THE FOUNDATIONS OF
EINSTEIN'S THEORY OF
GRAVITATION

ERWIN FREUNDLICH

DIRECTOR OF THE EINSTEIN TOWER

WITH A PREFACE BY

ALBERT EINSTEIN

TRANSLATED FROM THE FOURTH GERMAN EDITION,
WITH TWO ESSAYS, BY

HENRY L. BROSE

CHRIST CHURCH, OXFORD

WITH AN INTRODUCTION BY

H. H. TURNER, D.Sc., F.R.S.

SAVILIAN PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF OXFORD

WITH FIVE DIAGRAMS

NEW YORK
E. P. DUTTON AND COMPANY
PUBLISHERS

PREFACE

DR. FREUNDLICH has undertaken in the following essay to illumine the ideas and observations which gave rise to the general theory of relativity so as to make them available to a wider circle of readers.

I have gained the impression in perusing these pages that the author has succeeded in rendering the fundamental ideas of the theory accessible to all who are to some extent conversant with the methods of reasoning of the exact sciences. The relations of the problem to mathematics, to the theory of knowledge, physics and astronomy are expounded in a fascinating style, and the depth of thought of Riemann, a mathematician so far in advance of his time, has in particular received warm appreciation.

Dr. Freundlich is not only highly qualified as a specialist in the various branches of knowledge involved to demonstrate the subject; he is also the first amongst fellow-scientists who has taken pains to put the theory to the test.

May his booklet prove a source of pleasure to many!

A. EINSTEIN

[INTRODUCTION]

THE Universe is limited by the properties of light. Until half a century ago it was strictly true that we depended upon our eyes for all our knowledge of the universe, which extended no further than we could see. Even the invention of the telescope did not disturb this proposition, but it is otherwise with the invention of the photographic plate. It is now conceivable that a blind man, by taking photographs and rendering their records in some way decipherable by his fingers, could investigate the universe; but still it would remain true, that all his knowledge of anything outside the earth would be derived from the use of light and would therefore be limited by its properties. On this little earth there is, indeed, a tiny corner of the universe accessible to other senses: but feeling and taste act only at those minute distances which separate particles of matter when "in contact:" smell ranges over, at the utmost, a mile or two; and the greatest distance which sound is ever known to have travelled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth's girdle. The scale of phenomena manifested through agencies other than light is so small that we are unlikely to reach any noteworthy precision by their study.

Few people who are not astronomers have spent much thought on the limitations introduced by the news agency to which we are so profoundly indebted. Light comes speedily but has far to travel, and some of the news is thousands of years old before we get it. Hence our universe is not co-existent: the part close around us belongs to the peaceful present, but the nearest star is still in the midst of the late War, for our news of him is three years old; other stars are Elizabethan, others belong to the time of the Pharaohs; and we have alongside our modern civilization yet others of prehistoric date. The electric telegraph has accustomed us to a world in which the news is approximately of even date: but our forefathers must have been better able, from their daily experience of getting news many months old, to realize the unequal age of the universe we know. Nowadays the inequality is almost entirely the concern of the astronomer, and even he often neglects or forgets it. But when fundamental issues are at stake, the time taken by the messenger is an essential part of the discussion, and we must be careful to take account of it, with the utmost precision.

Our knowledge that light had a finite velocity followed on the invention of the telescope and the discovery of Jupiter's satellites: the news of their eclipses came late at times and these times were identified as those when Jupiter was unusually far away from us. But the full consequences of the discovery were not realized at first. One such consequence is that the stars are not seen in their true places, that is in the places which they truly held when the light left them (for what may have happened to them since we do not know at all—they may have gone out or exploded). Our earth is only moving slowly compared with the great haste of light: but still she is moving, and consequently there is "aberration"—a displacement due to the ratio of the two velocities, easy enough to recognize now, but so difficult to apprehend for the first time that Bradley spent two years in worrying over the conundrum presented by his observations before he thought of the solution. It came to him unexpectedly, as often happens in such cases. In his own words—"at last when he despaired of being able to account for the phenomena which he had observed, a satisfactory explanation of them occurred to him all at once when he was not in search of it." He accompanied a pleasure party in a sail upon the river Thames. The boat in which they were was provided with a mast which had a vane at the top of it. It blew a moderate wind, and the party sailed up and down the river for a considerable time. Dr. Bradley remarked that every time the boat put about, the vane at the top of the boat's mast shifted a little, as if there had been a slight change in the direction of the wind. The sailors told him that this was due to the change in the boat, not the wind: and at once the solution of his problem was suggested. The earth running hither and thither round the sun resembles the boat sailing up and down the river: and the apparent changes of wind correspond to the apparent changes in direction of the light of a star. But now comes a point of detail—does the vane itself affect the wind just round it? And, similarly, does the earth itself by its movement affect the ether just round it, or the apparent direction of the light waves? This question suggested the famous Michelson and Morley experiment (Phil. Mag., Dec. 1887). It is curious to think that in the little corner of the universe represented by the space available in a laboratory an experiment should be possible which alters our whole conceptions of what happens in the profoundest depths of space known to us, but so it is. The laboratory experiment of Michelson and Morley was the first step in the great advance recently made. It discredited the existence of the virtual stream of ether which is the natural antithesis to the earth's actual motion. It was, indeed, open to question whether restrictions of a laboratory might not be responsible for the result: for the ether stream might exist, but the laboratory in which it was hoped to detect it might be in a sheltered eddy. When bodies move through the air, they encounter an apparent stream of opposing air, as all motorists know: but by using a glass screen shelter from the stream can be found. And even without such special screening, there may be shelter. When a pendulum is set swinging in ordinary air, it is found from experiments on clocks that it carries a certain amount of air along with it in its movement, although the portion carried probably clings closely to the surface of the pendulum. A very small insect placed in the region might be unable to detect the streaming of the air further out. In a similar way it seemed possible that as the earth moved through the ether such tiny insects as the physicists in their laboratories might be in a part of the ether carried along with the earth, in which they could not detect the streaming outside. But another laboratory experiment, this time by Sir Oliver Lodge, discredited this explanation, and it was then suggested as an alternative that distances were automatically altered by movement.

