APPENDIXNote 1(p. 4). So long as the universal significance of the velocity of light remained unknown, two conjectures were possible in the question as to whether, under certain circumstances, the motion of the source of light would make itself observable in the velocity of propagation of light. It might be surmised that the velocity of the source simply added itself to that velocity of light which ischaracteristic for the propagation of the lightfrom a source at rest. Or, it might be conceived that the motion of the source has no influence at all on the velocity of the light emitted by it. In the second case it was imagined that the source of light only excites the periodically changing states of the luminiferous ether, which is at rest, that is, which does not share in the motion of the matter (source of light), and that these states then propagate themselves with avelocity that is characteristic of the ether, and with a velocity that makes these states perceptible to us as light waves. This view had finally apparently won the day. It was the advent of the special theory of relativity and the quantum hypothesis that made this view impossible. For the special theory of relativity, in robbing the assertion: "the ether is at rest" of its significance, since we may arbitrarily define any system as being at rest in the ether, as far as uniform translations are concerned, and in depriving the luminiferousether of its existence, deprived light-waves of their carrying or transmitting medium. The quantum hypothesis, in raising light-quanta to the rank of self-supporting individuals, deprived thevelocity of lightof its character as a constant that is characteristic of theether. Thus, our view of light-quanta again leads to a kind of emission theory of light. According to classical mechanics it would have been typical of a theory of emission for the velocity of the source in motion to have added itself to the velocity of the light from the source at rest. We thus revert to the conjecture which we quoted first above. Now, such a superposition of velocities would necessarily cause quite remarkable phenomena in the case of spectroscopic binary stars (de Sitter, "Phys. Zeitschrift,"14, 429). For if two stars move in circular Kepler orbits around each other, and if our line of sight lies in the common plane of the orbits, then we should necessarily perceive the following: ifis the time of revolution of the system,the orbital velocity of the one (bright) component,the distance of the whole system from the earth, and, finally,, the velocity in vacuo of the light from the source which is at rest, then the velocity of light at the epoch of greatest positive velocity in the direction of vision is, andin the other direction, respectively. Consequently the time-interval between two such successive positions would have the valuesandalternately, for the observer on the earth, as a simple calculation shows. Since, on account of the gigantic distances between the fixed stars, the membermay become very great, indeed, greater than, we should be able to observe definiteanomalies in the case of the spectroscopic binaries. For the time intervals between two such successive epochs in the orbit should be able to contract to nil, indeed, even become negative, and we should not be able to interpret the measured Doppler effects by means of motions in the Kepler ellipses. In reality, however, these anomalies have never manifested themselves. Observation of these very sensitive subjects of test (spectroscopic binaries) teaches us that the motion of the source of light does not make itself remarked in the propagation of the light. This renders our first view likewise untenable. The special principle of relativity, alone in postulating the constancy of the velocity of light, and in putting forward a new addition theorem of velocities, has led us to an attitude in this question that is free from inner contradictions and compatible with experience. (Cf.Note 2.)Note 2(p. 5). There are essentially two fundamental optical experiments on which our view of the distinctive significance of the velocity of light in physical nature is founded: Fizeau's experiment concerning the velocity of light in flowing water, and the Michelson-Morley experiment. Aberration, on the other hand, has nothing to do directly with the question whether it is possible to prove by means of optical experiments in the laboratory a motion of the earth relative to the ether. The aberration in the case of stars states merely that the motion of the earthrelativelyto the star under considerationchangesperiodically in the course of a year. If, however, we hold the view that an all-pervading ether is the carrier for the propagation of the light, the phenomenon of aberration may be satisfactorily explained only if we assume that this ether does not participate in the motion of the earth.Fizeau's experiment was designed to decide finally whether movingmatter influences the ether and to determine the value of the velocity of light in moving matter with respect to the observer. Michelson and Morley repeated the experiment in the following improved form. A beam of light from a source on the earth is sent through a-shaped tube, through which water flows, in the direction of both limbs. After each part of the beam has traversed the flowing water, the one in the direction of the current, the other contrary to it, the two beams, are made to interfere. The light and the water move in the same direction in the one limb, and oppositely in the other.fig01Fig. 1.Now, there immediately appear to be two possibilities. Either the water that flows with a velocitywith respect to the walls of the tube drags along the carrier that effects the transmission of the light, namely the ether; in this case, the velocity of the light isin the one limb, andin the other, for, on account of the coefficient of refractionof the water,is the velocity of light in resting water. Or, the motion of the water has no influence at all on theether which transmits light and which permeates the water. In this case the velocity of light isin both limbs. According, as the one or the other of these two assumptions is valid, the interference fringes would have to become displaced or remain at rest when the direction of the current is reversed. The experiment decided in favour ofneitherof these possibilities. The interference fringes did, indeed, become shifted, not to the expected amount, however, but only to an amount that would result if the ether assumes the velocityin water, and not the full value. This value of the convection of the ether is calledFresnel's convection coefficient. Yet this term is capable of being misunderstood inasmuch as in the electrodynamics developed by Lorentz, the result of Fizeau's experiment speaks in favour of an ether that isabsolutely at rest, and the so-called convection coefficient is only a consequence of the structure of matter, in particular of the interaction between electrons and matter, a question into which we cannot enter here. At any rate, at the time preceding the Michelson-Morley experiment aberration, as well as Fizeau's experiment, appeared to speak in favour of an ether that was absolutely at rest.Now, the Michelson-Morley experiment was to establish the existence of the current of ether (ether "wind") through which the earth continually moves, since the ether is supposed not to participate in the motion of the earth. The scheme of the experiment is as onp. 74.A ray of light, starting out from, traverses the course: hereand, are two mirrors, on to which the ray falls perpendicularly;is a glassplate that reflects one half of the light and allows the remainder to pass through;is the telescope of the observer. Another ray of light traverses the course. Let. Further, let, be in the direction of the earth's motion. Our assumption is that the ether does not share in the earth's motion. Let the velocity of the earth be.fig02Fig. 2.Then the velocity of the light relative to the instrument (earth) is as follows in the directions specified:Consequently, the courseis traversed in the timeand the coursein the timeThe difference of these two times isIf we exchange the positions of, and, by turning the whole apparatus through 90°, thenIf we make these tyro rays of light interfere atthen, when the apparatus is turned through 90°, the interference fringes should become shifted. The amount of this displacement may easily be calculated. If we denote bythe vibration frequency of the light-ray used in the experiment, thenis the corresponding wave-length. Thus, expressed in fractions of the interval between the fringes, the expected displacement becomes equal toBy causing the light to be reflected many timeswas magnified to such an extent thatbecame of the order. If, for example,,= the wavelength of sodium light, then. On the other handis of the orderthat is,. The expected displacement of the fringes would thus have to be about 0·56 of the breadth of a fringe. Actually, an amount of the order 0·02 of the breadth of a fringe was observed. Thus, the ether wind did not make itself remarked optically in the motion of the earth. By carrying out the experiment at different times of the year the possible objection that the motion of translation of the entire solar system might have counterbalanced the motion of the earth in her orbit was removed.The Michelson-Morley experiment has shown conclusively that there is no physical sense in talking of absolute rest or of a translation relative to absolute space, since all systems that move rectilinearly and uniformly with respect to one another are of equal value for describing natural phenomena. It is thus a matter of convention which system we are to regard as at rest and which as being in motion. We may assign the same value to the velocity of light in all systems. A detailed theory of these fundamental experiments may be found in all comprehensive accounts of the special theory of relativity. We here merely mention the original paper by A. Einstein (Annalen der Physik, Bd. 17, 1905, p. 891), and the booklet, "Einführung in die Relativitätstheorie," by Dr. W. Block, out of the series "Aus Natur und Geisteswelt," Teubner, 1918.Note 3(p. 9). Abolishing the transformations of Newton's principle of relativity and replacing them by the so-called Lorentz-Einstein transformations signified a step of extraordinarily far-reaching consequence. It was justified in that the new theory of relativity which followed as a result of it, confirmed, without difficulty, the results of all the fundamental experiments of optics and electrodynamics. Concerning the Michelson-Morley experiment,Lorentz, to account for its negative result within the realm of electrodynamics, had been compelled to set up the hypothesis that the dimensions of all bodies contract in the direction of their motion. But Einstein now showed that if we define the conception of simultaneity rigorously, taking into account the postulate of the constancy of the velocity of light, the Lorentz-transformations, which had been found empirically, followed necessarily as those equations of transformation that must hold between the co-ordinates of two systems moving uniformly and rectilinearly with respect to each other. And without the help of any further hypothesis there appears as a direct consequence of this transformation just that contraction of lengths which Lorentz had adduced to explain the result of the Michelson-Morley experiment. This contraction of a lengthin the direction of motion of an object to the valueis, however, in the new theory the expression of the general fact that the dimensions of a body have only a relative meaning, that is, that their values depend on the state of motion of the observer, which determines the dimensions of the body in question. This holds for the extension of bodies in time as well as in space. From the point of view of the new principle of relativity the negative result of the Michelson-Morley experiment was self-explained. But what was the position with regard to the other fundamental facts of optics and electrodynamics? The result of Fizeau's experiment concerning the velocity of light in flowing water became a direct test of the kinematics arising out of the new formulæ. According to the Lorentz transformation the two velocities,andwith which, for example, two locomotives approach each other, do not merelybecome added, so thatwould be the relative velocity of each with respect to the other, but rather each engine-driver will find as the velocity with which he passes the other driver, the valueaccording to the new formulæ. This is the addition theorem of velocities according to the new theory. It gives us immediately the amount observed in Fizeau's experiment for the velocity of light in flowing water. Aberration and the Doppler effect follow just as readily to the correct amount. A detailed discussion of these questions is to be found in every account of the "special" theory of relativity (cf. the references given inNote 2).Note 4(p. 12). Ph. Frank and H. Rothe,Ann. d. Phys., 4 Folge, Bd. 34, p. 825.The assumptions for the general equations of transformation by which two systemsand' that moveuniformlyandrectilinearlywith the velocity q with respect to each other sure connected are as follows:—1. The equations of transformation form a linear homogeneous group in the variable parameter. This means that the successive application of two equations of transformation, of which