THEORY OF GRAVITATION§ 1THE "SPECIAL" THEORY OF RELATIVITY AS A STEPPING STONE TO THE "GENERAL" THEORY OF RELATIVITYTHE 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 purelymathematical science, similar to geometry, in spite of the fact that it is founded upon hypotheses which are essentiallyphysical: 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 bodiesat 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 ofmovingmedia. All optical occurrences—and according to Maxwell's theory all these also belong to the sphere ofelectrodynamics—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 (videNote 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, thatof 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 thispostulateof 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 theobserver 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-ordinatedenotes 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 velocityin 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 makesanyangle with the co-ordinate axes, by the circumstance that the time-co-ordinatealways becomes transformed by the identityinto 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 formif 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 intalking 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 arenotequivalent 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 ofclassical mechanics with its group of equations of transformation does not represent theonlypossible 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 areequivalent.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 onlyonesuch 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 thepostulateof 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 beremodelled 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 formwhere= velocity of lightin 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):—butThus 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 toeach 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 velocityof each system with respect to the other to be very small compared with the velocity of light, so that the quotientorrespectively, 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 systemmeasure the velocityof thepropagation 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 forevery finitevelocity. 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 theadvent 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 purelymathematicalproblem 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 moregeneralthan those written down onp. 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 holdphysically, we must find out the singular velocity that plays this fundamental part. To determine itis the task of the experimental physicist. If he sets up thehypothesisthat a finite velocity can never be such an invariant, thegeneralequations 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 thevelocity of lightplays the part of an invariant velocity. Then thegeneral equationsof 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 theabsolute 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 massof 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 inNote 5for 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 vesselplusthe 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 newstarting-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 withmatter, but it tells us thateverykind ofenergyhas 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 definedwith reference to a gravitational field(Note 18): in our case, with reference to the earth. The idea ofinertialmass of a body is, however, introduced as an attribute of matterwithout 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 theequality 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 aninertiaof energy, it makes none for agravitationof 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 astepping-stoneto 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, butneither 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.
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 purelymathematical science, similar to geometry, in spite of the fact that it is founded upon hypotheses which are essentiallyphysical: up to the present, certainly, these hypotheses have not been recognized to be such.
[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 purelymathematical science, similar to geometry, in spite of the fact that it is founded upon hypotheses which are essentiallyphysical: 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 bodiesat 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 ofmovingmedia. All optical occurrences—and according to Maxwell's theory all these also belong to the sphere ofelectrodynamics—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 (videNote 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, thatof 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 thispostulateof 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 theobserver 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-ordinatedenotes 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 velocityin 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 makesanyangle with the co-ordinate axes, by the circumstance that the time-co-ordinatealways becomes transformed by the identityinto 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 formif 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 intalking 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 arenotequivalent 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 ofclassical mechanics with its group of equations of transformation does not represent theonlypossible 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 areequivalent.
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 onlyonesuch 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 thepostulateof 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 beremodelled 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 formwhere= velocity of lightin 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 toeach 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 velocityof each system with respect to the other to be very small compared with the velocity of light, so that the quotientorrespectively, 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 systemmeasure the velocityof thepropagation 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 forevery finitevelocity. 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 theadvent 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 purelymathematicalproblem 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 moregeneralthan those written down onp. 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 holdphysically, we must find out the singular velocity that plays this fundamental part. To determine itis the task of the experimental physicist. If he sets up thehypothesisthat a finite velocity can never be such an invariant, thegeneralequations 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 thevelocity of lightplays the part of an invariant velocity. Then thegeneral equationsof 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 theabsolute 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."
[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 massof 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 inNote 5for 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 vesselplusthe 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 newstarting-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 withmatter, but it tells us thateverykind ofenergyhas 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 definedwith reference to a gravitational field(Note 18): in our case, with reference to the earth. The idea ofinertialmass of a body is, however, introduced as an attribute of matterwithout 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 theequality 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 aninertiaof energy, it makes none for agravitationof 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 astepping-stoneto 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, butneither 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.