It may be well to explain briefly the significance of this alternative. The Michelson-Morley experiment depended on the difference between travelling up and down stream, and across it. To use a few figures may be the quickest way of making the point clear. Suppose a very wide, perfectly smooth stream running at 3 miles an hour, and that oarsmen are to start from a fixed point

in midstream, row out in any direction to a distance of 4 miles from

, and back again to the starting-point

. Which is the best direction to choose? We shall probably all agree that it will be either directly up and down stream, or directly across it, and we may confine attention to these two directions. First suppose an oarsman

starts straight across stream. To keep straight he must set his boat at an angle to the stream. If he reaches his 4 mile limit in an hour, the stream has been virtually carrying him down 3 miles in a direction at right angles to his course: and the well-known relation between the sides of a right-angled triangle tells us that he has effectively pulled 5 miles in the hour. It will take him similarly an hour to come back, and the total journey will involve an effective pull of 10 miles.

Now suppose another oarsman,

, of equal skill elects to row up stream. In two hours he could pull 10 miles if there were no stream; but since meantime the stream has pulled him back 6 miles by "direct action" he will have only just reached the 4 mile limit from the start, and has still his return journey to go. No doubt he will accomplish this pretty quickly with the stream to help him, but his antagonist has already got home before he begins the return. We might have let him do his quick journey down stream first, but it is easy to see that this would gain him no ultimate advantage.

Michelson and Morley sent two rays of light on two journeys similar to those of the oarsmen

and

. The stream was the supposed stream of ether from east to west which should result from the earth's movement of rotation from west to east. They confidently expected the return of

before that of

, and were quite taken aback to find the two reaching the goal together. In the aquatic analogy of which we have made use, it would no doubt be suspected that

was really the faster oar, which might be tested by interchanging the courses; but there are no known differences in the velocity of light which would allow of a parallel explanation. There was, however, the possibility that the distances had been marked wrongly, and this was tested by interchanging them, without altering the "dead-heat."

Now there are several alternative explanations of this result. One is that the ether does not itself exist, and therefore there is no stream of it, actual or apparent; and it is to this sweeping conclusion that modern reasoning, following recent experiments and observations, is tending. The possibility of saving the ether by endowing it with four dimensions instead of three is scarcely calculated to satisfy those who believed (until recently) that we knew more about the ether than about matter itself. They saved the ether for a time by an automatic shortening of all bodies in the direction of their movement, which explained the dead-heat puzzle. With the velocities used above, the goal attained by

must be automatically moved

of a mile nearer the starting-point, so that

only rows

miles out and back instead of 4 miles. So gross a piece of cheating would enable

to make his dead-heat, but could scarcely escape detection. The shortening of the course required in the case of light is very minute indeed, because the velocities of the heavenly bodies are so small compared with that of light. If they could be multiplied a thousand times we might see some curious things, but we have no actual experience to guide a forecast.

It is a great triumph for Pure Mathematics that it should have devised a forecast for us in its own peculiar way. Starting from axioms or postulates, Einstein, by sheer mathematical skill, making full use of the beautiful theoretical apparatus inherited from his predecessors, pointed ultimately to three observational tests, three things which must happen if the axioms and postulates were well founded. One of the tests—the movement of the perihelion of Mercury's orbit—had already been made and was awaiting explanation as a standing puzzle. Another—a displacement of lines in the spectrum of the sun—is still being made, the issue being not yet clear.

