differential laws, contain only the distances , of infinitely near points, so-called line-elements. We must, therefore, inquire whether our two postulates of § 2 have any influence upon the analytical expression for the line-element , and, if so, which expression for the latter is compatible with both. Riemann demands of a line-element in the first place that it can be compared in respect to its length with every other line-element independently of its position and direction. This is a distinguishing characteristic of the metric conditions ("measure relations") prevalent in space; in practice it denotes that the rods must be freely movable. This peculiarity does not exist, for instance, in the manifold of tones or in that of colours (vide Note 7). Riemann formulates this condition in the words, "that lines must have a length independent of their position and that every line is to be measurable by means of any other." He then discovers that: if , , and , , respectively denote two infinitely near points in space and if the continuously variable numbers , , are any co-ordinates whatsoever (not e.g. necessarily rectilinear), then the square root of an always positive, integral, homogeneous function of the second degree in the differentials , , has all the properties[5] which the line-element, being the expression for the length of an infinitely small rigid measuring rod, must exhibit. We thus find that in which the coefficients are continuous functions of the three variables , , , gives us an expression for the line-element at the point , , .
In this expression no assumptions are made concerning the nature of the co-ordinates that are represented by the three variables, , , , that is, concerning particular metrical properties of the manifold that go beyond the postulate of the freedom of movement of the measuring rods. But, if we demand, in particular, that each point of the manifold may be fixed by means of rectangular Cartesian co-ordinates, whereby particular assumptions are made concerning the possible ways of placing the measuring rods, then the line-element, expressed in these special variables, assumes the form Hitherto this expression has always been introduced for the length of the line-element in all physical laws. It is contained in the more general expression of Riemann's line-element as the special case By restricting ourselves to this special form of the line-element we are enabled to use the measure laws of Euclidean geometry in all our space-measurements.
But this particular assumption concerning the metrical constitution of space contains the hypothesis, as Helmholtz has shown in a detailed discussion, that finite rigid point-systems, i.e. finite fixed distances, are capable of unrestrained motion in space, and can be made (by superposition) to coincide with other (congruent) point-systems. With respect to the postulate of continuity, this hypothesis seems inconsistent, in so far as it introduces implicit statements about finite distances into purely differential laws, in which only line-elements occur; but it does not contradict the postulate.
The postulate of the relativity of all motion adopts a different attitude towards the possibility of giving the line-element the Euclidean form in particular.[6]
[6]Strictly speaking, I should at this juncture state in anticipation that the above investigations can manifestly also be so generalized as to be valid for the four-dimensional space-time manifold, in which all events actually take place, and that the transformation-formulæ apply to the four variables of this manifold. In these general remarks the neglect of the fourth dimension is of no importance. This statement will be justified later in § 3(b).
According to the principle of the relativity of all motions, all systems, which come about owing to relative motions of bodies towards one another, may be regarded as fully equivalent. The laws of physics must, therefore, preserve their form in passing from one such system to another; i.e. the transformation-formulæ of the variables , , which perform this transition to another system, must not alter the analytical expression for the physical law under consideration.
This leads us to set up a principle of relativity which will be called the general principle of relativity in the sequel. It demands the invariance of physical laws with respect to arbitrary continuous substitutions of the four variables. Moreover, the line-element that occurs in it must preserve its form when subjected to any arbitrary transformations whatsoever. This condition is fully satisfied by the line-element in which no restrictive reservations of any description are made as to what the co-ordinates , , are to signify. The Euclidean line-element on the other hand, preserves its form only for transformations of the special theory of relativity, which confine themselves to systems moving uniformly and rectilinearly. Consequently, the element of arc must be adapted to the further requirements of a general theory of relativity so that it preserves its form after any substitutions whatsoever.
