 This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recorded by Annie Coleman, www.anniecolman.com Relativity The Special and General Theory by Albert Einstein Continuing Part 2 Sections 21 through 23 Section 21 In what respects are the foundations of classical mechanics and of the special theory of relativity unsatisfactory? We have already stated several times that classical mechanics starts out from the following law. Material particles sufficiently far removed from other material particles continue to move uniformly in a straight line or continue in a state of rest. We have also repeatedly emphasized that this fundamental law can only be valid for bodies of reference k which possess certain unique states of motion and which are in uniform translational motion relative to each other. Relative to other reference bodies, k the law is not valid. Both in classical mechanics and in the special theory of relativity we therefore differentiate between reference bodies k relative to which the recognized laws of nature can be said to hold and reference bodies k relative to which these laws do not hold. But no person whose mode of thought is logical can rest satisfied with this condition of things. He asks, how does it come that certain reference bodies or their states of motion are given priority over other reference bodies or their states of motion? What is the reason for this preference? In order to show clearly what I mean by this question I shall make use of a comparison. I am standing in front of a gas range. Standing alongside of each other on the range are two pans so much alike that one may be mistaken for the other. Both are half full of water. I notice that steam is being emitted continuously from one pan but not from the other. I am surprised at this even if I have never seen either a gas range or a pan before. But if I now notice a luminous something of bluish color under the first pan but not under the other, I cease to be astonished even if I have never before seen a gas flame. For I can only say that this bluish something will cause the emission of the steam or at least possibly it may do so. If, however, I notice the bluish something in neither case and if I observe that the one continuously emits steam whilst the other does not then I shall remain astonished and dissatisfied until I have discovered some circumstance to which I can attribute the different behavior of the two pans. Analogously I seek in vain for a real something in classical mechanics or in the special theory of relativity to which I can attribute the different behavior of bodies compared with respect to the reference systems K and K prime. Begin footnote. The objection is of importance more especially when the state of motion of the reference body is of such a nature that it does not require any external agency for its maintenance. For example, in the case when the reference body is rotating uniformly. End footnote. Newton saw this objection and attempted to invalidate it but without success. But E. Mach recognized it most clearly of all and because of this objection he claimed that mechanics must be placed on a new basis. It can only be got rid of by means of a physics which is conformable to the general principle of relativity since the equations of such a theory hold for everybody of reference whatever may be its state of motion. Section 22. A few inferences from the general principle of relativity. The considerations of section 20 show that the general principle of relativity puts us in a position to derive properties of the gravitational field in a purely theoretical manner. Let us suppose for instance that we know the space-time course for any natural process whatsoever as regards the manner in which it takes place in the Galilean domain relative to a Galilean body of reference K. By means of purely theoretical operations i.e. simply by calculation we are then able to find how this known natural process appears as seen from a reference body K' which is accelerated relatively to K. But since the gravitational field exists with respect to this new body of reference K' our consideration also teaches us how the gravitational field influences the process studied. For example, we learn that a body which is in a state of uniform rectilinear motion with respect to K in accordance with the law of Galilei is executing an accelerated and in general curvilinear motion with respect to the accelerated reference body K'. Chest. This acceleration or curvature corresponds to the influence on the moving body of the gravitational field prevailing relatively to K'. It is known that a gravitational field influences the movement of bodies in this way. So that our consideration supplies us with nothing essentially new. However, we obtain a new result of fundamental importance when we carry out the analogous consideration for a ray of light. With respect to the Galilean reference body K' such a ray of light is transmitted rectilinearly with the velocity C'. It can easily be shown that the path of the same ray of light is no longer a straight line when we consider it with reference to the accelerated chest reference body K'. From this we conclude that in general rays of light are propagated curvilinearly in gravitational fields. In two respects this result is of great importance and the first place it can be compared with the reality. Although a detailed examination of the question shows that the curvature of light rays required by the general theory of relativity is only exceedingly small for the gravitational fields at our disposal in practice, its estimated magnitude for light rays passing the sun at grazing incidence is nevertheless 1.7 seconds of arc. This ought to manifest itself in the following way. As seen from the earth certain fixed stars appear to be in the neighborhood of the sun and are thus capable of observation during a total eclipse of the sun. At such times these stars ought to