APPENDIX

APPENDIXNote1 (page21)“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.Note2 (page27)Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.Note3 (page30)Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”And again:“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.Note4 (page51)For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)and this immediately provides us with a means of transforming the laws of one system to those of another.With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.Note5 (page55)The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.Note6 (page56)“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”(Dr. Walker)Note7 (page57)Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).Note8 (page59)That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.Note9 (page67)How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)TIME, SPACE, AND GRAVITATION1BYProf. Albert EinsteinThere are several kinds of theory in physics. Most of them are constructive. These attempt to build a picture of complex phenomena out of some relatively simple proposition. The kinetic theory of gases, for instance, attempts to refer to molecular movement the mechanical thermal, and diffusional properties of gases. When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.Theories of Principle.—But in addition to this most weighty group of theories, there is another group consisting of what I call theories of principle. These employ the analytic, not the synthetic method. Their starting-point and foundation are not hypothetical constituents, but empirically observed general properties of phenomena, principles from which mathematical formulæ are deduced of such a kind that they apply to every case which presents itself. Thermodynamics, for instance, starting from the fact that perpetual motion never occurs in ordinary experience, attempts to deduce from this, by analytic processes, a theory which will apply in every case. The merit of constructive theories is their comprehensiveness, adaptability, and clarity, that of the theories of principle, their logical perfection, and the security of their foundation.The theory of relativity is a theory of principle. To understand it, the principles on which it rests must be grasped. But before stating these it is necessary to point out that the theory of relativity is like a house with two separatestories, the special relativity theory and the general theory of relativity.Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer to another body. The motion of a railway train is described with reference to the ground, of a planet with reference to the total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termed systems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system of coordinates.The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system. The state of motion of an inertia-system, so far as mechanics are concerned, is not restricted by nature to one condition. The condition in the following proposition suffices; a system of coordinates moving in the same direction and at the same rate as a system of inertia is itself a system of inertia. The special relativity theory is therefore the application of the following proposition to any natural process: “Every law of nature which holds good with respect to a coordinate systemKmust also hold good for any other systemK′provided thatKandK′are in uniform movement of translation.”The second principle on which the special relativity theory rests is that of the constancy of the velocity of light in a vacuum. Light in a vacuum has a definite and constant velocity, independent of the velocity of its source. Physicists owe their confidence in this proposition to the Maxwell-Lorentz theory of electro-dynamics.The two principles which I have mentioned have received strong experimental confirmation, but do not seem to be logically compatible. The special relativity theory achieved their logical reconciliation by making a change in kinematics, that is to say, in the doctrine of the physical laws of space and time. It became evident that a statement of the coincidence of two events could have a meaning only in connection with a system of coordinates, that the mass of bodies and the rate of movement of clocks must depend on their state of motion with regard to the coordinates.The Older Physics.—But the older physics, including the laws of motion of Galileo and Newton, clashed with the relativistic kinematics that I have indicated. The latter gave origin to certain generalized mathematical conditions with which the laws of nature would have to conform if the two fundamental principles were compatible. Physics had to be modified. The most notable change was a new law of motion for (very rapidly) moving mass-points, and this soon came to be verified in the case of electrically-laden particles. The most important result of the special relativity system concerned the inert mass of a material system. It became evident that the inertia of such a system must depend on its energy-content, so that we were driven to the conception that inert mass was nothing else than latent energy. The doctrine of the conservation of mass lost its independence and became merged in the doctrine of conservation of energy.The special relativity theory which was simply a systematic extension of the electro-dynamics of Maxwell and Lorentz, had consequences which reached beyond itself. Must the independence of physical laws with regard to a system of coordinates be limited to systems of coordinates in uniform movement of translation with regard to one another? Whathas nature to do with the coordinate systems that we propose and with their motions? Although it may be necessary for our descriptions of nature to employ systems of coordinates that we have selected arbitrarily, the choice should not be limited in any way so far as their state of motion is concerned. (General theory of relativity.) The application of this general theory of relativity was found to be in conflict with a well-known experiment, according to which it appeared that the weight and the inertia of a body depended on the same constants (identity of inert and heavy masses). Consider the case of a system of coordinates which is conceived as being in stable rotation relative to a system of inertia in the Newtonian sense. The forces which, relatively to this system, are centrifugal must, in the Newtonian sense, be attributed to inertia. But these centrifugal forces are, like gravitation, proportional to the mass of the bodies. Is it not, then, possible to regard the system of coordinates as at rest, and the centrifugal forces as gravitational? The interpretation seemed obvious, but classical mechanics forbade it.This slight sketch indicates how a generalized theory of relativity must include the laws of gravitation, and actual pursuit of the conception has justified the hope. But the way was harder than was expected, because it contradicted Euclidian geometry. In other words, the laws according to which material bodies are arranged in space do not exactly agree with the laws of space prescribed by the Euclidian geometry of solids. This is what is meant by the phrase “a warp in space.” The fundamental concepts “straight,” “plane,” etc., accordingly lose their exact meaning in physics.In the generalized theory of relativity, the doctrine of space and time, kinematics, is no longer one of the absolute foundations of general physics. The geometrical states of bodiesand the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material system concerned.Thus the new theory of gravitation diverges widely from that of Newton with respect to its basal principle. But in practical application the two agree so closely that it has been difficult to find cases in which the actual differences could be subjected to observation. As yet only the following have been suggested:1. The distortion of the oval orbits of planets round the sun (confirmed in the case of the planet Mercury).2. The deviation of light-rays in a gravitational field (confirmed by the English Solar Eclipse expedition).3. The shifting of spectral lines towards the red end of the spectrum in the case of light coming to us from stars of appreciable mass (not yet confirmed).The great attraction of the theory is its logical consistency. If any deduction from it should prove untenable, it must be given up. A modification of it seems impossible without destruction of the whole.No one must think that Newton’s great creation can be overthrown in any real sense by this or by any other theory. His clear and wide ideas will for ever retain their significance as the foundation on which our modern conceptions of physics have been built.1Republished by permission from “Science.”↑EINSTEIN’S LAW OF GRAVITATION1BYProf. J. S. AmesJohns Hopkins University… In the treatment of Maxwell’s equations of the electromagnetic field, several investigators realized the importance of deducing the form of the equations when applied to a system moving with a uniform velocity. One object of such an investigation would be to determine such a set of transformation formulæ as would leave the mathematical form of the equations unaltered. The necessary relations between the new space-coordinates, those applying to the moving system, and the original set were of course obvious; and elementary methods led to the deduction of a new variable which should replace the time coordinate. This step was taken by Lorentz and also, I believe, by Larmor and by Voigt. The mathematical deductions and applications in the hands of these men were extremely beautiful, and are probably well known to you all.Lorentz’ paper on this subject appeared in the Proceedings of the Amsterdam Academy in 1904. In the following year there was published in theAnnalen der Physika paper by Einstein, written without any knowledge of the work of Lorentz, in which he arrived at the same transformation equations as did the latter, but with an entirely different and fundamentally new interpretation. Einstein called attention in his paper to the lack of definiteness in the concepts of time and space, as ordinarily stated and used. He analyzed clearly the definitions and postulates which were necessarybefore one could speak with exactness of a length or of an interval of time. He disposed forever of the propriety of speaking of the “true” length of a rod or of the “true” duration of time, showing, in fact, that the numerical values which we attach to lengths or intervals of time depend upon the definitions and postulates which we adopt. The words “absolute” space or time intervals are devoid of meaning. As an illustration of what is meant Einstein discussed two possible ways of measuring the length of a rod when it is moving in the direction of its own length with a uniform velocity, that is, after having adopted a scale of length, two ways of assigning a number to the length of the rod concerned. One method is to imagine the observer moving with the rod, applying along its length the measuring scale, and reading off the positions of the ends of the rod. Another method would be to have two observers at rest on the body with reference to which the rod has the uniform velocity, so stationed along the line of motion of the rod that as the rod moves past them they can note simultaneously on a stationary measuring scale the positions of the two ends of the rod. Einstein showed that, accepting two postulates which need no defense at this time, the two methods of measurements would lead to different numerical values, and, further, that the divergence of the two results would increase as the velocity of the rod was increased. In assigning a number, therefore, to the length of a moving rod, one must make a choice of the method to be used in measuring it. Obviously the preferable method is to agree that the observer shall move with the rod, carrying his measuring instrument with him. This disposes of the problem of measuring space relations. The observed fact that, if we measure the length of the rod on different days, or when the rod is lying in different positions, we alwaysobtain the same value offers no information concerning the “real” length of the rod. It may have changed, or it may not. It must always be remembered that measurement of the length of a rod is simply a process of comparison between it and an arbitrary standard,e.g., a meter-rod or yard-stick. In regard to the problem of assigning numbers to intervals of time, it must be borne in mind that, strictly speaking, we do not “measure” such intervals,i.e., that we do not select a unit interval of time and find how many times it is contained in the interval in question. (Similarly, we do not “measure” the pitch of a sound or the temperature of a room.) Our practical instruments for assigning numbers to time-intervals depend in the main upon our agreeing to believe that a pendulum swings in a perfectly uniform manner, each vibration taking the same time as the next one. Of course we cannotprovethat this is true, it is, strictly speaking, a definition of what we mean by equal intervals of time; and it is not a particularly good definition at that. Its limitations are sufficiently obvious. The best way to proceed is to consider the concept of uniform velocity, and then, using the idea of some entity having such a uniform velocity, to define equal intervals of time as such intervals as are required for the entity to traverse equal lengths. These last we have already defined. What is required in addition is to adopt some moving entity as giving our definition of uniform velocity. Considering our known universe it is self-evident that we should choose in our definition of uniform velocity the velocity of light, since this selection could be made by an observer anywhere in our universe. Having agreed then to illustrate by the words “uniform velocity” that of light, our definition of equal intervals of time is complete. This implies, of course, that there is no uncertainty on our part as to the fact that the velocity oflight always has the same value at any one point in the universe to any observer, quite regardless of the source of light. In other words, the postulate that this is true underlies our definition. Following this method Einstein developed a system of measuring both space and time intervals. As a matter of fact his system is identically that which we use in daily life with reference to events here on the earth. He further showed that if a man were to measure the length of a rod, for instance, on the earth and then were able to carry the rod and his measuring apparatus to Mars, the sun, or to Arcturus he would obtain the same numerical value for the length in all places and at all times. This doesn’t mean that any statement is implied as to whether the length of the rod has remained unchanged or not; such words do not have any meaning—remember that we can not speak of true length. It is thus clear that an observer living on the earth would have a definite system of units in terms of which to express space and time intervals,i.e., he would have a definite system of space coordinates (x,y,z) and a definite time coordinate (t); and similarly an observer living on Mars would have his system of coordinates (x′,y′,z′,t′). Provided that one observer has a definite uniform velocity with reference to the other, it is a comparatively simple matter to deduce the mathematical relations between the two sets of coordinates. When Einstein did this, he arrived at the same transformation formulæ as those used by Lorentz in his development of Maxwell’s equations. The latter had shown that, using these formulæ, the form of the laws for all electromagnetic phenomena maintained the same form; so Einstein’s method proves that using his system of measurement an observer, anywhere in the universe, would as the result of his own investigation of electromagnetic phenomena arrive at thesame mathematical statement of them as any other observer, provided only that the relative-velocity of the two observers was uniform.Einstein discussed many other most important questions at this time; but it is not necessary to refer to them in connection with the present subject. So far as this is concerned, the next important step to note is that taken in the famous address of Minkowski, in 1908, on the subject of “Space and Time.” It would be difficult to overstate the importance of the concepts advanced by Minkowski. They marked the beginning of a new period in the philosophy of physics. I shall not attempt to explain his ideas in detail, but shall confine myself to a few general statements. His point of view and his line of development of the theme are absolutely different from those of Lorentz or of Einstein; but in the end he makes use of the same transformation formulæ. His great contribution consists in giving us a new geometrical picture of their meaning. It is scarcely fair to call Minkowski’s development a picture; for to us a picture can never have more than three dimensions, our senses limit us; while his picture calls for perception of four dimensions. It is this fact that renders any even semi-popular discussion of Minkowski’s work so impossible. We can all see that for us to describe any event a knowledge of four coordinates is necessary, three for the space specification and one for the time. A complete picture could be given then by a point in four dimensions. All four coordinates are necessary: we never observe an event except at a certain time, and we never observe an instant of time except with reference to space. Discussing the laws of electromagnetic phenomena, Minkowski showed how in a space of four dimensions, by a suitable definition of axes, the mathematical transformation of Lorentz and Einsteincould be described by a rotation of the set of axes. We are all accustomed to a rotation of our ordinary cartesian set of axes describing the position of a point. We ordinarily choose our axes at any location on the earth as follows: one vertical, one east and west, one north and south. So if we move from any one laboratory to another, we change our axes; they are always orthogonal, but in moving from place to place there is a rotation. Similarly, Minkowski showed that if we choose four orthogonal axes at any point on the earth, according to his method, to represent a space-time point using the method of measuring space and time intervals as outlined by Einstein; and, if an observer on Arcturus used a similar set of axes and the method of measurement which he naturally would, the set of axes of the latter could be obtained from those of the observer on the earth by a pure rotation (and naturally a transfer of the origin). This is a beautiful geometrical result. To complete my statement of the method, I must add that instead of using as his fourth axis one along which numerical values of time are laid off, Minkowski defined his fourth coordinate as the product of time and the imaginary constant, the square root of minus one. This introduction of imaginary quantities might be expected, possibly, to introduce difficulties; but, in reality, it is the very essence of the simplicity of the geometrical description just given of the rotation of the sets of axes. It thus appears that different observers situated at different points in the universe would each have their own set of axes, all different, yet all connected by the fact that any one can be rotated so as to coincide with any other. This means that there is no one direction in the four-dimensional space that corresponds to time for all observers. Just as with reference to the earth there is no direction which can be called vertical for all observers living on the earth. In thesense of anabsolutemeaning the words “up and down,” “before and after,” “sooner or later,” are entirely meaningless.This concept of Minkowski’s may be made clearer, perhaps, by the following process of thought. If we take a section through our three-dimensional space, we have a plane,i.e., a two-dimensional space. Similarly, if a section is made through a four-dimensional space, one of three dimensions is obtained. Thus, for an observer on the earth a definite section of Minkowski’s four-dimensional space will give us our ordinary three-dimensional one; so that this section will, as it were, break up Minkowski’s space into our space and give us our ordinary time. Similarly, a different section would have to be used to the observer on Arcturus; but by a suitable selection he would get his own familiar three-dimensional space and his own time. Thus the space defined by Minkowski is completely isotropic in reference to measured lengths and times, there is absolutely no difference between any two directions in an absolute sense; for any particular observer, of course, a particular section will cause the space to fall apart so as to suit his habits of measurement; any section, however, taken at random will do the same thing for some observer somewhere. From another point of view, that of Lorentz and Einstein, it is obvious that, since this four-dimensional space is isotropic, the expression of the laws of electromagnetic phenomena take identical mathematical forms when expressed by any observer.The question of course must be raised as to what can be said in regard to phenomena which so far as we know do not have an electromagnetic origin. In particular what can be done with respect to gravitational phenomena? Before, however, showing how this problem was attacked by Einstein;and the fact that the subject of my address is Einstein’s work on gravitation shows that ultimately I shall explain this, I must emphasize another feature of Minkowski’s geometry. To describe the space-time characteristics of any event a point, defined by its four coordinates, is sufficient; so, if one observes the life-history of any entity,e.g., a particle of matter, a light-wave, etc., he observes a sequence of points in the space-time continuum; that is, the life-history of any entity is described fully by a line in this space. Such a line was called by Minkowski a “world-line.” Further, from a different point of view, all of our observations of nature are in reality observations of coincidences,e.g., if one reads a thermometer, what he does is to note the coincidence of the end of the column of mercury with a certain scale division on the thermometer tube. In other words, thinking of the world-line of the end of the mercury column and the world-line of the scale division, what we have observed was the intersection or crossing of these lines. In a similar manner any observation may be analyzed; and remembering that light rays, a point on the retina of the eye, etc., all have their world-lines, it will be recognized that it is a perfectly accurate statement to say that every observation is the perception of the intersection of world-lines. Further, since all we know of a world-line is the result of observations, it is evident that we do not know a world-line as a continuous series of points, but simply as a series of discontinuous points, each point being where the particular world-line in question is crossed by another world-line.It is clear, moreover, that for the description of a world-line we are not limited to the particular set of four orthogonal axes adopted by Minkowski. We can choose any set of four-dimensional axes we wish. It is further evident that the mathematical expression for the coincidence of two points isabsolutely independent of our selection of reference axes. If we change our axes, we will change the coordinates of both points simultaneously, so that the question of axes ceases to be of interest. But our so-called laws of nature are nothing but descriptions in mathematical language of our observations; we observe only coincidences; a sequence of coincidences when put in mathematical terms takes a form which is independent of the selection of reference axes; therefore the mathematical expression of our laws of nature, of every character, must be such that their form does not change if we make a transformation of axes. This is a simple but far-reaching deduction.There is a geometrical method of picturing the effect of a change of axes of reference,i.e., of a mathematical transformation. To a man in a railway coach the path of a drop of water does not appear vertical,i.e., it is not parallel to the edge of the window; still less so does it appear vertical to a man performing manœuvres in an airplane. This means that whereas with reference to axes fixed to the earth the path of the drop is vertical; with reference to other axes, the path is not. Or, stating the conclusion in general language, changing the axes of reference (or effecting a mathematical transformation) in general changes the shape of any line. If one imagines the line forming a part of the space, it is evident that if the space is deformed by compression or expansion the shape of the line is changed, and if sufficient care is taken it is clearly possible, by deforming the space, to make the line take any shape desired, or better stated, any shape specified by the previous change of axes. It is thus possible to picture a mathematical transformation as a deformation of space. Thus I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great varietyof shapes; each of these new shapes is a picture of a suitable transformation.Now, consider world-lines in our four-dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections. So in the space of our world-lines, the space may be deformed in any imaginable manner without introducing any new intersections or changing the sequence of the existing intersections. It is this sequence which gives us the mathematical expression of our so-called experimental laws; a deformation of our space is equivalent mathematically to a transformation of axes, consequently we see why it is that the form of our laws must be the same when referred to any and all sets of axes, that is, must remain unaltered by any mathematical transformation.Now, at last we come to gravitation. We can not imagine any world-line simpler than that of a particle of matter left to itself; we shall therefore call it a “straight” line. Our experience is that two particles of matter attract one another.Expressed in terms of world-lines, this means that, if the world-lines of two isolated