The third suggestion was that the rays of light from a star would be bent on passing near the sun by a particular amount, and this test has just provided a sensational triumph for Einstein. The application was particularly interesting because it was not known which of at least three results might be attained. If light were composed of material particles as Newton suggested, then in passing the sun they would suffer a natural deflection (the use of the adjective is an almost automatic consequence of modes of thought which we must now abandon) which we may call

. On Einstein's theory the deflection would be just twice this amount,

. But it was thought quite possible that the result might be neither

nor

but zero, and Professor Eddington remarked before setting out on the recent expedition that a zero result, however disappointing immediately, might ultimately turn out the most fruitful of all. That was less than a year ago. Perhaps a few dates are worth remembering. Einstein's theory was fully developed and stated in November, 1915, but news of it did not reach England (owing to the War) for some months. In 1917 the Astronomer Royal pointed out the special suitability of the Total Solar Eclipse of May, 1919, as an occasion for testing Einstein's Theory. Preparations for two Expeditions were commenced—Mr. Hinks described the geographical conditions on the central line in November, 1917—but could not be fully in earnest until the Armistice of November, 1918. In November, 1919, the entirely satisfactory outcome was announced to the Royal Society and characterized by the President as necessitating a veritable revolution in scientific thought.

But when Mr. Brose brought me his translation of the pamphlet in the spring of 1919, the issue was still in doubt. He had become deeply interested in the new theory while interned in Germany as a civilian prisoner and had there made this translation. I encouraged him to publish it and opened negotiations to that end, but it was not until we enlisted the sympathy of Professor Eddington (on his return from the Expedition) and approached the Cambridge Press that a feasible plan of publication was found. Professor Eddington would have been a far more appropriate introducer; and it is only in deference to his own express wish that I have ventured to take up the pen that he would have used to much better purpose. One advantage I reap from the decision: I can express the thanks of Mr. Brose and myself to him for his practical help, and perhaps I may add those of a far wider circle for his own able expositions of an intricate theory, which have done so much to make it known in England.

H. H. TURNER

UNIVERSITY OBSERVATORY,
OXFORD.
November 30, 1919

[INTRODUCTION. By Professor H. H. Turner, F.R.S.]
[BIOGRAPHICAL NOTE]
SECT.
1. [THE SPECIAL THEORY OF RELATIVITY AS A STEPPING-STONE TO THE GENERAL THEORY OF RELATIVITY]
2. [TWO FUNDAMENTAL POSTULATES IN THE MATHEMATICAL FORMULATION OF PHYSICAL LAWS]
3. [CONCERNING THE FULFILMENT OF THE TWO POSTULATES]
(а) The line-element in the three-dimensional manifold of points in space, expressed in a form compatible with the two postulates
(b) The line-element in the four-dimensional manifold of space-time points, expressed in a form compatible with the two postulates
4. [THE DIFFICULTIES IN THE PRINCIPLES OF CLASSICAL MECHANICS]
5. [EINSTEIN'S THEORY OF GRAVITATION]
(a) The fundamental law of motion and the principle of equivalence of the new theory
(b) Retrospect
6. [THE VERIFICATION OF THE NEW THEORY BY ACTUAL EXPERIENCE]
[APPENDIX:]
Explanatory notes and bibliographical references
[ON THE THEORY OF RELATIVITY. By Henry L. Brose]
[SOME ASPECTS OF RELATIVITY. THE THIRD TEST. By Henry L. Brose]

[BIOGRAPHICAL NOTE]

Albert Einstein was born in March, 1879, the town Ulm, situated on the banks of the Danube in Würtemberg, Germany. He attended school at Munich, where he remained till his sixteenth year.

His university studies extended over the period 1896-1900 at Zürich, Switzerland. He became a citizen of Zürich in 1901. During the following seven years he filled the post of engineer in the Patent Office, Bern. He accepted a call to Zürich as Professor Extraordinarius in 1910, which he, however, soon resigned in favour of a permanent chair in Prague University. In 1911 he decided to accept a similar post in Zürich. Since 1914 he has continued his researches in Berlin as a member of the Berlin Academy of Sciences.

His most important achievements are:

1905. The Special Theory of Relativity.
The discovery that all forms of energy possess
inertia.
The law underlying the Brownian movement.
The Quantum-Law of the emission and absorption of light.