appear to be displaced outwards from the sun by an amount indicated above as compared with their apparent position in the sky when the sun is situated at another part of the heavens. The examination of the correctness or otherwise of this deduction is a problem of the greatest importance, the early solution of which is to be expected of astronomers. Begin footnote. By means of the star photographs of two expeditions equipped by a joint committee of the Royal and Royal Astronomical Societies the existence of the deflection of light demanded by theory was first confirmed during the solar eclipse of 29th May 1919. End footnote. In the second place our result shows that according to the general theory of relativity the law of the constancy of the velocity of light in vacuo which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity. Its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena. For example of light. Since it has often been contended by opponents of the theory of relativity that the special theory of relativity is overthrown by the general theory of relativity it is perhaps advisable to make the facts of the case clearer by means of an appropriate comparison. Before the development of electrodynamics the laws of electrostatics were looked upon as the laws of electricity. At the present time we know that electric fields can be derived correctly from electrostatic considerations only for the case which is never strictly realized in which the electrical masses are quite at rest relatively to each other and to the coordinate system. Should we be justified in saying that for this reason electrostatics is overthrown by the field equations of Maxwell and electrodynamics? Not in the least. Electrostatics is contained in electrodynamics as a limiting case. The laws of the latter lead directly to those of the former for the case in which the fields are invariable with regard to time. No fairer destiny could be allotted to any physical theory than that it should of itself point out the way to the introduction of a more comprehensive theory in which it lives on as a limiting case. In the example of the transmission of light just dealt with we have seen that the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes the laws of which are already known when a gravitational field is absent but the most attractive problem to the solution of which the general theory of relativity supplies the key concerns the investigation of the laws satisfied by the gravitational field itself. Let us consider this for a moment. We are acquainted with space-time domains which behave approximately in a Galilean fashion under suitable choice of reference body i.e. domains in which gravitational fields are absent. If we now refer such a domain to a reference body k' possessing any kind of motion then relative to k' there exists a gravitational field which is variable with respect to space and time. Begin footnote. This follows from a generalization of the discussion in section 20. End footnote. The character of this field will of course depend on the motion chosen for k' according to the general theory of relativity the general law of the gravitational field must be satisfied for all gravitational fields obtainable in this way. Even though by no means all gravitational fields can be produced in this way yet we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind. This hope has been realized in the most beautiful manner but between the clear vision of this goal and its actual realization it was necessary to surmount a serious difficulty and as this lies deep at the root of things I dare not withhold it from the reader. We require to extend our ideas of the space-time continuum still farther. Section 23. Behavior of clocks and measuring rods on a rotating body of reference. Hitherto I have purposefully refrained from speaking about the physical interpretation of space and time data in the case of this general theory of relativity. As a consequence I am guilty of a certain slovenliness of treatment which as we know from the special theory of relativity is far from being unimportant and pardonable. It is now high time that we remedy this defect but I would mention at the outset that this matter lays no small claims on the patience and on the power of abstraction of the reader. We start off again from quite special cases which we have frequently used before. Let us consider a space-time domain in which no gravitational field exists relative to a reference body K whose state of motion has been suitably chosen. K is then a Galilean reference body as regards the domain considered and the results of the special theory of relativity hold relative to K. Let us suppose the same domain referred to a second body of reference K' which is rotating uniformly with respect to K. In order to fix our ideas we shall imagine K' to be in the form of a plain circular disc which rotates uniformly in its own plane about its center. An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outward in a radial direction and which would be interpreted as an effect of inertia, centrifugal force by an observer who was at rest with respect to the original reference body K. But the observer on the disc may regard his disc as a reference body which is at rest on the basis of the general principle of relativity he is justified in doing this. The force acting on himself and in fact on all other bodies which are at rest relative to the disc he regards as the effect of a gravitational field. Nevertheless the space distribution of this gravitational field is of a kind that would not be possible on Newton's theory of gravitation. Begin footnote The field disappears at the center of the disc and increases proportionally to the distance from the center as we proceed outwards. And footnote But since the observer believes in the general theory of relativity this does not disturb him. He is quite in the right when he believes that