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XIV; Einstein's Theory of Relativity

The theory of space time and gravitation propounded by Albert Einstein is one of the small class of scientific theories which have at once succeeded in captivating the attention not only of the scientific world but also in a marked degree of a very large number of persons whose primary interests are not scientific. The interest of Physicists and Mathematicians is sufficiently explained by the ambitious character of a theory of which the aim is to combine in a single scheme temporal and spatial measurements together with gravitational phenomena and by the fact that the theory includes a new law of gravitation and a new Mechanics involving a breach with certain assumptions formerly supposed to be axiomatic if indeed they were ever explicitly recognized. But in the case of Philosophers and of others to whom Natural Science is only of mediate or of secondary interest another factor enters into the explanation of the amount of attention they have given to the theory. This is the prevalence of the idea that Einstein's theory of relativity has implications which reach beyond the purely scientific domain; and that it may serve to throw light upon and perhaps to lead to changes in our general philosophical views of the nature of reality. The term “relativity” is a very general one capable of being employed in various directions in philosophic thought. Meanings may be assigned to it of more general scope and perhaps of a less definitely circumscribed character than the rigidly defined meaning which is assigned to the term by Einstein and other Physicists who have concerned themselves with the development and exposition of the theory. There has thus been exhibited in some quarters a disposition to make this theory a starting point for the development of relativistic views outside and beyond the scope of the scientific theory itself.

It is a question of general epistemological interest whether apart from the undoubtedly great importance of the theory in its purely scientific aspect there is anything in the nature of Einstein's theory which should properly give it in the eyes of Philosophers and of the educated public a unique position in relation to general thought of a kind which other previously existing physical theories do not occupy. Is Einstein's theory not only technically but also generically different from earlier physical theories? Does it rest upon a philosophically different basis? The answer to be given to this question cannot be considered in complete independence of the diverging views which are held as to the true character and functions of scientific theories in general. To those who regard Science as a means of penetrating to the inner nature of reality the theory so far as it is regarded as established will appear to have given new knowledge of the inner nature of the real world at least as regards spatial temporal and material relations. But some of those who hold this view of the functions of Science have been disconcerted by the highly abstract mathematical form of the theory in accordance with which matter at least in connection with its gravitational phenomena exhibits itself no longer as substance but only in the guise of specializations in a spatio-temporal metric. This attitude of mind is well illustrated by the utterance of Sir Oliver Lodge a propos of this theory which I quoted (p. 60) in my third lecture. The apparent impossibility of denying the descriptive efficiency of the abstract scheme and the admiration excited by the constructive genius manifested in its creation struggle With a reluctance to being drawn away from what in accordance with pre-conceived ideas and especially the old hankering after an underlying sub-stratum of matter is regarded as concrete reality. On the other hand to those whose view of the character and functions of scientific theories is in general agreement with that which I have advocated Einstein's theory of relativity however highly its scientific importance may be estimated will appear to be merely a new conceptual theory much more comprehensive no doubt in its scope than older theories which it aims to replace but epistemologically on the same tooting. Its extremely abstract character will occasion no great surprise since that must be a feature of every scientific theory which lays such a claim to precision and comprehensiveness of descriptive power as does the theory in question. This theory proposes certain changes in scientific hypotheses formerly accepted as a basis of physical theories of so striking a character that it may without exaggeration be described as revolutionary in its tendency. But if this revolution became stabilized and its prospects of becoming so appear to be bright although not at present fully assured it will so far as I can see be a revolution purely internal to Natural Science and will in no sense radically affect the external relations of Science with general Thought. In fact my answer to the question I have raised as to whether the Einstein theory is generically or epistemologically different from other scientific theories is that no such difference exists: that it has in fact the same independence of all special ontological assumptions and theories as has Natural Science in general.

One of the advantages which may be expected ultimately to accrue from the widespread discussion of the theory by Physicists Mathematicians and Philosophers is that it will lead to a clarification of ideas as to the nature and scope of scientific theories in general. The theory of Einstein is peculiarly well suited for this purpose because when it is clearly presented and its foundations are scrutinized the fact that its basal conception consists of an ideal scheme abstract in the highest degree incapable of representation by the sensuous imagination and in which nearly all elements derived from perceptive intuition have been removed by abstraction forces itself upon the attention in such wise that its essential character is less easy to disguise than in the case of many current physical theories. There can be no pretence that a fourfold ordered manifold in which the ordering is neither spatial nor temporal with a suitably chosen metric imposed upon it but not intrinsic to it is anything else than a most highly abstract conception. The metrical relations imposed upon this manifold can be analysed mathematically but the manifold has neither in whole nor in part perceptual actuality or presentability to the sensuous imagination. It can be conceived but not imagined.