1907. The fundamental notions of the general theory of
relativity.

1912. The recognition of the non-Euclidean nature of
space-determination and its connection with
gravitation.

1915. Gravitational field equations.
Explanation of the motion of Mercury's perihelion.

[INTRODUCTION]

TOWARDS the end of 1915 Albert Einstein brought to its conclusion a theory of gravitation on the basis of a general principle of relativity of all motions. His object was to create not a visual picture of the action of an attractive force between bodies, but rather a mechanics of the motions of the bodies relative to one another under the influence of inertia and gravity. To attain this difficult goal, it is true, many time-honoured views had to be sacrificed, but as a reward a standpoint was reached which had long seemed the highest aim of all who had occupied their minds with theoretical physics. The fact that these sacrifices are demanded by the new theory must, indeed, inspire confidence in it. For the unsuccessful attempts that have been made during the last centuries to fit the doctrine of gravitation satisfactorily into the scheme of natural science necessarily lead to the conclusion that this would not be possible without giving up many deeply-rooted ideas. As a matter of fact, Einstein reverted to the foundation pillars of mechanics as starting-points on which to build his theory, and he did not satisfy himself by merely reforming the Newtonian law in order to establish a link with the more recent views.

To get at an understanding of Einstein's ideas, we must compare the fundamental point of view adopted by Einstein with that of classical mechanics. We then recognize that a logical development leads from "the special" principle of relativity to the general theory, and simultaneously to a theory of gravitation.

THEORY OF GRAVITATION

§ 1
THE "SPECIAL" THEORY OF RELATIVITY AS A STEPPING STONE TO THE "GENERAL" THEORY OF RELATIVITY

THE complete upheaval which we are witnessing in the world of physics at the present time received its impulse from obstacles which were encountered in the progress of electrodynamics. Yet the important point in the later development was that an escape from these difficulties was possible [1] only by founding mechanics on a new basis.

[1]Note.—Most of the objections to the new development have, it is admitted, been raised because a branch of science which was not considered to have a just claim to deal with questions of mechanics, asserted the right of exercising a far-reaching influence upon the latter, extending even to its foundation. If, however, we trace these objections to their source, we discover that they are due to a wish to give mechanics the form of a purely mathematical science, similar to geometry, in spite of the fact that it is founded upon hypotheses which are essentially physical: up to the present, certainly, these hypotheses have not been recognized to be such.

The development of electrodynamics took place essentially without being influenced by the results of mechanics, and without itself exerting any influence upon the latter, so long as its range of investigation remained confined to the electrodynamic phenomena of bodies at rest. Only after Maxwell's equations had furnished a foundation for these did it become possible to take up the study of the electrodynamic phenomena of moving media. All optical occurrences—and according to Maxwell's theory all these also belong to the sphere of electrodynamics—take place either between stellar bodies which are in motion relatively to one another, or upon the earth, which revolves about the sun with a velocity of about 30 kilometres per second, and performs, together with the sun, a translational motion of about the same order of magnitude through the region of the stellar system. Hence questions of great fundamental importance at once asserted themselves. Does the motion of a light-source leave its trace on the velocity of the light emitted by it? And what is the influence of the earth's motion on the optical phenomena which occur on its surface, for example, in optical experiments in a laboratory? An endeavour was therefore to be made to find a theory of these phenomena in which electrodynamic and mechanical effects occurred simultaneously (vide [Note 1]). Mechanics, which had long stood as a structure complete in every detail, had to stand the test as to whether it was capable of supplying the fitting arguments for a description of such phenomena. Thus the problem of electrodynamic events in the case of moving matter became at the same time a decisive problem of mechanics.

The first outstanding attempt to describe these phenomena for moving bodies was made by H. Hertz. He extended Maxwell's equations by additional terms so as also to express the influence of the motion of matter on electrodynamic phenomena, and in his extensions he adopted the view, characteristic for his theory, that the carrier of the electromagnetic field, the ether, everywhere participates in the motion of matter. Consequently, in his equations the state of motion of the ether, as denoting the state of the ether, occurs as well as the electromagnetic field. As is well known, Hertz's extensions cannot be brought into harmony with the results of observation, for example, that of Fizeau's experiment ([Note 2]), so that they excite merely an historic interest as a land-mark on the road to an electrodynamics of moving matter. Lorentz was the first to derive from Maxwell's theory fundamental electrodynamic equations for moving matter which were in essential agreement with observation. He, indeed, succeeded in this only by renouncing a principle of fundamental importance, namely, by disallowing that Newton's and Galilei's principle of relativity of classical mechanics also holds for electrodynamics. The practical success of Lorentz's theory at first almost made us fail to see this sacrifice, but then the disintegration set in at this point which finally made the position of classical mechanics untenable. To understand this development we therefore require a detailed treatment of the principle of relativity in the fundamental equations of physics.