particles come near each other, the lines, instead of being straight, will be deflected or bent in towards each other. The world-line of any one particle is therefore deformed; and we have just seen that a deformation is the equivalent of a mathematical transformation. In other words, for any one particle it is possible to replace the effect of a gravitational field at any instant by a mathematical transformation of axes. The statement that this is always possiblefor any particle at any instant is Einstein’s famous “Principle of Equivalence.”Let us rest for a moment, while I call attention to a most interesting coincidence, not to be thought of as an intersection of world-lines. It is said that Newton’s thoughts were directed to the observation of gravitational phenomena by an apple falling on his head; from this striking event he passed by natural steps to a consideration of the universality of gravitation. Einstein in describing his mental process in the evolution of his law of gravitation says that his attention was called to a new point of view by discussing his experiences with a man whose fall from a high building he had just witnessed. The man fortunately suffered no serious injuries and assured Einstein that in the course of his fall he had not been conscious in the least of any pull downward on his body. In mathematical language, with reference to axes moving with the man the force of gravity had disappeared. This is a case where by the transfer of the axes from the earth itself to the man, the force of the gravitational field is annulled. The converse change of axes from the falling man to a point on the earth could be considered as introducing the force of gravity into the equations of motion. Another illustration of the introduction into our equations of a force by a means of a change of axes is furnished by the ordinary treatment of a body in uniform rotation about an axis. For instance, in the case of a so-called conical pendulum, that is, the motion of a bob suspended from a fixed point by string, which is so set in motion that the bob describes a horizontal circle and the string therefore describes a circular cone, if we transfer our axes from the earth and have them rotate around the vertical line through the fixed point with the same angular velocity as the bob, it is necessary to introduce into our equations ofmotion a fictitious “force” called the centrifugal force. No one ever thinks of this force other than as a mathematical quantity introduced into the equations for the sake of simplicity of treatment; no physical meaning is attached to it. Why should there be to any other so-called “force,” which like centrifugal force, is independent of the nature of the matter? Again, here on the earth our sensation of weight is interpreted mathematically by combining expressions for centrifugal force and gravity; we have no distinct sensation for either separately. Why then is there any difference in the essence of the two? Why not consider them both as brought into our equations by the agency of mathematical transformations? This is Einstein’s point of view.Granting, then, the principle of equivalence, we can so choose axes at any point at any instant that the gravitational field will disappear; these axes are therefore of what Eddington calls the “Galilean” type, the simplest possible. Consider, that is, an observer in a box, or compartment, which is falling with the acceleration of the gravitational field at that point. He would not be conscious of the field. If there were a projectile fired off in this compartment, the observer would describe its path as being straight. In this space the infinitesimal interval between two space-time points would then be given by the formulad s squared equals d x 1 squared plus d x 2 Subscript 2 Baseline plus d x 3 squared plus d x 2 Subscript 4 Baseline commawheredsis the interval andx 1 comma x 2 comma x 3 comma x 4are coordinates. If we make a mathematical transformation,i.e., use another set of axes, this interval would obviously take the formd s squared equals g 11 d x 33 squared plus g 22 d x 2 squared plus g 33 d x 3 squared plus g 44 d x 2 Subscript 4 Baseline plus 2 g 12 d x 1 d x 2 plus normal e normal t normal c period commawherex 1 comma x 2 comma x 3andx 4are now coordinates referring to the newaxes. This relation involves ten coefficients, the coefficients defining the transformation.But of course a certain dynamical value is also attached to theg’s, because by the transfer of our axes from the Galilean type we have made a change which is equivalent to the introduction of a gravitational field; and theg’s must specify the field. That is, theseg’s are the expressions of our experiences, and hence their values can not depend upon the use of any special axes; the values must be the same for all selections. In other words, whatever function of the coordinates any onegis for one set of axes, if other axes are chosen, thisgmust still be the same function of the new coordinates. There are teng’s defined by differential equations; so we have ten covariant equations. Einstein showed how theseg’s could be regarded as generalized potentials of the field. Our own experiments and observations upon gravitation have given us a certain knowledge concerning its potential; that is, we know a value for it which must be so near the truth that we can properly call it at least a first approximation. Or, stated differently, if Einstein succeeds in deducing the rigid value for the gravitational potential in any field, it must degenerate to the Newtonian value for the great majority of cases with which we have actual experience. Einstein’s method, then, was to investigate the functions (or equations) which would satisfy the mathematical conditions just described. A transformation from the axes used by the observer in the following box may be made so as to introduce into the equations the gravitational field recognized by an observer on the earth near the box; but this, obviously, would not be the general gravitational field, because the field changes as one moves over the surface of the earth. A solution found, therefore, as just indicated, would not be the one sought for the general field; and anothermust be found which is less stringent than the former but reduces to it as a special case. He found himself at liberty to make a selection from among several possibilities, and for several reasons chose the simplest solution. He then tested this decision by seeing if his formulæ would degenerate to Newton’s law for the limiting case of velocities small when compared with that of light, because this condition is satisfied in those cases to which Newton’s law applies. His formulæ satisfied this test, and he therefore was able to announce a “law of gravitation,” of which Newton’s was a special form for a simple case.To the ordinary scholar the difficulties surmounted by Einstein in his investigations appear stupendous. It is not improbable that the statement which he is alleged to have made to his editor, that only ten men in the world could understand his treatment of the subject, is true. I am fully prepared to believe it, and wish to add that I certainly am not one of the ten. But I can also say that, after a careful and serious study of his papers, I feel confident that there is nothing in them which I can not understand, given the time to become familiar with the special mathematical processes used. The more I work over Einstein’s papers, the more impressed I am, not simply by his genius in viewing the problem, but also by his great technical skill.Following the path outlined, Einstein, as just said, arrived at certain mathematical laws for a gravitational field, laws which reduced to Newton’s form in most cases where observations are possible, but which led to different conclusions in a few cases, knowledge concerning which we might obtain by careful observations. I shall mention a few deductions from Einstein’s formulæ.1. If a heavy particle is put at the center of a circle, and, ifthe length of the circumference and the length of the diameter are measured, it will be found that their ratio is not π (3.14159). In other words the geometrical properties of space in such a gravitational field are not those discussed by Euclid; the space is, then, non-Euclidean. There is no way by which this deduction can be verified, the difference between the predicted ratio and π is too minute for us to hope to make our measurements with sufficient exactness to determine the difference.2. All the lines in the solar spectrum should with reference to lines obtained by terrestrial sources be displaced slightly towards longer wave-lengths. The amount of displacement predicted for lines in the blue end of the spectrum is about one-hundredth of an Angstrom unit, a quantity well within experimental limits. Unfortunately, as far as the testing of this prediction is concerned, there are several physical causes which are also operating to cause displacement of the spectrum-lines; and so at present a decision can not be rendered as to the verification. St. John and other workers at the Mount Wilson Observatory have the question under investigation.3. According to Newton’s law an isolated planet in its motion around a central sun would describe, period after period, the same elliptical orbit; whereas Einstein’s laws lead to the prediction that the successive orbits traversed would not be identically the same. Each revolution would start the planet off on an orbit very approximately elliptical, but with the major axis of the ellipse rotated slightly in the plane of the orbit. When calculations were made for the various planets in our solar system, it was found that the only one which was of interest from the standpoint of verification of Einstein’s formulæ was Mercury. It has been known for a long timethat there was actually such a change as just described in the orbit of Mercury, amounting to 574″ of arc per century; and it has been shown that of this a rotation of 532″ was due to the direct action of other planets, thus leaving an unexplained rotation of 42″ per century. Einstein’s formulæ predicted a rotation of 43″, a striking agreement.4. In accordance with Einstein’s formulæ a ray of light passing close to a heavy piece of matter, the sun, for instance, should experience a sensible deflection in towards the sun. This might be expected from “general” consideration of energy in motion; energy and mass are generally considered to be identical in the sense that an amount of energyEhas the massupper E Baseline 1 c squaredwherecis the velocity of light; and consequently a ray of light might fall within the province of gravitation and the amount of deflection to be expected could be calculated by the ordinary formula for gravitation. Another point of view is to consider again the observer inside the compartment falling with the acceleration of the gravitational field. To him the path of a projectile and a ray of light would both appear straight; so that, if the projectile had a velocity equal to that of light, it and the light wave would travel side by side. To an observer outside the compartment,e.g., to one on the earth, both would then appear to have the same deflection owing to the sun. But how much would the path of the projectile be bent? What would be the shape of its parabola? One might apply Newton’s law; but, according to Einstein’s formulæ, Newton’s law should be used only for small velocities. In the case of a ray passing close to the sun it was decided that according to Einstein’s formula there should be a deflection of 1″.75 whereas Newton’s law of gravitation predicted half this amount. Careful plans were made by various astronomers, to investigate this question atthe solar eclipse last May, and the result announced by Dyson, Eddington and Crommelin, the leaders of astronomy in England, was that there was a deflection of 1″.9. Of course the detection of such a minute deflection was an extraordinarily difficult matter, so many corrections had to be applied to the original observations; but the names of the men who record the conclusions are such as to inspire confidence. Certainly any effect of refraction seems to be excluded.It is thus seen that the formulæ deduced by Einstein have been confirmed in a variety of ways and in a most brilliant manner. In connection with these formulæ one question must arise in the minds of everyone; by what process, where in the course of the mathematical development, does the idea of mass reveal itself? It was not in the equations at the beginning and yet here it is at the end. How does it appear? As a matter of fact it is first seen as a constant of integration in the discussion of the problem of the gravitational field due to a single particle; and the identity of this constant with mass is proved when one compares Einstein’s formulæ with Newton’s law which is simply its degenerated form. This mass, though, is the mass of which we become aware through our experiences with weight; and Einstein proceeded to prove that this quantity which entered as a constant of integration in his ideally simple problem also obeyed the laws of conservation of mass and conservation of momentum when he investigated the problems of two and more particles. Therefore Einstein deduced from his study of gravitational fields the well-known properties of matter which form the basis of theoretical mechanics. A further logical consequence of Einstein’s development is to show that energy has mass, a concept with which every one nowadays is familiar.The description of Einstein’s method which I have given sofar is simply the story of one success after another; and it is certainly fair to ask if we have at last reached finality in our investigation of nature, if we have attained to truth. Are there no outstanding difficulties? Is there no possibility of error? Certainly, not until all the predictions made from Einstein’s formulæ have been investigated can much be said; and further, it must be seen whether any other lines of argument will lead to the same conclusions. But without waiting for all this there is at least one difficulty which is apparent at this time. We have discussed the laws of nature as independent in their form of reference axes, a concept which appeals strongly to our philosophy; yet it is not at all clear, at first sight, that we can be justified in our belief. We can not imagine any way by which we can become conscious of the translation of the earth in space; but by means of gyroscopes we can learn a great deal about its rotation on its axis. We could locate the positions of its two poles, and by watching a Foucault pendulum or a gyroscope we can obtain a number which we interpret as the angular velocity of rotation of axes fixed in the earth; angular velocity with reference to what? Where is the fundamental set of axes? This is a real difficulty. It can be surmounted in several ways. Einstein himself has outlined a method which in the end amounts to assuming the existence on the confines of space of vast quantities of matter, a proposition which is not attractive. deSitter has suggested a peculiar quality of the space to which we refer our space-time coordinates. The consequences of this are most interesting, but no decision can as yet be made as to the justification of the hypothesis. In any case we can say that the difficulty raised is not one that destroys the real value of Einstein’s work.In conclusion I wish to emphasize the fact, which shouldbe obvious, that Einstein has not attempted any explanation of gravitation; he has been occupied with the deduction of its laws. These laws, together with those of electromagnetic phenomena, comprise our store of knowledge. There is not the slightest indication of a mechanism, meaning by that a picture in terms of our senses. In fact what we have learned has been to realize that our desire to use such mechanisms is futile.1Presidential address delivered at the St. Louis meeting of the Physical Society, December 30, 1919. Republished by permission from “Science.”↑THEDEFLECTIONOF LIGHT BY GRAVITATION AND THE EINSTEIN THEORY OF RELATIVITY.1Sir Frank Dysonthe Astronomer RoyalThe purpose of the expedition was to determine whether any displacement is caused to a ray of light by the gravitational field of the sun, and if so, the amount of the displacement. Einstein’s theory predicted a displacement varying inversely as the distance of the ray from the sun’s center, amounting to 1″.75 for a star seen just grazing the sun.…A study of the conditions of the 1919 eclipse showed that the sun would be very favorably placed among a group of bright stars—in fact, it would be in the most favorable possible position. A study of the conditions at various points on the path of the eclipse, in which Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an island off the west coast of Africa, as the most favorable stations.…The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil in ample time to prepare for the eclipse, and the usual preliminary focusing by photographing stellar fields was carried out. The day of the eclipse opened cloudy, but cleared later, and the observations were carried out with almost complete success. With the astrographic telescope Mr. Davidson secured 15 out of 18 photographs showing the required stellar images. Totality lasted 6 minutes, and the average exposure of the plates was 5 to 6 seconds. Dr.Crommelin with the other lens had 7 successful plates out of 8. The unsuccessful plates were spoiled for this purpose by the clouds, but show the remarkable prominence very well.When the plates were developed the astrographic images were found to be out of focus. This is attributed to the effect of the sun’s heat on the coelostat mirror. The images were fuzzy and quite different from those on the check-plates secured at night before and after the eclipse. Fortunately the mirror which fed the 4-inch lens was not affected, and the star images secured with this lens were good and similar to those got by the night-plates. The observers stayed on in Brazil until July to secure the field in the night sky at the altitude of the eclipse epoch and under identical instrumental conditions.The plates were measured at Greenwich immediately after the observers’ return. Each plate was measured twice over by Messrs. Davidson and Furner, and I am satisfied that such faults as lie in the results are in the plates themselves and not in the measures. The figures obtained may be briefly summarized as follows: The astrographic plates gave 0″.97 for the displacement at the limb when the scale-value was determined from the plates themselves, and 1″.40 when the scale-value was assumed from the check plates. But the much better plates gave for the displacement at the limb 1″.98, Einstein’s predicted value being 1″.75. Further, for these plates the agreement was all that could be expected.…After a careful study of the plates I am prepared to say that there can be no doubt that they confirm Einstein’s prediction. A very definite result has been obtained that light is deflected according to Einstein’s law of gravitation.Professor A. S. EddingtonRoyal ObservatoryMr. Cottingham and I left the other observers at Madeira and arrived atPrincipeon April 23.… We soon realized that the prospect of a clear sky at the end of May was not very good. Not even a heavy thunderstorm on the morning of the eclipse, three weeks after the end of the wet season, saved the situation. The sky was completely cloudy at first contact, but about half an hour before totality we began to see glimpses of the sun’s crescent through the clouds. We carried out our program exactly as arranged, and the sky must have been clearer towards the end of totality. Of the 16 plates taken during the five minutes of totality the first ten showed no stars at all; of the later plates two showed five stars each, from which a result could be obtained. Comparing them with the check-plates secured at Oxford before we went out, we obtained as the final result from the two plates for the value of the displacement of the limb 1″.6 ± 0.3.… This result supports the figures obtained at Sobral.…I will pass now to a few words on the meaning of the result. It points to the larger of the two possible values of the deflection. The simplest interpretation of the bending of the ray is to consider it as an effect of the weight of light. We know that momentum is carried along on the path of a beam of light. Gravity in acting creates momentum in a direction different from that of the path of the ray and so causes it to bend. For the half-effect we have to assume that gravity obeys Newton’s law; for the full effect which has been obtained we must assume that gravity obeys the new law proposed by Einstein. This is one of the most crucial tests between Newton’s law and the proposed new law. Einstein’s law had already indicated a perturbation, causing the orbit of Mercury to revolve.That confirms it for relatively small velocities. Going to the limit, where the speed is that of light, the perturbation is increased in such a way as to double the curvature of the path, and this is now confirmed.This effect may be taken as proving Einstein’slawrather than histheory. It is not affected by the failure to detect the displacement of Fraunhofer lines on the sun. If this latter failure is confirmed it will not affect Einstein’s law of gravitation, but it will affect the views on which the law was arrived at. The law is right, though the fundamental ideas underlying it may yet be questioned.…One further point must be touched upon. Are we to attribute the displacement to the gravitational field and not to the refracting matter around the sun? The refractive index required to produce the result at a distance of 15′ from the sun would be that given by gases at a pressure of 1⁄60​ to 1⁄200​ of an atmosphere. This is of too great a density considering the depth through which the light would have to pass.Sir J. J. ThomsonPresident of the Royal Society… If the results obtained had been only that light was affected by gravitation, it would have been of the greatest importance. Newton, did, in fact, suggest this very point in his “Optics,” and his suggestion would presumably have led to the half-value. But this result is not an isolated one; it is part of a whole continent of scientific ideas affecting the most fundamental concepts of physics.… This is the most important result obtained in connection with the theory of gravitation since Newton’s day, and it is fitting that it should beannounced at a meeting of the society so closely connected with him.The difference between the laws of gravitation of Einstein and Newton come only in special cases. The real interest of Einstein’s theory lies not so much in his results as in the method by which he gets them. If his theory is right, it makes us take an entirely new view of gravitation. If it is sustained that Einstein’s reasoning holds good—and it has survived two very severe tests in connection with the perihelion of mercury and the present eclipse—then it is the result of one of the highest achievements of human thought. The weak point in the theory is the great difficulty in expressing it. It would seem that no one can understand the new law of gravitation without a thorough knowledge of the theory of invariants and of the calculus of variations.One other point of physical interest arises from the discussion. Light is deflected in passing near huge bodies of matter. This involves alterations in the electric and magnetic field. This, again, implies the existence of electric and magnetic forces outside matter—forces at present unknown, though some idea of their nature may be got from the results of this expedition.1From a report inThe Observatory, of the Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, November 6, 1919.↑BY THE SAME AUTHOREMINENT CHEMISTS OF OUR TIMEA non-technical account of the more remarkable achievements in the realm of chemistry as exemplified by the life and work of the more modern chemists. There is hardly a chemist of note whose work is not mentioned in connection with one or another of the eleven following: Perkin and Coal Tar Dyes; Mendeléeff and the Periodic Law; Richards and Atomic Weights; Ramsay and the Gases of the Atmosphere;Van’t Hoff and Physical Chemistry; Arrhenius and The Theory of Electrolytic Dissociation; Moissan and the Electric Furnace; Curie and Radium; Victor Meyer and the Rise of Organic Chemistry; Remsen and the Rise of Chemistry in America; Fischer and the Chemistry of Foods.250 Pages 5 × 7½ Illustrated Cloth $2.50BOOKS OF INTERESTTHE NATURE OF MATTER AND ELECTRICITYAN OUTLINE OF MODERN VIEWSBy DANIEL F. COMSTOCK, S.B., Ph.D.Engineer and Associate Professor of Theoretical Physics In the Massachusetts Institute of Technology, andLEONARD T. TROLAND, S.B., A.M., Ph.D.Instructor In Harvard University225 Pages 5½ × 8 Cloth 22 Illustrations 11 Plates Postpaid $2.50THE MYSTERY OF MATTER AND ENERGYRECENT PROGRESS AS TO THE STRUCTURE OF MATTERBy A. C. CREHORE, Ph.D.172 Pages 4½ × 6½ Cloth 8 Plates and Folding Charts Postpaid $1.00THE ATOMBy A. C. CREHORE, Ph.D.250 Pages 5 × 7½ 6 Illustrations $2.00ColophonAvailabilityThis eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online atwww.gutenberg.org.This eBook is produced by the Online Distributed Proofreading Team atwww.pgdp.net.MetadataTitle:From Newton to EinsteinAuthor:Benjamin Harrow (1888–1970)InfoContributor:Joseph Sweetman Ames (1864–1943)InfoContributor:Frank Watson Dyson (1868–1939)InfoContributor:Arthur Stanley Eddington (1882–1944)InfoContributor:Albert Einstein (1879–1955)InfoContributor:Joseph John Thomson (1856–1940)InfoLanguage:EnglishOriginal publication date:1920EncodingRevision History2019-08-19 Started.External ReferencesThis Project Gutenberg eBook contains external references. These links may not work for you.CorrectionsThe following corrections have been applied to the text:PageSourceCorrectionEdit distanceN.A..,1viReflectionDeflection131electro-magneticelectromagnetic173,117vanVan176[Not in source])177-177UeberÜber2 / 177Relativitats-theorieRelativitätstheorie2 / 178have[Deleted]581LeibnitzLeibniz187sytemsystem1102[Not in source].1112REFLECTIONDEFLECTION1112“[Deleted]1AbbreviationsOverview of abbreviations used.AbbreviationExpansionPh.D.Philosophiae Doctor