I propose to explain in general terms the main characteristics of Einstein's theory of space time and gravitation so far as is possible in the short time at my disposal for the purpose. Extensions of the theory have been suggested with a view to combining in one whole not only a theory of space time and gravitation but also a theory of electromagnetic phenomena in general; but with such extensions I do not propose here to deal. I must say however at once that the theory is only capable of complete expression in mathematical terminology and is of such a character that only a trained Mathematician who is prepared to spend a considerable amount of time and energy upon the detailed study of its foundations can obtain a complete grasp of it. Fortunately however it is possible without making such a complete study of the theory to obtain a general knowledge of the points in which it differs from the older physical theories. These older theories have proved adequate for the purpose of measurement of spatial and temporal magnitudes and of gravitational effects not only in ordinary life but for nearly all scientific purposes; and they will always retain that adequacy. It is only for certain special scientific purposes that Einstein's theory will in case it survives the tests which are being applied to it be applied in future; it will never be applied in any of the measurements made in ordinary life.

When it is said that Einstein's theory of gravitation has overthrown and superseded Newton's theory that is in any case only true in a very limited sense. In all ordinary astronomical cases Einstein's theory gives the same results as Newton's; it is only in very special cases that the difference between the results they lead to is sufficiently large to be discernible by the means of measurement at our disposal. That Newton's theory has its limits of applicability will not surprise those of us who contemplate the discovery of such limits in the case of all conceptual theories. The claim is made on behalf of Einstein's theory that the limits of its applicability are wider than those of Newton's theory. Before Einstein's theory can be regarded as an established and indispensable part of our stock of scientific conceptual apparatus it must not only be shown to be logically self-consistent and to be capable of being applied for the purpose of providing an adequate symbolical description of the range of percepts to deal with which it has been constructed but it must further be shown that no conceptual scheme of a simpler character is adequate for the representation of the same range of physical percepts.

The first point in which Einstein's theory has made a new departure has reference to the measurement of space and time only. In order to explain the character of this new departure I must refer to the observations I made in my lecture on “Time and Space” having reference to the pre-Einstein theory of their measurements. The private spaces of individuals were I pointed out correlated with a single public or physical space in which all physical objects with which we have to deal either in ordinary life or in Science are regarded as located and in which all actual spatial measurements are made Similarly the private times of individuals were correlated with a single public time in which all measurements of time by clocks by the rotation of the earth and by other sufficiently approximatively equivalent processes are made. That these constructs of a single physical space and a single public time each independent to the other are sufficient to represent by process of correlation the private spatial and temporal experiences of all individuals whatever might be their relative positions and motions has been the universal assumption made before the rise or the theory of relativity; and it has been regarded as axiomatic. For all the purposes of ordinary life and also for the purposes of Science it had always been found sufficient until attention was directed to certain facts of observation to which I shall presently refer. The first great breach of the Einstein theory with the previously universally accepted tradition consists in a denial of the sufficiency of these constructs a single physical space and a single public time independent of each other as affording the basis of a system of spatial and temporal measurements which will completely describe the spatio-temporal experiences of all observers. The measurements in the single physical space were all made in approximate accordance with an abstract geometrical scheme of which the basis was a manifold of elements with a three-fold order into which manifold a Euclidean metric was introduced. This three-fold ordered manifold with the imposed Euclidean metric is the space of abstract Euclidean Geometry each element or point of which manifold is representable by a triplet of numbers. Its properties were developed deductively as a scheme of Euclidean Geometry and all its metric properties correspond to facts of measurement which can be verified in physical space with a degree of approximation dependent on the fineness of our senses when reinforced by instruments of precision. In the same manner public time was correlated with an abstract scheme which consists of a singly ordered manifold of elements represented by the numbers of the arithmetic continuum. Thus the theory of the measurements of physical space and public time rested upon the postulation of two abstract ordered manifolds one with a three-fold order and the other singly-ordered entirely independent to one another. In accordance with the theory developed by Minkowski and Einstein instead of these two abstract manifolds a single manifold with a tour-told order of its elements is postulated to serve as the theoretical basis for all actual spatial and temporal measurements. In order that it may serve both for spatial and for temporal measurements it must be whenever it is applied in some manner split up into two manifolds the one with a three-fold order and the other with a single order. An essential element in the theory is that this cannot be done in a unique manner; indeed if the division were unique the system would be reduced to the former scheme in which there are two independent manifolds one to be correlated with physical space and the other with public time. An essential point of the theory is that although each element of the fundamental manifold is represented by four numbers or coordinates it cannot be said straight off that three of these are for the purpose of spatial and the other for the purpose of temporal representation. The mode in which the sets of four numbers are to be applied to represent both spatial and temporal measurements will depend upon the observer that is upon the physical frame of reference which he employs. Thus space and time cannot be immediately separated out from one another in any absolute way in the fundamental four-fold manifold. An element of this manifold specified by tour numbers may be regarded as an abstract event that is an ex tensionless object at an instant of abstract time but the actual event to which it may be made to correspond will be represented by three spatial measurements representing the position of a point and a clock-time both of which will be dependent upon the physical frame of reference employed by the observer. Thus the position and time of a single elementary actual event will be measured in wholly different ways temporally and spatially when referred to different physical frames of reference. In order that the fundamental fourfold manifold may be effective for its purpose a suitable metric must be introduced into it; and the whole value of the theory depends upon making such a choice of this metric that the manifold may serve its purpose of providing an abstract representation of spatio-temporal measurements which will succeed in resuming the actual facts in the physical domain as measured with reference to any actual physical frame which a particular observer may employ. Einstein's success so far as it is established consists in his having discovered by the employment of a refined mathematical Analysis involving the use of the Calculus of Tensors how this choice could be made. On the observational side the leading feature of the new scheme is that there is no longer a single physical space and a single public time common to all observers but that two physical frames of reference in motion relatively to one another will have different physical spaces and different time-measurements. As I have already remarked for ordinary purposes not only of everyday life but of Science the differences in question are negligible; it is only for certain purposes to which I shall presently refer that the Einstein scheme gives results which are sensibly different from those obtained on the basis of the older theory and thus becomes effective.