a general law of gravitation can be formulated a law which not only explains the motion of the stars correctly but also the field of force experienced by himself. The observer performs experiments on his circular disc with clocks and measuring rods. In doing so it is his intention to arrive at exact definitions for the significance of time and space data with reference to the circular disc K prime. These definitions being based on his observations. What will be his experience in this enterprise? To start with he places one of two identically constructed clocks at the center of the circular disc and the other on the edge of the disc so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the non-rotating Galilean reference body K. As judged from this body the clock at the center of the disc has no velocity whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation. According to a result obtained in section 12 it follows that the latter clock goes at a rate permanently slower than that of the clock at the center of the circular disc i.e. as observed from K. It is obvious that the same effect would be noted by an observer who will imagine sitting alongside his clock at the center of the circular disc. Thus on our circular disc or to make the case more general in every gravitational field a clock will go more quickly or less quickly according to the position in which the clock is situated at rest. For this reason it is not possible to obtain a reasonable definition of time which are arranged at rest with respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case but I do not wish to go any farther into this question. Moreover at this stage the definition of the space coordinates also presents insurmountable difficulties. If the observer applies his standard measuring rod a rod which is short as compared with the radius of the disc tangentially to the edge of the disc then as judged from the Galilean system the length of this rod will be less than i since according to section 12 moving bodies suffer a shortening in the direction of the motion. On the other hand the measuring rod will not experience a shortening in length as judged from K if it is applied to the disc in the direction of the radius. If then the observer first measures the circumference of the disc with his measuring rod and then the diameter of the disc on dividing the one by the other he will not obtain as quotient the familiar number pi equals 3.14 etc. but a larger number. Begin footnote. Throughout this consideration we have to use the Galilean non-rotating system K as reference body since we may only assume the validity of the results of the special theory of relativity relative to K. Relative to K' a gravitational field prevails. End footnote. But a larger number whereas of course for a disc which is at rest with respect to K this operation would yield pi exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc nor in general in a gravitational field at least if we attribute the length I to the rod in all positions in every orientation. Hence the idea of a straight line also loses its meaning. We are therefore not in a position to define exactly the coordinates x, y, z relative to the disc by means of the method used in discussing the special theory and as long as the coordinates and times of events have not been defined we cannot assign an exact meaning to the natural laws in which these occur. Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour in order to be able to apply the postulate of general relativity exactly. I shall prepare the reader for this in the following paragraphs. End of sections 21 to 23 Read by Annie Coleman in St. Louis, Missouri on August 13, 2006 For more information or to volunteer please visit LibriVox.org Recording by Meredith Hughes, Cambridge, Massachusetts Relativity The Special and General Theory by Albert Einstein Continuing Part 2 The General Theory of Relativity Sections 24 through 26 Section 24 Euclidean and Non-Euclidean Continuum The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a neighboring one and repeating this process a large number of times or in other words by going from point to point without executing jumps. I am sure the reader will appreciate with sufficient clearness what I mean here by neighboring and by jumps if he is not too pedantic. We express this property of the surface by describing the latter as a continuum. Let us now imagine that a large number of little rods of equal length have been made their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length I mean that one can be laid on any other one without the ends overlapping. We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure, a square, the diagonals of which are equally long. To ensure the equality of the diagonals we make use of a little testing rod. To this square we add similar ones each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such that each side of a square belongs to two squares and each corner to four squares. It is a veritable wonder that we can carry out this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares needed a corner then two sides of the fourth square are already laid and as a consequence the arrangement of the remaining two sides of the square is already completely determined. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord then this is in a special favor of the marble slab and of the little rods about which I can only be thankfully surprised. We must need to experience many such surprises if the construction is to be successful. If everything has really gone smoothly then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod which has been used as a distance, line interval. By choosing one corner of a square as origin I can characterize every other corner of a square with reference to this origin by means of two numbers. I only need to state how many rods I must pass over when, starting from the origin, I proceed towards the right and then upwards in order to arrive at the corner of the square under consideration. These two numbers are then the Cartesian coordinates with reference to the Cartesian coordinate system which is determined by the arrangement of little rods. By making use of the following modification of this abstract experiment we recognize that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods expand by an amount proportional to the increase of temperature. We heat the central part of the marble slab but not the periphery in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating because the little rods on the central region of the table expand whereas those on the outer part do not. With reference to our little rods defined as unit lengths the marble slab is no longer a Euclidean continuum and we are also no longer in the position of defining Cartesian coordinates directly with their aid since the above construction can no longer be carried out. But since there are other things which are not influenced in a similar manner to the little rods or perhaps not at all by the temperature of the table it is possible, quite naturally to maintain the point of view that the marble slab is a Euclidean continuum. This can be done in a satisfactory manner by making a more subtle stipulation about the measurement or the comparison of lengths. But if rods of every kind i.e. of every material were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab and if we had no other means of detecting the effect of temperature than the geometrical behavior of our rods in experiments analogous to the one described above then our best plan would be to assign the distance 1 to 2 points on the slab provided that the ends of one of our rods could be made to coincide with these two points. For how else should we define the distance without our proceeding being in the highest measure grossly arbitrary? The method of Cartesian coordinates must then be discarded and replaced by another which does not assume the validity of Euclidean geometry for rigid bodies. Begin footnote. Mathematicians have been confronted with our problem in the following form. If we are given a surface e.g. an ellipsoid in Euclidean three-dimensional space then there exists for this surface a two-dimensional geometry just as much as for a plain surface. Gauss undertook the task of treating this two-dimensional geometry from first principles without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If we imagine constructions to be made with rigid rods in the surface similar to that above with the marble slab we should find that different laws hold for these from those resulting basis of Euclidean plain geometry. The surface is not a Euclidean continuum with respect to the rods and we cannot define Cartesian coordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface and thus pointed out the way to the method of Riemann of treating multi-dimensional non-Euclidean continuum. Thus it is that mathematicians long ago solved the formal problems to which the general postulate of relativity end footnote. The reader will notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity section 23 end of section 24 section 25 Gaussian coordinates According to Gauss this combined analytical and geometrical mode of handling the problem can be arrived at in the following way We imagine a system of arbitrary curves C figure 4 drawn on the surface of the table These we designate as U curves and we indicate each of them by means of a number The curves U equals 1 U equals 2 and U equals 3 are drawn in the diagram Between the curves U equals 1 and U equals 2 we must imagine an infinitely large number to be drawn which correspond to the real numbers lying between 1 and 2 We have then a system of U curves and this infinitely dense system covers the whole surface of the table These U curves must not intersect each other and through each point of the surface one and only one curve must pass Thus a perfectly definite value of U belongs to every point on the surface of the marble slab In like manner we imagine a system of V curves drawn on the surface These satisfy the same conditions as the U curves, they are provided with numbers in a corresponding manner and they may likewise be of arbitrary shape It follows that a value of U and a value of V belong to every point on the surface of the table We call these two numbers the coordinates of the surface of the table Gaussian coordinates For example, the point capital P in the diagram Gaussian coordinates U equals 3 V equals 1 Two neighboring points, capital P and capital P prime on the surface then correspond to the coordinates capital P colon U comma V capital P prime colon U plus du comma V plus dV Where dU and dV signify very small numbers In a similar manner we may indicate the distance interval between capital P and capital P prime as measured with a little rod by means of the very small number dS Then according to Gauss we have dS squared equals g sub 1 1 dU squared plus 2 g sub 1 2 dU dV plus g sub 2 2 dV squared Where g sub 1 1 g sub 1 2 g sub 2 2 are magnitudes which depend in a perfectly definite way on U and V The magnitudes g sub 1 1 g sub 1 2 and g sub 2 2 determine the behavior of the rods relative to the U curves and V curves and thus also relative to the surface of the table For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring rods but only in this case it is possible to draw the U curves and V curves and to attach numbers to them in such a manner that we simply have dS squared equals dU squared plus dV squared Under these conditions the U curves and V curves are straight lines in the sense of Euclidean geometry and they are perpendicular to each other Here the Gaussian coordinates are simply Cartesian ones It is clear that Gauss coordinates are nothing more than an association of two sets of numbers with the points of the surface considered of such a nature that