the one refers the systemto the system', and the second' to''(is to have the constant velocitywith respect to', and' the constant velocity' with respect to'') again leads to an equation of transformation of the same form as that of the initial equations. The parameter''that occurs in the new equation depends in a definite way on' and.2. The contractions of the lengths depend only on the value of theparameter. We must, of course, from the very outset reckon with the possibility that the length of a rod measured in the system that is at rest comes out differently when measured in the moving system. Now, condition 2 requires that if contractions occur (that is, changes of length in these various methods of determination) values are to depend only on themagnitudeof the velocity of both systems and not on the direction of their motion in space. Thus this postulate endows space with the property of isotropy, and is in fair correspondence with the postulate ofsection 3a, which states that it must be possible to compare each line-element with every other in length independently of its position in space, and its direction.An essential feature is that the constancy of the velocity of light isnotdemanded in either of the postulates 1 and 2. Rather, the distinguishing property of adefinitevelocity in virtue of which it preserves its value inallsystems that emerge out of one another through such transformations is a direct corollary to these two general postulates, and the result of the Michelson-Morley experiment merely determines the value of this special velocity which could, of course, be found only from observation.Note 5(p. 15). Einstein has shown in a simple example how, on the basis of the formulæ of the special theory of relativity, a point-mass loses inertial mass when it radiates out energy.We assume that a point-mass emits a light-wave of energyin a certain direction, and a light-wave of the same energyin the opposite direction. Then, in view of the symmetry of the process of emission with respect to the system of reference of the co-ordinates,,,originally chosen, the point-mass remains at rest. Let the total energyof the point-mass bereferred to this system, butreferred to a second system which we suppose moving with the uniform velocitywith respect to the first. We shall apply the principle of energy to this process. Ifandare the frequency and amplitude of the light-wave in the initial system,',',',',',' the frequency, amplitude, and co-ordinates in the second (the moving) system, further,the angle between the wave-normals and the line connecting the point-mass with the observer, then Doppler's principle gives for the frequency of the light-wave in the moving system:The formulæ of the special theory of relativity give us, correspondingly, for the amplitude in the moving systemAccording to Maxwell's theory the energy of the light-wave per unit volume is. We now wish to calculate the corresponding energy-density also with respect to the moving system. We must here take into account that, in consequence of the contraction of the lengths according to the Lorentz-Einstein transformation formulæ, the volumeof a sphere in the resting system becomes transformed into that of am ellipsoid as measured from the moving system; indeed, this volume of the ellipsoid isHence the energy-densities in the accented and unaccented system are in the ratio:If we now designate the energy-content of the point-massafterthe emission by, and the corresponding quantity referred to the moving system by, then we have:whereasFrom this we get directly thatWhat does this equation assert?andare the energy-values of the same point-mass, in the first place referred to a system with respect to which the point-mass moves, and in the second related to a system in which the point-massis at rest. Hence the difference, except for an additive constant, must be equal to thekineticenergy of the point-mass referred to the moving system. Thus, we may writewhereindenotes a constant which does not alter during the light-emission of the point-mass, since, owing to the symmetry of the process, the point-mass remains at rest with respect to the initial system. So we arrive at the relation:In words this equation states that owing to the point-mass emitting the energyas light, its kinetic energy referred to a moving system sinks from the valueto the value, corresponding to a loss in inertial mass of the amount. For, according to classical mechanics, the expressionin whichis the inertial mass of the observed body, is a measure of the kinetic energy of this body referred to a system with respect to which it moves with the velocity. Thusmust be taken as standing for the inertial mass of an amount of energy.Note 6(p. 29). The facts thateverypair of points (point-pair) in space have the same magnitude-relation (viz. the same expression for the mutualdistancebetween them) and that with the aid of this relation, every point-pair can be compared with every other, constitute the characteristic feature which distinguishes spacefrom the remaining continuous manifolds which are known to us. We measure the mutual distance between two points on the floor of a room, and the mutual distance between two points which he vertically above one another on the wall, with the same measuring-scale, which we thus apply in any direction at pleasure. This enables us to "compare" the mutual distance of a point-pair on the floor with the mutual distance of any other pair of points on the wall.In the system of tones, on the contrary, quite different conditions prevail. The system of tones represents a manifold of two dimensions, if one distinguishes every tone from the remaining tones by its pitch and its intensity. It is, however, not possible to compare the "distance" between two tones of thesame pitchbutdifferent intensity(analogous to the two points on the floor) with the "distance" between two tones ofdifferent pitchbutequal intensity(analogous to the two points on the wall). The measure-conditions are thus quite different in this manifold.In the system of colours, too, the measure-relations have their own peculiarity. The dimensions of the manifold of colours are the same as those of space, as each colour can be produced by mixing thethree"primary" colours. But there is no relation between two arbitrary colours, which would correspond to the distance between two points in space. Only when a third colour is derived by mixing these two, does one obtain an equation between these three colours similar to that which connects three points in space lying in one straight line.These examples, which are borrowed from Helmholtz's essays, serve to show that the measure-relations of a continuous manifold are not already given in its definition as acontinuousmanifold.nor by fixing its dimensions. A continuous manifold generally allows of various measure-relations. It is only experience which enables us to derive the measure-laws which are valid for each particular manifold. The fact, discovered by experience, that the dimensions of bodies are independent of their particular position and motion, led to the laws of Euclidean geometry wherecongruenceis the deciding factor in comparing various portions of space. These questions have been exhaustively treated by Helmholtz in various essays. References:—Riemann, "Über die Hypothesen, welche der Geometrie zugrunde liegen" (1854). Newly published and annotated by H. Weyl, Berlin, 1919.Helmholtz. "Ueber die tatsächlichen Grundlagen der Geometrie,"Wiss. Abh.2, S. 10.Helmholtz. "Ueber die Tatsachen, welche der Geometrie zugrunde liegen,"Wiss. Abh.2, S. 618.Helmholtz. "Ueber den Ursprung und die Bedeutung der geometrischen Axiome,"Vorträge und Reden, Bd. 2, S. 1.Note 7(p. 26). The postulate that finite rigid bodies are to be capable of free motions, can be most strikingly illustrated in the realm of two-dimensions. Let us imagine a triangle to be drawn upon asphere, and also upon aplane: the former being bounded by arcs of great circles and the latter by straight lines; one can then slide these triangles over their respective surfaces at will, and can make them coincide with other triangles, without thereby altering the lengths of the sides or the angles. Gauss has shown that this is possible because thecurvatureat every point of the sphere (or the plane, respectively) has exactly the same value. And yet the geometry of curves traced upon a sphere is different from that of curvestraced upon aplane, for the reason that these two configurations cannot be deformed into one another without tearing (videNote 27). But uponbothof them planimetrical figures can be freely shifted about, and, therefore, theorems of congruence hold upon them. If, however, we were to define a curvilinear triangle upon an egg-shaped surface by the three shortest lines connecting three given points upon it, we should find that triangles could be constructed at different places on this surface, having the same lengths for the sides; but these sides would enclose angles different from those included by the corresponding sides of the initial triangle, and, consequently, such triangles would not be congruent, in spite of the fact that corresponding sides are equal. Figures upon an egg-shaped surface cannot, therefore, be made to slide over the surface without altering their dimensions: and in studying the geometrical conditions upon such a surface, we do not arrive at the usual theorems of congruence. Quite analogous arguments can be applied to three- and four-dimensional realms: but the latter casesoffer no corresponding pictures to the mind. If we demand that bodies are to be freely movable in space without suffering a change of dimensions, the "curvature" of the space must be the same at every point. The conception of curvature, as applied to any manifold of more than two dimensions, allows of strict mathematical formulation; the term itself only hints at its analogous meaning, as compared with the conception of curvature of a surface. In three-dimensional space, too, various cases can be distinguished, similarly to plane- and spherical-geometry in two-dimensional space. Corresponding to the sphere, we have a non-Euclidean space with constant positive curvature; corresponding to the plane wehave Euclidean space with curvature zero. In both these spaces bodies can be moved about without their dimensions altering; but Euclidean space is furthermore infinitely extended: whereas "spherical" space, though unbounded, like the surface of a sphere, is not infinitely extended. These questions are to be found extensively treated in a very attractive fashion in Helmholtz's familiar essay: "Ueber den Ursprung und die Bedeutung der geometrischen Axiome" (Vorträge und Reden, Bd. 2, S. 1).Note 8(p. 26). The properties, which the analytical expression for the length of the line-element must have, may be understood from the following:Let the numbers,denote any point of any continuous two-dimensional manifold, e.g. a surface. Then, together with this point, a certain "domain" around the point is given, which includes points all of which lie in the plane.—D. Hilbert has strictly defined the conception of a multiply-extended magnitude (i.e. a manifold) upon the basis of the theory of aggregates in his "Grundlagen der Geometrie" (p. 177). In this definition the conception of the "domain" encircling a point is made to give Riemann's postulate of thecontinuousconnection existing between the elements of a manifold and a strict form.Setting out from the point,we can continuously pass into its domain, and at any point, e.g.,, inquire as to the "distance" of this point from the starting-point. The function whichmeasuresthis distance will depend upon the values of,,,, and for everyintermediatepoint of the path which has conducted us from,to the point,will successively assume certain continually changing, and, as we may suppose, continuallyincreasing, values. At the point,itself it will assume the value zero, and for every other point of the domain its value must be positive. Moreover, we shall expect to find that, for any intermediate point, denoted by,,andthe required function which measures the distance of this point from,, will, at this point, have a value half that of its value for the point,. Under these assumptions, the function will be homogeneous and of the first degree in the's; its value will then appear multiplied by that factor in proportion to which the's were increased. In addition, it must itself vanish if all the's are zero; and if they all change their sign it must not alter its value, which always remains positive. It will immediately be evident that the functionfulfils all these requirements; but it is by no means the only function of this kind.Note 9(p. 29). But the expression of the fourth degree for the fine element would not permit of any geometrical interpretation of the formula, such as is possible with the expressionwhich latter may be regarded as a general case of Pythagoras' theorem.Note 10(p. 30). By a "discrete" manifold we mean one in which no continuous transition of the single elements from one to another is possible, but each element to a certain extent represents an independent entity. The aggregate of all whole numbers, for instance, is a manifold of this type, or the aggregate of allplanetsin our solar system, etc., and many other examples may be found; andindeed all finite aggregates in the theory of aggregates are such discrete manifolds. "Measuring," in the case of discrete