APPENDIXNote1 (page21)“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.Note2 (page27)Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.Note3 (page30)Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”And again:“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.Note4 (page51)For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)and this immediately provides us with a means of transforming the laws of one system to those of another.With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.Note5 (page55)The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.Note6 (page56)“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”(Dr. Walker)Note7 (page57)Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).Note8 (page59)That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.Note9 (page67)How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)

APPENDIX

Note1 (page21)“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.Note2 (page27)Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.Note3 (page30)Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”And again:“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.Note4 (page51)For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)and this immediately provides us with a means of transforming the laws of one system to those of another.With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.Note5 (page55)The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.Note6 (page56)“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”(Dr. Walker)Note7 (page57)Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).Note8 (page59)That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.Note9 (page67)How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)

Note1 (page21)“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.

Note1 (page21)

“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.

“On this earth there is indeed a tiny corner of the universe accessible to other senses [than the sense of sight]: but feeling and taste act only at those minute distances which separate particles of matter when ‘in contact:’ smell ranges over, at the utmost, a mile or two, and the greatest distance which sound is ever known to have traveled (when Krakatoa exploded in 1883) is but a few thousand miles—a mere fraction of the earth’s girdle.”—Prof. H. H. Turner of Oxford.

Note2 (page27)Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.

Note2 (page27)

Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.

Huyghens andLeibnizboth objected to Newton’s inverse square law because it postulated “action at a distance,”—for example, the attractive force of the sun and the earth. This desire for “continuity” in physical laws led to the supposition of an “ether.” We may here anticipate and state that the reason which prompted Huyghens to object to Newton’s law led Einstein in our own day to raise objections to the “ether” theory. “In the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed.” And the “ether” has not been “actually observed.”

The idea of “continuity” implies distances between adjacent points that are infinitesimal in extent; hence the idea of “continuity” comes in direct opposition with the finite distances of Newton.

The statement relating to causal connection—the refusal to accept an “ether” as an absolute base of reference—leads to the principle of the relativity of motion.

Note3 (page30)Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”And again:“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.

Note3 (page30)

Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”And again:“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.

Sir Oliver Lodge goes to the extreme of pinning his faith in the reality of this ether rather than in that of matter. Witness the following statement he made recently before a New York audience:

“To my mind the ether of space is a substantial reality with extraordinarily perfect properties, with an immense amount of energy stored up in it, with a constitution which we must discover, but a substantial reality far more impressive than that of matter. Empty space, as we call it, is full of ether, but it makes no appeal to our senses. The appearance is as if it were nothing. It is the most important thing in the material universe. I believe that matter is a modification of ether, a very porous substance, a thing more analogous to a cobweb or the Milky Way or something very slight and unsubstantial, as compared to ether.”

And again:

“The properties of ether seem to be perfect. Matter is less so; it has friction and elasticity. No imperfection has been discovered in the ether space. It doesn’t wear out; there is no dissipation of energy; there is no friction. Ether is material, yet it is not matter; both are substantial realities in physics, but it is the ether of space that holds things together and acts as a cement. My business is to call attention to the whole world of etherealness of things, and I have made it a subject of thirty years’ study, but we must admit that there is no getting hold of ether except indirectly.”

“I consider the ether of space,” says Lodge, in conclusion,“the one substantial thing in the universe.” And Lodge is certainly entitled to his opinion.

Note4 (page51)For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)and this immediately provides us with a means of transforming the laws of one system to those of another.With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.

Note4 (page51)

For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)and this immediately provides us with a means of transforming the laws of one system to those of another.With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.

For the benefit of those readers who wish to gain a deeper insight into the relativity principle, we shall here discuss it very briefly.

Newton and Galileo had developed a relativity principle in mechanics which may be stated as follows: If one system of reference is in uniform rectilinear motion with respect to another system of reference, then whatever physical laws are deduced from the first system hold true for the second system. The two systems areequivalent. If the two systems be represented byxyzandx prime y prime z prime, and if they move with the velocity ofvalong thex-axis with respect to one another, then the two systems are mathematically related thus:(1)x prime equals x minus v t comma y prime equals y comma z prime equals z comma t prime equals t comma(1)

and this immediately provides us with a means of transforming the laws of one system to those of another.

With the development of electrodynamics (which we may call electricity in motion) difficulties arose which equations in mechanics of type (1) could no longer solve. These difficulties merely increased when Maxwell showed that light must be regarded as an electromagnetic phenomenon. For suppose we wish to investigate the motion of a source of light (which may be the equivalent of the motion of the earth with reference to the sun) with respect to the velocity of the light it emits—a typical example of the study ofmoving systems—how are we to coordinate the electrodynamical and mechanical elements? Or, again, suppose we wish toinvestigate the velocity of electrons shot out from radium with a speed comparable to that of light, how are we to coordinate the two branches in tracing the course of these negative particles of electricity?

It was difficulties such as these that led to the Lorentz-Einstein modifications of the Newton-Galileo relativity equations (1). The Lorentz-Einstein equations are expressed in the form:(2)x prime equals StartFraction x minus v t Over StartRoot 1 minus StartFraction v squared Over c EndFraction EndRoot EndFraction comma y prime equals y comma z prime equals z comma t prime equals StartStartFraction t minus StartFraction v Over c squared EndFraction dot x OverOver StartRoot 1 minus StartFraction v squared Over c squared EndFraction EndRoot EndEndFraction comma(2)

cdenoting the velocity of lightin vacuo(which, according to all observations, is the same, irrespective of the observer’s state of motion). Here, you see, electrodynamical systems (light and therefore “ray” velocities such as those due to electrons) are brought into play.

This gives us Einstein’sspecial theory of relativity. From it Einstein deduced some startling conceptions of time and space.

Note5 (page55)The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.

Note5 (page55)

The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.

The velocity (v) of an object in one system will have a different velocity (v′) if referred to another system in uniform motion relative to the first. It had been supposed that only a “something” endowed withinfinitevelocity would show thesamevelocity inallsystems, irrespective of the motions of the latter. Michelson and Morley’s results actually point to the velocity of light as showing the properties of the imaginary “infinite velocity.” The velocity of light possesses universal significance; and this is the basis for much of Einstein’s earlier work.

Note6 (page56)“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”(Dr. Walker)

Note6 (page56)

“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”(Dr. Walker)

“Euclid assumes that parallel lines never meet, which they cannot do of course if they be defined as equidistant. But are there such lines? And if not, why not assume that all lines drawn through a point outside a given line will eventually intersect it? Such an assumption leads to a geometry in which all lines are conceived as being drawn on the surface of a sphere or an ellipse, and in it the three angles of a triangle are never quite equal to two right angles, nor the circumference of a circle quite π times its diameter.But that is precisely what the contraction effect due to motion requires.”

(Dr. Walker)

Note7 (page57)Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).

Note7 (page57)

Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).

Einstein had become tired of assumptions. He had no particular objection to the “ether” theory beyond the fact that this “ether” did not come within the range of our senses; it could not be “observed.” “The consistent fulfilment of the two postulates—‘action by contact’ and causal relationship between only such things as lie within the realm of observation [seeNote 2] combined together is, I believe, the mainspring of Einstein’s method of investigation.…” (Prof. Freundlich).

Note8 (page59)That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.

Note8 (page59)

That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.

That the conception of the “simultaneity” of events is devoid of meaning can be deduced from equation (2) [seeNote4]. We owe the proof to Einstein. “It is possible to select a suitable time-coordinate in such a way that a time-measurement enters into physical laws in exactly the same manner as regards its significance as a space measurement (that is, they are fully equivalent symbolically), and haslikewise a definite coordinate direction.… It never occurred to anyone that the use of a light-signal as a means of connection between the moving-body and the observer, which is necessary in practice in order to determinesimultaneity, might affect the final result,i.e., of time measurements in different systems.” (Freundlich). But that is just what Einstein shows, because time-measurements are based on “simultaneity of events,” and this, as pointed out above, is devoid of meaning.

Had the older masters the occasion to study enormous velocities, such as the velocity of light, rather than relatively small ones—and even the velocity of the earth around the sun is small as compared to the velocity of light—discrepancies between theory and experiment would have become apparent.

Note9 (page67)How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)

Note9 (page67)

How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)

How thespecialtheory of relativity (seeNote 4) led to thegeneraltheory of relativity (which included gravitation) may now be briefly traced.

When we speak of electrons, or negative particles of electricity, in motion, we are speaking ofenergyin motion. Now these electrons when in motion exhibit properties that are very similar to matter in motion. Whatever deviations there are are due to the enormous velocity of these electrons, and this velocity, as has already been pointed out, is comparable to that of light; whereas before the advent of the electron, the velocity of no particles comparable to that of light had ever been measured.

According to present views “all inertia of matter consists only of the inertia of the latent energy in it; … everythingthat we know of the inertia of energy holds without exception for the inertia ofmatter.”

Now it is on the assumption that inertial mass and gravitational “pull” are equivalent that the mass of a body is determined by itsweight. What is true of matter should be true of energy.

The special theory of relativity, however, takes into account only inertia (“inertialmass”) but not gravitation (gravitational pull orweight) of energy. When a body absorbs energy equation 2 (seeNote 4) will record a gain in inertia but not in weight—which is contrary to one of the fundamental facts in mechanics.

This means that a moregeneraltheory of relativity is required to include gravitational phenomena. Hence Einstein’sGeneral Theory of Relativity. Hence the approach to a new theory of gravitation. Hence “the setting up of a differential equation which comprises the motion of a body under the influence of bothinertia and gravity, and which symbolically expresses the relativity of motions.… The differential law must always preserve the same form, irrespective of thesystemof coordinates to which it is referred, so that no system of coordinates enjoys a preference to any other.” (For the general form of the equation and for an excellent discussion of its significance, see Freundlich’s monograph, pages 27–33.)

TIME, SPACE, AND GRAVITATION1BYProf. Albert EinsteinThere are several kinds of theory in physics. Most of them are constructive. These attempt to build a picture of complex phenomena out of some relatively simple proposition. The kinetic theory of gases, for instance, attempts to refer to molecular movement the mechanical thermal, and diffusional properties of gases. When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.Theories of Principle.—But in addition to this most weighty group of theories, there is another group consisting of what I call theories of principle. These employ the analytic, not the synthetic method. Their starting-point and foundation are not hypothetical constituents, but empirically observed general properties of phenomena, principles from which mathematical formulæ are deduced of such a kind that they apply to every case which presents itself. Thermodynamics, for instance, starting from the fact that perpetual motion never occurs in ordinary experience, attempts to deduce from this, by analytic processes, a theory which will apply in every case. The merit of constructive theories is their comprehensiveness, adaptability, and clarity, that of the theories of principle, their logical perfection, and the security of their foundation.The theory of relativity is a theory of principle. To understand it, the principles on which it rests must be grasped. But before stating these it is necessary to point out that the theory of relativity is like a house with two separatestories, the special relativity theory and the general theory of relativity.Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer to another body. The motion of a railway train is described with reference to the ground, of a planet with reference to the total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termed systems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system of coordinates.The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system. The state of motion of an inertia-system, so far as mechanics are concerned, is not restricted by nature to one condition. The condition in the following proposition suffices; a system of coordinates moving in the same direction and at the same rate as a system of inertia is itself a system of inertia. The special relativity theory is therefore the application of the following proposition to any natural process: “Every law of nature which holds good with respect to a coordinate systemKmust also hold good for any other systemK′provided thatKandK′are in uniform movement of translation.”The second principle on which the special relativity theory rests is that of the constancy of the velocity of light in a vacuum. Light in a vacuum has a definite and constant velocity, independent of the velocity of its source. Physicists owe their confidence in this proposition to the Maxwell-Lorentz theory of electro-dynamics.The two principles which I have mentioned have received strong experimental confirmation, but do not seem to be logically compatible. The special relativity theory achieved their logical reconciliation by making a change in kinematics, that is to say, in the doctrine of the physical laws of space and time. It became evident that a statement of the coincidence of two events could have a meaning only in connection with a system of coordinates, that the mass of bodies and the rate of movement of clocks must depend on their state of motion with regard to the coordinates.The Older Physics.—But the older physics, including the laws of motion of Galileo and Newton, clashed with the relativistic kinematics that I have indicated. The latter gave origin to certain generalized mathematical conditions with which the laws of nature would have to conform if the two fundamental principles were compatible. Physics had to be modified. The most notable change was a new law of motion for (very rapidly) moving mass-points, and this soon came to be verified in the case of electrically-laden particles. The most important result of the special relativity system concerned the inert mass of a material system. It became evident that the inertia of such a system must depend on its energy-content, so that we were driven to the conception that inert mass was nothing else than latent energy. The doctrine of the conservation of mass lost its independence and became merged in the doctrine of conservation of energy.The special relativity theory which was simply a systematic extension of the electro-dynamics of Maxwell and Lorentz, had consequences which reached beyond itself. Must the independence of physical laws with regard to a system of coordinates be limited to systems of coordinates in uniform movement of translation with regard to one another? Whathas nature to do with the coordinate systems that we propose and with their motions? Although it may be necessary for our descriptions of nature to employ systems of coordinates that we have selected arbitrarily, the choice should not be limited in any way so far as their state of motion is concerned. (General theory of relativity.) The application of this general theory of relativity was found to be in conflict with a well-known experiment, according to which it appeared that the weight and the inertia of a body depended on the same constants (identity of inert and heavy masses). Consider the case of a system of coordinates which is conceived as being in stable rotation relative to a system of inertia in the Newtonian sense. The forces which, relatively to this system, are centrifugal must, in the Newtonian sense, be attributed to inertia. But these centrifugal forces are, like gravitation, proportional to the mass of the bodies. Is it not, then, possible to regard the system of coordinates as at rest, and the centrifugal forces as gravitational? The interpretation seemed obvious, but classical mechanics forbade it.This slight sketch indicates how a generalized theory of relativity must include the laws of gravitation, and actual pursuit of the conception has justified the hope. But the way was harder than was expected, because it contradicted Euclidian geometry. In other words, the laws according to which material bodies are arranged in space do not exactly agree with the laws of space prescribed by the Euclidian geometry of solids. This is what is meant by the phrase “a warp in space.” The fundamental concepts “straight,” “plane,” etc., accordingly lose their exact meaning in physics.In the generalized theory of relativity, the doctrine of space and time, kinematics, is no longer one of the absolute foundations of general physics. The geometrical states of bodiesand the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material system concerned.Thus the new theory of gravitation diverges widely from that of Newton with respect to its basal principle. But in practical application the two agree so closely that it has been difficult to find cases in which the actual differences could be subjected to observation. As yet only the following have been suggested:1. The distortion of the oval orbits of planets round the sun (confirmed in the case of the planet Mercury).2. The deviation of light-rays in a gravitational field (confirmed by the English Solar Eclipse expedition).3. The shifting of spectral lines towards the red end of the spectrum in the case of light coming to us from stars of appreciable mass (not yet confirmed).The great attraction of the theory is its logical consistency. If any deduction from it should prove untenable, it must be given up. A modification of it seems impossible without destruction of the whole.No one must think that Newton’s great creation can be overthrown in any real sense by this or by any other theory. His clear and wide ideas will for ever retain their significance as the foundation on which our modern conceptions of physics have been built.1Republished by permission from “Science.”↑

TIME, SPACE, AND GRAVITATION1BYProf. Albert Einstein

BYProf. Albert Einstein

There are several kinds of theory in physics. Most of them are constructive. These attempt to build a picture of complex phenomena out of some relatively simple proposition. The kinetic theory of gases, for instance, attempts to refer to molecular movement the mechanical thermal, and diffusional properties of gases. When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.Theories of Principle.—But in addition to this most weighty group of theories, there is another group consisting of what I call theories of principle. These employ the analytic, not the synthetic method. Their starting-point and foundation are not hypothetical constituents, but empirically observed general properties of phenomena, principles from which mathematical formulæ are deduced of such a kind that they apply to every case which presents itself. Thermodynamics, for instance, starting from the fact that perpetual motion never occurs in ordinary experience, attempts to deduce from this, by analytic processes, a theory which will apply in every case. The merit of constructive theories is their comprehensiveness, adaptability, and clarity, that of the theories of principle, their logical perfection, and the security of their foundation.The theory of relativity is a theory of principle. To understand it, the principles on which it rests must be grasped. But before stating these it is necessary to point out that the theory of relativity is like a house with two separatestories, the special relativity theory and the general theory of relativity.Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer to another body. The motion of a railway train is described with reference to the ground, of a planet with reference to the total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termed systems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system of coordinates.The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system. The state of motion of an inertia-system, so far as mechanics are concerned, is not restricted by nature to one condition. The condition in the following proposition suffices; a system of coordinates moving in the same direction and at the same rate as a system of inertia is itself a system of inertia. The special relativity theory is therefore the application of the following proposition to any natural process: “Every law of nature which holds good with respect to a coordinate systemKmust also hold good for any other systemK′provided thatKandK′are in uniform movement of translation.”The second principle on which the special relativity theory rests is that of the constancy of the velocity of light in a vacuum. Light in a vacuum has a definite and constant velocity, independent of the velocity of its source. Physicists owe their confidence in this proposition to the Maxwell-Lorentz theory of electro-dynamics.The two principles which I have mentioned have received strong experimental confirmation, but do not seem to be logically compatible. The special relativity theory achieved their logical reconciliation by making a change in kinematics, that is to say, in the doctrine of the physical laws of space and time. It became evident that a statement of the coincidence of two events could have a meaning only in connection with a system of coordinates, that the mass of bodies and the rate of movement of clocks must depend on their state of motion with regard to the coordinates.The Older Physics.—But the older physics, including the laws of motion of Galileo and Newton, clashed with the relativistic kinematics that I have indicated. The latter gave origin to certain generalized mathematical conditions with which the laws of nature would have to conform if the two fundamental principles were compatible. Physics had to be modified. The most notable change was a new law of motion for (very rapidly) moving mass-points, and this soon came to be verified in the case of electrically-laden particles. The most important result of the special relativity system concerned the inert mass of a material system. It became evident that the inertia of such a system must depend on its energy-content, so that we were driven to the conception that inert mass was nothing else than latent energy. The doctrine of the conservation of mass lost its independence and became merged in the doctrine of conservation of energy.The special relativity theory which was simply a systematic extension of the electro-dynamics of Maxwell and Lorentz, had consequences which reached beyond itself. Must the independence of physical laws with regard to a system of coordinates be limited to systems of coordinates in uniform movement of translation with regard to one another? Whathas nature to do with the coordinate systems that we propose and with their motions? Although it may be necessary for our descriptions of nature to employ systems of coordinates that we have selected arbitrarily, the choice should not be limited in any way so far as their state of motion is concerned. (General theory of relativity.) The application of this general theory of relativity was found to be in conflict with a well-known experiment, according to which it appeared that the weight and the inertia of a body depended on the same constants (identity of inert and heavy masses). Consider the case of a system of coordinates which is conceived as being in stable rotation relative to a system of inertia in the Newtonian sense. The forces which, relatively to this system, are centrifugal must, in the Newtonian sense, be attributed to inertia. But these centrifugal forces are, like gravitation, proportional to the mass of the bodies. Is it not, then, possible to regard the system of coordinates as at rest, and the centrifugal forces as gravitational? The interpretation seemed obvious, but classical mechanics forbade it.This slight sketch indicates how a generalized theory of relativity must include the laws of gravitation, and actual pursuit of the conception has justified the hope. But the way was harder than was expected, because it contradicted Euclidian geometry. In other words, the laws according to which material bodies are arranged in space do not exactly agree with the laws of space prescribed by the Euclidian geometry of solids. This is what is meant by the phrase “a warp in space.” The fundamental concepts “straight,” “plane,” etc., accordingly lose their exact meaning in physics.In the generalized theory of relativity, the doctrine of space and time, kinematics, is no longer one of the absolute foundations of general physics. The geometrical states of bodiesand the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material system concerned.Thus the new theory of gravitation diverges widely from that of Newton with respect to its basal principle. But in practical application the two agree so closely that it has been difficult to find cases in which the actual differences could be subjected to observation. As yet only the following have been suggested:1. The distortion of the oval orbits of planets round the sun (confirmed in the case of the planet Mercury).2. The deviation of light-rays in a gravitational field (confirmed by the English Solar Eclipse expedition).3. The shifting of spectral lines towards the red end of the spectrum in the case of light coming to us from stars of appreciable mass (not yet confirmed).The great attraction of the theory is its logical consistency. If any deduction from it should prove untenable, it must be given up. A modification of it seems impossible without destruction of the whole.No one must think that Newton’s great creation can be overthrown in any real sense by this or by any other theory. His clear and wide ideas will for ever retain their significance as the foundation on which our modern conceptions of physics have been built.