It is sometimes said that Einstein's theory involves an obliteration of all distinction between time and space and between past and future since their measurements cannot be disentangled from one another in any absolute way. It must however be remembered that the original qualitative distinctions in our spatial and temporal intuitions are untouched by Einstein's theory or by any other theory of the measurement of space and time. These qualitative distinctions as also the qualitative intuitional distinction between future and past are removed by abstraction even in accordance with the older traditional scheme when we pass from intuitional space and time to abstract space and abstract time; moreover all our measurements of public time by means of clocks or by any other method in which some regular physical process is employed are spatial measurements; and thus the qualitative distinction between time and space has already been removed by abstraction. This has been spoken of by Bergson as the spatialization of time. The obliteration of the distinction in Einstein's scheme consists in an abstraction of all qualities of intuitional space and time except the element of order which appertains to both a three-fold order in the one case and a linear order in the other. The fundamental four-fold manifold involves the notion of order that is of an abstract order neither specifically spatial nor specifically temporal but obtained by abstraction from both and generalized into a single four-fold order. Both extension and duration have been removed by abstraction. That this abstract manifold is spoken of by many expositors of the theory as the “world” may perhaps have a certain convenience but it is I think somewhat unfortunate especially when it is stated or suggested that this abstract construct is the “real world” because such terminology has at least the appearance of involving a prejudgment of metaphysical theories.

The next point in which Einstein's theory has introduced new conceptions is connected not only with the measurement of space and time but also brings those measurements into relation with physical phenomena especially with the gravitational phenomena of matter and with the electromagnetic phenomenon of light. The ancient idea that space and time are objects with definite metric properties of their own independent of matter but forming a kind of framework into which material bodies at rest or in motion could be fitted and that these metric properties of space and time are independent of all physical laws had become moribund after the investigations of the foundations of Geometry due to Riemann and Helmholtz. It was shown by them that the abstract Geometry that is applicable to describe our measurements in physical space has a metric which is not fixed a priori but is dependent on the fact of experience that there exist bodies which are approximately rigid and are freely movable in physical space; the metric is then so determined that the numerical measures of the distances of pairs of points of such bodies remain unaltered during their motions. So far the choice was left open either of employing a Euclidean metric or a non-Euclidean metric with either positive or negative space-constant as the basis of measurements in physical space. But as I described in my lecture on “Time and Space” in connection with the suggested experiment or measuring the angles of a triangle with very large sides in order to determine whether a Euclidean or a non-Euclidean metric would accord with facts of observation in physical space the interpretation of any result obtained in such an experiment would depend upon the mode in which we formulate the laws of Optics. This shows that the system of Geometry adapted to form the basis of measurements in physical space is dependent upon the form in which we state physical laws; and thus that Geometry and Physics are in our experience inseparable! from one another. The choice of the particular system of metrical Geometry best adapted to describe our spatial measurements will be such as to be consistent with the simplest formulation of physical laws especially those of Optics. This fact of the interdependence of spatial measurements and physical laws is not a convention but an unalterable datum of observation. In Einstein's theory this interdependence is developed to a further degree so that spatial and temporal measurements are brought into relation with the phenomenon of gravitation. In accordance with the theory any portion of matter exhibits itself in the presence of a gravitational field surrounding the matter in which spatio-temporal measurements must be made in accordance with a metric varying from one part to another of the field and differing from the metric which would be applicable at places very remote from gravitating matter. This gravitational phenomenon considered as involving a special metric distribution in the field of the body will exhibit itself in the orbital motion of a particle or in the path of a ray of light in the field. Thus for example the orbital motion of a planet round the sun is represented not as due to a supposed attractive force towards the sun but simply as exhibiting the spatio-temporal metric throughout the gravitational field of the sun. Thus in Einstein's theory what we have been accustomed to regard as the effect of gravitational forces is included in a scheme of spatio-temporal measurements. In a sense gravitation is included in Geometry only that Geometry is four dimensional non-Euclidean and it has a metric of a kind more complicated than in the older schemes of non-Euclidean Geometry; moreover matter at least in its gravitational aspect only exhibits itself in and through the spatio-temporal metric. The fact that measurements of time and space are not independent of the physical phenomena of gravitation and of light leads to the idea that all the laws of physics and the laws which govern spatio-temporal measurements must be regarded as belonging to one interconnected whole and should be conceptually represented by a single unitary scheme. Einstein's theory is an attempt to attain to such a unitary scheme at least so far as the gravitational phenomena are concerned. Extensions of Einstein's conception have been already suggested by Weyl and others with a view to embracing electromagnetic phenomena in general as well as gravitation in a single conceptual scheme.