The principle of relativity of classical mechanics is understood to signify the consequence, which arises out of Newton's equations of motion, that two systems of co-ordinates, moving with uniform motion in a straight line with respect to one another, are to be regarded as fully equivalent for the description of events in the domain of mechanics. For our observations on the earth this means that any mechanical event on the surface of the earth—for example, the motion of a projected body—does not become modified by the circumstance that the earth is not at rest, but, as is approximately the case, is moving rectilinearly and uniformly. Yet this postulate of relativity does not fully characterize the Newtonian principle of relativity, even if it expresses that experimental fact which constitutes the essence of the principle of relativity. The postulate of relativity has yet to be supplemented by those formulæ of transformation by means of which the observer is able to transform the co-ordinates

that occur in Newton's equations of motion into those of a system of reference which is moving uniformly and rectilinearly with respect to his own and which has the co-ordinates

'. Here the co-ordinates,

, that occur in the Newtonian equations denote throughout the results of measurement (obtained by means of rigid measuring rods according to the rules of Euclidean geometry), of the spatial positions of the bodies during the event in question, and the fourth co-ordinate

denotes the point of time assigned to the same event given by the position of the hands of a clock placed at the point at which the event occurs. Classical mechanics now supplemented the postulate of relativity above formulated by equations of transformation of the form:

for the cases in which we are dealing with the co-ordinate relations of two systems of reference moving with the uniform velocity

in the direction of the

-axis with respect to each other. This group of so-called Galilei-transformations is distinguished, even in the case in which the direction of motion makes any angle with the co-ordinate axes, by the circumstance that the time-co-ordinate

always becomes transformed by the identity

into the time-values of the second system of reference; it is in this that the absolute character of the time-measures manifests itself in the classical theory. Newton's equations of mechanics do not alter their form if we substitute the co-ordinates

' in them for

by means of these equations of transformation. So long as we restrict ourselves to those systems of reference among all others that emerge out of each other as a result of transformations of the above type, there is no sense in talking of absolute rest or absolute motion. For we may freely decide to regard either of two systems moving in such a way as the one that is at rest or in motion. According to classical mechanics it was, indeed, believed that only the Galilei-transformations could come into question when we were concerned with referring equivalent systems of reference to each other according to the principle of relativity. This, however, is not the case. The recognition of the fact that other equations of transformation may come into question for this purpose, and, indeed, may be chosen to suit the facts of observation which are to be accounted for, the recognition of this fact is the characteristic feature of the "special" theory of relativity of Lorentz-Einstein which replaced that of Galilei-Newton. Lorentz's fundamental equations of the electrodynamics of moving matter led to it. This system of electrodynamics, which is in satisfactory agreement with observation, is founded, in contradistinction to Hertz's theory, on the view of an absolutely rigid ether at rest. Its fundamental equations assume as its favoured system the co-ordinate system that is at rest in the ether.

These fundamental electrodynamical equations of Lorentz, however, change their form if, in them, we replace the co-ordinates

of a system of reference, initially chosen, by the co-ordinates

' of a system moving uniformly and rectilinear with respect to the former by means of the transformation relationships. Must we infer from this that systems of reference which are moving uniformly and rectilinearly with respect to each other are not equivalent as regards electrodynamic events, and that there is no relativity principle of electrodynamics? No, this inference is not necessary, because, as remarked, the principle of relativity of classical mechanics with its group of equations of transformation does not represent the only possible way of expressing the equivalence of systems of reference that are moving uniformly and rectilinearly with respect to each other. As we shall show in the sequel, the same postulate of relativity may be associated with another group of transformations. Nor did experiment seem to offer a reason for answering the above question in the affirmative. For all attempts to prove by optical experiments in our laboratories on the earth the progressive motion of the latter gave a negative result ([Note 2]). According to our observations of electrodynamic events in the laboratory the earth may be regarded equally well as at rest or in motion; these two assumptions are equivalent.

This led to the definite conviction that in fact a principle of relativity holds for all phenomena, be their character mechanical or electrodynamic. But there can be only one such principle, and not one for mechanics and another for electrodynamics. For two such principles would annul each other's effects because we should be able to derive a favoured system from them in the case of events in which mechanical and electrodynamical events occur in conjunction, and this favoured system would allow us to talk with sense of absolute rest or motion with regard to it.

The one escape from this difficulty is that opened up by Einstein. In place of the relativity principle of Galilei and Newton we have to set another which comprehends the events of mechanics and electrodynamics. This may be done, without altering the postulate of relativity formulated above, by setting up a new group of transformations, which refer the co-ordinates of equivalent systems of reference to one another. The fundamental equations of mechanics must, certainly, then be remodelled so that they preserve their form when subjected to such a transformation. Starting-points for this remodelling were already given. For it had been found empirically that Lorentz's fundamental equations of electrodynamics allowed new kinds of transformations of co-ordinates, namely, those of the form

where

= velocity of light in vacuo.

The new principle of relativity set up by Einstein is as follows: Systems that are moving uniformly and rectilinearly with respect to each other are completely equivalent for the description of physical events. The equations of transformation that allow us to pass from the co-ordinates of one such system to those of another possible system, however, are not then (for the case when both systems are moving parallel to their

-axes with the constant velocity

):—

but

Thus the Galilei-Newton principle of relativity of classical mechanics and the Lorentz-Einstein "special" principle of relativity differ only in the form of the equations of transformation that effect the transition to equivalent systems of reference ([Note 3]).