There are several kinds of theory in physics. Most of them are constructive. These attempt to build a picture of complex phenomena out of some relatively simple proposition. The kinetic theory of gases, for instance, attempts to refer to molecular movement the mechanical thermal, and diffusional properties of gases. When we say that we understand a group of natural phenomena, we mean that we have found a constructive theory which embraces them.

Theories of Principle.—But in addition to this most weighty group of theories, there is another group consisting of what I call theories of principle. These employ the analytic, not the synthetic method. Their starting-point and foundation are not hypothetical constituents, but empirically observed general properties of phenomena, principles from which mathematical formulæ are deduced of such a kind that they apply to every case which presents itself. Thermodynamics, for instance, starting from the fact that perpetual motion never occurs in ordinary experience, attempts to deduce from this, by analytic processes, a theory which will apply in every case. The merit of constructive theories is their comprehensiveness, adaptability, and clarity, that of the theories of principle, their logical perfection, and the security of their foundation.

The theory of relativity is a theory of principle. To understand it, the principles on which it rests must be grasped. But before stating these it is necessary to point out that the theory of relativity is like a house with two separatestories, the special relativity theory and the general theory of relativity.

Since the time of the ancient Greeks it has been well known that in describing the motion of a body we must refer to another body. The motion of a railway train is described with reference to the ground, of a planet with reference to the total assemblage of visible fixed stars. In physics the bodies to which motions are spatially referred are termed systems of coordinates. The laws of mechanics of Galileo and Newton can be formulated only by using a system of coordinates.

The state of motion of a system of coordinates can not be chosen arbitrarily if the laws of mechanics are to hold good (it must be free from twisting and from acceleration). The system of coordinates employed in mechanics is called an inertia-system. The state of motion of an inertia-system, so far as mechanics are concerned, is not restricted by nature to one condition. The condition in the following proposition suffices; a system of coordinates moving in the same direction and at the same rate as a system of inertia is itself a system of inertia. The special relativity theory is therefore the application of the following proposition to any natural process: “Every law of nature which holds good with respect to a coordinate systemKmust also hold good for any other systemK′provided thatKandK′are in uniform movement of translation.”

The second principle on which the special relativity theory rests is that of the constancy of the velocity of light in a vacuum. Light in a vacuum has a definite and constant velocity, independent of the velocity of its source. Physicists owe their confidence in this proposition to the Maxwell-Lorentz theory of electro-dynamics.

The two principles which I have mentioned have received strong experimental confirmation, but do not seem to be logically compatible. The special relativity theory achieved their logical reconciliation by making a change in kinematics, that is to say, in the doctrine of the physical laws of space and time. It became evident that a statement of the coincidence of two events could have a meaning only in connection with a system of coordinates, that the mass of bodies and the rate of movement of clocks must depend on their state of motion with regard to the coordinates.

The Older Physics.—But the older physics, including the laws of motion of Galileo and Newton, clashed with the relativistic kinematics that I have indicated. The latter gave origin to certain generalized mathematical conditions with which the laws of nature would have to conform if the two fundamental principles were compatible. Physics had to be modified. The most notable change was a new law of motion for (very rapidly) moving mass-points, and this soon came to be verified in the case of electrically-laden particles. The most important result of the special relativity system concerned the inert mass of a material system. It became evident that the inertia of such a system must depend on its energy-content, so that we were driven to the conception that inert mass was nothing else than latent energy. The doctrine of the conservation of mass lost its independence and became merged in the doctrine of conservation of energy.

The special relativity theory which was simply a systematic extension of the electro-dynamics of Maxwell and Lorentz, had consequences which reached beyond itself. Must the independence of physical laws with regard to a system of coordinates be limited to systems of coordinates in uniform movement of translation with regard to one another? Whathas nature to do with the coordinate systems that we propose and with their motions? Although it may be necessary for our descriptions of nature to employ systems of coordinates that we have selected arbitrarily, the choice should not be limited in any way so far as their state of motion is concerned. (General theory of relativity.) The application of this general theory of relativity was found to be in conflict with a well-known experiment, according to which it appeared that the weight and the inertia of a body depended on the same constants (identity of inert and heavy masses). Consider the case of a system of coordinates which is conceived as being in stable rotation relative to a system of inertia in the Newtonian sense. The forces which, relatively to this system, are centrifugal must, in the Newtonian sense, be attributed to inertia. But these centrifugal forces are, like gravitation, proportional to the mass of the bodies. Is it not, then, possible to regard the system of coordinates as at rest, and the centrifugal forces as gravitational? The interpretation seemed obvious, but classical mechanics forbade it.

This slight sketch indicates how a generalized theory of relativity must include the laws of gravitation, and actual pursuit of the conception has justified the hope. But the way was harder than was expected, because it contradicted Euclidian geometry. In other words, the laws according to which material bodies are arranged in space do not exactly agree with the laws of space prescribed by the Euclidian geometry of solids. This is what is meant by the phrase “a warp in space.” The fundamental concepts “straight,” “plane,” etc., accordingly lose their exact meaning in physics.

In the generalized theory of relativity, the doctrine of space and time, kinematics, is no longer one of the absolute foundations of general physics. The geometrical states of bodiesand the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material system concerned.

Thus the new theory of gravitation diverges widely from that of Newton with respect to its basal principle. But in practical application the two agree so closely that it has been difficult to find cases in which the actual differences could be subjected to observation. As yet only the following have been suggested:

1. The distortion of the oval orbits of planets round the sun (confirmed in the case of the planet Mercury).

2. The deviation of light-rays in a gravitational field (confirmed by the English Solar Eclipse expedition).

3. The shifting of spectral lines towards the red end of the spectrum in the case of light coming to us from stars of appreciable mass (not yet confirmed).

The great attraction of the theory is its logical consistency. If any deduction from it should prove untenable, it must be given up. A modification of it seems impossible without destruction of the whole.

No one must think that Newton’s great creation can be overthrown in any real sense by this or by any other theory. His clear and wide ideas will for ever retain their significance as the foundation on which our modern conceptions of physics have been built.

1Republished by permission from “Science.”↑

1Republished by permission from “Science.”↑

1Republished by permission from “Science.”↑

EINSTEIN’S LAW OF GRAVITATION1BYProf. J. S. AmesJohns Hopkins University… In the treatment of Maxwell’s equations of the electromagnetic field, several investigators realized the importance of deducing the form of the equations when applied to a system moving with a uniform velocity. One object of such an investigation would be to determine such a set of transformation formulæ as would leave the mathematical form of the equations unaltered. The necessary relations between the new space-coordinates, those applying to the moving system, and the original set were of course obvious; and elementary methods led to the deduction of a new variable which should replace the time coordinate. This step was taken by Lorentz and also, I believe, by Larmor and by Voigt. The mathematical deductions and applications in the hands of these men were extremely beautiful, and are probably well known to you all.Lorentz’ paper on this subject appeared in the Proceedings of the Amsterdam Academy in 1904. In the following year there was published in theAnnalen der Physika paper by Einstein, written without any knowledge of the work of Lorentz, in which he arrived at the same transformation equations as did the latter, but with an entirely different and fundamentally new interpretation. Einstein called attention in his paper to the lack of definiteness in the concepts of time and space, as ordinarily stated and used. He analyzed clearly the definitions and postulates which were necessarybefore one could speak with exactness of a length or of an interval of time. He disposed forever of the propriety of speaking of the “true” length of a rod or of the “true” duration of time, showing, in fact, that the numerical values which we attach to lengths or intervals of time depend upon the definitions and postulates which we adopt. The words “absolute” space or time intervals are devoid of meaning. As an illustration of what is meant Einstein discussed two possible ways of measuring the length of a rod when it is moving in the direction of its own length with a uniform velocity, that is, after having adopted a scale of length, two ways of assigning a number to the length of the rod concerned. One method is to imagine the observer moving with the rod, applying along its length the measuring scale, and reading off the positions of the ends of the rod. Another method would be to have two observers at rest on the body with reference to which the rod has the uniform velocity, so stationed along the line of motion of the rod that as the rod moves past them they can note simultaneously on a stationary measuring scale the positions of the two ends of the rod. Einstein showed that, accepting two postulates which need no defense at this time, the two methods of measurements would lead to different numerical values, and, further, that the divergence of the two results would increase as the velocity of the rod was increased. In assigning a number, therefore, to the length of a moving rod, one must make a choice of the method to be used in measuring it. Obviously the preferable method is to agree that the observer shall move with the rod, carrying his measuring instrument with him. This disposes of the problem of measuring space relations. The observed fact that, if we measure the length of the rod on different days, or when the rod is lying in different positions, we alwaysobtain the same value offers no information concerning the “real” length of the rod. It may have changed, or it may not. It must always be remembered that measurement of the length of a rod is simply a process of comparison between it and an arbitrary standard,e.g., a meter-rod or yard-stick. In regard to the problem of assigning numbers to intervals of time, it must be borne in mind that, strictly speaking, we do not “measure” such intervals,i.e., that we do not select a unit interval of time and find how many times it is contained in the interval in question. (Similarly, we do not “measure” the pitch of a sound or the temperature of a room.) Our practical instruments for assigning numbers to time-intervals depend in the main upon our agreeing to believe that a pendulum swings in a perfectly uniform manner, each vibration taking the same time as the next one. Of course we cannotprovethat this is true, it is, strictly speaking, a definition of what we mean by equal intervals of time; and it is not a particularly good definition at that. Its limitations are sufficiently obvious. The best way to proceed is to consider the concept of uniform velocity, and then, using the idea of some entity having such a uniform velocity, to define equal intervals of time as such intervals as are required for the entity to traverse equal lengths. These last we have already defined. What is required in addition is to adopt some moving entity as giving our definition of uniform velocity. Considering our known universe it is self-evident that we should choose in our definition of uniform velocity the velocity of light, since this selection could be made by an observer anywhere in our universe. Having agreed then to illustrate by the words “uniform velocity” that of light, our definition of equal intervals of time is complete. This implies, of course, that there is no uncertainty on our part as to the fact that the velocity oflight always has the same value at any one point in the universe to any observer, quite regardless of the source of light. In other words, the postulate that this is true underlies our definition. Following this method Einstein developed a system of measuring both space and time intervals. As a matter of fact his system is identically that which we use in daily life with reference to events here on the earth. He further showed that if a man were to measure the length of a rod, for instance, on the earth and then were able to carry the rod and his measuring apparatus to Mars, the sun, or to Arcturus he would obtain the same numerical value for the length in all places and at all times. This doesn’t mean that any statement is implied as to whether the length of the rod has remained unchanged or not; such words do not have any meaning—remember that we can not speak of true length. It is thus clear that an observer living on the earth would have a definite system of units in terms of which to express space and time intervals,i.e., he would have a definite system of space coordinates (x,y,z) and a definite time coordinate (t); and similarly an observer living on Mars would have his system of coordinates (x′,y′,z′,t′). Provided that one observer has a definite uniform velocity with reference to the other, it is a comparatively simple matter to deduce the mathematical relations between the two sets of coordinates. When Einstein did this, he arrived at the same transformation formulæ as those used by Lorentz in his development of Maxwell’s equations. The latter had shown that, using these formulæ, the form of the laws for all electromagnetic phenomena maintained the same form; so Einstein’s method proves that using his system of measurement an observer, anywhere in the universe, would as the result of his own investigation of electromagnetic phenomena arrive at thesame mathematical statement of them as any other observer, provided only that the relative-velocity of the two observers was uniform.Einstein discussed many other most important questions at this time; but it is not necessary to refer to them in connection with the present subject. So far as this is concerned, the next important step to note is that taken in the famous address of Minkowski, in 1908, on the subject of “Space and Time.” It would be difficult to overstate the importance of the concepts advanced by Minkowski. They marked the beginning of a new period in the philosophy of physics. I shall not attempt to explain his ideas in detail, but shall confine myself to a few general statements. His point of view and his line of development of the theme are absolutely different from those of Lorentz or of Einstein; but in the end he makes use of the same transformation formulæ. His great contribution consists in giving us a new geometrical picture of their meaning. It is scarcely fair to call Minkowski’s development a picture; for to us a picture can never have more than three dimensions, our senses limit us; while his picture calls for perception of four dimensions. It is this fact that renders any even semi-popular discussion of Minkowski’s work so impossible. We can all see that for us to describe any event a knowledge of four coordinates is necessary, three for the space specification and one for the time. A complete picture could be given then by a point in four dimensions. All four coordinates are necessary: we never observe an event except at a certain time, and we never observe an instant of time except with reference to space. Discussing the laws of electromagnetic phenomena, Minkowski showed how in a space of four dimensions, by a suitable definition of axes, the mathematical transformation of Lorentz and Einsteincould be described by a rotation of the set of axes. We are all accustomed to a rotation of our ordinary cartesian set of axes describing the position of a point. We ordinarily choose our axes at any location on the earth as follows: one vertical, one east and west, one north and south. So if we move from any one laboratory to another, we change our axes; they are always orthogonal, but in moving from place to place there is a rotation. Similarly, Minkowski showed that if we choose four orthogonal axes at any point on the earth, according to his method, to represent a space-time point using the method of measuring space and time intervals as outlined by Einstein; and, if an observer on Arcturus used a similar set of axes and the method of measurement which he naturally would, the set of axes of the latter could be obtained from those of the observer on the earth by a pure rotation (and naturally a transfer of the origin). This is a beautiful geometrical result. To complete my statement of the method, I must add that instead of using as his fourth axis one along which numerical values of time are laid off, Minkowski defined his fourth coordinate as the product of time and the imaginary constant, the square root of minus one. This introduction of imaginary quantities might be expected, possibly, to introduce difficulties; but, in reality, it is the very essence of the simplicity of the geometrical description just given of the rotation of the sets of axes. It thus appears that different observers situated at different points in the universe would each have their own set of axes, all different, yet all connected by the fact that any one can be rotated so as to coincide with any other. This means that there is no one direction in the four-dimensional space that corresponds to time for all observers. Just as with reference to the earth there is no direction which can be called vertical for all observers living on the earth. In thesense of anabsolutemeaning the words “up and down,” “before and after,” “sooner or later,” are entirely meaningless.This concept of Minkowski’s may be made clearer, perhaps, by the following process of thought. If we take a section through our three-dimensional space, we have a plane,i.e., a two-dimensional space. Similarly, if a section is made through a four-dimensional space, one of three dimensions is obtained. Thus, for an observer on the earth a definite section of Minkowski’s four-dimensional space will give us our ordinary three-dimensional one; so that this section will, as it were, break up Minkowski’s space into our space and give us our ordinary time. Similarly, a different section would have to be used to the observer on Arcturus; but by a suitable selection he would get his own familiar three-dimensional space and his own time. Thus the space defined by Minkowski is completely isotropic in reference to measured lengths and times, there is absolutely no difference between any two directions in an absolute sense; for any particular observer, of course, a particular section will cause the space to fall apart so as to suit his habits of measurement; any section, however, taken at random will do the same thing for some observer somewhere. From another point of view, that of Lorentz and Einstein, it is obvious that, since this four-dimensional space is isotropic, the expression of the laws of electromagnetic phenomena take identical mathematical forms when expressed by any observer.The question of course must be raised as to what can be said in regard to phenomena which so far as we know do not have an electromagnetic origin. In particular what can be