numerical values differing slightly from each other are associated with neighboring points in space So far these considerations hold for a continuum of two dimensions but the Gaussian method can be applied also to a continuum of three, four, or more dimensions If for instance a continuum of four dimensions be supposed available we may represent it in the following way With every point of the continuum we associate arbitrarily four numbers X sub 1, X sub 2, X sub 3, X sub 4 which are known as coordinates Adjacent points correspond to adjacent values of the coordinates If a distance dS is associated with the adjacent points P and P prime this distance being measurable and well defined from a physical point of view then the following formula holds dS squared equals G sub 1, 1 dX sub 1 squared plus 2G sub 1, 2 dX sub 1 dX sub 2 dot dot dot plus G sub 4, 4 dX sub 4 squared where the magnitudes have values which vary with the position in the continuum Only when the continuum is a Euclidean one is it possible to associate the coordinates X sub 1 to X sub 4 with the points of the continuum so that we have simply dS squared equals dX sub 1 squared plus dX sub 2 squared plus dX sub 3 squared plus dX sub 4 squared In this case relations hold in the four dimensional continuum which are analogous to those holding in our three dimensional measurements However, the Gauss treatment for dS squared which we have given above is not always possible It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean Continua For example this obviously holds in the case of the marble slab of the table and local variation of the temperature The temperature is practically constant for a small part of the slab and thus the geometrical behavior of the rods is almost as it ought to be according to the rules of Euclidean geometry Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table We can sum this up as follows Gauss invented a method for the mathematical treatment of Continua in general in which size relations distances between neighboring points are defined To every point of a continuum are assigned as many numbers Gaussian coordinates as the continuum has dimensions This is done in such a way that only one meaning can be attached to the assignment and that numbers Gaussian coordinates which differ by an indefinitely small amount are assigned to adjacent points The Gaussian coordinate system is a logical generalization of the Gaussian coordinate system It is also applicable to non-Euclidean Continua but only one with respect to the defined size or distance Small parts of the continuum under consideration behave more nearly like Euclidean system the smaller the part of the continuum under our notice End of section 25 Section 26 The space-time continuum of the special theory of relativity considered as Euclidean Continuum We are now in a position to formulate more exactly the idea of Minkowski which was only vaguely indicated in section 17 In accordance with the special theory of relativity certain coordinate systems are given preference for the description of the four-dimensional space-time continuum We called these Galilean coordinate systems For these systems the four coordinates X, Y, Z, T which determine an event in other words, a point of the four-dimensional continuum are defined physically in a simple manner as set forth in detail in the first part of this book For the transition from one Galilean system to another which is moving uniformly with reference to the first the equations of the Lorentz transformation are valid These last form the basis for the derivation of deductions from the special theory of relativity and in themselves they are nothing more than the expression of the universal validity of the law of transmission of light for all Galilean systems of reference Minkowski found that the Lorentz transformations satisfy the following simple conditions Let us consider two neighboring events the relative position of which in the four-dimensional continuum is given with respect to a Galilean reference body capital K by the space coordinate differences dx, dy, dz and dt with reference to a second Galilean system we shall suppose that the corresponding differences for these two events are dx prime, dy prime dz prime, dt prime then these magnitudes always fulfill the condition begin footnote cf appendices one and two the relations which are derived there for the coordinates themselves are valid also for coordinate differences and thus also for coordinate differentials indefinitely small differences end footnote dx squared plus dy squared plus dz squared minus c squared dt squared equals dx prime squared plus dy prime squared plus dz prime squared minus c squared dt prime squared the validity of the Lorentz transformation follows from this condition we can express this as follows the magnitude ds squared equals dx squared plus dy squared plus dz squared minus c squared dt squared which belongs to two adjacent points of the four-dimensional space time continuum has the same value for all selected Galilean reference bodies if we replace x, y, z square root of quantity minus one and quantity ct by x sub one x sub two x sub three x sub four we also obtain the result that ds squared equals dx sub one squared plus dx sub two squared plus dx sub three squared plus dx sub four squared is independent of the choice of the body of reference we call the magnitude ds the distance apart of the two events or four-dimensional points is a variable square root quantity minus one and quantity ct instead of the real quantity t we can regard the space time continuum in accordance with the special theory of relativity as a Euclidean four-dimensional continuum a result which follows from the considerations of the preceding section end of section 26 this is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org relativity the special and general theory by Albert Einstein continuing part two the general