Moreover, the relation of these two different transformation formulæ to each other comes out clearly in the circumstance that the equations of transformation of Galilei and Newton may be derived by a simple passage to the limit from the new equations of Lorentz and Einstein. For if we assume the velocity

of each system with respect to the other to be very small compared with the velocity of light

, so that the quotient

respectively, may be neglected in comparison with the remaining terms—an admissible assumption in all cases with which classical mechanics had so far dealt—the Lorentz-Einstein transformations pass over into those of Newton and Galilei.

It immediately suggests itself to us to ask what it is that compels us to give up the principle of relativity of classical mechanics, that is, what are the physical assumptions in its equations of transformation that stand, in contradiction with experience? The answer is that the principle of relativity of Newton and Galilei does not account for the facts of experience that emerge from Fizeau's and the Michelson-Morley experiment, and from which it may be inferred that the velocity of light has the particular character of a universal constant in the transformation relationships of the principle of relativity. In how far this peculiar property of the velocity of light receives expression in the new equations of transformation requires the following detailed explanation.

The equations of transformation of the principle of relativity of Galilei and Newton contain a hypothesis (which had hitherto not been recognized as such). For it had been tacitly assumed that the following assumption was fulfilled quite naturally: if an observer in a co-ordinate system

measure the velocity

of the propagation of some effect or other, for example, a sound wave, then an observer in another co-ordinate system

' which is moving relatively to

, necessarily obtains a different measure for the velocity of propagation of the same action. This was to hold for every finite velocity

. Only infinite velocity was to be distinguished by the singular property that it was to come out in every system independently of its state of motion as having exactly the same value in all the measurements, namely, the value infinity.

This hypothesis—for we are here, of course, dealing only with a purely physical hypothesis—immediately suggested itself. Without further test there was no support for supposing that also a finite velocity, namely, the velocity of light, which the naïve point of view is inclined to endow with infinitely great velocity, would manifest the same singular property.

The fact, however, which the Michelson-Morley experiment helped us to become aware of was that the law of propagation for light is, for the observer, independent of any progressive motion of his system of reference, and has the property of isotropy (that is, equivalence of all systems) (cf. [Note 2]), so that it immediately suggests itself to us that the velocity of light is to be considered as having the same value for all systems of reference. The recognition of the fact thus arrived at was, without doubt, a surprise, but it will appear less strange to those who bear in mind the particular rôle of the velocity of light in the equations of Maxwell, the foundation of our theory of matter.

In consequence of this peculiarity, the velocity of light occurs in the equations of kinematics as a universal constant. To understand this better we pursue the following argument. Long before the advent of the questions of electrodynamic phenomena in moving bodies we might, on grounds of principle, have suggested quite generally the question: how are the co-ordinates in two systems of reference that are moving uniformly and rectilinearly with respect to each other to be referred to each other? We should have been able to attack the purely mathematical problem with a full consciousness of the assumptions contained in the hypotheses, as was actually done later by Frank and Rothe ([Note 4]). We then arrive at equations of transformation that are much more general than those written down on [p. 9]. By taking into account the special conditions that nature manifests to us, for example the isotropy of space, we may derive from them particular forms, the hypothetical assumptions contained in which come clearly to view. Now, in these general equations of transformation a quantity occurs that deserves special notice. There are "invariants" of these equations of transformation, that is, quantities that preserve their value even when such a transformation is carried out. Among these invariants there is a velocity. This signifies the following: if an effect propagates itself in one system with the velocity

, then in general the velocity of propagation of the same effect in another system is other than

, if the second system is moving relatively to the first. Only the invariant velocity preserves its value in all systems, no matter with what velocity they be moving relatively to one another. The value of this invariant velocity enters as a characteristic constant into the equations of transformation. Hence, if we wish to find those transformation relations that hold physically, we must find out the singular velocity that plays this fundamental part. To determine it is the task of the experimental physicist. If he sets up the hypothesis that a finite velocity can never be such an invariant, the general equations of transformation degenerate into the transformation-relationships of the principle of relativity of Galilei and Newton. (This hypothesis was made, albeit unconsciously, in Newtonian mechanics.) It had to be discarded after the results of the Michelson-Morley and Fizeau's experiment had justified the view that the velocity of light

plays the part of an invariant velocity. Then the general equations of transformation degenerate into those of the "special" principle of relativity of Lorentz and Einstein.

This remodelling of the co-ordinate-transformations of the principle of relativity led to discoveries of fundamental importance, as, for example, to the surprising fact that the conception of the "simultaneity" of events at different points of space, the conception on which all time-measurements are based, has only a relative meaning, that is, that two events that are simultaneous for one observer will not, in general, be simultaneous for another. [2] This deprived time-values of the absolute character which had previously been a great point of distinction between them and space co-ordinates. So much has been written in recent years about this question that we need not treat it in detail here.