In order to describe in more detail the nature of the theory of relativity it is necessary to sketch the history of its origin. The theory has been developed in two stages the first of these culminating in what is known as the special theory of relativity and the second in the general theory of relativity. The special theory of relativity is accepted by some Physicists and Mathematicians who are sceptic al as regards the general theory at least in the particular form in which it has been developed by Einstein. The special theory takes no account of the phenomenon of gravitation; it is applicable strictly only in localities very remote from large gravitating masses; but it applies also very approximately to optical phenomena in weak fields of gravitation such as that of the earth. It had its origin in experimental investigations undertaken in connection with the electromagnetic theory of light in accordance with which light was regarded as an electromagnetic disturbance propagated through the ether. After various attempts to elucidate the structure of the ether and its relations with material bodies it came to be regarded for the most part as a substance which is undisturbed by the motion of material bodies through it and as freely interpenetrating such bodies. On this theory the ether would form a natural frame of reference with respect to which all motions of material bodies might at least ideally be measured; and it became a matter of importance as a test of the theory to detect by experimental observation the existence of the velocities of material bodies relative to the ether. It is known that light is propagated with a velocity approximately of 300000 kms. or 186000 miles a second. If a body is moving in the direction of or in the opposite direction to a ray of light with a velocity through the ether which is an' appreciable fraction of the velocity of light the ray of light would appear to have a velocity relative to the body less in the first case and greater in the second than the velocity of propagation of light through the ether; the defect or excess being the velocity of the body through the ether. The earth in its orbital motion round the sun has a velocity relatively to a frame fixed by the sun and stars of about 30 kms. or 18 miles a second. Although the velocity of the earth with respect to the ether was unknown if it were assumed that it is of the same order of magnitude as the orbital velocity of the earth it seemed certain that the velocity relative to the ether might be detected and its magnitude determined by comparing the velocity relatively to the earth of a ray of light in the direction of the earth's orbital motion with that of a ray in the perpendicular direction provided sufficiently precise measurements were made in an apparatus devised for the purpose. The celebrated experiment of Michelson and Morley carried out in 1887 was devised in order to establish the existence of the expected effect due to the velocity of the earth relative to the ether; the effect of such velocity was expected to be apparent if the velocity of the earth in the ether was only a quarter of its orbital velocity. A beam of light from a single source was divided by partial reflection at a mirror into two portions one in the original direction of the beam and the other at right angles to the first. These two portions were reflected back by mirrors and struck the first mirror again when portions of them would be re-united. The whole apparatus could be rotated into any position and could be fixed so that the original beam was in any required direction. The two portions of the beam which are reunited having required different times to pass from the first mirror to the reflecting mirrors and back again exhibit interference fringes. A rotation of the instrument was expected to show a displacement of these fringes owing to a change in the retardation of one portion relatively to the other. But when various positions of the apparatus were explored no trace of such displacement of the fringes was observed whereas the expected displacements would have been easily capable of measurement had they existed. The same negative result was reached by a still more refined experiment conducted by Morley and Miller in 1905. This and other experiments have led to the conclusion that it is impossible to detect by any observation the motion of a material body relatively to the ether. The first attempt to account for this inability to detect any effect of the motion of matter through the ether was made by Fitzgerald and independently by Lorentz. It was suggested that when a material body is in motion through the ether its dimensions in the direction of the motion are shortened by an amount depending upon the square of the ratio of its velocity to the velocity of light; and thus that the whole apparatus in the Michelson-Morley experiment is shortened in this way in the direction of its velocity through the ether thus annulling the displacements of the interference fringes. This so-called Fitzgerald contraction of a body moving through the ether could only be conceived as due to some interaction of the ether with the constituents of the moving body; the hypothesis of its existence was much discussed and led to various difficulties.