The choice of the expression to represent the line-element in physical laws is, in spite of its very general character, still to be regarded as a hypothesis, as Riemann has already pointed out. For there are other functions of the differentials , , —such as e.g. the fourth root of a homogeneous differential expression of the fourth degree in these variables—which could provide a measure for the length of the line-element (vide Note 9). But at present there is no ground for abandoning the simplest general expression for the line-element (viz. that of the second degree), and adopting more complicated functions. Within the range (of fulfilment) of the two postulates, which we have imposed upon every description of physical events, the former expression for satisfies all requirements. Nevertheless, it must never be forgotten that the choice of an analytical expression for the line-element always contains a hypothetical factor; and it is the duty of the physicist to remain fully conscious of this fact at all times, without being in any way prejudiced. It is for this reason that Riemann closes his essay with the following remarks, which impress one particularly with their great importance for the present time:[7]
[7]B. Riemann, Über die Hypothesen, welche der Geometrie zugrunde liegen. New edn., annotated by H. Weyl, Berlin: Springer & Co., 1919.
"The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. In this question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete[8] manifold, the principle or character of its metric relations is already given in the notion of the manifold, whereas in a continuous manifold this ground has to be found elsewhere, i.e. has to come from outside.
"Either, therefore, the reality which underlies space must form a discrete[9] manifold, or we must seek the ground of its metric relations (measure-conditions) outside it, in binding forces which act upon it.
"A decisive answer to these questions can be obtained only by starting from the conception of phenomena which has hitherto been justified by experience, and of which Newton laid the foundation, and then making in this conception the successive changes required by facts which admit of no explanation on the old theory; researches of this kind, which commence with general notions, cannot be other than useful in preventing the work from being hampered by too narrow views, and in keeping progress in the knowledge of the inter-connections of things from being checked by traditional prejudices.
"This carries us over into the sphere of another science, that of physics, into which the character and purpose of the present discussion will not allow us to enter."
That is to say: according to Riemann's view these questions are to be solved by starting from Newton's view of physical phenomena, and compelled by facts which do not allow of any explanation by it, gradually remoulding it. This is what Einstein has done. The "binding forces," to which Riemann points, will be found again in Einstein's theory. As we shall see in the fifth chapter, Einstein's theory of gravitation is based upon the view that the gravitational forces are the "binding forces," i.e. they represent the "inner ground" of the metric conditions (measure-relations) in space.
(b) THE LINE-ELEMENT IN THE FOUR-DIMENSIONAL MANIFOLD OF SPACE-TIME POINTS, EXPRESSED IN A FORM COMPATIBLE WITH THE TWO POSTULATES
The measure-conditions, which we were to take as a basis for the formulation of physical laws, could have been treated immediately in connection with the four-dimensional manifold of space-time points. For the special theory of relativity has led us to make the important discovery that the space-time-manifold has uniform measure-relations in its four dimensions. Nevertheless, I wish to treat time-measurements separately; for one reason that it is just this result of the relativity-theory which has experienced the greatest opposition at the hands of supporters of classical mechanics; and for another that classical mechanics is also obliged to establish certain conditions about time-measurement, but that it never succeeded in establishing agreement on this point. The difficulties with which classical mechanics had to contend are contained in its fundamental conceptions. The law of inertia, particularly, was a permanent factor of discord that caused the foundations of mechanics to be incessantly criticised. And since the foundations of time-measurement had been brought into close relationship with the law of inertia, these critical attacks applied to them likewise.
In Galilei's law of inertia, a body which is not subject to external influences continues to move with uniform motion in a straight line. Two determining elements are lacking, viz. the reference of the motion to a definite system of co-ordinates, and a definite time-measure. Without a time-measure one cannot speak of a uniform velocity.
Following a suggestion by C. Neumann,[10] the law of inertia has itself been adduced to give a definition of a time-measure in the form: "Two material points, both left to themselves, move in such a way that equal lengths of path of the one correspond to equal lengths of path of the other." On this principle, into which time-measure does not enter explicitly, we can define "equal intervals of time as such, within which a point, when left to itself, traverses equal lengths of path."
This is the attitude which was also taken up by L. Lange, H. Seeliger, and others, in later researches. Maxwell selected this definition too (in "Matter and Motion"). On the other hand, H. Streintz[11] (following Poisson and d'Alembert) has demanded the disconnection and independence of the time-measure from the law of inertia, on the ground that the roots of the time-concept have a deeper and more general foundation than the law of inertia. According to his opinion, every physical event, which can be made to take place again under exactly the same conditions, can serve for the determination of a time-measure, inasmuch as every identical event must claim precisely the same duration of time; otherwise, an ordered description of physical events would be out of the question. In point of fact, the clock is constructed on this principle. It is this principle which enables an observer to undertake a time-measurement at least for his place of observation.