manifolds, is performed merely by "counting," and does not present any special difficulties, as all manifolds of this type are subject to the same principle of measurement. When Riemann then proceeds to say: "Either, therefore, the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it," he only wishes to hint at a possibility, which is at present still remote, but which must, in principle, always be left open. In just the last few years a similar change of view has actually occurred in the case of another manifold which plays a very important part in physics, viz. "energy"; the meaning of the hint Riemann gives will become clearer if we consider this example.Up till a few years ago, the energy which a body emanates by radiation was regarded as a continuously variable quantity: and attempts were therefore made to measure its amount at any particular moment by means of a continuously varying sequence of numbers. The researches of Max Planck have, however, led to the view that this energy is emitted in "quanta," and that therefore the "measuring" of its amount is performed by counting the number of "quanta." The reality underlying radiant energy, according to this, is a discrete andnota continuous manifold. If we now suppose that the view were gradually to take root that, on the one hand, all measurements in space only have to do with distances between ether-atoms; and that, on the other hand, the distances of single ether-atoms from one another can only assume certaindefinite values, all distances in space would be obtained by "counting" these values, and we should have to regard space as a discrete manifold.Note 11(p. 32). C. Neumann. "Ueber die Prinzipien der Galilei-Newtonschen Theorie," Leipzig 1870, S. 18.Note 12(p. 32). H. Streintz. "Die physikalischen Grundlagen der Mechanik," Leipzig, 1883.Note 13(p. 33). A. Einstein. "Annalen der Physik," 4 Folge, Bd. 17, S. 891.Note 14(p. 35). Minkowski was the first to call particular attention to this deduction of the special principle of relativity.Note 15(p, 38). The term "inertial system" was originally not associated with the system, which Neumann attached to the hypothetical body. Nowadays it is generally understood to signify a rectilinear system of co-ordinates, relatively to which a point-mass, which is only subject to its own inertia, moves uniformly in a straight line. Whereas C. Neumann only invented the body, as an absolutely hypothetical configuration, in order to be able to formulate the law of inertia, later researches, especially those of Lange, tended to show that, on the basis of rigorous kinematical considerations, a co-ordinate system could be derived, which would possess the properties of such an inertial system. However, as C. Neumann and J. Petzoldt have demonstrated, these developments contain faulty assumptions, and give the law of inertia no firmer basis than the bodyintroduced by Neumann.Such an inertial system is determined by the straight lines which connect three point-masses infinitely distant from one another (and thus unable to exert a mutual influence upon one another) and which are not subject to any other forces. This definition makes it evident why noinertial system will be discoverable in nature, and why, consequently, the law of inertia will never be able to be formulated so as to satisfy the physicist. References:—C. Neumann. "Ueber die Prinzipien der Galilei-Newtonschen Theorie," Leipzig, 1870.L. Lange. "Berichte der Kgl. Sächs. Ges. d. Wissenschaften. Math.-phil. Klasse," 1885.L. Lange. "Die Geschichte der Entwickelung des Bewegungsbegriffes," Leipzig, 1886.H. Seeliger. "Ber. der Bayr. Akademie," 1906, Heft 1.C. Neumann. "Ber der Kgl. Sächs. Ges. d. Wiss. Math.-phys. Klasse," 1910, Bd. 62, S. 69 and 383.J. Petzoldt. "Ann. der Naturphilosophie," Bd. 7.Note 16(p. 38). E. Mach. "Die Mechanik in in ihrer Entwickelung," 4 Aufl. S. 244.Note 17(p. 40). The new points of view as to the nature of inertia are based upon the study of the electromagnetic phenomena of radiation. The special theory of relativity, by stating the theorem of the inertia of energy, organically grafted these views on to the existing structure of theoretical physics. The dynamics of cavity-radiation, i.e. the dynamics of a space enclosed by walls without mass, and filled with electromagnetic radiation, taught us that a system of this kind opposes a resistance to every change of its motion, just like a heavy body in motion. The study of electrons (free electric charges) in a state of free motion, e.g. in a cathode-tube, taught us likewise that these exceedingly small particles behave like inert bodies; that their inertia is not, however, conditioned by the matter to which they might happen to be attached, but rather by the electromagnetic effects of the field to which the moving electron is subject. This gave rise to the conception of the apparent(electromagnetic) mass of an electron. The special theory of relativity finally led to the conclusion that toallenergy must be accorded the property of inertia.Every body contains energy (e.g. a certain definite amount in the form of heat-radiation internally). The inertia, which the body reveals, is thus partly to be debited to the account of this contained energy. As this share of inertia is, according to the special theory of relativity, relative (i.e. represents a quantity which depends upon the choice of the system of reference), the whole amount of the inertial mass of the body has no absolute value, but only a relative one. This energy-content of radiant heat is distributed throughout the whole volume of each particular body; one can thus speak of the energy-content of unit volume. This enables us to derive the notion of density of energy. The density of the energy (i.e. amount per unit volume) is thus a quantity, the value of which is also dependent upon the system of reference. References:—M. Planck. "Ann. der Phys.," 4 Folge, Bd. 26.M. Abraham. "Electromagnetische Energie der Strahlung," 4 Aufl., 1908.Note 18(p. 40). The determination of the inertial mass of a body by measuring its weight is rendered possible only by the experimental fact that all bodies fall with equal acceleration in the gravitational field at the earth's surface. Ifand' denote the pressures of two bodies upon the same support (i.e. their respective weights), anddenote the acceleration due to the earth's gravitational field at the point in question, thendynes anddynes, respectively, whereand' are the factors of proportionality, and are called themassesof the two bodies, respectively. Ashas the same value in both equations, we haveand we can accordingly measure themassesof two bodies at the same place, by determining theirweights.Although Galilei and Newton had already known that all bodies at the same place fall with the same velocity (if the resistance of the air be eliminated), this very remarkable fact has not received any recognition in the foundations of mechanics. Einstein's principle of equivalence is the first to assign to it the position to which it is, beyond doubt, entitled.Note 19(p. 41). Arguing along the same lines B. and J. Friedländer have suggested an experiment to show the relativity of rotational motions, and, accordingly, the reversibility of centrifugal phenomena ("Absolute and Relative Motion," Berlin, Leonhard Simion, 1896). On account of the smallness of the effect, the experiment cannot, at present, be performed successfully; but it is quite appropriate for making the physical content of this postulate more evident. The following remarks may be quoted:"The torsion-balance is the most sensitive of all instruments. The largest rotating-masses, with which we can experiment, are probably the large fly-wheels in rolling-mills and other big factories. The centrifugal forces assert themselves as a pressure which tends from the axis of rotation. If, therefore, we set up a torsion-balance in somewhat close proximity to one of these large fly-wheels, in such a position that the point of suspension of the movable part of the torsion-balance (the needle) lies exactly, or as nearly as possible, in the continuation of the axis of the fly-wheel, the needle should endeavourto set itself parallel to the plane of the fly-wheel, if it is not originally so, and should register a corresponding displacement. For centrifugal force acts upon every portion of mass which does not lie exactly in the axis of rotation, in such a way as to tend to increase the distance of the mass from the axis. It is immediately apparent that the greatest possible displacement-effect is attained when the needle is parallel to the plane of the wheel."This proposed experiment of B. and J. Friedländer is only a variation of the experiment which persuaded Newton to his view of the absolute character of rotation. Newton suspended a cylindrical vessel filled with water by a thread, and turned it about the axis defined by the thread till the thread became quite stiff. After the vessel and the contained liquid had completely come to rest, he allowed the thread to untwist itself again, whereby the vessel and the liquid started to rotate rapidly. He thereby made the following observations. Immediately after its release thevessel aloneassumed a motion of rotation, since the friction (viscosity) of the water was not sufficient to transmit the rotation immediately to the water. So long as this state of affairs prevailed, the surface of the water remained a horizontal plane. But the more rapidly the water was carried along by the rotating walls of the vessel, the more definitely did the centrifugal forces assert themselves, and drive the water up the walls, so that finally its free surface assumed the form of a paraboloid of revolution. From these observations Newton concluded that therotationof the walls of the vesselrelativeto the water does not call up forces in the latter. Only when the water itself shares in the rotation, do the centrifugal forces make their appearance. From this he came to his conclusion of the absolute character of rotations.This experiment became a subject of frequent discussion later: and E. Mach long ago objected to Newton's deduction, and pointed out that it cannot be straightway affirmed that the rotation of the walls of the vessel relative to the water is entirely without effect upon the latter. He regards it as quite conceivable that, provided the mass of the vessel were large enough, e.g. if its walls were many kilometres thick, then the free surface of the water which is at rest in the rotating vessel would not remain plane. This objection is quite in keeping with the view entailed by the general theory of relativity. According to the latter, the centrifugal forces can also be regarded as gravitational forces, which the total sum of the masses rotating around the water exerts upon it. The gravitational effect of the walls of the vessel upon the enclosed liquid is, of course, vanishingly small compared with that of all the masses in the universe. It is only when the water is in rotation relatively to all these masses that perceptible centrifugal forces are to be expected. The experiment of B. and J. Friedländer was intended to refine the experiment performed by Newton, by using a sensitive torsion-balance susceptible to exceedingly small forces in place of the water, and by substituting a huge fly-wheel for the vessel which contained the water. But this arrangement, too, can lead to no positive result, as even the greatest fly-wheel at present available represents only a vanishingly small mass compared with the sum-total of masses in the universe.Note 20(p. 42). We use the term "field of force" to denote a field in which the force in question varies continuously from place to place, and is given for each point in the field by the value of some function of the place. The centrifugal forces in theinterior and on the outer surface of a rotating body are so distributed as to compose a field of this kind throughout the whole volume of the body, and there is nothing to hinder us from imagining this field to extend outwards beyond the outer surface of the body, e.g. beyond the surface of the earth into its own atmosphere. We can thus briefly speak of the whole field as the centrifugal field of the earth; and, as the centrifugal field, according to the older views, is conditioned only by the inertia of bodies, and not by their gravitation, we can further speak of it as an inertial field, in contradistinction to the gravitational field, under the influence of which all bodies which are not suspended or supported fall to earth.Accordingly the effects of various fields of force are superposed at the earth's surface: (1) the effect of the gravitational field, due to the gravitation of the particles of the earth's mass towards one another, and which is directed towards the centre of the earth; (2) the effect of the centrifugal field, which, according to Einstein's view, can be regarded as a gravitational field, and the direction of action of which is outwards and parallel to the plane of the meridian of latitude; finally (3) the effect of the gravitational field, due to the various heavenly bodies, foremost amongst them, the sun and the moon.Note 21(p. 42). Eötvös has published the results of his measurements in the "Mathematische und Naturwissenschaftliche Berichte aus Ungarn," Bd. 8, S. 64, 1891. A detailed account is given by D. Pekár, "Das Gesetz der Proportionalität von Trägheit und Gravitation." "Die Naturwissenschaft," 1919, 7, p. 327.Whereas the earlier investigations of Newton and Bessel ("Astr. Nachr." 