The interpretation of the observed facts that was given by Einstein was of a radically different character; it was in this connection that he propounded the theory which is now called the special theory of relativity. The first postulate of this theory is that all physical phenomena as observed from a material body as frame of reference appear to be completely independent of any uniform translational motion which that body may have relatively to another physical frame of reference. The second postulate of the theory is that the velocity of light is independent of the motion of the source of light. If these postulates be accepted the negative result of the Michelson-Morley experiment is immediately accounted for. In accordance with these postulates the velocity of light is the same when referred to any two frames of reference in uniform translational motion with respect to one another and is independent of the origin of the light. It should be observed that the traditional Newtonian Dynamics is in full accord with the foregoing postulate of relativity because the Newtonian equations of motion of a dynamical system referred to a given material frame of reference are unaltered when a new frame of reference is employed which is in uniform translational motion with respect to the given frame. But that is not the case with Maxwell's equations for the representation of electromagnetic phenomena relatively to a frame of reference fixed in the ether. It was shown by Larmor and Lorentz that these equations are unaltered in form when the coordinates and the time are transformed by means of a certain linear transformation of a less simple character than the transformation of the coordinates in a dynamical system from one frame of reference to another in uniform translational motion with respect to the first. In this so-called Lorentz-transformation the new coordinates are expressed in terms of the old in a form which involves the time as well as the translational velocity of the new frame of reference and the new time-measurement involves not only the original time-measurement but also the original spatial coordinates. It thus appears that the Newtonian dynamical scheme and the Maxwellian electrodynamical scheme which represents optical phenomena when taken together do not satisfy the postulates of the restricted principle of Relativity; and consequently one of these must be changed if that principle is to be accepted. The bold step taken by Einstein consists of a rejection of the Newtonian system of Dynamics and the substitution of a new dynamical scheme in which the Lorentz-transformation is applicable not only in the electromagnetic equations but also in Dynamics. The consequences and implications of this step are of a far-reaching character. A complete revision of the old ideas about spatial and temporal measurements is involved in the change. When we pass from the measures of space and time which an observer with his physical frame of reference employs to the corresponding measurements of time and space employed for the same physical event by another observer with a frame of reference in uniform translational motion with respect to that of the first observer the scales both of spatial and temporal measurements of one and the same event are different for the two observers. The measure of time for the one observer depends not only upon the measure of time of the other observer but also upon his spatial measurements as well as upon the relative velocities of the two observers; thus there exists no single system of measurement of time which is common to all observers Neither can the two observers employ one and the same system of spatial measurement. If two events occur at different places the interval of time between them will be measured differently by two observers in motion relatively to one another; for one observer the two events may be simultaneous whilst for the other observer the same events may occur at different times. The distance between the places at which the two events occur will be in general different for the two observers. It might appear that as the intervals both of time and of space which distinguish two events depend upon the observer there is no invariant relation between the two events; that is no relation which is common to all the observers. When however the complete scheme in an abstract form is set up it appears that this is not the case. The important step was taken by Minkowski of establishing that the Lorentz-transformation is capable of simple representation and interpretation if an abstract four-fold ordered continuous manifold which he called the “world” with a certain metric system imposed upon it is taken as the conceptual basis upon which all actual spatio-temporal measurements are made to rest. In view of the use which is to be made of the elements of this manifold any one of which is to be regarded as correlated with an actual observable event which for any observer is extremely small both in extension and in time-measure these elements of the manifold may be spoken of as abstract elementary events; they are frequently also spoken of as points in a four-dimensional geometrical space but such language ought not to be taken to imply that our intuitional notions of space and of time are applicable to this fundamental manifold. The notion of continuous order taken in the abstract is the sole remaining element of our spatial and temporal intuitions which is a constituent of the conception of this manifold. The order of the elements is assigned by correlating each element with four real numbers often spoken of analogically as the coordinates of the element. The metric is imposed upon the manifold by means of a definition of what may be called the “separation of two elements.” It is taken to be the square root of the sum of the squares of the differences of the corresponding coordinates of the two elements; but in this sum three of the squared differences are taken to have the negative sign and the metric is not therefore strictly an extension of the ordinary Euclidean metric. If the velocity of light is not taken to be unity the positive squared difference in the expression for the “separation” must be multiplied by the square of the constant numerical measure of the velocity of light. If now by means of the linear Lorentz-transformation a new system of coordinates is introduced into the manifold so that each element is represented by a new quadruplet of numbers determined from the original quadruplet by means of the Lorentz-transformation it will be found that the I separation” between two elements has exactly the same value when expressed in terms of the new coordinates as it had originally. It is this fact upon which the utility of the abstract scheme depends. The “separation” between two particular events is an invariant for all observers. A change of coordinates in the manifold of the kind described corresponds to and represents a change from the spatio-temporal measurements of one observer with his own scale of measurements to the measurements of another observer in uniform motion relatively to the former one. Thus this fundamental manifold forms the ultimate conceptual basis for Einstein's restricted theory of relativity which satisfies the two postulates I have already specified. Of the four coordinates of an element of the manifold for a particular observer in the physical domain that one of the coordinates which appears with the positive sign in the expression for the “separation” of two events may be taken to represent by correlation his measure of time and the other three his spatial measurements relatively to his rectangular frame of reference. But we cannot say that of the four coordinates of an element of the fundamental manifold one is to be always correlated with temporal measurements and thus represents abstractly a time and the other three represent abstractly spatial coordinates. This is the case for one particular observer but another observer in motion relatively to the former will have to employ an entirely different set of four coordinates in the manifold for correlation with his spatio-temporal measures; this set being related with the former set by means of the Lorentz-transformation. Thus in general each one of the four coordinates in the manifold is correlated with a mixture of spatial and temporal measures in the physical domain.