done with respect to gravitational phenomena? Before, however, showing how this problem was attacked by Einstein;and the fact that the subject of my address is Einstein’s work on gravitation shows that ultimately I shall explain this, I must emphasize another feature of Minkowski’s geometry. To describe the space-time characteristics of any event a point, defined by its four coordinates, is sufficient; so, if one observes the life-history of any entity,e.g., a particle of matter, a light-wave, etc., he observes a sequence of points in the space-time continuum; that is, the life-history of any entity is described fully by a line in this space. Such a line was called by Minkowski a “world-line.” Further, from a different point of view, all of our observations of nature are in reality observations of coincidences,e.g., if one reads a thermometer, what he does is to note the coincidence of the end of the column of mercury with a certain scale division on the thermometer tube. In other words, thinking of the world-line of the end of the mercury column and the world-line of the scale division, what we have observed was the intersection or crossing of these lines. In a similar manner any observation may be analyzed; and remembering that light rays, a point on the retina of the eye, etc., all have their world-lines, it will be recognized that it is a perfectly accurate statement to say that every observation is the perception of the intersection of world-lines. Further, since all we know of a world-line is the result of observations, it is evident that we do not know a world-line as a continuous series of points, but simply as a series of discontinuous points, each point being where the particular world-line in question is crossed by another world-line.It is clear, moreover, that for the description of a world-line we are not limited to the particular set of four orthogonal axes adopted by Minkowski. We can choose any set of four-dimensional axes we wish. It is further evident that the mathematical expression for the coincidence of two points isabsolutely independent of our selection of reference axes. If we change our axes, we will change the coordinates of both points simultaneously, so that the question of axes ceases to be of interest. But our so-called laws of nature are nothing but descriptions in mathematical language of our observations; we observe only coincidences; a sequence of coincidences when put in mathematical terms takes a form which is independent of the selection of reference axes; therefore the mathematical expression of our laws of nature, of every character, must be such that their form does not change if we make a transformation of axes. This is a simple but far-reaching deduction.There is a geometrical method of picturing the effect of a change of axes of reference,i.e., of a mathematical transformation. To a man in a railway coach the path of a drop of water does not appear vertical,i.e., it is not parallel to the edge of the window; still less so does it appear vertical to a man performing manœuvres in an airplane. This means that whereas with reference to axes fixed to the earth the path of the drop is vertical; with reference to other axes, the path is not. Or, stating the conclusion in general language, changing the axes of reference (or effecting a mathematical transformation) in general changes the shape of any line. If one imagines the line forming a part of the space, it is evident that if the space is deformed by compression or expansion the shape of the line is changed, and if sufficient care is taken it is clearly possible, by deforming the space, to make the line take any shape desired, or better stated, any shape specified by the previous change of axes. It is thus possible to picture a mathematical transformation as a deformation of space. Thus I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great varietyof shapes; each of these new shapes is a picture of a suitable transformation.Now, consider world-lines in our four-dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections. So in the space of our world-lines, the space may be deformed in any imaginable manner without introducing any new intersections or changing the sequence of the existing intersections. It is this sequence which gives us the mathematical expression of our so-called experimental laws; a deformation of our space is equivalent mathematically to a transformation of axes, consequently we see why it is that the form of our laws must be the same when referred to any and all sets of axes, that is, must remain unaltered by any mathematical transformation.Now, at last we come to gravitation. We can not imagine any world-line simpler than that of a particle of matter left to itself; we shall therefore call it a “straight” line. Our experience is that two particles of matter attract one another.Expressed in terms of world-lines, this means that, if the world-lines of two isolated particles come near each other, the lines, instead of being straight, will be deflected or bent in towards each other. The world-line of any one particle is therefore deformed; and we have just seen that a deformation is the equivalent of a mathematical transformation. In other words, for any one particle it is possible to replace the effect of a gravitational field at any instant by a mathematical transformation of axes. The statement that this is always possiblefor any particle at any instant is Einstein’s famous “Principle of Equivalence.”Let us rest for a moment, while I call attention to a most interesting coincidence, not to be thought of as an intersection of world-lines. It is said that Newton’s thoughts were directed to the observation of gravitational phenomena by an apple falling on his head; from this striking event he passed by natural steps to a consideration of the universality of gravitation. Einstein in describing his mental process in the evolution of his law of gravitation says that his attention was called to a new point of view by discussing his experiences with a man whose fall from a high building he had just witnessed. The man fortunately suffered no serious injuries and assured Einstein that in the course of his fall he had not been conscious in the least of any pull downward on his body. In mathematical language, with reference to axes moving with the man the force of gravity had disappeared. This is a case where by the transfer of the axes from the earth itself to the man, the force of the gravitational field is annulled. The converse change of axes from the falling man to a point on the earth could be considered as introducing the force of gravity into the equations of motion. Another illustration of the introduction into our equations of a force by a means of a change of axes is furnished by the ordinary treatment of a body in uniform rotation about an axis. For instance, in the case of a so-called conical pendulum, that is, the motion of a bob suspended from a fixed point by string, which is so set in motion that the bob describes a horizontal circle and the string therefore describes a circular cone, if we transfer our axes from the earth and have them rotate around the vertical line through the fixed point with the same angular velocity as the bob, it is necessary to introduce into our equations ofmotion a fictitious “force” called the centrifugal force. No one ever thinks of this force other than as a mathematical quantity introduced into the equations for the sake of simplicity of treatment; no physical meaning is attached to it. Why should there be to any other so-called “force,” which like centrifugal force, is independent of the nature of the matter? Again, here on the earth our sensation of weight is interpreted mathematically by combining expressions for centrifugal force and gravity; we have no distinct sensation for either separately. Why then is there any difference in the essence of the two? Why not consider them both as brought into our equations by the agency of mathematical transformations? This is Einstein’s point of view.Granting, then, the principle of equivalence, we can so choose axes at any point at any instant that the gravitational field will disappear; these axes are therefore of what Eddington calls the “Galilean” type, the simplest possible. Consider, that is, an observer in a box, or compartment, which is falling with the acceleration of the gravitational field at that point. He would not be conscious of the field. If there were a projectile fired off in this compartment, the observer would describe its path as being straight. In this space the infinitesimal interval between two space-time points would then be given by the formulad s squared equals d x 1 squared plus d x 2 Subscript 2 Baseline plus d x 3 squared plus d x 2 Subscript 4 Baseline commawheredsis the interval andx 1 comma x 2 comma x 3 comma x 4are coordinates. If we make a mathematical transformation,i.e., use another set of axes, this interval would obviously take the formd s squared equals g 11 d x 33 squared plus g 22 d x 2 squared plus g 33 d x 3 squared plus g 44 d x 2 Subscript 4 Baseline plus 2 g 12 d x 1 d x 2 plus normal e normal t normal c period commawherex 1 comma x 2 comma x 3andx 4are now coordinates referring to the newaxes. This relation involves ten coefficients, the coefficients defining the transformation.But of course a certain dynamical value is also attached to theg’s, because by the transfer of our axes from the Galilean type we have made a change which is equivalent to the introduction of a gravitational field; and theg’s must specify the field. That is, theseg’s are the expressions of our experiences, and hence their values can not depend upon the use of any special axes; the values must be the same for all selections. In other words, whatever function of the coordinates any onegis for one set of axes, if other axes are chosen, thisgmust still be the same function of the new coordinates. There are teng’s defined by differential equations; so we have ten covariant equations. Einstein showed how theseg’s could be regarded as generalized potentials of the field. Our own experiments and observations upon gravitation have given us a certain knowledge concerning its potential; that is, we know a value for it which must be so near the truth that we can properly call it at least a first approximation. Or, stated differently, if Einstein succeeds in deducing the rigid value for the gravitational potential in any field, it must degenerate to the Newtonian value for the great majority of cases with which we have actual experience. Einstein’s method, then, was to investigate the functions (or equations) which would satisfy the mathematical conditions just described. A transformation from the axes used by the observer in the following box may be made so as to introduce into the equations the gravitational field recognized by an observer on the earth near the box; but this, obviously, would not be the general gravitational field, because the field changes as one moves over the surface of the earth. A solution found, therefore, as just indicated, would not be the one sought for the general field; and anothermust be found which is less stringent than the former but reduces to it as a special case. He found himself at liberty to make a selection from among several possibilities, and for several reasons chose the simplest solution. He then tested this decision by seeing if his formulæ would degenerate to Newton’s law for the limiting case of velocities small when compared with that of light, because this condition is satisfied in those cases to which Newton’s law applies. His formulæ satisfied this test, and he therefore was able to announce a “law of gravitation,” of which Newton’s was a special form for a simple case.To the ordinary scholar the difficulties surmounted by Einstein in his investigations appear stupendous. It is not improbable that the statement which he is alleged to have made to his editor, that only ten men in the world could understand his treatment of the subject, is true. I am fully prepared to believe it, and wish to add that I certainly am not one of the ten. But I can also say that, after a careful and serious study of his papers, I feel confident that there is nothing in them which I can not understand, given the time to become familiar with the special mathematical processes used. The more I work over Einstein’s papers, the more impressed I am, not simply by his genius in viewing the problem, but also by his great technical skill.Following the path outlined, Einstein, as just said, arrived at certain mathematical laws for a gravitational field, laws which reduced to Newton’s form in most cases where observations are possible, but which led to different conclusions in a few cases, knowledge concerning which we might obtain by careful observations. I shall mention a few deductions from Einstein’s formulæ.1. If a heavy particle is put at the center of a circle, and, ifthe length of the circumference and the length of the diameter are measured, it will be found that their ratio is not π (3.14159). In other words the geometrical properties of space in such a gravitational field are not those discussed by Euclid; the space is, then, non-Euclidean. There is no way by which this deduction can be verified, the difference between the predicted ratio and π is too minute for us to hope to make our measurements with sufficient exactness to determine the difference.2. All the lines in the solar spectrum should with reference to lines obtained by terrestrial sources be displaced slightly towards longer wave-lengths. The amount of displacement predicted for lines in the blue end of the spectrum is about one-hundredth of an Angstrom unit, a quantity well within experimental limits. Unfortunately, as far as the testing of this prediction is concerned, there are several physical causes which are also operating to cause displacement of the spectrum-lines; and so at present a decision can not be rendered as to the verification. St. John and other workers at the Mount Wilson Observatory have the question under investigation.3. According to Newton’s law an isolated planet in its motion around a central sun would describe, period after period, the same elliptical orbit; whereas Einstein’s laws lead to the prediction that the successive orbits traversed would not be identically the same. Each revolution would start the planet off on an orbit very approximately elliptical, but with the major axis of the ellipse rotated slightly in the plane of the orbit. When calculations were made for the various planets in our solar system, it was found that the only one which was of interest from the standpoint of verification of Einstein’s formulæ was Mercury. It has been known for a long timethat there was actually such a change as just described in the orbit of Mercury, amounting to 574″ of arc per century; and it has been shown that of this a rotation of 532″ was due to the direct action of other planets, thus leaving an unexplained rotation of 42″ per century. Einstein’s formulæ predicted a rotation of 43″, a striking agreement.4. In accordance with Einstein’s formulæ a ray of light passing close to a heavy piece of matter, the sun, for instance, should experience a sensible deflection in towards the sun. This might be expected from “general” consideration of energy in motion; energy and mass are generally considered to be identical in the sense that an amount of energyEhas the massupper E Baseline 1 c squaredwherecis the velocity of light; and consequently a ray of light might fall within the province of gravitation and the amount of deflection to be expected could be calculated by the ordinary formula for gravitation. Another point of view is to consider again the observer inside the compartment falling with the acceleration of the gravitational field. To him the path of a projectile and a ray of light would both appear straight; so that, if the projectile had a velocity equal to that of light, it and the light wave would travel side by side. To an observer outside the compartment,e.g., to one on the earth, both would then appear to have the same deflection owing to the sun. But how much would the path of the projectile be bent? What would be the shape of its parabola? One might apply Newton’s law; but, according to Einstein’s formulæ, Newton’s law should be used only for small velocities. In the case of a ray passing close to the sun it was decided that according to Einstein’s formula there should be a deflection of 1″.75 whereas Newton’s law of gravitation predicted half this amount. Careful plans were made by various astronomers, to investigate this question atthe solar eclipse last May, and the result announced by Dyson, Eddington and Crommelin, the leaders of astronomy in England, was that there was a deflection of 1″.9. Of course the detection of such a minute deflection was an extraordinarily difficult matter, so many corrections had to be applied to the original observations; but the names of the men who record the conclusions are such as to inspire confidence. Certainly any effect of refraction seems to be excluded.It is thus seen that the formulæ deduced by Einstein have been confirmed in a variety of ways and in a most brilliant manner. In connection with these formulæ one question must arise in the minds of everyone; by what process, where in the course of the mathematical development, does the idea of mass reveal itself? It was not in the equations at the beginning and yet here it is at the end. How does it appear? As a matter of fact it is first seen as a constant of integration in the discussion of the problem of the gravitational field due to a single particle; and the identity of this constant with mass is proved when one compares Einstein’s formulæ with Newton’s law which is simply its degenerated form. This mass, though, is the mass of which we become aware through our experiences with weight; and Einstein proceeded to prove that this quantity which entered as a constant of integration in his ideally simple problem also obeyed the laws of conservation of mass and conservation of momentum when he investigated the problems of two and more particles. Therefore Einstein deduced from his study of gravitational fields the well-known properties of matter which form the basis of theoretical mechanics. A further logical consequence of Einstein’s development is to show that energy has mass, a concept with which every one nowadays is familiar.The description of Einstein’s method which I have given sofar is simply the story of one success after another; and it is certainly fair to ask if we have at last reached finality in our investigation of nature, if we have attained to truth. Are there no outstanding difficulties? Is there no possibility of error? Certainly, not until all the predictions made from Einstein’s formulæ have been investigated can much be said; and further, it must be seen whether any other lines of argument will lead to the same conclusions. But without waiting for all this there is at least one difficulty which is apparent at this time. We have discussed the laws of nature as independent in their form of reference axes, a concept which appeals strongly to our philosophy; yet it is not at all clear, at first sight, that we can be justified in our belief. We can not imagine any way by which we can become conscious of the translation of the earth in space; but by means of gyroscopes we can learn a great deal about its rotation on its axis. We could locate the positions of its two poles, and by watching a Foucault pendulum or a gyroscope we can obtain a number which we interpret as the angular velocity of rotation of axes fixed in the earth; angular velocity with reference to what? Where is the fundamental set of axes? This is a real difficulty. It can be surmounted in several ways. Einstein himself has outlined a method which in the end amounts to assuming the existence on the confines of space of vast quantities of matter, a proposition which is not attractive. deSitter has suggested a peculiar quality of the space to which we refer our space-time coordinates. The consequences of this are most interesting, but no decision can as yet be made as to the justification of the hypothesis. In any case we can say that the difficulty raised is not one that destroys the real value of Einstein’s work.In conclusion I wish to emphasize the fact, which shouldbe obvious, that Einstein has not attempted any explanation of gravitation; he has been occupied with the deduction of its laws. These laws, together with those of electromagnetic phenomena, comprise our store of knowledge. There is not the slightest indication of a mechanism, meaning by that a picture in terms of our senses. In fact what we have learned has been to realize that our desire to use such mechanisms is futile.1Presidential address delivered at the St. Louis meeting of the Physical Society, December 30, 1919. Republished by permission from “Science.”↑