10, S. 97, and "Abhandlungen von Bessel," Bd. 3, S. 217), about theattractive effect of the earth upon various substances, are based upon observations with a pendulum, Eötvös worked with sensitive torsion-balances.The force, in consequence of which bodies fall, is composed of two components: first the attractive force of the earth, which (except for deviations which may, for the present, be neglected) is directed towards the centre of the earth; and, second, the centrifugal force, which is directed outwards parallel to the meridians of latitude. If the attraction of the earth upon two bodies of equal mass but of different substance were different, the resultant of the attractive and centrifugal forces would point in a different direction for each body. Eötvös then states: "By calculation we find that if the attractive effect of the earth upon two bodies of equal mass, but composed of different substance, differed by a thousandth, the directions of the gravitational forces acting upon the two bodies respectively would make an angle of 0.356 second, i.e. about a third of a second with one another; "and if the difference in the attractive force were to amount to a twenty-millionth, this angle would have to beth seconds; that is, slightly more thanth of a second; and later:"I attached separate bodies of about 30 grms. weight to the end of a balance-beam about 25 to 30 cms. long, suspended by a thin platinum wire in my torsion-balance. After the beam had been placed in a position perpendicular to the meridian, I determined its position exactly by means of two mirrors, one fixed to it and another fastened to the case of the instrument. I then turned the instrument, together with the case, through 180°, so that the body which was originally at the east end of the beam now arrived at the west end: I then determined the position ofthe beam again, relative to the instrument. If the resultant weights of the bodies attached to both sides pointed in different directions, a torsion of the suspending wire should ensue. But this did not occur in the cases in which a brass sphere was constantly attached to the one side, and glass, cork, or crystal antimony was attached to the other; and yet a deviation ofth of a second in the direction of the gravitational force would have produced a torsion of one minute, and this would have been observed accurately."Eötvös thus attained a degree of accuracy, such as is approximately reached in weighing; and this was his aim: for his method of determining the mass of bodies by weighing is founded upon the axiom that the attraction exerted by the earth upon various bodies depends only upon their mass, and not upon the substance composing them. This axiom had, therefore, to be verified with the same degree of accuracy as is attained in weighing. If a difference of this kind in the gravitation of various bodies having the same mass but being composed of different substance exists at all, it is, according to Eötvös, less than a twenty-millionth for brass, glass, antimonite, cork, and less than a hundred-thousandth for air.Note 22(p. 44).Videalso A. Einstein, "Grundlagen der allgemeinen Relativitätstheorie," "Ann. d. Phys.," 4 Folge, Bd. 49, S. 769.Note 23(p. 46). The equationasserts that the variation in the length of path between two sufficiently near points of the path vanishes for the path actually traversed; i.e. the path actually chosen between two such points is the shortest of all possible ones. If we retain the view of classical mechanics for a moment, the following example will give us the sense ofthe principle clearly: In the case of the motion of a point-mass, free to move about in space, the straight line is always the shortest connecting line between two points in space: and the point-mass will move from the one point to the other along this straight line, provided no other disturbing influences come into play (Law of inertia). If the point-mass is constrained to move over any curved surface, it will pass from one point to another along a geodetic line to the surface, since the geodetic lines represent the shortest connecting lines between points on the surface. In Einstein's theory there is a fully corresponding principle, but of a much more general form. Under the influence ofinertia and gravitationevery point-mass passes along the geodetic lines of the space-time-manifold. The fact of these lines not, in general, being straight lines, is due to the gravitational field, in a certain sense, putting the point-mass under a sort of constraint, similar to that imposed upon the freedom of motion of the point-mass by a curved surface. A principle in every way corresponding had already been installed in mechanics as a fundamental principle for all motions by Heinrich Hertz.Note 24(p. 48).VideA. Einstein, "Ann. d. Phys.," 4 Folge, Bd. 35, S. 898.Note 25(p. 48). The expression "acceleration-transformation" means that the equations giving the transformation from the variables,,,to the system of variables,,,, which is the basis of our discussion, can be regarded as giving the relations between two systems of reference which are moving with anacceleratedmotion relatively to one another. The nature of the state of motion of two systems of reference relative to one another finds its expression in the analytical form of the equations of transformation of their co-ordinates.Note 26(p. 51). Two things are to be undertaken in the following: (1) the fundamental equations of the new theory are to be written in an explicit form, and (2) the transition to Newton's fundamental equations is to be performed.1. From the equation of variationwherewe have, after carrying out the operation of variation, the four total differential equations:These are the equations of motion of a material point in the gravitational field defined by the's.The symbolhere denotes the expressionThe symboldenotes the minor ofin the determinantdivided by the determinant itself.The ten differential equations for the "gravitational potentials" are:The quantitiesandare expressions which are related in a simple manner to the components of the stress-energy-tensor(which plays the part of the quantity exciting the field in the new theory in place of the density of mass).is essentially equal to the gravitational constant of Newton's theory.The differential equations(1)and(2)are the fundamental equations of the new theory. The derivation of these equations is carried out in detail in the tract by A. Einstein, "The Foundations of the General Principle of Relativity," J. A. Barth, Leipzig, 1916.2. In order to obtain a connection between these equations and Newton's theory, we must make several simplifying assumptions. We shall first assume that the's differ only by quantities which are small compared with unity from the values given by the scheme:These values for the's characterize the case of the special theory of relativity, i.e. the case of the condition free of gravitation. We shall also assume that, at infinite distances, the's tend to, and do finally, assume the above values; that is, that matter does not extend into infinite space.Secondly, we shall assume that the velocities of matter are small compared with the velocity of light, and can be regarded as small quantities of the first order. The quantitieswill then be infinitely small quantities of the first order, andwill equal 1, except for quantities ofthe second order. From the equations defining theit will then be seen that these quantities will be infinitely small, of the first order. If we neglect quantities of the second order, and finally assume that, for small velocities of matter, the changes of the gravitational field with respect to time are small (i.e. that the derivatives of the s with respect to time may be neglected in comparison with the derivatives taken with regard to the space-co-ordinates) the system of equations (1) assumes the form:This would be the equation of motion of a point-mass as already given by Newton's mechanics, ifbe taken as representing the ordinary gravitational potential. It still remains to be seen what the differential equation forbecomes in the new theory under the simplifying assumptions we have chosen.The stress-energy-tensor, which excites the field, degenerates, as a result of our quite special assumptions, into the density of mass:In the differential equations (2) the second term on the left-hand side is the product of two magnitudes, which, according to the above arguments, are to be regarded as infinitely small quantities of the first order. Thus the second term, being of the second order of small quantities, may be dismissed. The first term, on the other hand, if we omit the terms differentiated with respect to time, as above (i.e. if we regard the gravitational field as "stationary"), reduces to:The differential equation forthus degenerates into Poisson's equation:Thus, to a first approximation (i.e. if one regards the velocity of light as infinitely great, and this is a characteristic feature of the classical theory, as was explained in detail in§ 3(b): if certain simple assumptions are made about the behaviour of the's at infinity; and if the time-changes of the gravitational field are neglected) the well-known equations of Newtonian mechanics emerge out of the differential equations of Einstein's theory, which were obtained from perfectly general beginnings.Note 27(p. 53). The theory of surfaces, i.e. the study of geometry upon surfaces, makes it immediately apparent that the theorems, which have been established for any surface, also hold for any surface which can be generated by distorting the firstwithout tearing. For if two surfaces have a point-to-point correspondence, such that the line-elements are equalat corresponding points, then corresponding finite arcs, angles, and areas, etc., will be equal. One thus arrives at the same planimetrical theorems for the two surfaces. Such surfaces are called "deformable" surfaces. The necessary and sufficient condition that surfaces be continuously deformable is that the expression for the line-element of the one surfacecan be transformed into that for the other,According to Gauss, it isnecessarythat both surfaces have equalmeasures of curvature. If the latter is constant over the whole surface, as e.g. in the case of a cylinder or a plane, all conditions for the deformability of the surfaces are fulfilled. In other cases, special equations offer a criterion as to whether surfaces, or portions of surfaces, are deformable into one another. The numerous subsidiary problems, which result out of these questions, are discussed at length in every book dealing with differential geometry (e.g. Bianchi-Lukat).[17]This branch of training, which was hitherto of interest only to mathematicians, now assumes very considerable importance for the physicist too.[17]Forsyth's "Differential Geometry."—H. L. B.Note 28(p. 61). One must avoid being deceived into the belief that Newton's fundamental law is in any way to be regarded as anexplanationof gravitation. The conception of attractive force is borrowed from our muscular sensations, and has therefore no meaning when applied to dead matter. C. Neumann, who took great pains to place Newton's mechanics on a solid basis, glosses upon this point himself in a drastic fashion, in the following narrative, which shows up the weaknesses of the former view:"Let us suppose an explorer to narrate to us his experiences in yonder mysterious ocean. He had succeeded in gaining access to it, and a remarkable sight had greeted his eyes. In the middle of the sea he had observed two floating icebergs, a larger and a smaller one, at a considerable distance from one another. Out of the interior of the larger one, a voice had resounded, issuing the following command in a peremptory tone: 'Ten feet nearer!' The little iceberg had immediately carried out the order, approaching ten feet nearer the larger one. Again, the larger gave out the order: 'Six feet nearer!' The other hadagain immediately executed it. And in this manner order after order had echoed out: and the little iceberg had continually been in motion, eager to put every command immediately and implicitly into action."We should certainly consign such a report to the realm of fables. But let us not scoff too soon! The ideas, which appear so extraordinary to us in this case, are exactly the same as those which lie at the base of the most complete branch of natural science, and to which the most famous of physicists owes the glory attached to his name."For in cosmic space such commands are continually resounding, proceeding from each of the heavenly bodies—from the sun, planets, moons, and comets. Every single body in space hearkens to the orders which the other bodies give it, always striving to carry them out punctiliously. Our earth would dash through space in a straight line, if she were not controlled and guided by the voice of command, issuing from moment to moment, from the sun, in which the instructions of the remaining cosmic bodies are less audibly mingled."These commands are certainly given just assilentlyas they are obeyed; and Newton has denominated this play of interchange between commanding and obeying by another name. He talks quite briefly of a mutual attractive force, which exists between cosmic bodies. But the fact remains the same. For this mutual influence consists in one body dealing out orders, and the other obeying them."