A set of events in the physical domain represented by a material particle which never impinges on any other particle will be correlated with a continuous set of elements in the fundamental manifold forming what may by analogy be called a straight line and this is spoken of by Minkowski as the world-line of the particle. This purely statical abstract object in the man fold represents conceptually the whole history of the material particle past and future; its interpretation spatially and temporally will vary according to the circumstances of the observer involving the way in which he judges from his own relative standpoint the spatial and temporal circumstances of the particle of matter in question. The principle of relativity as embodied in the two postulates is incompatible with the conception of ether as a substantial medium for the transmission of light. There have at all times been great difficulties in formulating the properties of an ether which should satisfy the conflicting demands which fact and theory seemed to require. Notwithstanding these difficulties there has been much reluctance on the part of Physicists to give up a conception which was designed to afford a pictorial representation of electromagnetic phenomena and which appealed strongly to those who regard the ether as a concrete though not directly perceptible reality. The idea has however gained ground that it is in the equations of the electrodynamic theory that the really effective formulation of that theory is to be found and thus the ground has been prepared for that final removal of the formerly useful notion of the ether which is involved in the acceptance of Einstein's theory even in its restricted form.

The restricted principle of relativity which was completely stated by Einstein in 1905 changes radically the notions of the measurement of time and space which were employed in the Newtonian Dynamics and in ordinary affairs but the principle suffers from the defect that the relativity is applicable only to material frames of reference in uniform motion of translation with respect to one another and that it takes no account of the phenomena of gravitation. The decade after 1905 was spent by Einstein in an endeavour to remedy these defects; and in 1915 he found himself able to propound a complete principle in which the relativity is applicable to all frames of reference in motion of any kind with respect to one another. The general principle of relativity includes the conception that all actual spatial and temporal measurements are dependent upon and vary in a definite manner with a material frame of reference employed by an observer. The scheme includes a mode of taking into account and measuring by means of spatio-temporal measurements gravitational phenomena as exhibited in gravitational fields. This mode of treating gravitation is fundamentally different from that of Newton; there is in it not even a suggestion of anything that could be regarded as a causal law of gravitation in accordance with the older traditional meaning attached to the term causation. The special spatio-temporal peculiarities of a field of gravitation are taken to give the only theory of gravitation that Science requires or can attain. Newton's law of gravitation in accordance with which the gravitation between two material particles is represented by a stress proportional in magnitude to the product of their masses and inversely as the square of their distance from one another being independent at any instant of the motions of the particles and of all other matter had become quite indefinite in meaning. In the first place the mass of a given particle has in modern Physics lost the characteristic property of having a constant value independent of the motion of the particle relative to an observer. In accordance with the electron theory of matter which rests upon the observed facts of radiation and radioactivity a material particle is constituted in part at least of electrons of which the effective mass increases when their velocity is increased. Thus the mass of a particle may be sensibly changed if it is set in motion as a whole with a velocity comparable with that of light or when the motions of the electrons within it are considerably altered by the receipt of energy from without. In fact in accordance with the special principle of relativity the mass of a system is the total energy in it divided by the square of the velocity of light; this energy being measured relatively to the observer. It is then no longer clear how the magnitudes of the masses in Newton's law are to be fixed. Again the distance between the two particles has no longer an absolute measure independent of their motions relative to an observer. Further the fact that gravitational force in Newton's theory depends at any instant only on the positions of the particles at that instant is not consistent with the conception of the propagation of gravitation with finite velocity through a medium; and thus Newtonian gravitation was never linked up with other physical phenomena in any unitary scheme.

As I explained in my lecture on Dynamics in the Newtonian system all actual phenomena of motion are described upon the basis of a conceptual scheme in which there is an absolute frame of reference; but in physical space it is possible with a degree of approximation sufficient for the purposes of any special case to determine so-called Newtonian frames of reference which can be correlated with the absolute conceptual frame. Any frame of reference which is in rotational motion or in accelerated translational motion with respect to a Newtonian frame can only be employed if certain “fictitious forces” among which is the so-called centrifugal force in the case of rotation are introduced into the equations of motion. An essential element in Einstein's scheme consists in his principle of equivalence which involves a denial of any distinction between such fictitious forces and gravitational forces. All of them alike are regarded as due to the gravitational field and the mode in which this gravitational field exhibits itself is in the spatio-temporal measurements suitable to the particular frame of reference employed. In any sufficiently small region it is impossible by any experiment to distinguish between a fictitious and a gravitational field of force. The Einstein scheme does not depend as does the Newtonian upon the selection of any specially suitable frames of reference; all such frames are on a parity and in this the completeness of the relativity consists.