EINSTEIN’S LAW OF GRAVITATION1BYProf. J. S. AmesJohns Hopkins University

BYProf. J. S. AmesJohns Hopkins University

… In the treatment of Maxwell’s equations of the electromagnetic field, several investigators realized the importance of deducing the form of the equations when applied to a system moving with a uniform velocity. One object of such an investigation would be to determine such a set of transformation formulæ as would leave the mathematical form of the equations unaltered. The necessary relations between the new space-coordinates, those applying to the moving system, and the original set were of course obvious; and elementary methods led to the deduction of a new variable which should replace the time coordinate. This step was taken by Lorentz and also, I believe, by Larmor and by Voigt. The mathematical deductions and applications in the hands of these men were extremely beautiful, and are probably well known to you all.Lorentz’ paper on this subject appeared in the Proceedings of the Amsterdam Academy in 1904. In the following year there was published in theAnnalen der Physika paper by Einstein, written without any knowledge of the work of Lorentz, in which he arrived at the same transformation equations as did the latter, but with an entirely different and fundamentally new interpretation. Einstein called attention in his paper to the lack of definiteness in the concepts of time and space, as ordinarily stated and used. He analyzed clearly the definitions and postulates which were necessarybefore one could speak with exactness of a length or of an interval of time. He disposed forever of the propriety of speaking of the “true” length of a rod or of the “true” duration of time, showing, in fact, that the numerical values which we attach to lengths or intervals of time depend upon the definitions and postulates which we adopt. The words “absolute” space or time intervals are devoid of meaning. As an illustration of what is meant Einstein discussed two possible ways of measuring the length of a rod when it is moving in the direction of its own length with a uniform velocity, that is, after having adopted a scale of length, two ways of assigning a number to the length of the rod concerned. One method is to imagine the observer moving with the rod, applying along its length the measuring scale, and reading off the positions of the ends of the rod. Another method would be to have two observers at rest on the body with reference to which the rod has the uniform velocity, so stationed along the line of motion of the rod that as the rod moves past them they can note simultaneously on a stationary measuring scale the positions of the two ends of the rod. Einstein showed that, accepting two postulates which need no defense at this time, the two methods of measurements would lead to different numerical values, and, further, that the divergence of the two results would increase as the velocity of the rod was increased. In assigning a number, therefore, to the length of a moving rod, one must make a choice of the method to be used in measuring it. Obviously the preferable method is to agree that the observer shall move with the rod, carrying his measuring instrument with him. This disposes of the problem of measuring space relations. The observed fact that, if we measure the length of the rod on different days, or when the rod is lying in different positions, we alwaysobtain the same value offers no information concerning the “real” length of the rod. It may have changed, or it may not. It must always be remembered that measurement of the length of a rod is simply a process of comparison between it and an arbitrary standard,e.g., a meter-rod or yard-stick. In regard to the problem of assigning numbers to intervals of time, it must be borne in mind that, strictly speaking, we do not “measure” such intervals,i.e., that we do not select a unit interval of time and find how many times it is contained in the interval in question. (Similarly, we do not “measure” the pitch of a sound or the temperature of a room.) Our practical instruments for assigning numbers to time-intervals depend in the main upon our agreeing to believe that a pendulum swings in a perfectly uniform manner, each vibration taking the same time as the next one. Of course we cannotprovethat this is true, it is, strictly speaking, a definition of what we mean by equal intervals of time; and it is not a particularly good definition at that. Its limitations are sufficiently obvious. The best way to proceed is to consider the concept of uniform velocity, and then, using the idea of some entity having such a uniform velocity, to define equal intervals of time as such intervals as are required for the entity to traverse equal lengths. These last we have already defined. What is required in addition is to adopt some moving entity as giving our definition of uniform velocity. Considering our known universe it is self-evident that we should choose in our definition of uniform velocity the velocity of light, since this selection could be made by an observer anywhere in our universe. Having agreed then to illustrate by the words “uniform velocity” that of light, our definition of equal intervals of time is complete. This implies, of course, that there is no uncertainty on our part as to the fact that the velocity oflight always has the same value at any one point in the universe to any observer, quite regardless of the source of light. In other words, the postulate that this is true underlies our definition. Following this method Einstein developed a system of measuring both space and time intervals. As a matter of fact his system is identically that which we use in daily life with reference to events here on the earth. He further showed that if a man were to measure the length of a rod, for instance, on the earth and then were able to carry the rod and his measuring apparatus to Mars, the sun, or to Arcturus he would obtain the same numerical value for the length in all places and at all times. This doesn’t mean that any statement is implied as to whether the length of the rod has remained unchanged or not; such words do not have any meaning—remember that we can not speak of true length. It is thus clear that an observer living on the earth would have a definite system of units in terms of which to express space and time intervals,i.e., he would have a definite system of space coordinates (x,y,z) and a definite time coordinate (t); and similarly an observer living on Mars would have his system of coordinates (x′,y′,z′,t′). Provided that one observer has a definite uniform velocity with reference to the other, it is a comparatively simple matter to deduce the mathematical relations between the two sets of coordinates. When Einstein did this, he arrived at the same transformation formulæ as those used by Lorentz in his development of Maxwell’s equations. The latter had shown that, using these formulæ, the form of the laws for all electromagnetic phenomena maintained the same form; so Einstein’s method proves that using his system of measurement an observer, anywhere in the universe, would as the result of his own investigation of electromagnetic phenomena arrive at thesame mathematical statement of them as any other observer, provided only that the relative-velocity of the two observers was uniform.Einstein discussed many other most important questions at this time; but it is not necessary to refer to them in connection with the present subject. So far as this is concerned, the next important step to note is that taken in the famous address of Minkowski, in 1908, on the subject of “Space and Time.” It would be difficult to overstate the importance of the concepts advanced by Minkowski. They marked the beginning of a new period in the philosophy of physics. I shall not attempt to explain his ideas in detail, but shall confine myself to a few general statements. His point of view and his line of development of the theme are absolutely different from those of Lorentz or of Einstein; but in the end he makes use of the same transformation formulæ. His great contribution consists in giving us a new geometrical picture of their meaning. It is scarcely fair to call Minkowski’s development a picture; for to us a picture can never have more than three dimensions, our senses limit us; while his picture calls for perception of four dimensions. It is this fact that renders any even semi-popular discussion of Minkowski’s work so impossible. We can all see that for us to describe any event a knowledge of four coordinates is necessary, three for the space specification and one for the time. A complete picture could be given then by a point in four dimensions. All four coordinates are necessary: we never observe an event except at a certain time, and we never observe an instant of time except with reference to space. Discussing the laws of electromagnetic phenomena, Minkowski showed how in a space of four dimensions, by a suitable definition of axes, the mathematical transformation of Lorentz and Einsteincould be described by a rotation of the set of axes. We are all accustomed to a rotation of our ordinary cartesian set of axes describing the position of a point. We ordinarily choose our axes at any location on the earth as follows: one vertical, one east and west, one north and south. So if we move from any one laboratory to another, we change our axes; they are always orthogonal, but in moving from place to place there is a rotation. Similarly, Minkowski showed that if we choose four orthogonal axes at any point on the earth, according to his method, to represent a space-time point using the method of measuring space and time intervals as outlined by Einstein; and, if an observer on Arcturus used a similar set of axes and the method of measurement which he naturally would, the set of axes of the latter could be obtained from those of the observer on the earth by a pure rotation (and naturally a transfer of the origin). This is a beautiful geometrical result. To complete my statement of the method, I must add that instead of using as his fourth axis one along which numerical values of time are laid off, Minkowski defined his fourth coordinate as the product of time and the imaginary constant, the square root of minus one. This introduction of imaginary quantities might be expected, possibly, to introduce difficulties; but, in reality, it is the very essence of the simplicity of the geometrical description just given of the rotation of the sets of axes. It thus appears that different observers situated at different points in the universe would each have their own set of axes, all different, yet all connected by the fact that any one can be rotated so as to coincide with any other. This means that there is no one direction in the four-dimensional space that corresponds to time for all observers. Just as with reference to the earth there is no direction which can be called vertical for all observers living on the earth. In thesense of anabsolutemeaning the words “up and down,” “before and after,” “sooner or later,” are entirely meaningless.This concept of Minkowski’s may be made clearer, perhaps, by the following process of thought. If we take a section through our three-dimensional space, we have a plane,i.e., a two-dimensional space. Similarly, if a section is made through a four-dimensional space, one of three dimensions is obtained. Thus, for an observer on the earth a definite section of Minkowski’s four-dimensional space will give us our ordinary three-dimensional one; so that this section will, as it were, break up Minkowski’s space into our space and give us our ordinary time. Similarly, a different section would have to be used to the observer on Arcturus; but by a suitable selection he would get his own familiar three-dimensional space and his own time. Thus the space defined by Minkowski is completely isotropic in reference to measured lengths and times, there is absolutely no difference between any two directions in an absolute sense; for any particular observer, of course, a particular section will cause the space to fall apart so as to suit his habits of measurement; any section, however, taken at random will do the same thing for some observer somewhere. From another point of view, that of Lorentz and Einstein, it is obvious that, since this four-dimensional space is isotropic, the expression of the laws of electromagnetic phenomena take identical mathematical forms when expressed by any observer.The question of course must be raised as to what can be said in regard to phenomena which so far as we know do not have an electromagnetic origin. In particular what can be done with respect to gravitational phenomena? Before, however, showing how this problem was attacked by Einstein;and the fact that the subject of my address is Einstein’s work on gravitation shows that ultimately I shall explain this, I must emphasize another feature of Minkowski’s geometry. To describe the space-time characteristics of any event a point, defined by its four coordinates, is sufficient; so, if one observes the life-history of any entity,e.g., a particle of matter, a light-wave, etc., he observes a sequence of points in the space-time continuum; that is, the life-history of any entity is described fully by a line in this space. Such a line was called by Minkowski a “world-line.” Further, from a different point of view, all of our observations of nature are in reality observations of coincidences,e.g., if one reads a thermometer, what he does is to note the coincidence of the end of the column of mercury with a certain scale division on the thermometer tube. In other words, thinking of the world-line of the end of the mercury column and the world-line of the scale division, what we have observed was the intersection or crossing of these lines. In a similar manner any observation may be analyzed; and remembering that light rays, a point on the retina of the eye, etc., all have their world-lines, it will be recognized that it is a perfectly accurate statement to say that every observation is the perception of the intersection of world-lines. Further, since all we know of a world-line is the result of observations, it is evident that we do not know a world-line as a continuous series of points, but simply as a series of discontinuous points, each point being where the particular world-line in question is crossed by another world-line.It is clear, moreover, that for the description of a world-line we are not limited to the particular set of four orthogonal axes adopted by Minkowski. We can choose any set of four-dimensional axes we wish. It is further evident that the mathematical expression for the coincidence of two points isabsolutely independent of our selection of reference axes. If we change our axes, we will change the coordinates of both points simultaneously, so that the question of axes ceases to be of interest. But our so-called laws of nature are nothing but descriptions in mathematical language of our observations; we observe only coincidences; a sequence of coincidences when put in mathematical terms takes a form which is independent of the selection of reference axes; therefore the mathematical expression of our laws of nature, of every character, must be such that their form does not change if we make a transformation of axes. This is a simple but far-reaching deduction.There is a geometrical method of picturing the effect of a change of axes of reference,i.e., of a mathematical transformation. To a man in a railway coach the path of a drop of water does not appear vertical,i.e., it is not parallel to the edge of the window; still less so does it appear vertical to a man performing manœuvres in an airplane. This means that whereas with reference to axes fixed to the earth the path of the drop is vertical; with reference to other axes, the path is not. Or, stating the conclusion in general language, changing the axes of reference (or effecting a mathematical transformation) in general changes the shape of any line. If one imagines the line forming a part of the space, it is evident that if the space is deformed by compression or expansion the shape of the line is changed, and if sufficient care is taken it is clearly possible, by deforming the space, to make the line take any shape desired, or better stated, any shape specified by the previous change of axes. It is thus possible to picture a mathematical transformation as a deformation of space. Thus I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great varietyof shapes; each of these new shapes is a picture of a suitable transformation.Now, consider world-lines in our four-dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections. So in the space of our world-lines, the space may be deformed in any imaginable manner without introducing any new intersections or changing the sequence of the existing intersections. It is this sequence which gives us the mathematical expression of our so-called experimental laws; a deformation of our space is equivalent mathematically to a transformation of axes, consequently we see why it is that the form of our laws must be the same when referred to any and all sets of axes, that is, must remain unaltered by any mathematical transformation.Now, at last we come to gravitation. We can not imagine any world-line simpler than that of a particle of matter left to itself; we shall therefore call it a “straight” line. Our experience is that two particles of matter attract one another.Expressed in terms of world-lines, this means that, if the world-lines of two isolated particles come near each other, the lines, instead of being straight, will be deflected or bent in towards each other. The world-line of any one particle is therefore deformed; and we have just seen that a deformation is the equivalent of a mathematical transformation. In other words, for any one particle it is possible to replace the effect of a gravitational field at any instant by a mathematical transformation of axes. The statement that this is always possiblefor any particle at any instant is Einstein’s famous “Principle of Equivalence.”Let us rest for a moment, while I call attention to a most interesting coincidence, not to be thought of as an intersection of world-lines. It is said that Newton’s thoughts were directed to the observation of gravitational phenomena by an apple falling on his head; from this striking event he passed by natural steps to a consideration of the universality of gravitation. Einstein in describing his mental process in the evolution of his law of gravitation says that his attention was called to a new point of view by discussing his experiences with a man whose fall from a high building he had just witnessed. The man fortunately suffered no serious injuries and assured Einstein that in the course of his fall he had not been conscious in the least of any pull downward on his body. In mathematical language, with reference to axes moving with the man the force of gravity had disappeared. This is a case where by the transfer of the axes from the earth itself to the man, the force of the gravitational field is annulled. The converse change of axes from the falling man to a point on the earth could be considered as introducing the force of gravity into the equations of motion. Another illustration of the introduction into our equations of a force by a means of a change of axes is furnished by the ordinary treatment of a body in uniform rotation about an axis. For instance, in the case of a so-called conical pendulum, that is, the motion of a bob suspended from a fixed point by string, which is so set in motion that the bob describes a horizontal circle and the string therefore describes a circular cone, if we transfer our axes from the earth and have them rotate around the vertical line through the fixed point with the same angular velocity as the bob, it is necessary to introduce into our equations ofmotion a fictitious “force” called the centrifugal force. No one ever thinks of this force other than as a mathematical quantity introduced into the equations for the sake of simplicity of treatment; no physical meaning is attached to it. Why should there be to any other so-called “force,” which like centrifugal force, is independent of the nature of the matter? Again, here on the earth our sensation of weight is interpreted mathematically by combining expressions for centrifugal force and gravity; we have no distinct sensation for either separately. Why then is there any difference in the essence of the two? Why not consider them both as brought into our equations by the agency of mathematical transformations? This is Einstein’s point of view.Granting, then, the principle of equivalence, we can so choose axes at any point at any instant that the gravitational field will disappear; these axes are therefore of what Eddington calls the “Galilean” type, the simplest possible. Consider, that is, an observer in a box, or compartment, which is falling with the acceleration of the gravitational field at that point. He would not be conscious of the field. If there were a projectile fired off in this compartment, the observer would describe its path as being straight. In this space the infinitesimal interval between two space-time points would then be given by the formulad s squared equals d x 1 squared plus d x 2 Subscript 2 Baseline plus d x 3 squared plus d x 2 Subscript 4 Baseline commawheredsis the interval andx 1 comma x 2 comma x 3 comma x 4are coordinates. If we make a mathematical transformation,i.e., use another set of axes, this interval would obviously take the formd s squared equals g 11 d x 33 squared plus g 22 d x 2 squared plus g 33 d x 3 squared plus g 44 d x 2 Subscript 4 Baseline plus 2 g 12 d x 1 d x 2 plus normal e normal t normal c period commawherex 1 comma x 2 comma x 3andx 4are now coordinates referring to the newaxes. This relation involves ten coefficients, the coefficients defining the transformation.But of course a certain dynamical value is also attached to theg’s, because by the transfer of our axes from the Galilean type we have made a change which is equivalent to the introduction of a gravitational field; and theg’s must specify the field. That is, theseg’s are the expressions of our experiences, and hence their values can not depend upon the use of any special axes; the values must be the same for all selections. In other words, whatever function of the coordinates any onegis for one set of axes, if other axes are chosen, thisgmust still be the same function of the new coordinates. There are teng’s defined by differential equations; so we have ten covariant equations. Einstein showed how theseg’s could be regarded as generalized potentials of the field. Our own experiments and observations upon gravitation have given us a certain knowledge concerning its potential; that is, we know a value for it which must be so near the truth that we can properly call it at least a first approximation. Or, stated differently, if Einstein succeeds in deducing the rigid value for the gravitational potential in any field, it must degenerate to the Newtonian value for the great majority of cases with which we have actual experience. Einstein’s method, then, was to investigate the functions (or equations) which would satisfy the mathematical conditions just described. A transformation from the axes used by the observer in the following box may be made so as to introduce into the equations the gravitational field recognized by an observer on the earth near the box; but this, obviously, would not be the general gravitational field, because the field changes as one moves over the surface of the earth. A solution found, therefore, as just indicated, would not be the one sought for the general field; and anothermust be found which is less stringent than the former but reduces to it as a special case. He found himself at liberty to make a selection from among several possibilities, and for several reasons chose the simplest solution. He then tested this decision by seeing if his formulæ would degenerate to Newton’s law for the limiting case of velocities small when compared with that of light, because this condition is satisfied in those cases to which Newton’s law applies. His formulæ satisfied this test, and he therefore was able to announce a “law of gravitation,” of which Newton’s was a special form for a simple case.To the ordinary scholar the difficulties surmounted by Einstein in his investigations appear stupendous. It is not improbable that the statement which he is alleged to have made to his editor, that only ten men in the world could understand his treatment of the subject, is true. I am fully prepared to believe it, and wish to add that I certainly am not one of the ten. But I can also say that, after a careful and serious study of his papers, I feel confident that there is nothing in them which I can not understand, given the time to become familiar with the special mathematical processes used. The more I work over Einstein’s papers, the more impressed I am, not simply by his genius in viewing the problem, but also by his great technical skill.Following the path outlined, Einstein, as just said, arrived at certain mathematical laws for a gravitational field, laws which reduced to Newton’s form in most cases where observations are possible, but which led to different conclusions in a few cases, knowledge concerning which we might obtain by careful observations. I shall mention a few deductions from Einstein’s formulæ.1. If a heavy particle is put at the center of a circle, and, ifthe length of the circumference and the length of the diameter are measured, it will be found that their ratio is not π (3.14159). In other words the geometrical properties of space in such a gravitational field are not those discussed by Euclid; the space is, then, non-Euclidean. There is no way by which this deduction can be verified, the difference between the predicted ratio and π is too minute for us to hope to make our measurements with sufficient exactness to determine the difference.2. All the lines in the solar spectrum should with reference to lines obtained by terrestrial sources be displaced slightly towards longer wave-lengths. The amount of displacement predicted for lines in the blue end of the spectrum is about one-hundredth of an Angstrom unit, a quantity well within experimental limits. Unfortunately, as far as the testing of this prediction is concerned, there are several physical causes which are also operating to cause displacement of the spectrum-lines; and so at present a decision can not be rendered as to the verification. St. John and other workers at the Mount Wilson Observatory have the question under investigation.3. According to Newton’s law an isolated planet in its motion around a central sun would describe, period after period, the same elliptical orbit; whereas Einstein’s laws lead to the prediction that the successive orbits traversed would not be identically the same. Each revolution would start the planet off on an orbit very approximately elliptical, but with the major axis of the ellipse rotated slightly in the plane of the orbit. When calculations were made for the various planets in our solar system, it was found that the only one which was of interest from the standpoint of verification of Einstein’s formulæ was Mercury. It has been known for a long timethat there was actually such a change as just described in the orbit of Mercury, amounting to 574″ of arc per century; and it has been shown that of this a rotation of 532″ was due to the direct action of other planets, thus leaving an unexplained rotation of 42″ per century. Einstein’s formulæ predicted a rotation of 43″, a striking agreement.4. In accordance with Einstein’s formulæ a ray of light passing close to a heavy piece of matter, the sun, for instance, should experience a sensible deflection in towards the sun. This might be expected from “general” consideration of energy in motion; energy and mass are generally considered to be identical in the sense that an amount of energyEhas the massupper E Baseline 1 c squaredwherecis the velocity of light; and consequently a ray of light might fall within the province of gravitation and the amount of deflection to be expected could be calculated by the ordinary formula for gravitation. Another point of view is to consider again the observer inside the compartment falling with the acceleration of the gravitational field. To him the path of a projectile and a ray of light would both appear straight; so that, if the projectile had a velocity equal to that of light, it and the light wave would travel side by side. To an observer outside the compartment,e.g., to one on the earth, both would then appear to have the same deflection owing to the sun. But how much would the path of the projectile be bent? What would be the shape of its parabola? One might apply Newton’s law; but, according to Einstein’s formulæ, Newton’s law should be used only for small velocities. In the case of a ray passing close to the sun it was decided that according to Einstein’s formula there should be a deflection of 1″.75 whereas Newton’s law of gravitation predicted half this amount. Careful plans were made by various astronomers, to investigate this question atthe solar eclipse last May, and the result announced by Dyson, Eddington and Crommelin, the leaders of astronomy in England, was that there was a deflection of 1″.9. Of course the detection of such a minute deflection was an extraordinarily difficult matter, so many corrections had to be applied to the original observations; but the names of the men who record the conclusions are such as to inspire confidence. Certainly any effect of refraction seems to be excluded.It is thus seen that the formulæ deduced by Einstein have been confirmed in a variety of ways and in a most brilliant manner. In connection with these formulæ one question must arise in the minds of everyone; by what process, where in the course of the mathematical development, does the idea of mass reveal itself? It was not in the equations at the beginning and yet here it is at the end. How does it appear? As a matter of fact it is first seen as a constant of integration in the discussion of the problem of the gravitational field due to a single particle; and the identity of this constant with mass is proved when one compares Einstein’s formulæ with Newton’s law which is simply its degenerated form. This mass, though, is the mass of which we become aware through our experiences with weight; and Einstein proceeded to prove that this quantity which entered as a constant of integration in his ideally simple problem also obeyed the laws of conservation of mass and conservation of momentum when he investigated the problems of two and more particles. Therefore Einstein deduced from his study of gravitational fields the well-known properties of matter which form the basis of theoretical mechanics. A further logical consequence of Einstein’s development is to show that energy has mass, a concept with which every one nowadays is familiar.The description of Einstein’s method which I have given sofar is simply the story of one success after another; and it is certainly fair to ask if we have at last reached finality in our investigation of nature, if we have attained to truth. Are there no outstanding difficulties? Is there no possibility of error? Certainly, not until all the predictions made from Einstein’s formulæ have been investigated can much be said; and further, it must be seen whether any other lines of argument will lead to the same conclusions. But without waiting for all this there is at least one difficulty which is apparent at this time. We have discussed the laws of nature as independent in their form of reference axes, a concept which appeals strongly to our philosophy; yet it is not at all clear, at first sight, that we can be justified in our belief. We can not imagine any way by which we can become conscious of the translation of the earth in space; but by means of gyroscopes we can learn a great deal about its rotation on its axis. We could locate the positions of its two poles, and by watching a Foucault pendulum or a gyroscope we can obtain a number which we interpret as the angular velocity of rotation of axes fixed in the earth; angular velocity with reference to what? Where is the fundamental set of axes? This is a real difficulty. It can be surmounted in several ways. Einstein himself has outlined a method which in the end amounts to assuming the existence on the confines of space of vast quantities of matter, a proposition which is not attractive. deSitter has suggested a peculiar quality of the space to which we refer our space-time coordinates. The consequences of this are most interesting, but no decision can as yet be made as to the justification of the hypothesis. In any case we can say that the difficulty raised is not one that destroys the real value of Einstein’s work.In conclusion I wish to emphasize the fact, which shouldbe obvious, that Einstein has not attempted any explanation of gravitation; he has been occupied with the deduction of its laws. These laws, together with those of electromagnetic phenomena, comprise our store of knowledge. There is not the slightest indication of a mechanism, meaning by that a picture in terms of our senses. In fact what we have learned has been to realize that our desire to use such mechanisms is futile.

… In the treatment of Maxwell’s equations of the electromagnetic field, several investigators realized the importance of deducing the form of the equations when applied to a system moving with a uniform velocity. One object of such an investigation would be to determine such a set of transformation formulæ as would leave the mathematical form of the equations unaltered. The necessary relations between the new space-coordinates, those applying to the moving system, and the original set were of course obvious; and elementary methods led to the deduction of a new variable which should replace the time coordinate. This step was taken by Lorentz and also, I believe, by Larmor and by Voigt. The mathematical deductions and applications in the hands of these men were extremely beautiful, and are probably well known to you all.

Lorentz’ paper on this subject appeared in the Proceedings of the Amsterdam Academy in 1904. In the following year there was published in theAnnalen der Physika paper by Einstein, written without any knowledge of the work of Lorentz, in which he arrived at the same transformation equations as did the latter, but with an entirely different and fundamentally new interpretation. Einstein called attention in his paper to the lack of definiteness in the concepts of time and space, as ordinarily stated and used. He analyzed clearly the definitions and postulates which were necessarybefore one could speak with exactness of a length or of an interval of time. He disposed forever of the propriety of speaking of the “true” length of a rod or of the “true” duration of time, showing, in fact, that the numerical values which we attach to lengths or intervals of time depend upon the definitions and postulates which we adopt. The words “absolute” space or time intervals are devoid of meaning. As an illustration of what is meant Einstein discussed two possible ways of measuring the length of a rod when it is moving in the direction of its own length with a uniform velocity, that is, after having adopted a scale of length, two ways of assigning a number to the length of the rod concerned. One method is to imagine the observer moving with the rod, applying along its length the measuring scale, and reading off the positions of the ends of the rod. Another method would be to have two observers at rest on the body with reference to which the rod has the uniform velocity, so stationed along the line of motion of the rod that as the rod moves past them they can note simultaneously on a stationary measuring scale the positions of the two ends of the rod. Einstein showed that, accepting two postulates which need no defense at this time, the two methods of measurements would lead to different numerical values, and, further, that the divergence of the two results would increase as the velocity of the rod was increased. In assigning a number, therefore, to the length of a moving rod, one must make a choice of the method to be used in measuring it. Obviously the preferable method is to agree that the observer shall move with the rod, carrying his measuring instrument with him. This disposes of the problem of measuring space relations. The observed fact that, if we measure the length of the rod on different days, or when the rod is lying in different positions, we alwaysobtain the same value offers no information concerning the “real” length of the rod. It may have changed, or it may not. It must always be remembered that measurement of the length of a rod is simply a process of comparison between it and an arbitrary standard,e.g., a meter-rod or yard-stick. In regard to the problem of assigning numbers to intervals of time, it must be borne in mind that, strictly speaking, we do not “measure” such intervals,i.e., that we do not select a unit interval of time and find how many times it is contained in the interval in question. (Similarly, we do not “measure” the pitch of a sound or the temperature of a room.) Our practical instruments for assigning numbers to time-intervals depend in the main upon our agreeing to believe that a pendulum swings in a perfectly uniform manner, each vibration taking the same time as the next one. Of course we cannotprovethat this is true, it is, strictly speaking, a definition of what we mean by equal intervals of time; and it is not a particularly good definition at that. Its limitations are sufficiently obvious. The best way to proceed is to consider the concept of uniform velocity, and then, using the idea of some entity having such a uniform velocity, to define equal intervals of time as such intervals as are required for the entity to traverse equal lengths. These last we have already defined. What is required in addition is to adopt some moving entity as giving our definition of uniform velocity. Considering our known universe it is self-evident that we should choose in our definition of uniform velocity the velocity of light, since this selection could be made by an observer anywhere in our universe. Having agreed then to illustrate by the words “uniform velocity” that of light, our definition of equal intervals of time is complete. This implies, of course, that there is no uncertainty on our part as to the fact that the velocity oflight always has the same value at any one point in the universe to any observer, quite regardless of the source of light. In other words, the postulate that this is true underlies our definition. Following this method Einstein developed a system of measuring both space and time intervals. As a matter of fact his system is identically that which we use in daily life with reference to events here on the earth. He further showed that if a man were to measure the length of a rod, for instance, on the earth and then were able to carry the rod and his measuring apparatus to Mars, the sun, or to Arcturus he would obtain the same numerical value for the length in all places and at all times. This doesn’t mean that any statement is implied as to whether the length of the rod has remained unchanged or not; such words do not have any meaning—remember that we can not speak of true length. It is thus clear that an observer living on the earth would have a definite system of units in terms of which to express space and time intervals,i.e., he would have a definite system of space coordinates (x,y,z) and a definite time coordinate (t); and similarly an observer living on Mars would have his system of coordinates (x′,y′,z′,t′). Provided that one observer has a definite uniform velocity with reference to the other, it is a comparatively simple matter to deduce the mathematical relations between the two sets of coordinates. When Einstein did this, he arrived at the same transformation formulæ as those used by Lorentz in his development of Maxwell’s equations. The latter had shown that, using these formulæ, the form of the laws for all electromagnetic phenomena maintained the same form; so Einstein’s method proves that using his system of measurement an observer, anywhere in the universe, would as the result of his own investigation of electromagnetic phenomena arrive at thesame mathematical statement of them as any other observer, provided only that the relative-velocity of the two observers was uniform.