Note 1(p. 4). So long as the universal significance of the velocity of light remained unknown, two conjectures were possible in the question as to whether, under certain circumstances, the motion of the source of light would make itself observable in the velocity of propagation of light. It might be surmised that the velocity of the source simply added itself to that velocity of light which ischaracteristic for the propagation of the lightfrom a source at rest. Or, it might be conceived that the motion of the source has no influence at all on the velocity of the light emitted by it. In the second case it was imagined that the source of light only excites the periodically changing states of the luminiferous ether, which is at rest, that is, which does not share in the motion of the matter (source of light), and that these states then propagate themselves with avelocity that is characteristic of the ether, and with a velocity that makes these states perceptible to us as light waves. This view had finally apparently won the day. It was the advent of the special theory of relativity and the quantum hypothesis that made this view impossible. For the special theory of relativity, in robbing the assertion: "the ether is at rest" of its significance, since we may arbitrarily define any system as being at rest in the ether, as far as uniform translations are concerned, and in depriving the luminiferousether of its existence, deprived light-waves of their carrying or transmitting medium. The quantum hypothesis, in raising light-quanta to the rank of self-supporting individuals, deprived thevelocity of lightof its character as a constant that is characteristic of theether. Thus, our view of light-quanta again leads to a kind of emission theory of light. According to classical mechanics it would have been typical of a theory of emission for the velocity of the source in motion to have added itself to the velocity of the light from the source at rest. We thus revert to the conjecture which we quoted first above. Now, such a superposition of velocities would necessarily cause quite remarkable phenomena in the case of spectroscopic binary stars (de Sitter, "Phys. Zeitschrift,"14, 429). For if two stars move in circular Kepler orbits around each other, and if our line of sight lies in the common plane of the orbits, then we should necessarily perceive the following: ifis the time of revolution of the system,the orbital velocity of the one (bright) component,the distance of the whole system from the earth, and, finally,, the velocity in vacuo of the light from the source which is at rest, then the velocity of light at the epoch of greatest positive velocity in the direction of vision is, andin the other direction, respectively. Consequently the time-interval between two such successive positions would have the valuesandalternately, for the observer on the earth, as a simple calculation shows. Since, on account of the gigantic distances between the fixed stars, the membermay become very great, indeed, greater than, we should be able to observe definiteanomalies in the case of the spectroscopic binaries. For the time intervals between two such successive epochs in the orbit should be able to contract to nil, indeed, even become negative, and we should not be able to interpret the measured Doppler effects by means of motions in the Kepler ellipses. In reality, however, these anomalies have never manifested themselves. Observation of these very sensitive subjects of test (spectroscopic binaries) teaches us that the motion of the source of light does not make itself remarked in the propagation of the light. This renders our first view likewise untenable. The special principle of relativity, alone in postulating the constancy of the velocity of light, and in putting forward a new addition theorem of velocities, has led us to an attitude in this question that is free from inner contradictions and compatible with experience. (Cf.Note 2.)
Note 2(p. 5). There are essentially two fundamental optical experiments on which our view of the distinctive significance of the velocity of light in physical nature is founded: Fizeau's experiment concerning the velocity of light in flowing water, and the Michelson-Morley experiment. Aberration, on the other hand, has nothing to do directly with the question whether it is possible to prove by means of optical experiments in the laboratory a motion of the earth relative to the ether. The aberration in the case of stars states merely that the motion of the earthrelativelyto the star under considerationchangesperiodically in the course of a year. If, however, we hold the view that an all-pervading ether is the carrier for the propagation of the light, the phenomenon of aberration may be satisfactorily explained only if we assume that this ether does not participate in the motion of the earth.
Fizeau's experiment was designed to decide finally whether movingmatter influences the ether and to determine the value of the velocity of light in moving matter with respect to the observer. Michelson and Morley repeated the experiment in the following improved form. A beam of light from a source on the earth is sent through a-shaped tube, through which water flows, in the direction of both limbs. After each part of the beam has traversed the flowing water, the one in the direction of the current, the other contrary to it, the two beams, are made to interfere. The light and the water move in the same direction in the one limb, and oppositely in the other.
fig01Fig. 1.
Fig. 1.
Fig. 1.
Now, there immediately appear to be two possibilities. Either the water that flows with a velocitywith respect to the walls of the tube drags along the carrier that effects the transmission of the light, namely the ether; in this case, the velocity of the light isin the one limb, andin the other, for, on account of the coefficient of refractionof the water,is the velocity of light in resting water. Or, the motion of the water has no influence at all on theether which transmits light and which permeates the water. In this case the velocity of light isin both limbs. According, as the one or the other of these two assumptions is valid, the interference fringes would have to become displaced or remain at rest when the direction of the current is reversed. The experiment decided in favour ofneitherof these possibilities. The interference fringes did, indeed, become shifted, not to the expected amount, however, but only to an amount that would result if the ether assumes the velocityin water, and not the full value. This value of the convection of the ether is calledFresnel's convection coefficient. Yet this term is capable of being misunderstood inasmuch as in the electrodynamics developed by Lorentz, the result of Fizeau's experiment speaks in favour of an ether that isabsolutely at rest, and the so-called convection coefficient is only a consequence of the structure of matter, in particular of the interaction between electrons and matter, a question into which we cannot enter here. At any rate, at the time preceding the Michelson-Morley experiment aberration, as well as Fizeau's experiment, appeared to speak in favour of an ether that was absolutely at rest.
Now, the Michelson-Morley experiment was to establish the existence of the current of ether (ether "wind") through which the earth continually moves, since the ether is supposed not to participate in the motion of the earth. The scheme of the experiment is as onp. 74.
A ray of light, starting out from, traverses the course: hereand, are two mirrors, on to which the ray falls perpendicularly;is a glassplate that reflects one half of the light and allows the remainder to pass through;is the telescope of the observer. Another ray of light traverses the course. Let. Further, let, be in the direction of the earth's motion. Our assumption is that the ether does not share in the earth's motion. Let the velocity of the earth be.
fig02Fig. 2.
Fig. 2.
Fig. 2.
Then the velocity of the light relative to the instrument (earth) is as follows in the directions specified:Consequently, the courseis traversed in the timeand the coursein the timeThe difference of these two times isIf we exchange the positions of, and, by turning the whole apparatus through 90°, then
If we make these tyro rays of light interfere atthen, when the apparatus is turned through 90°, the interference fringes should become shifted. The amount of this displacement may easily be calculated. If we denote bythe vibration frequency of the light-ray used in the experiment, thenis the corresponding wave-length. Thus, expressed in fractions of the interval between the fringes, the expected displacement becomes equal toBy causing the light to be reflected many timeswas magnified to such an extent thatbecame of the order. If, for example,,= the wavelength of sodium light, then. On the other handis of the orderthat is,. The expected displacement of the fringes would thus have to be about 0·56 of the breadth of a fringe. Actually, an amount of the order 0·02 of the breadth of a fringe was observed. Thus, the ether wind did not make itself remarked optically in the motion of the earth. By carrying out the experiment at different times of the year the possible objection that the motion of translation of the entire solar system might have counterbalanced the motion of the earth in her orbit was removed.
The Michelson-Morley experiment has shown conclusively that there is no physical sense in talking of absolute rest or of a translation relative to absolute space, since all systems that move rectilinearly and uniformly with respect to one another are of equal value for describing natural phenomena. It is thus a matter of convention which system we are to regard as at rest and which as being in motion. We may assign the same value to the velocity of light in all systems. A detailed theory of these fundamental experiments may be found in all comprehensive accounts of the special theory of relativity. We here merely mention the original paper by A. Einstein (Annalen der Physik, Bd. 17, 1905, p. 891), and the booklet, "Einführung in die Relativitätstheorie," by Dr. W. Block, out of the series "Aus Natur und Geisteswelt," Teubner, 1918.
Note 3(p. 9). Abolishing the transformations of Newton's principle of relativity and replacing them by the so-called Lorentz-Einstein transformations signified a step of extraordinarily far-reaching consequence. It was justified in that the new theory of relativity which followed as a result of it, confirmed, without difficulty, the results of all the fundamental experiments of optics and electrodynamics. Concerning the Michelson-Morley experiment,Lorentz, to account for its negative result within the realm of electrodynamics, had been compelled to set up the hypothesis that the dimensions of all bodies contract in the direction of their motion. But Einstein now showed that if we define the conception of simultaneity rigorously, taking into account the postulate of the constancy of the velocity of light, the Lorentz-transformations, which had been found empirically, followed necessarily as those equations of transformation that must hold between the co-ordinates of two systems moving uniformly and rectilinearly with respect to each other. And without the help of any further hypothesis there appears as a direct consequence of this transformation just that contraction of lengths which Lorentz had adduced to explain the result of the Michelson-Morley experiment. This contraction of a lengthin the direction of motion of an object to the valueis, however, in the new theory the expression of the general fact that the dimensions of a body have only a relative meaning, that is, that their values depend on the state of motion of the observer, which determines the dimensions of the body in question. This holds for the extension of bodies in time as well as in space. From the point of view of the new principle of relativity the negative result of the Michelson-Morley experiment was self-explained. But what was the position with regard to the other fundamental facts of optics and electrodynamics? The result of Fizeau's experiment concerning the velocity of light in flowing water became a direct test of the kinematics arising out of the new formulæ. According to the Lorentz transformation the two velocities,andwith which, for example, two locomotives approach each other, do not merelybecome added, so thatwould be the relative velocity of each with respect to the other, but rather each engine-driver will find as the velocity with which he passes the other driver, the valueaccording to the new formulæ. This is the addition theorem of velocities according to the new theory. It gives us immediately the amount observed in Fizeau's experiment for the velocity of light in flowing water. Aberration and the Doppler effect follow just as readily to the correct amount. A detailed discussion of these questions is to be found in every account of the "special" theory of relativity (cf. the references given inNote 2).
Note 4(p. 12). Ph. Frank and H. Rothe,Ann. d. Phys., 4 Folge, Bd. 34, p. 825.
The assumptions for the general equations of transformation by which two systemsand' that moveuniformlyandrectilinearlywith the velocity q with respect to each other sure connected are as follows:—
1. The equations of transformation form a linear homogeneous group in the variable parameter. This means that the successive application of two equations of transformation, of which the one refers the systemto the system', and the second' to''(is to have the constant velocitywith respect to', and' the constant velocity' with respect to'') again leads to an equation of transformation of the same form as that of the initial equations. The parameter''that occurs in the new equation depends in a definite way on' and.