This independence of any particular frame of reference is essentially connected with the fact that all actual processes of measuring space or time consist in the establishment of the coincidence of two points belonging to two material bodies; and such a coincidence is a fact which is unaltered by any change in the frame of reference employed in making the measurements.

In Newtonian Dynamics the equality of the inertial mass and the gravitational mass of a material body remained as a bare tact derived from observation which appeared from the theoretical point of view to be merely an accident. In Einstein's conceptual scheme the two are indistinguishable or rather identical in accordance with the fundamental postulates of the theory. An essential consequence of Einstein's scheme is that any phenomena which an observer perceives in his neighborhood to be due to a gravitational field would be perceived unaltered if the gravitational field were not present provided that the observer makes his frame of reference move with the acceleration characteristic of the gravitational field at the place at which he makes the observation. This amounts to the assertion that it is possible to eliminate a gravitational field at a particular place by a proper choice of the frame of reference. The fundamental four-fold ordered manifold is employed as the conceptual basis of the general theory of relativity as in the case of the special theory except that the metric imposed upon it is of a more complicated character designed to take into account by correlation the existence of gravitational fields and to exhibit their presence in spatio-temporal measurements. The determination of the precise mode of correlating the conceptual scheme with actual measurements would appear to be one of the most difficult points in the whole theory.

The precise mode in which the theory of this metric was developed by Einstein is of a so highly technically mathematical character that I can only give the briefest and most superficial indication of its nature. In this matter the geometrical investigations of Gauss and Riemann were the original source of inspiration. Riemann's theory of a continuous ordered manifold designed by him for the purpose of investigating the foundations of Geometry led in the hands of Ricci Christoffel and others to a mathematical development known as the theory of tensors and this theory was utilized by Einstein and developed for his purpose of determining the nature of a metric system to be imposed upon the fundamental manifold of such a character that it could be applied to the spatio-temporal metrical characterization of observed gravitational fields. Two elements of the fundamental manifold are regarded as neighboring elements when each of the four coordinates of one of them differs by a very small number (more exactly a differential) from the corresponding coordinate of the other. The “separation” of two such neighboring elements is defined as the square root of a quadratic function of the differentials of the four coordinates. The ten coefficients in this quadratic function are in general functions of the coordinates of the element and are taken to be the potentials characterizing a gravitational field. In order that this definition of the differential “separation” may be the basis of a definite metric in the manifold these coefficients must satisfy a certain number of conditions which involve also the gradients of these coefficients. The determination of these conditions is made in accordance with the developed Riemannian theory which had its origin in Gauss' theory of the curvature of surfaces in three-dimensional space. Einstein succeeded in overcoming the great difficulty of showing how all these conditions could be satisfied so as at the same time to make the metric available for the representation of actual spatio-temporal measurements in gravitational fields such as those which are given us by experience.

The history of a material particle which does not impinge upon any other particle is completely represented by a set of elements in the fundamental manifold which may by analogy be spoken of as a geodesic in that manifold and is obtained as an extremal of the integral of the differential separations. This geodesic is the world-line of the particular particle and it is analogous to a curved line in ordinary geometry whenever the particle is in a gravitational field; when there is no such held the world-line is the analogue of a straight line as in the special theory. The world-line forms the conceptual basis of a spatio-temporal description of the motion of the particle as estimated by an observer who chooses arbitrarily his material frame of reference. All such frames may be employed indifferently the observed path of the particle being relative to the particular frame employed; but the world-line is absolute for a particular material particle.

One of the most important applications which Einstein has made of his theory is to the motion of a planet in the gravitational field of the sun. His result contains a correction to the Newtonian law of force upon the planet which is in any actual case very small but the effect of which in one case that of the planet Mercury is sufficient to render it capable of being observed. This has formed one of the tests of the applicability of Einstein's theory and the theory appears to have proved itself able to stand the test. In accordance with Newton's theory of gravitation a planet would move over and over again in a fixed ellipse with the sun in one of the foci in accordance with Kepler's law if it were not subject to the disturbing effects due to the other planets. Thus the line joining the sun to the planet when in perihelion that is when nearest to the sun would preserve for all time a direction fixed relatively to the stars. However one of the effects upon the orbit produced by the gravitation of the other planets is to give this line a very small change of direction which steadily accumulates so as to make the change in a century sufficient to be observed. The orbit of the planet Mercury is more elongated than in the case of other planets so that it can be observed at what times it is in perihelion more accurately than in the case of a planet whose orbit is more nearly circular. The motion of the line due to the disturbance of the other planets was calculated by astronomers to amount to 532″ in a century; but the actually observed amount was found to be 574″ so that the excess of 42″ remained to be accounted for. It was calculated by Einstein that the effect of his amendment to Newton's law would be that there would be an advance of the line of perihelion amounting to 43″ in a century which differs only by a very minute amount from that of the discrepancy to be accounted for. It thus appears that Einstein's law leads in this crucial case to a result which is in close agreement with observation and in which Newton's law is in default. It is not possible to apply a similar test in the case of the other planets either because the corresponding advance is too small or because the orbits are too nearly circular to make sufficiently accurate observations possible.