The reduction of time-measurements to a dependence upon the law of inertia, on the other hand, leads to an unobjectionable definition of equal lengths of time; but the measurement of the equal paths traversed by uniformly moving bodies, and the establishment of a unit of time involved therein, are only then possible for a place of observation, when the observer and the moving body are in constant connection, e.g. by light-signals. It cannot, however, be straightway assumed that two observers, who are in rectilinear motion relatively to one another, and, therefore, according to the law of inertia, equivalent as reference systems, would in this manner gain identical results in their time-measurements. Poisson's idea thus leads to a satisfactory time-measurement for a given place of observation itself; i.e. in a certain sense it allows the construction of a clock for that place. But it does not broach the question of the time-relations of different places with one another at all; whereas Neumann's suggestion leads directly to those questions which have been a centre of discussion since Einstein's enunciation of the relativity-principle.
In the endeavour to reduce classical mechanics to as small a number of principles as possible, in perfect agreement with one another, writers resorted to ideal-constructions and imaginary experiments.
Yet no one conceived the idea that in fixing a unit of time on the basis of the law of inertia, that is, by measuring a length (the distance traversed), the state of motion of the observer might exert an influence. It was assumed that the data obtained from the necessary observations had an absolute meaning quite independent of the conditions of observation when simultaneous moments were chosen and a length was evaluated. As Einstein has shown, however, this is not the case. Rather, this recognition of the relativity of space- and time-measurements formed the starting point of his principle of relativity (Note 13). It is a necessary consequence of the universal significance of the velocity of light, of which we spoke in the first section. Its recognition furnished us at once with the correct formulae of transformation, allowing us to relate the space-time measurements of systems moving uniformly and rectilinearly with respect to each other, and this is what we are concerned with in Neumann's suggestion of fixing a measure of time with the aid of the law of inertia. In the new equations of transformation, ' is not identically equal to , but rather The time-measurements in the second system which is moving relatively to the first are thus essentially conditioned by the velocity of each relative to the other. Consequently, the fixing of a measure of time on the basis of the law of inertia, as proposed by Neumann, does not at all lead to the result that the time-measurements are entirely independent of the state of motion of the systems with respect to each other, as assumed in classical mechanics. Only when the researches of Einstein concerning the special theory of relativity had been carried out, did the fundamental assumptions of our time-measurements become fully cleared up, and thus a serious shortcoming in classical mechanics was made good.
That such a fundamental revision of the assumptions made regarding time-measurements became necessary only after so great a lapse of time, is to be explained by the fact that even the velocities which occur in astronomy are so small, in comparison with the velocity of light, that no serious discrepancies could arise between theory and observation. So it occurred that the weaknesses of the theory—in particular, those due to the motional relations of various systems to one another—did not come to light until the study of electronic motions, in which velocities of the order of that of light occur, proved the insufficiency of the existing theory.
The details of the effects, which result from the relativity of space-time measurements, have so frequently been discussed in recent years that it is only possible to repeat what has already often been said. The essential point in the discussion of this section is the recognition of the fact that space and time represent a homogeneous manifold of "four" dimensions, with homogeneous measure-relations (vide Note 14). Consequently, to be consistent, we must apply .the arguments of the preceding § 3(a) about the measure-relations to the four-dimensional space-time-manifold; and, in view of the two fundamental postulates (1) of continuity and (2) of relativity, and including the time-measurement as the fourth dimension, we must select for our line-element the expression: in which the () are functions of the variables , , , .
Hitherto we have been led to adopt this much more general attitude towards the questions of the metric laws involved in physical formulæ merely by the desire not to introduce, from the very outset, more assumptions into the formulations of physical laws than are compatible with both postulates, and to bring about a deeper appreciation of the points of view, to which the special theory of relativity has led us.
We can briefly summarize by saying: the adoption of Euclidean metric-conditions (measure-relations) is compatible with the postulate of continuity; though the special assumptions thereby involved appear as restrictive or limiting hypotheses, which need not be made. But the second postulate, the reduction of all motions to relative motions, compels us to abandon the Euclidean measure-determination (cf. p. 43). A description of the difficulties still remaining in mechanics will make this step clear.