[2]The assertion, "At a particular point of the earth the sun rises at 5 o'clock 10'6"," denotes that "the rising of the sun at a particular point of the earth is simultaneous with the arrival of the hands of the clock at the position 5 o'clock 10'6" at that point of the earth." In short, the determination of the point of time for the occurrence of an event is the determination of the simultaneity of happening of two events, of which one is the arrival of the hands of a clock at a definite position at the point of observation. The comparison of the points of time at which one and the same event occurs, as noted by several observers situated at different points, requires a convention concerning the times noted at the different points. The analysis of the necessary conventions led Einstein to the fundamental discovery that the conception "simultaneous" is only "relative inasmuch as the relation of time-measurements to one another in systems that are moving relatively to one another is dependent on their state of motion. This was the starting-point for the arguments that led to the enunciation of the "special principle of relativity."

The new form of the equations of transformation by no means exhausts the whole effect of the principle of relativity upon classical mechanics. The change which it brought about in the conception of mass was almost still more marked.

Newtonian mechanics attributes to every body a certain inertial mass, as a property that is in no wise influenced by the physical conditions to which the body is subject. Consequently, the Principle of the Conservation of Mass also appears in classical mechanics as independent from the Principle of the Conservation of Energy. The special principle of relativity shed an entirely new light on these circumstances when it led to the discovery that energy also manifests inertial mass, and it hereby fused together the two laws of conservation, that of mass and that of energy, to a single principle. The following circumstance moves us to adopt this new view of the conception of mass.

The equations of motion of Newtonian mechanics do not preserve their form when new co-ordinates have been introduced with the help of the Lorentz-Einstein transformations. Consequently, the fundamental equation of mechanics had to be remodelled accordingly. It was then found that Newton's Second Law of Motion: force = mass x accel. cannot be retained, and that the expression for the kinetic energy of a body may no longer be furnished by the simple expression

, which involves the mass and the velocity. Both these results are consequences of the change which we found necessary to make in our view of the nature of the mass of matter. The new principle of relativity and the equations of electrodynamics led, rather, to the fundamentally new discovery that inertial mass is a property of every kind of energy, and that a point-mass, in emitting or absorbing energy, decreases or increases, respectively, in inertial mass, as is shown in [Note 5] for a simple case. The new kinematics thereby disposes of the simple relation between the kinetic energy of a body and its velocity relatively to the system of reference. The simplicity of the expression for the kinetic energy in Newtonian mechanics rendered possible the revolution of the energy of a body into that (kinetic) of its motion and of the internal energy of the body, which is independent of the former. Let us consider, for example, a vessel containing material particles, no matter of what kind, in motion. If we resolve the velocity of each particle into two components, namely, into the velocity, common to all, of the centre of gravity and the accidental velocity of a particle relative to the centre of gravity of the system, then, according to the formulæ of classical mechanics, the kinetic energy divides up into two parts: one that contains exclusively the velocity of the centre of gravity and that represents the usual expression for the kinetic energy of the whole system (mass of the vessel plus the mass of the particles), and a second component that involves only the inner velocities of the system. This category of internal energy is no longer possible so long as the expression for the kinetic energy contains the velocity not merely as a quadratic factor; so we are led to the view that the internal energy of the body comes into expression in the energy due to its progressive motion, and, indeed, as an increase in the inertial mass of the body.

This discovery of the inertia of energy created an entirely new starting-point for erecting the structure of mechanics. Classical mechanics regards the inertial mass of a body as an absolute, invariable, characteristic quantity. The special theory of relativity, it is true, makes no direct mention of the inertial mass associated with matter, but it tells us that every kind of energy has also inertia. But, as every kind of matter has at all times a probably enormous amount of latent energy, its inertia is composed of two components; the inertia of the matter and the inertia of its contained energy, which consequently alters with the amount of the energy-content. This view leads us naturally to ascribe the phenomenon of inertia in bodies to their energy-content altogether.

Thus, there arose the important task of absorbing these new discoveries concerning the nature of inert mass into the principles of mechanics. A difficulty hereby arose which, in a certain sense, pointed out the limits of achievement of the special theory of relativity. One of the fundamental facts of mechanics is the equality of the inertial and gravitational mass of a body. It is on the supposition that this is true that we determine the mass of a body by measuring its weight. The weight of a body is, however, only defined with reference to a gravitational field ([Note 18]): in our case, with reference to the earth. The idea of inertial mass of a body is, however, introduced as an attribute of matter without any reference whatsoever to physical conditions external to the body. How does the mysterious coincidence in the values of the inertial and gravitational mass of a body come about?