Einstein discussed many other most important questions at this time; but it is not necessary to refer to them in connection with the present subject. So far as this is concerned, the next important step to note is that taken in the famous address of Minkowski, in 1908, on the subject of “Space and Time.” It would be difficult to overstate the importance of the concepts advanced by Minkowski. They marked the beginning of a new period in the philosophy of physics. I shall not attempt to explain his ideas in detail, but shall confine myself to a few general statements. His point of view and his line of development of the theme are absolutely different from those of Lorentz or of Einstein; but in the end he makes use of the same transformation formulæ. His great contribution consists in giving us a new geometrical picture of their meaning. It is scarcely fair to call Minkowski’s development a picture; for to us a picture can never have more than three dimensions, our senses limit us; while his picture calls for perception of four dimensions. It is this fact that renders any even semi-popular discussion of Minkowski’s work so impossible. We can all see that for us to describe any event a knowledge of four coordinates is necessary, three for the space specification and one for the time. A complete picture could be given then by a point in four dimensions. All four coordinates are necessary: we never observe an event except at a certain time, and we never observe an instant of time except with reference to space. Discussing the laws of electromagnetic phenomena, Minkowski showed how in a space of four dimensions, by a suitable definition of axes, the mathematical transformation of Lorentz and Einsteincould be described by a rotation of the set of axes. We are all accustomed to a rotation of our ordinary cartesian set of axes describing the position of a point. We ordinarily choose our axes at any location on the earth as follows: one vertical, one east and west, one north and south. So if we move from any one laboratory to another, we change our axes; they are always orthogonal, but in moving from place to place there is a rotation. Similarly, Minkowski showed that if we choose four orthogonal axes at any point on the earth, according to his method, to represent a space-time point using the method of measuring space and time intervals as outlined by Einstein; and, if an observer on Arcturus used a similar set of axes and the method of measurement which he naturally would, the set of axes of the latter could be obtained from those of the observer on the earth by a pure rotation (and naturally a transfer of the origin). This is a beautiful geometrical result. To complete my statement of the method, I must add that instead of using as his fourth axis one along which numerical values of time are laid off, Minkowski defined his fourth coordinate as the product of time and the imaginary constant, the square root of minus one. This introduction of imaginary quantities might be expected, possibly, to introduce difficulties; but, in reality, it is the very essence of the simplicity of the geometrical description just given of the rotation of the sets of axes. It thus appears that different observers situated at different points in the universe would each have their own set of axes, all different, yet all connected by the fact that any one can be rotated so as to coincide with any other. This means that there is no one direction in the four-dimensional space that corresponds to time for all observers. Just as with reference to the earth there is no direction which can be called vertical for all observers living on the earth. In thesense of anabsolutemeaning the words “up and down,” “before and after,” “sooner or later,” are entirely meaningless.

This concept of Minkowski’s may be made clearer, perhaps, by the following process of thought. If we take a section through our three-dimensional space, we have a plane,i.e., a two-dimensional space. Similarly, if a section is made through a four-dimensional space, one of three dimensions is obtained. Thus, for an observer on the earth a definite section of Minkowski’s four-dimensional space will give us our ordinary three-dimensional one; so that this section will, as it were, break up Minkowski’s space into our space and give us our ordinary time. Similarly, a different section would have to be used to the observer on Arcturus; but by a suitable selection he would get his own familiar three-dimensional space and his own time. Thus the space defined by Minkowski is completely isotropic in reference to measured lengths and times, there is absolutely no difference between any two directions in an absolute sense; for any particular observer, of course, a particular section will cause the space to fall apart so as to suit his habits of measurement; any section, however, taken at random will do the same thing for some observer somewhere. From another point of view, that of Lorentz and Einstein, it is obvious that, since this four-dimensional space is isotropic, the expression of the laws of electromagnetic phenomena take identical mathematical forms when expressed by any observer.

The question of course must be raised as to what can be said in regard to phenomena which so far as we know do not have an electromagnetic origin. In particular what can be done with respect to gravitational phenomena? Before, however, showing how this problem was attacked by Einstein;and the fact that the subject of my address is Einstein’s work on gravitation shows that ultimately I shall explain this, I must emphasize another feature of Minkowski’s geometry. To describe the space-time characteristics of any event a point, defined by its four coordinates, is sufficient; so, if one observes the life-history of any entity,e.g., a particle of matter, a light-wave, etc., he observes a sequence of points in the space-time continuum; that is, the life-history of any entity is described fully by a line in this space. Such a line was called by Minkowski a “world-line.” Further, from a different point of view, all of our observations of nature are in reality observations of coincidences,e.g., if one reads a thermometer, what he does is to note the coincidence of the end of the column of mercury with a certain scale division on the thermometer tube. In other words, thinking of the world-line of the end of the mercury column and the world-line of the scale division, what we have observed was the intersection or crossing of these lines. In a similar manner any observation may be analyzed; and remembering that light rays, a point on the retina of the eye, etc., all have their world-lines, it will be recognized that it is a perfectly accurate statement to say that every observation is the perception of the intersection of world-lines. Further, since all we know of a world-line is the result of observations, it is evident that we do not know a world-line as a continuous series of points, but simply as a series of discontinuous points, each point being where the particular world-line in question is crossed by another world-line.

It is clear, moreover, that for the description of a world-line we are not limited to the particular set of four orthogonal axes adopted by Minkowski. We can choose any set of four-dimensional axes we wish. It is further evident that the mathematical expression for the coincidence of two points isabsolutely independent of our selection of reference axes. If we change our axes, we will change the coordinates of both points simultaneously, so that the question of axes ceases to be of interest. But our so-called laws of nature are nothing but descriptions in mathematical language of our observations; we observe only coincidences; a sequence of coincidences when put in mathematical terms takes a form which is independent of the selection of reference axes; therefore the mathematical expression of our laws of nature, of every character, must be such that their form does not change if we make a transformation of axes. This is a simple but far-reaching deduction.

There is a geometrical method of picturing the effect of a change of axes of reference,i.e., of a mathematical transformation. To a man in a railway coach the path of a drop of water does not appear vertical,i.e., it is not parallel to the edge of the window; still less so does it appear vertical to a man performing manœuvres in an airplane. This means that whereas with reference to axes fixed to the earth the path of the drop is vertical; with reference to other axes, the path is not. Or, stating the conclusion in general language, changing the axes of reference (or effecting a mathematical transformation) in general changes the shape of any line. If one imagines the line forming a part of the space, it is evident that if the space is deformed by compression or expansion the shape of the line is changed, and if sufficient care is taken it is clearly possible, by deforming the space, to make the line take any shape desired, or better stated, any shape specified by the previous change of axes. It is thus possible to picture a mathematical transformation as a deformation of space. Thus I can draw a line on a sheet of paper or of rubber and by bending and stretching the sheet, I can make the line assume a great varietyof shapes; each of these new shapes is a picture of a suitable transformation.

Now, consider world-lines in our four-dimensional space. The complete record of all our knowledge is a series of sequences of intersections of such lines. By analogy I can draw in ordinary space a great number of intersecting lines on a sheet of rubber; I can then bend and deform the sheet to please myself; by so doing I do not introduce any new intersections nor do I alter in the least the sequence of intersections. So in the space of our world-lines, the space may be deformed in any imaginable manner without introducing any new intersections or changing the sequence of the existing intersections. It is this sequence which gives us the mathematical expression of our so-called experimental laws; a deformation of our space is equivalent mathematically to a transformation of axes, consequently we see why it is that the form of our laws must be the same when referred to any and all sets of axes, that is, must remain unaltered by any mathematical transformation.

Now, at last we come to gravitation. We can not imagine any world-line simpler than that of a particle of matter left to itself; we shall therefore call it a “straight” line. Our experience is that two particles of matter attract one another.Expressed in terms of world-lines, this means that, if the world-lines of two isolated particles come near each other, the lines, instead of being straight, will be deflected or bent in towards each other. The world-line of any one particle is therefore deformed; and we have just seen that a deformation is the equivalent of a mathematical transformation. In other words, for any one particle it is possible to replace the effect of a gravitational field at any instant by a mathematical transformation of axes. The statement that this is always possiblefor any particle at any instant is Einstein’s famous “Principle of Equivalence.”

Let us rest for a moment, while I call attention to a most interesting coincidence, not to be thought of as an intersection of world-lines. It is said that Newton’s thoughts were directed to the observation of gravitational phenomena by an apple falling on his head; from this striking event he passed by natural steps to a consideration of the universality of gravitation. Einstein in describing his mental process in the evolution of his law of gravitation says that his attention was called to a new point of view by discussing his experiences with a man whose fall from a high building he had just witnessed. The man fortunately suffered no serious injuries and assured Einstein that in the course of his fall he had not been conscious in the least of any pull downward on his body. In mathematical language, with reference to axes moving with the man the force of gravity had disappeared. This is a case where by the transfer of the axes from the earth itself to the man, the force of the gravitational field is annulled. The converse change of axes from the falling man to a point on the earth could be considered as introducing the force of gravity into the equations of motion. Another illustration of the introduction into our equations of a force by a means of a change of axes is furnished by the ordinary treatment of a body in uniform rotation about an axis. For instance, in the case of a so-called conical pendulum, that is, the motion of a bob suspended from a fixed point by string, which is so set in motion that the bob describes a horizontal circle and the string therefore describes a circular cone, if we transfer our axes from the earth and have them rotate around the vertical line through the fixed point with the same angular velocity as the bob, it is necessary to introduce into our equations ofmotion a fictitious “force” called the centrifugal force. No one ever thinks of this force other than as a mathematical quantity introduced into the equations for the sake of simplicity of treatment; no physical meaning is attached to it. Why should there be to any other so-called “force,” which like centrifugal force, is independent of the nature of the matter? Again, here on the earth our sensation of weight is interpreted mathematically by combining expressions for centrifugal force and gravity; we have no distinct sensation for either separately. Why then is there any difference in the essence of the two? Why not consider them both as brought into our equations by the agency of mathematical transformations? This is Einstein’s point of view.

Granting, then, the principle of equivalence, we can so choose axes at any point at any instant that the gravitational field will disappear; these axes are therefore of what Eddington calls the “Galilean” type, the simplest possible. Consider, that is, an observer in a box, or compartment, which is falling with the acceleration of the gravitational field at that point. He would not be conscious of the field. If there were a projectile fired off in this compartment, the observer would describe its path as being straight. In this space the infinitesimal interval between two space-time points would then be given by the formulad s squared equals d x 1 squared plus d x 2 Subscript 2 Baseline plus d x 3 squared plus d x 2 Subscript 4 Baseline comma

wheredsis the interval andx 1 comma x 2 comma x 3 comma x 4are coordinates. If we make a mathematical transformation,i.e., use another set of axes, this interval would obviously take the formd s squared equals g 11 d x 33 squared plus g 22 d x 2 squared plus g 33 d x 3 squared plus g 44 d x 2 Subscript 4 Baseline plus 2 g 12 d x 1 d x 2 plus normal e normal t normal c period comma

wherex 1 comma x 2 comma x 3andx 4are now coordinates referring to the newaxes. This relation involves ten coefficients, the coefficients defining the transformation.

But of course a certain dynamical value is also attached to theg’s, because by the transfer of our axes from the Galilean type we have made a change which is equivalent to the introduction of a gravitational field; and theg’s must specify the field. That is, theseg’s are the expressions of our experiences, and hence their values can not depend upon the use of any special axes; the values must be the same for all selections. In other words, whatever function of the coordinates any onegis for one set of axes, if other axes are chosen, thisgmust still be the same function of the new coordinates. There are teng’s defined by differential equations; so we have ten covariant equations. Einstein showed how theseg’s could be regarded as generalized potentials of the field. Our own experiments and observations upon gravitation have given us a certain knowledge concerning its potential; that is, we know a value for it which must be so near the truth that we can properly call it at least a first approximation. Or, stated differently, if Einstein succeeds in deducing the rigid value for the gravitational potential in any field, it must degenerate to the Newtonian value for the great majority of cases with which we have actual experience. Einstein’s method, then, was to investigate the functions (or equations) which would satisfy the mathematical conditions just described. A transformation from the axes used by the observer in the following box may be made so as to introduce into the equations the gravitational field recognized by an observer on the earth near the box; but this, obviously, would not be the general gravitational field, because the field changes as one moves over the surface of the earth. A solution found, therefore, as just indicated, would not be the one sought for the general field; and anothermust be found which is less stringent than the former but reduces to it as a special case. He found himself at liberty to make a selection from among several possibilities, and for several reasons chose the simplest solution. He then tested this decision by seeing if his formulæ would degenerate to Newton’s law for the limiting case of velocities small when compared with that of light, because this condition is satisfied in those cases to which Newton’s law applies. His formulæ satisfied this test, and he therefore was able to announce a “law of gravitation,” of which Newton’s was a special form for a simple case.

To the ordinary scholar the difficulties surmounted by Einstein in his investigations appear stupendous. It is not improbable that the statement which he is alleged to have made to his editor, that only ten men in the world could understand his treatment of the subject, is true. I am fully prepared to believe it, and wish to add that I certainly am not one of the ten. But I can also say that, after a careful and serious study of his papers, I feel confident that there is nothing in them which I can not understand, given the time to become familiar with the special mathematical processes used. The more I work over Einstein’s papers, the more impressed I am, not simply by his genius in viewing the problem, but also by his great technical skill.

Following the path outlined, Einstein, as just said, arrived at certain mathematical laws for a gravitational field, laws which reduced to Newton’s form in most cases where observations are possible, but which led to different conclusions in a few cases, knowledge concerning which we might obtain by careful observations. I shall mention a few deductions from Einstein’s formulæ.

1. If a heavy particle is put at the center of a circle, and, ifthe length of the circumference and the length of the diameter are measured, it will be found that their ratio is not π (3.14159). In other words the geometrical properties of space in such a gravitational field are not those discussed by Euclid; the space is, then, non-Euclidean. There is no way by which this deduction can be verified, the difference between the predicted ratio and π is too minute for us to hope to make our measurements with sufficient exactness to determine the difference.

2. All the lines in the solar spectrum should with reference to lines obtained by terrestrial sources be displaced slightly towards longer wave-lengths. The amount of displacement predicted for lines in the blue end of the spectrum is about one-hundredth of an Angstrom unit, a quantity well within experimental limits. Unfortunately, as far as the testing of this prediction is concerned, there are several physical causes which are also operating to cause displacement of the spectrum-lines; and so at present a decision can not be rendered as to the verification. St. John and other workers at the Mount Wilson Observatory have the question under investigation.

3. According to Newton’s law an isolated planet in its motion around a central sun would describe, period after period, the same elliptical orbit; whereas Einstein’s laws lead to the prediction that the successive orbits traversed would not be identically the same. Each revolution would start the planet off on an orbit very approximately elliptical, but with the major axis of the ellipse rotated slightly in the plane of the orbit. When calculations were made for the various planets in our solar system, it was found that the only one which was of interest from the standpoint of verification of Einstein’s formulæ was Mercury. It has been known for a long timethat there was actually such a change as just described in the orbit of Mercury, amounting to 574″ of arc per century; and it has been shown that of this a rotation of 532″ was due to the direct action of other planets, thus leaving an unexplained rotation of 42″ per century. Einstein’s formulæ predicted a rotation of 43″, a striking agreement.

4. In accordance with Einstein’s formulæ a ray of light passing close to a heavy piece of matter, the sun, for instance, should experience a sensible deflection in towards the sun. This might be expected from “general” consideration of energy in motion; energy and mass are generally considered to be identical in the sense that an amount of energyEhas the massupper E Baseline 1 c squaredwherecis the velocity of light; and consequently a ray of light might fall within the province of gravitation and the amount of deflection to be expected could be calculated by the ordinary formula for gravitation. Another point of view is to consider again the observer inside the compartment falling with the acceleration of the gravitational field. To him the path of a projectile and a ray of light would both appear straight; so that, if the projectile had a velocity equal to that of light, it and the light wave would travel side by side. To an observer outside the compartment,e.g., to one on the earth, both would then appear to have the same deflection owing to the sun. But how much would the path of the projectile be bent? What would be the shape of its parabola? One might apply Newton’s law; but, according to Einstein’s formulæ, Newton’s law should be used only for small velocities. In the case of a ray passing close to the sun it was decided that according to Einstein’s formula there should be a deflection of 1″.75 whereas Newton’s law of gravitation predicted half this amount. Careful plans were made by various astronomers, to investigate this question atthe solar eclipse last May, and the result announced by Dyson, Eddington and Crommelin, the leaders of astronomy in England, was that there was a deflection of 1″.9. Of course the detection of such a minute deflection was an extraordinarily difficult matter, so many corrections had to be applied to the original observations; but the names of the men who record the conclusions are such as to inspire confidence. Certainly any effect of refraction seems to be excluded.

It is thus seen that the formulæ deduced by Einstein have been confirmed in a variety of ways and in a most brilliant manner. In connection with these formulæ one question must arise in the minds of everyone; by what process, where in the course of the mathematical development, does the idea of mass reveal itself? It was not in the equations at the beginning and yet here it is at the end. How does it appear? As a matter of fact it is first seen as a constant of integration in the discussion of the problem of the gravitational field due to a single particle; and the identity of this constant with mass is proved when one compares Einstein’s formulæ with Newton’s law which is simply its degenerated form. This mass, though, is the mass of which we become aware through our experiences with weight; and Einstein proceeded to prove that this quantity which entered as a constant of integration in his ideally simple problem also obeyed the laws of conservation of mass and conservation of momentum when he investigated the problems of two and more particles. Therefore Einstein deduced from his study of gravitational fields the well-known properties of matter which form the basis of theoretical mechanics. A further logical consequence of Einstein’s development is to show that energy has mass, a concept with which every one nowadays is familiar.

The description of Einstein’s method which I have given sofar is simply the story of one success after another; and it is certainly fair to ask if we have at last reached finality in our investigation of nature, if we have attained to truth. Are there no outstanding difficulties? Is there no possibility of error? Certainly, not until all the predictions made from Einstein’s formulæ have been investigated can much be said; and further, it must be seen whether any other lines of argument will lead to the same conclusions. But without waiting for all this there is at least one difficulty which is apparent at this time. We have discussed the laws of nature as independent in their form of reference axes, a concept which appeals strongly to our philosophy; yet it is not at all clear, at first sight, that we can be justified in our belief. We can not imagine any way by which we can become conscious of the translation of the earth in space; but by means of gyroscopes we can learn a great deal about its rotation on its axis. We could locate the positions of its two poles, and by watching a Foucault pendulum or a gyroscope we can obtain a number which we interpret as the angular velocity of rotation of axes fixed in the earth; angular velocity with reference to what? Where is the fundamental set of axes? This is a real difficulty. It can be surmounted in several ways. Einstein himself has outlined a method which in the end amounts to assuming the existence on the confines of space of vast quantities of matter, a proposition which is not attractive. deSitter has suggested a peculiar quality of the space to which we refer our space-time coordinates. The consequences of this are most interesting, but no decision can as yet be made as to the justification of the hypothesis. In any case we can say that the difficulty raised is not one that destroys the real value of Einstein’s work.