2. The contractions of the lengths depend only on the value of theparameter. We must, of course, from the very outset reckon with the possibility that the length of a rod measured in the system that is at rest comes out differently when measured in the moving system. Now, condition 2 requires that if contractions occur (that is, changes of length in these various methods of determination) values are to depend only on themagnitudeof the velocity of both systems and not on the direction of their motion in space. Thus this postulate endows space with the property of isotropy, and is in fair correspondence with the postulate ofsection 3a, which states that it must be possible to compare each line-element with every other in length independently of its position in space, and its direction.
An essential feature is that the constancy of the velocity of light isnotdemanded in either of the postulates 1 and 2. Rather, the distinguishing property of adefinitevelocity in virtue of which it preserves its value inallsystems that emerge out of one another through such transformations is a direct corollary to these two general postulates, and the result of the Michelson-Morley experiment merely determines the value of this special velocity which could, of course, be found only from observation.
Note 5(p. 15). Einstein has shown in a simple example how, on the basis of the formulæ of the special theory of relativity, a point-mass loses inertial mass when it radiates out energy.
We assume that a point-mass emits a light-wave of energyin a certain direction, and a light-wave of the same energyin the opposite direction. Then, in view of the symmetry of the process of emission with respect to the system of reference of the co-ordinates,,,originally chosen, the point-mass remains at rest. Let the total energyof the point-mass bereferred to this system, butreferred to a second system which we suppose moving with the uniform velocitywith respect to the first. We shall apply the principle of energy to this process. Ifandare the frequency and amplitude of the light-wave in the initial system,',',',',',' the frequency, amplitude, and co-ordinates in the second (the moving) system, further,the angle between the wave-normals and the line connecting the point-mass with the observer, then Doppler's principle gives for the frequency of the light-wave in the moving system:The formulæ of the special theory of relativity give us, correspondingly, for the amplitude in the moving systemAccording to Maxwell's theory the energy of the light-wave per unit volume is. We now wish to calculate the corresponding energy-density also with respect to the moving system. We must here take into account that, in consequence of the contraction of the lengths according to the Lorentz-Einstein transformation formulæ, the volumeof a sphere in the resting system becomes transformed into that of am ellipsoid as measured from the moving system; indeed, this volume of the ellipsoid isHence the energy-densities in the accented and unaccented system are in the ratio:If we now designate the energy-content of the point-massafterthe emission by, and the corresponding quantity referred to the moving system by, then we have:whereasFrom this we get directly thatWhat does this equation assert?
andare the energy-values of the same point-mass, in the first place referred to a system with respect to which the point-mass moves, and in the second related to a system in which the point-massis at rest. Hence the difference, except for an additive constant, must be equal to thekineticenergy of the point-mass referred to the moving system. Thus, we may writewhereindenotes a constant which does not alter during the light-emission of the point-mass, since, owing to the symmetry of the process, the point-mass remains at rest with respect to the initial system. So we arrive at the relation:In words this equation states that owing to the point-mass emitting the energyas light, its kinetic energy referred to a moving system sinks from the valueto the value, corresponding to a loss in inertial mass of the amount. For, according to classical mechanics, the expressionin whichis the inertial mass of the observed body, is a measure of the kinetic energy of this body referred to a system with respect to which it moves with the velocity. Thusmust be taken as standing for the inertial mass of an amount of energy.
Note 6(p. 29). The facts thateverypair of points (point-pair) in space have the same magnitude-relation (viz. the same expression for the mutualdistancebetween them) and that with the aid of this relation, every point-pair can be compared with every other, constitute the characteristic feature which distinguishes spacefrom the remaining continuous manifolds which are known to us. We measure the mutual distance between two points on the floor of a room, and the mutual distance between two points which he vertically above one another on the wall, with the same measuring-scale, which we thus apply in any direction at pleasure. This enables us to "compare" the mutual distance of a point-pair on the floor with the mutual distance of any other pair of points on the wall.
In the system of tones, on the contrary, quite different conditions prevail. The system of tones represents a manifold of two dimensions, if one distinguishes every tone from the remaining tones by its pitch and its intensity. It is, however, not possible to compare the "distance" between two tones of thesame pitchbutdifferent intensity(analogous to the two points on the floor) with the "distance" between two tones ofdifferent pitchbutequal intensity(analogous to the two points on the wall). The measure-conditions are thus quite different in this manifold.
In the system of colours, too, the measure-relations have their own peculiarity. The dimensions of the manifold of colours are the same as those of space, as each colour can be produced by mixing thethree"primary" colours. But there is no relation between two arbitrary colours, which would correspond to the distance between two points in space. Only when a third colour is derived by mixing these two, does one obtain an equation between these three colours similar to that which connects three points in space lying in one straight line.
These examples, which are borrowed from Helmholtz's essays, serve to show that the measure-relations of a continuous manifold are not already given in its definition as acontinuousmanifold.nor by fixing its dimensions. A continuous manifold generally allows of various measure-relations. It is only experience which enables us to derive the measure-laws which are valid for each particular manifold. The fact, discovered by experience, that the dimensions of bodies are independent of their particular position and motion, led to the laws of Euclidean geometry wherecongruenceis the deciding factor in comparing various portions of space. These questions have been exhaustively treated by Helmholtz in various essays. References:—
Riemann, "Über die Hypothesen, welche der Geometrie zugrunde liegen" (1854). Newly published and annotated by H. Weyl, Berlin, 1919.
Helmholtz. "Ueber die tatsächlichen Grundlagen der Geometrie,"Wiss. Abh.2, S. 10.
Helmholtz. "Ueber die Tatsachen, welche der Geometrie zugrunde liegen,"Wiss. Abh.2, S. 618.
Helmholtz. "Ueber den Ursprung und die Bedeutung der geometrischen Axiome,"Vorträge und Reden, Bd. 2, S. 1.
Note 7(p. 26). The postulate that finite rigid bodies are to be capable of free motions, can be most strikingly illustrated in the realm of two-dimensions. Let us imagine a triangle to be drawn upon asphere, and also upon aplane: the former being bounded by arcs of great circles and the latter by straight lines; one can then slide these triangles over their respective surfaces at will, and can make them coincide with other triangles, without thereby altering the lengths of the sides or the angles. Gauss has shown that this is possible because thecurvatureat every point of the sphere (or the plane, respectively) has exactly the same value. And yet the geometry of curves traced upon a sphere is different from that of curvestraced upon aplane, for the reason that these two configurations cannot be deformed into one another without tearing (videNote 27). But uponbothof them planimetrical figures can be freely shifted about, and, therefore, theorems of congruence hold upon them. If, however, we were to define a curvilinear triangle upon an egg-shaped surface by the three shortest lines connecting three given points upon it, we should find that triangles could be constructed at different places on this surface, having the same lengths for the sides; but these sides would enclose angles different from those included by the corresponding sides of the initial triangle, and, consequently, such triangles would not be congruent, in spite of the fact that corresponding sides are equal. Figures upon an egg-shaped surface cannot, therefore, be made to slide over the surface without altering their dimensions: and in studying the geometrical conditions upon such a surface, we do not arrive at the usual theorems of congruence. Quite analogous arguments can be applied to three- and four-dimensional realms: but the latter casesoffer no corresponding pictures to the mind. If we demand that bodies are to be freely movable in space without suffering a change of dimensions, the "curvature" of the space must be the same at every point. The conception of curvature, as applied to any manifold of more than two dimensions, allows of strict mathematical formulation; the term itself only hints at its analogous meaning, as compared with the conception of curvature of a surface. In three-dimensional space, too, various cases can be distinguished, similarly to plane- and spherical-geometry in two-dimensional space. Corresponding to the sphere, we have a non-Euclidean space with constant positive curvature; corresponding to the plane wehave Euclidean space with curvature zero. In both these spaces bodies can be moved about without their dimensions altering; but Euclidean space is furthermore infinitely extended: whereas "spherical" space, though unbounded, like the surface of a sphere, is not infinitely extended. These questions are to be found extensively treated in a very attractive fashion in Helmholtz's familiar essay: "Ueber den Ursprung und die Bedeutung der geometrischen Axiome" (Vorträge und Reden, Bd. 2, S. 1).
Note 8(p. 26). The properties, which the analytical expression for the length of the line-element must have, may be understood from the following:
Let the numbers,denote any point of any continuous two-dimensional manifold, e.g. a surface. Then, together with this point, a certain "domain" around the point is given, which includes points all of which lie in the plane.—D. Hilbert has strictly defined the conception of a multiply-extended magnitude (i.e. a manifold) upon the basis of the theory of aggregates in his "Grundlagen der Geometrie" (p. 177). In this definition the conception of the "domain" encircling a point is made to give Riemann's postulate of thecontinuousconnection existing between the elements of a manifold and a strict form.
Setting out from the point,we can continuously pass into its domain, and at any point, e.g.,, inquire as to the "distance" of this point from the starting-point. The function whichmeasuresthis distance will depend upon the values of,,,, and for everyintermediatepoint of the path which has conducted us from,to the point,will successively assume certain continually changing, and, as we may suppose, continuallyincreasing, values. At the point,itself it will assume the value zero, and for every other point of the domain its value must be positive. Moreover, we shall expect to find that, for any intermediate point, denoted by,,andthe required function which measures the distance of this point from,, will, at this point, have a value half that of its value for the point,. Under these assumptions, the function will be homogeneous and of the first degree in the's; its value will then appear multiplied by that factor in proportion to which the's were increased. In addition, it must itself vanish if all the's are zero; and if they all change their sign it must not alter its value, which always remains positive. It will immediately be evident that the functionfulfils all these requirements; but it is by no means the only function of this kind.
Note 9(p. 29). But the expression of the fourth degree for the fine element would not permit of any geometrical interpretation of the formula, such as is possible with the expressionwhich latter may be regarded as a general case of Pythagoras' theorem.
Note 10(p. 30). By a "discrete" manifold we mean one in which no continuous transition of the single elements from one to another is possible, but each element to a certain extent represents an independent entity. The aggregate of all whole numbers, for instance, is a manifold of this type, or the aggregate of allplanetsin our solar system, etc., and many other examples may be found; andindeed all finite aggregates in the theory of aggregates are such discrete manifolds. "Measuring," in the case of discrete manifolds, is performed merely by "counting," and does not present any special difficulties, as all manifolds of this type are subject to the same principle of measurement. When Riemann then proceeds to say: "Either, therefore, the reality which underlies space must form a discrete manifold, or we must seek the ground of its metric relations outside it, in binding forces which act upon it," he only wishes to hint at a possibility, which is at present still remote, but which must, in principle, always be left open. In just the last few years a similar change of view has actually occurred in the case of another manifold which plays a very important part in physics, viz. "energy"; the meaning of the hint Riemann gives will become clearer if we consider this example.