Another crucial test of the theory was provided by observation of the deviation of direction in a ray of light which takes place when the ray passes very near to the sun. It has for some time been known that radiation has inertia as manifested in radiation-pressure; in accordance with Einstein's theory it consequently has gravitational mass or weight. As the result of a calculation made by Einstein the effect of the intense gravitational field near the surface of the sun would be that a ray of light passing very near that surface would appear to an observer on the earth to be deflected through an angle of 1″⋅74. If Newton's theory of gravitation were applicable the deflection would be just half this amount 0″⋅87. It was decided to put to the test of observation the question which of these two values represents the deflection that actually takes place. At the time of a total eclipse of the sun a deflection of the light from a star very near the sun would exhibit itself in an apparent displacement of the star from its true position in a direction away from the sun. The very delicate operation of measuring the displacements of position of stars near the sun at the time of the total eclipse on May 29 1919 was undertaken by astronomers in two expeditions one in Brazil and the other in the Gulf of Guinea. Although the observations of the latter expedition were very seriously hindered by cloudy weather in both cases the results of observation of several stars after the elaborate process of correction for various errors of observation had been carried out were found to be very fairly in accordance with Einstein's prediction and definitely to rule out the correctness of the deflection as calculated by Newton's theory. Thus the result of the observations was distinctly in favour of Einstein's theory as against the Newtonian. It must however be remembered that such observations are of an extremely delicate character and involve various sources of error. Accordingly it is necessary to await the results of further observations of the same kind before complete confidence may be placed upon the results of this test.

The third practical test of the theory which has been made has not as yet been brought to a decisive conclusion. It was expected that in accordance with the theory when the spectral lines of a chemical substance in the sun are observed they should show a displacement from the position of the spectral lines of the same substance on the earth. The results obtained by different observers in the course of their attempts to verify or disprove this predicted effect are discordant and contradictory. It appears therefore that up to the present time no decisive evidence has been obtained by means of which it can be definitely decided whether the theory satisfies this test or not.

Whatever be the ultimate fate of Einstein's theory and to whatever modifications it may in the future be subjected it is as it stands of the highest interest not only on account of its comprehensive character and on account of the novelty of its conceptions but also as a chapter in the history of Science. It could not have arisen apart from its two great roots the one the physical theory of Electromagnetism including the theory of light and the other the highly abstract theory of Geometry in its most generalized form. The latter includes a whole series of investigations into the foundations of Geometry which reach far back into the past culminating in the work of Riemann and Helmholtz who were the first to perceive that Geometry as applicable to actual measurements is not really independent of Physics and which have been continued in Mathematical detail by others.

All this line of mathematical work was carried out almost entirely by thinkers who had no hope that their labours would ever form an essential groundwork of a great physical theory. This piece of history illustrates in a most striking manner the fact that there is never any certainty that the most abstract mathematical theory which although it of course has its ultimate roots in the perceptual domain appears to have no direct relations with that domain may not turn out to be of the last importance in relation to some unforeseen theory of physical phenomena. That the further development of Einstein's theory so that it may embrace all the phenomena with which Physics has to deal may be subject to limitations which in its present form at least it cannot avoid without undergoing at least some modification is indicated by the rise and development of the theory of Quanta. It would appear that sub-atomic Dynamics is essentially dissimilar in character to that of systems which consist of large aggregations of atoms. This would appear to indicate that Einstein's scheme which like the Classical Dynamics is a continuous theory may prove to be inapplicable as the conceptual basis for the representation of sub-atomic phenomena. Suggestions have even been made recently that a discrete manifold may be requisite as the conceptual basis of spatio-temporal measurements for this purpose instead of the continuous manifold employed at present by the Einstein theory.

It is interesting to observe that Riemann himself made at the conclusion of his celebrated dissertation on the foundations of Geometry a remark which showed that he had a prophetic insight into the possibility of his conceptions being one day linked up with the physics of matter. The passage to which I refer states that1:

The question as regards the validity of Geometry in the region of the infinitesimally small is connected with the question relating to the inner ground of the metric of space. In connection with this question which can surely be reckoned as belonging to the doctrine of space the above remark can be applied that in a discrete manifold the principle of the metric is contained in the conception of the manifold itself but that in a continuous manifold it must come from outside. Thus the reality upon which space is based must either form a discrete manifold or else the basis of the metric must be sought outside in binding forces that act upon it.