In conclusion I wish to emphasize the fact, which shouldbe obvious, that Einstein has not attempted any explanation of gravitation; he has been occupied with the deduction of its laws. These laws, together with those of electromagnetic phenomena, comprise our store of knowledge. There is not the slightest indication of a mechanism, meaning by that a picture in terms of our senses. In fact what we have learned has been to realize that our desire to use such mechanisms is futile.

1Presidential address delivered at the St. Louis meeting of the Physical Society, December 30, 1919. Republished by permission from “Science.”↑

1Presidential address delivered at the St. Louis meeting of the Physical Society, December 30, 1919. Republished by permission from “Science.”↑

1Presidential address delivered at the St. Louis meeting of the Physical Society, December 30, 1919. Republished by permission from “Science.”↑

THEDEFLECTIONOF LIGHT BY GRAVITATION AND THE EINSTEIN THEORY OF RELATIVITY.1Sir Frank Dysonthe Astronomer RoyalThe purpose of the expedition was to determine whether any displacement is caused to a ray of light by the gravitational field of the sun, and if so, the amount of the displacement. Einstein’s theory predicted a displacement varying inversely as the distance of the ray from the sun’s center, amounting to 1″.75 for a star seen just grazing the sun.…A study of the conditions of the 1919 eclipse showed that the sun would be very favorably placed among a group of bright stars—in fact, it would be in the most favorable possible position. A study of the conditions at various points on the path of the eclipse, in which Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an island off the west coast of Africa, as the most favorable stations.…The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil in ample time to prepare for the eclipse, and the usual preliminary focusing by photographing stellar fields was carried out. The day of the eclipse opened cloudy, but cleared later, and the observations were carried out with almost complete success. With the astrographic telescope Mr. Davidson secured 15 out of 18 photographs showing the required stellar images. Totality lasted 6 minutes, and the average exposure of the plates was 5 to 6 seconds. Dr.Crommelin with the other lens had 7 successful plates out of 8. The unsuccessful plates were spoiled for this purpose by the clouds, but show the remarkable prominence very well.When the plates were developed the astrographic images were found to be out of focus. This is attributed to the effect of the sun’s heat on the coelostat mirror. The images were fuzzy and quite different from those on the check-plates secured at night before and after the eclipse. Fortunately the mirror which fed the 4-inch lens was not affected, and the star images secured with this lens were good and similar to those got by the night-plates. The observers stayed on in Brazil until July to secure the field in the night sky at the altitude of the eclipse epoch and under identical instrumental conditions.The plates were measured at Greenwich immediately after the observers’ return. Each plate was measured twice over by Messrs. Davidson and Furner, and I am satisfied that such faults as lie in the results are in the plates themselves and not in the measures. The figures obtained may be briefly summarized as follows: The astrographic plates gave 0″.97 for the displacement at the limb when the scale-value was determined from the plates themselves, and 1″.40 when the scale-value was assumed from the check plates. But the much better plates gave for the displacement at the limb 1″.98, Einstein’s predicted value being 1″.75. Further, for these plates the agreement was all that could be expected.…After a careful study of the plates I am prepared to say that there can be no doubt that they confirm Einstein’s prediction. A very definite result has been obtained that light is deflected according to Einstein’s law of gravitation.Professor A. S. EddingtonRoyal ObservatoryMr. Cottingham and I left the other observers at Madeira and arrived atPrincipeon April 23.… We soon realized that the prospect of a clear sky at the end of May was not very good. Not even a heavy thunderstorm on the morning of the eclipse, three weeks after the end of the wet season, saved the situation. The sky was completely cloudy at first contact, but about half an hour before totality we began to see glimpses of the sun’s crescent through the clouds. We carried out our program exactly as arranged, and the sky must have been clearer towards the end of totality. Of the 16 plates taken during the five minutes of totality the first ten showed no stars at all; of the later plates two showed five stars each, from which a result could be obtained. Comparing them with the check-plates secured at Oxford before we went out, we obtained as the final result from the two plates for the value of the displacement of the limb 1″.6 ± 0.3.… This result supports the figures obtained at Sobral.…I will pass now to a few words on the meaning of the result. It points to the larger of the two possible values of the deflection. The simplest interpretation of the bending of the ray is to consider it as an effect of the weight of light. We know that momentum is carried along on the path of a beam of light. Gravity in acting creates momentum in a direction different from that of the path of the ray and so causes it to bend. For the half-effect we have to assume that gravity obeys Newton’s law; for the full effect which has been obtained we must assume that gravity obeys the new law proposed by Einstein. This is one of the most crucial tests between Newton’s law and the proposed new law. Einstein’s law had already indicated a perturbation, causing the orbit of Mercury to revolve.That confirms it for relatively small velocities. Going to the limit, where the speed is that of light, the perturbation is increased in such a way as to double the curvature of the path, and this is now confirmed.This effect may be taken as proving Einstein’slawrather than histheory. It is not affected by the failure to detect the displacement of Fraunhofer lines on the sun. If this latter failure is confirmed it will not affect Einstein’s law of gravitation, but it will affect the views on which the law was arrived at. The law is right, though the fundamental ideas underlying it may yet be questioned.…One further point must be touched upon. Are we to attribute the displacement to the gravitational field and not to the refracting matter around the sun? The refractive index required to produce the result at a distance of 15′ from the sun would be that given by gases at a pressure of 1⁄60​ to 1⁄200​ of an atmosphere. This is of too great a density considering the depth through which the light would have to pass.Sir J. J. ThomsonPresident of the Royal Society… If the results obtained had been only that light was affected by gravitation, it would have been of the greatest importance. Newton, did, in fact, suggest this very point in his “Optics,” and his suggestion would presumably have led to the half-value. But this result is not an isolated one; it is part of a whole continent of scientific ideas affecting the most fundamental concepts of physics.… This is the most important result obtained in connection with the theory of gravitation since Newton’s day, and it is fitting that it should beannounced at a meeting of the society so closely connected with him.The difference between the laws of gravitation of Einstein and Newton come only in special cases. The real interest of Einstein’s theory lies not so much in his results as in the method by which he gets them. If his theory is right, it makes us take an entirely new view of gravitation. If it is sustained that Einstein’s reasoning holds good—and it has survived two very severe tests in connection with the perihelion of mercury and the present eclipse—then it is the result of one of the highest achievements of human thought. The weak point in the theory is the great difficulty in expressing it. It would seem that no one can understand the new law of gravitation without a thorough knowledge of the theory of invariants and of the calculus of variations.One other point of physical interest arises from the discussion. Light is deflected in passing near huge bodies of matter. This involves alterations in the electric and magnetic field. This, again, implies the existence of electric and magnetic forces outside matter—forces at present unknown, though some idea of their nature may be got from the results of this expedition.1From a report inThe Observatory, of the Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, November 6, 1919.↑

THEDEFLECTIONOF LIGHT BY GRAVITATION AND THE EINSTEIN THEORY OF RELATIVITY.1

Sir Frank Dysonthe Astronomer RoyalThe purpose of the expedition was to determine whether any displacement is caused to a ray of light by the gravitational field of the sun, and if so, the amount of the displacement. Einstein’s theory predicted a displacement varying inversely as the distance of the ray from the sun’s center, amounting to 1″.75 for a star seen just grazing the sun.…A study of the conditions of the 1919 eclipse showed that the sun would be very favorably placed among a group of bright stars—in fact, it would be in the most favorable possible position. A study of the conditions at various points on the path of the eclipse, in which Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an island off the west coast of Africa, as the most favorable stations.…The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil in ample time to prepare for the eclipse, and the usual preliminary focusing by photographing stellar fields was carried out. The day of the eclipse opened cloudy, but cleared later, and the observations were carried out with almost complete success. With the astrographic telescope Mr. Davidson secured 15 out of 18 photographs showing the required stellar images. Totality lasted 6 minutes, and the average exposure of the plates was 5 to 6 seconds. Dr.Crommelin with the other lens had 7 successful plates out of 8. The unsuccessful plates were spoiled for this purpose by the clouds, but show the remarkable prominence very well.When the plates were developed the astrographic images were found to be out of focus. This is attributed to the effect of the sun’s heat on the coelostat mirror. The images were fuzzy and quite different from those on the check-plates secured at night before and after the eclipse. Fortunately the mirror which fed the 4-inch lens was not affected, and the star images secured with this lens were good and similar to those got by the night-plates. The observers stayed on in Brazil until July to secure the field in the night sky at the altitude of the eclipse epoch and under identical instrumental conditions.The plates were measured at Greenwich immediately after the observers’ return. Each plate was measured twice over by Messrs. Davidson and Furner, and I am satisfied that such faults as lie in the results are in the plates themselves and not in the measures. The figures obtained may be briefly summarized as follows: The astrographic plates gave 0″.97 for the displacement at the limb when the scale-value was determined from the plates themselves, and 1″.40 when the scale-value was assumed from the check plates. But the much better plates gave for the displacement at the limb 1″.98, Einstein’s predicted value being 1″.75. Further, for these plates the agreement was all that could be expected.…After a careful study of the plates I am prepared to say that there can be no doubt that they confirm Einstein’s prediction. A very definite result has been obtained that light is deflected according to Einstein’s law of gravitation.Professor A. S. EddingtonRoyal ObservatoryMr. Cottingham and I left the other observers at Madeira and arrived atPrincipeon April 23.… We soon realized that the prospect of a clear sky at the end of May was not very good. Not even a heavy thunderstorm on the morning of the eclipse, three weeks after the end of the wet season, saved the situation. The sky was completely cloudy at first contact, but about half an hour before totality we began to see glimpses of the sun’s crescent through the clouds. We carried out our program exactly as arranged, and the sky must have been clearer towards the end of totality. Of the 16 plates taken during the five minutes of totality the first ten showed no stars at all; of the later plates two showed five stars each, from which a result could be obtained. Comparing them with the check-plates secured at Oxford before we went out, we obtained as the final result from the two plates for the value of the displacement of the limb 1″.6 ± 0.3.… This result supports the figures obtained at Sobral.…I will pass now to a few words on the meaning of the result. It points to the larger of the two possible values of the deflection. The simplest interpretation of the bending of the ray is to consider it as an effect of the weight of light. We know that momentum is carried along on the path of a beam of light. Gravity in acting creates momentum in a direction different from that of the path of the ray and so causes it to bend. For the half-effect we have to assume that gravity obeys Newton’s law; for the full effect which has been obtained we must assume that gravity obeys the new law proposed by Einstein. This is one of the most crucial tests between Newton’s law and the proposed new law. Einstein’s law had already indicated a perturbation, causing the orbit of Mercury to revolve.That confirms it for relatively small velocities. Going to the limit, where the speed is that of light, the perturbation is increased in such a way as to double the curvature of the path, and this is now confirmed.This effect may be taken as proving Einstein’slawrather than histheory. It is not affected by the failure to detect the displacement of Fraunhofer lines on the sun. If this latter failure is confirmed it will not affect Einstein’s law of gravitation, but it will affect the views on which the law was arrived at. The law is right, though the fundamental ideas underlying it may yet be questioned.…One further point must be touched upon. Are we to attribute the displacement to the gravitational field and not to the refracting matter around the sun? The refractive index required to produce the result at a distance of 15′ from the sun would be that given by gases at a pressure of 1⁄60​ to 1⁄200​ of an atmosphere. This is of too great a density considering the depth through which the light would have to pass.Sir J. J. ThomsonPresident of the Royal Society… If the results obtained had been only that light was affected by gravitation, it would have been of the greatest importance. Newton, did, in fact, suggest this very point in his “Optics,” and his suggestion would presumably have led to the half-value. But this result is not an isolated one; it is part of a whole continent of scientific ideas affecting the most fundamental concepts of physics.… This is the most important result obtained in connection with the theory of gravitation since Newton’s day, and it is fitting that it should beannounced at a meeting of the society so closely connected with him.The difference between the laws of gravitation of Einstein and Newton come only in special cases. The real interest of Einstein’s theory lies not so much in his results as in the method by which he gets them. If his theory is right, it makes us take an entirely new view of gravitation. If it is sustained that Einstein’s reasoning holds good—and it has survived two very severe tests in connection with the perihelion of mercury and the present eclipse—then it is the result of one of the highest achievements of human thought. The weak point in the theory is the great difficulty in expressing it. It would seem that no one can understand the new law of gravitation without a thorough knowledge of the theory of invariants and of the calculus of variations.One other point of physical interest arises from the discussion. Light is deflected in passing near huge bodies of matter. This involves alterations in the electric and magnetic field. This, again, implies the existence of electric and magnetic forces outside matter—forces at present unknown, though some idea of their nature may be got from the results of this expedition.

Sir Frank Dysonthe Astronomer Royal

The purpose of the expedition was to determine whether any displacement is caused to a ray of light by the gravitational field of the sun, and if so, the amount of the displacement. Einstein’s theory predicted a displacement varying inversely as the distance of the ray from the sun’s center, amounting to 1″.75 for a star seen just grazing the sun.…

A study of the conditions of the 1919 eclipse showed that the sun would be very favorably placed among a group of bright stars—in fact, it would be in the most favorable possible position. A study of the conditions at various points on the path of the eclipse, in which Mr. Hinks helped us, pointed to Sobral, in Brazil, and Principe, an island off the west coast of Africa, as the most favorable stations.…

The Greenwich party, Dr. Crommelin and Mr. Davidson, reached Brazil in ample time to prepare for the eclipse, and the usual preliminary focusing by photographing stellar fields was carried out. The day of the eclipse opened cloudy, but cleared later, and the observations were carried out with almost complete success. With the astrographic telescope Mr. Davidson secured 15 out of 18 photographs showing the required stellar images. Totality lasted 6 minutes, and the average exposure of the plates was 5 to 6 seconds. Dr.Crommelin with the other lens had 7 successful plates out of 8. The unsuccessful plates were spoiled for this purpose by the clouds, but show the remarkable prominence very well.

When the plates were developed the astrographic images were found to be out of focus. This is attributed to the effect of the sun’s heat on the coelostat mirror. The images were fuzzy and quite different from those on the check-plates secured at night before and after the eclipse. Fortunately the mirror which fed the 4-inch lens was not affected, and the star images secured with this lens were good and similar to those got by the night-plates. The observers stayed on in Brazil until July to secure the field in the night sky at the altitude of the eclipse epoch and under identical instrumental conditions.

The plates were measured at Greenwich immediately after the observers’ return. Each plate was measured twice over by Messrs. Davidson and Furner, and I am satisfied that such faults as lie in the results are in the plates themselves and not in the measures. The figures obtained may be briefly summarized as follows: The astrographic plates gave 0″.97 for the displacement at the limb when the scale-value was determined from the plates themselves, and 1″.40 when the scale-value was assumed from the check plates. But the much better plates gave for the displacement at the limb 1″.98, Einstein’s predicted value being 1″.75. Further, for these plates the agreement was all that could be expected.…

After a careful study of the plates I am prepared to say that there can be no doubt that they confirm Einstein’s prediction. A very definite result has been obtained that light is deflected according to Einstein’s law of gravitation.

Professor A. S. EddingtonRoyal Observatory

Mr. Cottingham and I left the other observers at Madeira and arrived atPrincipeon April 23.… We soon realized that the prospect of a clear sky at the end of May was not very good. Not even a heavy thunderstorm on the morning of the eclipse, three weeks after the end of the wet season, saved the situation. The sky was completely cloudy at first contact, but about half an hour before totality we began to see glimpses of the sun’s crescent through the clouds. We carried out our program exactly as arranged, and the sky must have been clearer towards the end of totality. Of the 16 plates taken during the five minutes of totality the first ten showed no stars at all; of the later plates two showed five stars each, from which a result could be obtained. Comparing them with the check-plates secured at Oxford before we went out, we obtained as the final result from the two plates for the value of the displacement of the limb 1″.6 ± 0.3.… This result supports the figures obtained at Sobral.…

I will pass now to a few words on the meaning of the result. It points to the larger of the two possible values of the deflection. The simplest interpretation of the bending of the ray is to consider it as an effect of the weight of light. We know that momentum is carried along on the path of a beam of light. Gravity in acting creates momentum in a direction different from that of the path of the ray and so causes it to bend. For the half-effect we have to assume that gravity obeys Newton’s law; for the full effect which has been obtained we must assume that gravity obeys the new law proposed by Einstein. This is one of the most crucial tests between Newton’s law and the proposed new law. Einstein’s law had already indicated a perturbation, causing the orbit of Mercury to revolve.That confirms it for relatively small velocities. Going to the limit, where the speed is that of light, the perturbation is increased in such a way as to double the curvature of the path, and this is now confirmed.

This effect may be taken as proving Einstein’slawrather than histheory. It is not affected by the failure to detect the displacement of Fraunhofer lines on the sun. If this latter failure is confirmed it will not affect Einstein’s law of gravitation, but it will affect the views on which the law was arrived at. The law is right, though the fundamental ideas underlying it may yet be questioned.…

One further point must be touched upon. Are we to attribute the displacement to the gravitational field and not to the refracting matter around the sun? The refractive index required to produce the result at a distance of 15′ from the sun would be that given by gases at a pressure of 1⁄60​ to 1⁄200​ of an atmosphere. This is of too great a density considering the depth through which the light would have to pass.

Sir J. J. ThomsonPresident of the Royal Society

… If the results obtained had been only that light was affected by gravitation, it would have been of the greatest importance. Newton, did, in fact, suggest this very point in his “Optics,” and his suggestion would presumably have led to the half-value. But this result is not an isolated one; it is part of a whole continent of scientific ideas affecting the most fundamental concepts of physics.… This is the most important result obtained in connection with the theory of gravitation since Newton’s day, and it is fitting that it should beannounced at a meeting of the society so closely connected with him.

The difference between the laws of gravitation of Einstein and Newton come only in special cases. The real interest of Einstein’s theory lies not so much in his results as in the method by which he gets them. If his theory is right, it makes us take an entirely new view of gravitation. If it is sustained that Einstein’s reasoning holds good—and it has survived two very severe tests in connection with the perihelion of mercury and the present eclipse—then it is the result of one of the highest achievements of human thought. The weak point in the theory is the great difficulty in expressing it. It would seem that no one can understand the new law of gravitation without a thorough knowledge of the theory of invariants and of the calculus of variations.

One other point of physical interest arises from the discussion. Light is deflected in passing near huge bodies of matter. This involves alterations in the electric and magnetic field. This, again, implies the existence of electric and magnetic forces outside matter—forces at present unknown, though some idea of their nature may be got from the results of this expedition.

1From a report inThe Observatory, of the Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, November 6, 1919.↑

1From a report inThe Observatory, of the Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, November 6, 1919.↑

1From a report inThe Observatory, of the Joint Eclipse Meeting of the Royal Society and the Royal Astronomical Society, November 6, 1919.↑

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ColophonAvailabilityThis eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online atwww.gutenberg.org.This eBook is produced by the Online Distributed Proofreading Team atwww.pgdp.net.MetadataTitle:From Newton to EinsteinAuthor:Benjamin Harrow (1888–1970)InfoContributor:Joseph Sweetman Ames (1864–1943)InfoContributor:Frank Watson Dyson (1868–1939)InfoContributor:Arthur Stanley Eddington (1882–1944)InfoContributor:Albert Einstein (1879–1955)InfoContributor:Joseph John Thomson (1856–1940)InfoLanguage:EnglishOriginal publication date:1920EncodingRevision History2019-08-19 Started.External ReferencesThis Project Gutenberg eBook contains external references. These links may not work for you.CorrectionsThe following corrections have been applied to the text:PageSourceCorrectionEdit distanceN.A..,1viReflectionDeflection131electro-magneticelectromagnetic173,117vanVan176[Not in source])177-177UeberÜber2 / 177Relativitats-theorieRelativitätstheorie2 / 178have[Deleted]581LeibnitzLeibniz187sytemsystem1102[Not in source].1112REFLECTIONDEFLECTION1112“[Deleted]1AbbreviationsOverview of abbreviations used.AbbreviationExpansionPh.D.Philosophiae Doctor

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