Up till a few years ago, the energy which a body emanates by radiation was regarded as a continuously variable quantity: and attempts were therefore made to measure its amount at any particular moment by means of a continuously varying sequence of numbers. The researches of Max Planck have, however, led to the view that this energy is emitted in "quanta," and that therefore the "measuring" of its amount is performed by counting the number of "quanta." The reality underlying radiant energy, according to this, is a discrete andnota continuous manifold. If we now suppose that the view were gradually to take root that, on the one hand, all measurements in space only have to do with distances between ether-atoms; and that, on the other hand, the distances of single ether-atoms from one another can only assume certaindefinite values, all distances in space would be obtained by "counting" these values, and we should have to regard space as a discrete manifold.
Note 11(p. 32). C. Neumann. "Ueber die Prinzipien der Galilei-Newtonschen Theorie," Leipzig 1870, S. 18.
Note 12(p. 32). H. Streintz. "Die physikalischen Grundlagen der Mechanik," Leipzig, 1883.
Note 13(p. 33). A. Einstein. "Annalen der Physik," 4 Folge, Bd. 17, S. 891.
Note 14(p. 35). Minkowski was the first to call particular attention to this deduction of the special principle of relativity.
Note 15(p, 38). The term "inertial system" was originally not associated with the system, which Neumann attached to the hypothetical body. Nowadays it is generally understood to signify a rectilinear system of co-ordinates, relatively to which a point-mass, which is only subject to its own inertia, moves uniformly in a straight line. Whereas C. Neumann only invented the body, as an absolutely hypothetical configuration, in order to be able to formulate the law of inertia, later researches, especially those of Lange, tended to show that, on the basis of rigorous kinematical considerations, a co-ordinate system could be derived, which would possess the properties of such an inertial system. However, as C. Neumann and J. Petzoldt have demonstrated, these developments contain faulty assumptions, and give the law of inertia no firmer basis than the bodyintroduced by Neumann.
Such an inertial system is determined by the straight lines which connect three point-masses infinitely distant from one another (and thus unable to exert a mutual influence upon one another) and which are not subject to any other forces. This definition makes it evident why noinertial system will be discoverable in nature, and why, consequently, the law of inertia will never be able to be formulated so as to satisfy the physicist. References:—
C. Neumann. "Ueber die Prinzipien der Galilei-Newtonschen Theorie," Leipzig, 1870.
L. Lange. "Berichte der Kgl. Sächs. Ges. d. Wissenschaften. Math.-phil. Klasse," 1885.
L. Lange. "Die Geschichte der Entwickelung des Bewegungsbegriffes," Leipzig, 1886.
H. Seeliger. "Ber. der Bayr. Akademie," 1906, Heft 1.
C. Neumann. "Ber der Kgl. Sächs. Ges. d. Wiss. Math.-phys. Klasse," 1910, Bd. 62, S. 69 and 383.
J. Petzoldt. "Ann. der Naturphilosophie," Bd. 7.
Note 16(p. 38). E. Mach. "Die Mechanik in in ihrer Entwickelung," 4 Aufl. S. 244.
Note 17(p. 40). The new points of view as to the nature of inertia are based upon the study of the electromagnetic phenomena of radiation. The special theory of relativity, by stating the theorem of the inertia of energy, organically grafted these views on to the existing structure of theoretical physics. The dynamics of cavity-radiation, i.e. the dynamics of a space enclosed by walls without mass, and filled with electromagnetic radiation, taught us that a system of this kind opposes a resistance to every change of its motion, just like a heavy body in motion. The study of electrons (free electric charges) in a state of free motion, e.g. in a cathode-tube, taught us likewise that these exceedingly small particles behave like inert bodies; that their inertia is not, however, conditioned by the matter to which they might happen to be attached, but rather by the electromagnetic effects of the field to which the moving electron is subject. This gave rise to the conception of the apparent(electromagnetic) mass of an electron. The special theory of relativity finally led to the conclusion that toallenergy must be accorded the property of inertia.
Every body contains energy (e.g. a certain definite amount in the form of heat-radiation internally). The inertia, which the body reveals, is thus partly to be debited to the account of this contained energy. As this share of inertia is, according to the special theory of relativity, relative (i.e. represents a quantity which depends upon the choice of the system of reference), the whole amount of the inertial mass of the body has no absolute value, but only a relative one. This energy-content of radiant heat is distributed throughout the whole volume of each particular body; one can thus speak of the energy-content of unit volume. This enables us to derive the notion of density of energy. The density of the energy (i.e. amount per unit volume) is thus a quantity, the value of which is also dependent upon the system of reference. References:—
M. Planck. "Ann. der Phys.," 4 Folge, Bd. 26.
M. Abraham. "Electromagnetische Energie der Strahlung," 4 Aufl., 1908.
Note 18(p. 40). The determination of the inertial mass of a body by measuring its weight is rendered possible only by the experimental fact that all bodies fall with equal acceleration in the gravitational field at the earth's surface. Ifand' denote the pressures of two bodies upon the same support (i.e. their respective weights), anddenote the acceleration due to the earth's gravitational field at the point in question, thendynes anddynes, respectively, whereand' are the factors of proportionality, and are called themassesof the two bodies, respectively. Ashas the same value in both equations, we haveand we can accordingly measure themassesof two bodies at the same place, by determining theirweights.
Although Galilei and Newton had already known that all bodies at the same place fall with the same velocity (if the resistance of the air be eliminated), this very remarkable fact has not received any recognition in the foundations of mechanics. Einstein's principle of equivalence is the first to assign to it the position to which it is, beyond doubt, entitled.
Note 19(p. 41). Arguing along the same lines B. and J. Friedländer have suggested an experiment to show the relativity of rotational motions, and, accordingly, the reversibility of centrifugal phenomena ("Absolute and Relative Motion," Berlin, Leonhard Simion, 1896). On account of the smallness of the effect, the experiment cannot, at present, be performed successfully; but it is quite appropriate for making the physical content of this postulate more evident. The following remarks may be quoted:
"The torsion-balance is the most sensitive of all instruments. The largest rotating-masses, with which we can experiment, are probably the large fly-wheels in rolling-mills and other big factories. The centrifugal forces assert themselves as a pressure which tends from the axis of rotation. If, therefore, we set up a torsion-balance in somewhat close proximity to one of these large fly-wheels, in such a position that the point of suspension of the movable part of the torsion-balance (the needle) lies exactly, or as nearly as possible, in the continuation of the axis of the fly-wheel, the needle should endeavourto set itself parallel to the plane of the fly-wheel, if it is not originally so, and should register a corresponding displacement. For centrifugal force acts upon every portion of mass which does not lie exactly in the axis of rotation, in such a way as to tend to increase the distance of the mass from the axis. It is immediately apparent that the greatest possible displacement-effect is attained when the needle is parallel to the plane of the wheel."
This proposed experiment of B. and J. Friedländer is only a variation of the experiment which persuaded Newton to his view of the absolute character of rotation. Newton suspended a cylindrical vessel filled with water by a thread, and turned it about the axis defined by the thread till the thread became quite stiff. After the vessel and the contained liquid had completely come to rest, he allowed the thread to untwist itself again, whereby the vessel and the liquid started to rotate rapidly. He thereby made the following observations. Immediately after its release thevessel aloneassumed a motion of rotation, since the friction (viscosity) of the water was not sufficient to transmit the rotation immediately to the water. So long as this state of affairs prevailed, the surface of the water remained a horizontal plane. But the more rapidly the water was carried along by the rotating walls of the vessel, the more definitely did the centrifugal forces assert themselves, and drive the water up the walls, so that finally its free surface assumed the form of a paraboloid of revolution. From these observations Newton concluded that therotationof the walls of the vesselrelativeto the water does not call up forces in the latter. Only when the water itself shares in the rotation, do the centrifugal forces make their appearance. From this he came to his conclusion of the absolute character of rotations.
This experiment became a subject of frequent discussion later: and E. Mach long ago objected to Newton's deduction, and pointed out that it cannot be straightway affirmed that the rotation of the walls of the vessel relative to the water is entirely without effect upon the latter. He regards it as quite conceivable that, provided the mass of the vessel were large enough, e.g. if its walls were many kilometres thick, then the free surface of the water which is at rest in the rotating vessel would not remain plane. This objection is quite in keeping with the view entailed by the general theory of relativity. According to the latter, the centrifugal forces can also be regarded as gravitational forces, which the total sum of the masses rotating around the water exerts upon it. The gravitational effect of the walls of the vessel upon the enclosed liquid is, of course, vanishingly small compared with that of all the masses in the universe. It is only when the water is in rotation relatively to all these masses that perceptible centrifugal forces are to be expected. The experiment of B. and J. Friedländer was intended to refine the experiment performed by Newton, by using a sensitive torsion-balance susceptible to exceedingly small forces in place of the water, and by substituting a huge fly-wheel for the vessel which contained the water. But this arrangement, too, can lead to no positive result, as even the greatest fly-wheel at present available represents only a vanishingly small mass compared with the sum-total of masses in the universe.
Note 20(p. 42). We use the term "field of force" to denote a field in which the force in question varies continuously from place to place, and is given for each point in the field by the value of some function of the place. The centrifugal forces in theinterior and on the outer surface of a rotating body are so distributed as to compose a field of this kind throughout the whole volume of the body, and there is nothing to hinder us from imagining this field to extend outwards beyond the outer surface of the body, e.g. beyond the surface of the earth into its own atmosphere. We can thus briefly speak of the whole field as the centrifugal field of the earth; and, as the centrifugal field, according to the older views, is conditioned only by the inertia of bodies, and not by their gravitation, we can further speak of it as an inertial field, in contradistinction to the gravitational field, under the influence of which all bodies which are not suspended or supported fall to earth.
Accordingly the effects of various fields of force are superposed at the earth's surface: (1) the effect of the gravitational field, due to the gravitation of the particles of the earth's mass towards one another, and which is directed towards the centre of the earth; (2) the effect of the centrifugal field, which, according to Einstein's view, can be regarded as a gravitational field, and the direction of action of which is outwards and parallel to the plane of the meridian of latitude; finally (3) the effect of the gravitational field, due to the various heavenly bodies, foremost amongst them, the sun and the moon.
Note 21(p. 42). Eötvös has published the results of his measurements in the "Mathematische und Naturwissenschaftliche Berichte aus Ungarn," Bd. 8, S. 64, 1891. A detailed account is given by D. Pekár, "Das Gesetz der Proportionalität von Trägheit und Gravitation." "Die Naturwissenschaft," 1919, 7, p. 327.