Nor does the special theory of relativity provide an answer to this question. The special theory of relativity does not even preserve the equality in the values of inertia and gravitational mass; a fact which is to be reckoned amongst the most firmly established facts in the whole of physics. For, although the special theory of relativity makes allowance for an inertia of energy, it makes none for a gravitation of energy. Consequently, a body which absorbs energy in any way will register a gain of inertia but not of weight, thereby transgressing the principle of the equality of inertial and gravitational mass; for this purpose a theory of gravitational phenomena, a theory of gravitation, is required. The special theory of relativity can, therefore, be regarded only as a stepping-stone to a more general principle, which orders gravitational phenomena satisfactorily into the principles of mechanics.

This is the point where Einstein's researches towards establishing a general theory of relativity set in. He has discovered that, by extending the application of the relativity-principle to accelerated motions, and by introducing gravitational phenomena into the consideration of the fundamental principles of mechanics, a new foundation for mechanics is made possible, in which all the difficulties occurring up to the present are solved. Although this theory represents a consistent development of the knowledge gathered by means of the special theory of relativity, it is so deeply rooted in the substructure of our principles of knowing, in their application to physical phenomena, that it is possible thoroughly to grasp the new theory only by clearly understanding its attitude toward these guiding lines provided by the theory of knowledge.

I shall, therefore, commence the account of his theory by discussing two general postulates, which should be fulfilled by every physical law, but neither of which is satisfied in classical mechanics: whereas their strict fulfilment is a characteristic feature of the new theory. Here we have thus a suitable point of entry into the essential outlines of the general theory of relativity.

§ 2
TWO FUNDAMENTAL POSTULATES IN THE MATHEMATICAL FORMULATION OF PHYSICAL LAWS

NEWTON had established the simple and fruitful law that two bodies, even when they are not visibly connected with one another, as in the case of the heavenly bodies, exert a mutual influence, attracting one another with a force directly proportional to the product of their masses, and inversely proportional to the square of the distance between them. But Huygens and Leibniz refused to acknowledge the validity of this law, on the ground that it did not satisfy a fundamental condition to which every physical law is subject, viz. that of continuity (continuity in the transmission of force, action "by contact" in contradistinction to action "at a distance"). How were two bodies to exert an influence upon one another without a medium between them to transmit the action? The demand for a satisfactory answer to this question became, in fact, so imperative that finally, in order to satisfy it, the existence of a substance which pervaded the whole of cosmic space and permeated all matter—the "luminiferous ether"—was assumed, although this substance seemed to be condemned to remain intangible and invisible (i.e. imperceptible to the senses for all time) and had to be endowed with all sorts of contradictory properties. In the course of time, however, there arose in opposition to such assumptions the more and more definite demand that, in the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed: a demand which doubtless originates from the same instinct in the search for knowledge as that of continuity, and which really gives the law of causality the true character of an empirical law, i.e. one of actual experience.

The consistent fulfilment of these two postulates combined together is, I believe, the mainspring of Einstein's method of investigation; this imbues his results with their far-reaching importance in the construction of a physical picture of the world. In this respect his endeavours will probably not encounter any opposition in the matter of principle on the part of scientists. For both postulates—(1) that of continuity and (2) that of causal relationship between only such things as lie within the realm of observation—are of an inherent nature, i.e. contained in the very nature of the problem. The only question that might be raised is whether it is expedient to abandon such useful working hypotheses as "forces at a distance."

The principle of continuity requires that all physical laws allow of formulation as differential laws, i.e. physical laws must be expressible in a form such that the physical state at any point is completely determined by that of the point in its immediate neighbourhood. Consequently, the distances between points, which are at finite distances from one another, must not occur in these laws, but only those between points infinitely near to one another. The law of attraction of Newton given above, inasmuch as it involves "action at a distance," disobeys the first postulate.

The second postulate, that of a stricter form of expression for causality in its occurrence in physical laws, is intimately connected with a general theory of relativity of motions. Such a general principle of relativity requires that all possible systems of reference in nature be equivalent for the description of physical phenomena, and hence it avoids the introduction of the very questionable conception of absolute space which, for reasons we know (see [§ 4]), could not be circumvented by Newtonian mechanics. A general theory of relativity would, in excluding the fictitious quantity "absolute space," reduce the laws of mechanics to motions of bodies relative to one another, which are actually and exclusively what we observe. Thus, its laws would be founded on observed facts more completely than are those of classical mechanics.

The rigorous application of the principles of continuity and relativity in their general form penetrates deeply into the problem of the mathematical formulation of physical laws. It will, therefore, be essential at the outset to enter into a consideration of the principles involved in the latter process.

§ 3
CONCERNING THE FULFILMENT OF THE TWO POSTULATES

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]

[3]Vide [Note 2].

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.

[4]Vide [Note 3].

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