APPENDIX

Fizeau's experiment was designed to decide finally whether moving matter 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.

Fig. 1.

Now, there immediately appear to be two possibilities. Either the water that flows with a velocity with 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 is in the one limb, and in the other, for, on account of the coefficient of refraction of the water, is the velocity of light in resting water. Or, the motion of the water has no influence at all on the ether which transmits light and which permeates the water. In this case the velocity of light is in 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 of neither of 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 velocity in water, and not the full value . This value of the convection of the ether is called Fresnel'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 is absolutely 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 on p. 74.

A ray of light, starting out from , traverses the course : here and , are two mirrors, on to which the ray falls perpendicularly; is a glass plate 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 .

Fig. 2.

Then the velocity of the light relative to the instrument (earth) is as follows in the directions specified: Consequently, the course is traversed in the time and the course in the time The difference of these two times is If we exchange the positions of , and , by turning the whole apparatus through 90°, then

If we make these tyro rays of light interfere at then, 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 by the vibration frequency of the light-ray used in the experiment, then is the corresponding wave-length. Thus, expressed in fractions of the interval between the fringes, the expected displacement becomes equal to By causing the light to be reflected many times was magnified to such an extent that became of the order . If, for example, , = the wavelength of sodium light, then . On the other hand is of the order that 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.

The assumptions for the general equations of transformation by which two systems and ' that move uniformly and rectilinearly with 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 system to the system ', and the second ' to '' ( is to have the constant velocity with 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 the parameter . 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 the magnitude of 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 of section 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 is not demanded in either of the postulates 1 and 2. Rather, the distinguishing property of a definite velocity in virtue of which it preserves its value in all systems 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.

We assume that a point-mass emits a light-wave of energy in a certain direction, and a light-wave of the same energy in 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 energy of the point-mass be referred to this system, but referred to a second system which we suppose moving with the uniform velocity with respect to the first. We shall apply the principle of energy to this process. If and are 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 system According 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 volume of 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 is Hence the energy-densities in the accented and unaccented system are in the ratio: If we now designate the energy-content of the point-mass after the emission by , and the corresponding quantity referred to the moving system by , then we have: whereas From this we get directly that What does this equation assert?

and are 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-mass is at rest. Hence the difference , except for an additive constant, must be equal to the kinetic energy of the point-mass referred to the moving system. Thus, we may write wherein denotes 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 energy as light, its kinetic energy referred to a moving system sinks from the value to the value , corresponding to a loss in inertial mass of the amount . For, according to classical mechanics, the expression in which is 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 . Thus must be taken as standing for the inertial mass of an amount of energy .

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 the same pitch but different intensity (analogous to the two points on the floor) with the "distance" between two tones of different pitch but equal 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 the three "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 a continuous manifold. 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 where congruence is 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.

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 the continuous connection 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 which measures this distance will depend upon the values of , , , , and for every intermediate point of the path which has conducted us from , to the point , will successively assume certain continually changing, and, as we may suppose, continually increasing, 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 , , and the 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 function fulfils all these requirements; but it is by no means the only function of this kind.

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 and not a 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 certain definite values, all distances in space would be obtained by "counting" these values, and we should have to regard space as a discrete manifold.

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 no inertial 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.

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.

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.

"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 endeavour to 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 the vessel alone assumed 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 the rotation of the walls of the vessel relative to 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.

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.

Whereas the earlier investigations of Newton and Bessel ("Astr. Nachr." 10, S. 97, and "Abhandlungen von Bessel," Bd. 3, S. 217), about the attractive 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 be th seconds; that is, slightly more than th 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 of the 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 of th 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.

1. From the equation of variation where we 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 symbol here denotes the expression The symbol denotes the minor of in the determinant divided by the determinant itself.

The ten differential equations for the "gravitational potentials" are:

The quantities and are 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 quantities will then be infinitely small quantities of the first order, and will equal 1, except for quantities of the second order. From the equations defining the it 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, if be taken as representing the ordinary gravitational potential. It still remains to be seen what the differential equation for becomes 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 for thus 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.

"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 had again 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 as silently as 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."