Whereas the earlier investigations of Newton and Bessel ("Astr. Nachr." 10, S. 97, and "Abhandlungen von Bessel," Bd. 3, S. 217), about theattractive effect of the earth upon various substances, are based upon observations with a pendulum, Eötvös worked with sensitive torsion-balances.
The force, in consequence of which bodies fall, is composed of two components: first the attractive force of the earth, which (except for deviations which may, for the present, be neglected) is directed towards the centre of the earth; and, second, the centrifugal force, which is directed outwards parallel to the meridians of latitude. If the attraction of the earth upon two bodies of equal mass but of different substance were different, the resultant of the attractive and centrifugal forces would point in a different direction for each body. Eötvös then states: "By calculation we find that if the attractive effect of the earth upon two bodies of equal mass, but composed of different substance, differed by a thousandth, the directions of the gravitational forces acting upon the two bodies respectively would make an angle of 0.356 second, i.e. about a third of a second with one another; "and if the difference in the attractive force were to amount to a twenty-millionth, this angle would have to beth seconds; that is, slightly more thanth of a second; and later:
"I attached separate bodies of about 30 grms. weight to the end of a balance-beam about 25 to 30 cms. long, suspended by a thin platinum wire in my torsion-balance. After the beam had been placed in a position perpendicular to the meridian, I determined its position exactly by means of two mirrors, one fixed to it and another fastened to the case of the instrument. I then turned the instrument, together with the case, through 180°, so that the body which was originally at the east end of the beam now arrived at the west end: I then determined the position ofthe beam again, relative to the instrument. If the resultant weights of the bodies attached to both sides pointed in different directions, a torsion of the suspending wire should ensue. But this did not occur in the cases in which a brass sphere was constantly attached to the one side, and glass, cork, or crystal antimony was attached to the other; and yet a deviation ofth of a second in the direction of the gravitational force would have produced a torsion of one minute, and this would have been observed accurately."
Eötvös thus attained a degree of accuracy, such as is approximately reached in weighing; and this was his aim: for his method of determining the mass of bodies by weighing is founded upon the axiom that the attraction exerted by the earth upon various bodies depends only upon their mass, and not upon the substance composing them. This axiom had, therefore, to be verified with the same degree of accuracy as is attained in weighing. If a difference of this kind in the gravitation of various bodies having the same mass but being composed of different substance exists at all, it is, according to Eötvös, less than a twenty-millionth for brass, glass, antimonite, cork, and less than a hundred-thousandth for air.
Note 22(p. 44).Videalso A. Einstein, "Grundlagen der allgemeinen Relativitätstheorie," "Ann. d. Phys.," 4 Folge, Bd. 49, S. 769.
Note 23(p. 46). The equationasserts that the variation in the length of path between two sufficiently near points of the path vanishes for the path actually traversed; i.e. the path actually chosen between two such points is the shortest of all possible ones. If we retain the view of classical mechanics for a moment, the following example will give us the sense ofthe principle clearly: In the case of the motion of a point-mass, free to move about in space, the straight line is always the shortest connecting line between two points in space: and the point-mass will move from the one point to the other along this straight line, provided no other disturbing influences come into play (Law of inertia). If the point-mass is constrained to move over any curved surface, it will pass from one point to another along a geodetic line to the surface, since the geodetic lines represent the shortest connecting lines between points on the surface. In Einstein's theory there is a fully corresponding principle, but of a much more general form. Under the influence ofinertia and gravitationevery point-mass passes along the geodetic lines of the space-time-manifold. The fact of these lines not, in general, being straight lines, is due to the gravitational field, in a certain sense, putting the point-mass under a sort of constraint, similar to that imposed upon the freedom of motion of the point-mass by a curved surface. A principle in every way corresponding had already been installed in mechanics as a fundamental principle for all motions by Heinrich Hertz.
Note 24(p. 48).VideA. Einstein, "Ann. d. Phys.," 4 Folge, Bd. 35, S. 898.
Note 25(p. 48). The expression "acceleration-transformation" means that the equations giving the transformation from the variables,,,to the system of variables,,,, which is the basis of our discussion, can be regarded as giving the relations between two systems of reference which are moving with anacceleratedmotion relatively to one another. The nature of the state of motion of two systems of reference relative to one another finds its expression in the analytical form of the equations of transformation of their co-ordinates.
Note 26(p. 51). Two things are to be undertaken in the following: (1) the fundamental equations of the new theory are to be written in an explicit form, and (2) the transition to Newton's fundamental equations is to be performed.
1. From the equation of variationwherewe have, after carrying out the operation of variation, the four total differential equations:These are the equations of motion of a material point in the gravitational field defined by the's.
The symbolhere denotes the expressionThe symboldenotes the minor ofin the determinantdivided by the determinant itself.
The ten differential equations for the "gravitational potentials" are:
The quantitiesandare expressions which are related in a simple manner to the components of the stress-energy-tensor(which plays the part of the quantity exciting the field in the new theory in place of the density of mass).is essentially equal to the gravitational constant of Newton's theory.
The differential equations(1)and(2)are the fundamental equations of the new theory. The derivation of these equations is carried out in detail in the tract by A. Einstein, "The Foundations of the General Principle of Relativity," J. A. Barth, Leipzig, 1916.
2. In order to obtain a connection between these equations and Newton's theory, we must make several simplifying assumptions. We shall first assume that the's differ only by quantities which are small compared with unity from the values given by the scheme:These values for the's characterize the case of the special theory of relativity, i.e. the case of the condition free of gravitation. We shall also assume that, at infinite distances, the's tend to, and do finally, assume the above values; that is, that matter does not extend into infinite space.
Secondly, we shall assume that the velocities of matter are small compared with the velocity of light, and can be regarded as small quantities of the first order. The quantitieswill then be infinitely small quantities of the first order, andwill equal 1, except for quantities ofthe second order. From the equations defining theit will then be seen that these quantities will be infinitely small, of the first order. If we neglect quantities of the second order, and finally assume that, for small velocities of matter, the changes of the gravitational field with respect to time are small (i.e. that the derivatives of the s with respect to time may be neglected in comparison with the derivatives taken with regard to the space-co-ordinates) the system of equations (1) assumes the form:This would be the equation of motion of a point-mass as already given by Newton's mechanics, ifbe taken as representing the ordinary gravitational potential. It still remains to be seen what the differential equation forbecomes in the new theory under the simplifying assumptions we have chosen.
The stress-energy-tensor, which excites the field, degenerates, as a result of our quite special assumptions, into the density of mass:In the differential equations (2) the second term on the left-hand side is the product of two magnitudes, which, according to the above arguments, are to be regarded as infinitely small quantities of the first order. Thus the second term, being of the second order of small quantities, may be dismissed. The first term, on the other hand, if we omit the terms differentiated with respect to time, as above (i.e. if we regard the gravitational field as "stationary"), reduces to:The differential equation forthus degenerates into Poisson's equation:Thus, to a first approximation (i.e. if one regards the velocity of light as infinitely great, and this is a characteristic feature of the classical theory, as was explained in detail in§ 3(b): if certain simple assumptions are made about the behaviour of the's at infinity; and if the time-changes of the gravitational field are neglected) the well-known equations of Newtonian mechanics emerge out of the differential equations of Einstein's theory, which were obtained from perfectly general beginnings.
Note 27(p. 53). The theory of surfaces, i.e. the study of geometry upon surfaces, makes it immediately apparent that the theorems, which have been established for any surface, also hold for any surface which can be generated by distorting the firstwithout tearing. For if two surfaces have a point-to-point correspondence, such that the line-elements are equalat corresponding points, then corresponding finite arcs, angles, and areas, etc., will be equal. One thus arrives at the same planimetrical theorems for the two surfaces. Such surfaces are called "deformable" surfaces. The necessary and sufficient condition that surfaces be continuously deformable is that the expression for the line-element of the one surfacecan be transformed into that for the other,According to Gauss, it isnecessarythat both surfaces have equalmeasures of curvature. If the latter is constant over the whole surface, as e.g. in the case of a cylinder or a plane, all conditions for the deformability of the surfaces are fulfilled. In other cases, special equations offer a criterion as to whether surfaces, or portions of surfaces, are deformable into one another. The numerous subsidiary problems, which result out of these questions, are discussed at length in every book dealing with differential geometry (e.g. Bianchi-Lukat).[17]This branch of training, which was hitherto of interest only to mathematicians, now assumes very considerable importance for the physicist too.
[17]Forsyth's "Differential Geometry."—H. L. B.
[17]Forsyth's "Differential Geometry."—H. L. B.
Note 28(p. 61). One must avoid being deceived into the belief that Newton's fundamental law is in any way to be regarded as anexplanationof gravitation. The conception of attractive force is borrowed from our muscular sensations, and has therefore no meaning when applied to dead matter. C. Neumann, who took great pains to place Newton's mechanics on a solid basis, glosses upon this point himself in a drastic fashion, in the following narrative, which shows up the weaknesses of the former view:
"Let us suppose an explorer to narrate to us his experiences in yonder mysterious ocean. He had succeeded in gaining access to it, and a remarkable sight had greeted his eyes. In the middle of the sea he had observed two floating icebergs, a larger and a smaller one, at a considerable distance from one another. Out of the interior of the larger one, a voice had resounded, issuing the following command in a peremptory tone: 'Ten feet nearer!' The little iceberg had immediately carried out the order, approaching ten feet nearer the larger one. Again, the larger gave out the order: 'Six feet nearer!' The other hadagain immediately executed it. And in this manner order after order had echoed out: and the little iceberg had continually been in motion, eager to put every command immediately and implicitly into action.
"We should certainly consign such a report to the realm of fables. But let us not scoff too soon! The ideas, which appear so extraordinary to us in this case, are exactly the same as those which lie at the base of the most complete branch of natural science, and to which the most famous of physicists owes the glory attached to his name.
"For in cosmic space such commands are continually resounding, proceeding from each of the heavenly bodies—from the sun, planets, moons, and comets. Every single body in space hearkens to the orders which the other bodies give it, always striving to carry them out punctiliously. Our earth would dash through space in a straight line, if she were not controlled and guided by the voice of command, issuing from moment to moment, from the sun, in which the instructions of the remaining cosmic bodies are less audibly mingled.
"These commands are certainly given just assilentlyas they are obeyed; and Newton has denominated this play of interchange between commanding and obeying by another name. He talks quite briefly of a mutual attractive force, which exists between cosmic bodies. But the fact remains the same. For this mutual influence consists in one body dealing out orders, and the other obeying them."