theory of relativity sections 27 to 29 section 27 the space time continuum of the general theory of relativity is not a Euclidean continuum in the first part of this book we were able to make use of space time coordinates which allowed of a simple and direct physical interpretation and which according to section 26 can be regarded as four-dimensional Cartesian coordinates this was possible on the basis of the law of the constancy of the velocity of light but according to section 21 the general theory of relativity cannot retain this law on the contrary we arrived at the result that according to this latter theory the velocity of light must always depend on the coordinates when a gravitational field is present in connection with a specific illustration in section 23 we found that the presence of a gravitational field invalidates the definition of the coordinates and the time which led us to our objective in the special theory of relativity in view of the results of these considerations we are led to the conviction that according to the general principle of relativity the space time continuum cannot be regarded as a Euclidean one but that here we have the general case corresponding to the Marble slab with local variations of temperature and with which we made acquaintance as an example of a two-dimensional continuum just as it was there impossible to construct a Cartesian coordinate system from equal rods so here it is impossible to build up a system or reference body from rigid bodies and clocks which shall be of such a nature that measuring rods and clocks arranged rigidly with respect to one another shall indicate position and time directly which was the essence of the difficulty with which we were confronted in section 23 but the considerations of section 25 and 26 show us the way to surmount this difficulty we refer the four-dimensional space time continuum in an arbitrary manner to Gauss coordinates we assign to every point of the continuum or event four numbers X sub 3 and X sub 4 coordinates which have not the least direct physical significance but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner this arrangement does not even need to be of such a kind that we must regard X sub 1, X sub 2 and X sub 3 as space coordinates and X sub 4 as a time coordinate the reader may think that such a description of the world would be quite inadequate what does it mean to assign to an event the particular coordinates X sub 1, X sub 2, X sub 3 and X sub 4 if in themselves these coordinates have no significance more careful consideration shows however that this anxiety is unfounded let us consider for instance a material point with any kind of motion if this point had only a momentary existence without duration then it would be described in space-time by a single system of values X sub 1, X sub 2 X sub 3 and X sub 4 thus its permanent existence must be characterised by an infinitely large number of such systems of values the coordinate values of which are so close together as to give continuity corresponding to the material point we thus have a unidimensional line in the four-dimensional continuum in the same way any such lines in our continuum correspond to many points in motion the only statements having regard to these points which can claim a physical existence in reality the statements about their encounters in our mathematical treatment such an encounter is expressed in the fact that the two lines which represent the motions of the points in question have a particular system of coordinate values X sub 1, X sub 2, X sub 3 and X sub 4 in common after mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements when we were describing the motion of a material point relative to a body of reference we stated nothing more than the encounters of this point with particular points of the reference body we can also determine the corresponding values of the time by the observation of encounters of the body with clocks in conjunction with the observation of the encounter of the hands of clocks with particular points on the dials it is just the same in the case of space measurements by means of measuring rods as a little consideration will show the following statements hold generally every physical description resolves itself into a number of statements of which refers to the space-time coincidence of two events A and B in terms of Gaussian coordinates every such statement is expressed by the agreement of their four coordinates X sub 1, X sub 2, X sub 3 and X sub 4 thus in reality the description of the time-space continuum by means of Gauss coordinates completely replaces the description with the aid of a body of reference without suffering from the defects of the latter mode of description it is not tied down to the Euclidean character of the continuum which has to be represented section 28 exact formulation of the general principle of relativity we are now in a position to replace the provisional formulation of the general principle of relativity given in section 18 by an exact formulation the form they're used quote all bodies of reference K, K prime etc are equivalent for the description of natural phenomena or formulation of the general laws of nature whatever may be their state of motion unquote cannot be maintained because the use of rigid reference bodies in the sense of the method of special theory of relativity is in general not possible in space-time description the Gauss coordinate system has to take the place of the body of reference the following statement corresponds to the fundamental idea of the general principle of relativity all Gaussian coordinate systems are essentially equivalent for the formulation of the general laws of nature we can state this general principle of relativity in still another form which renders it yet more clearly intelligible than it is when in the form of the natural extension of the special principle of relativity according to the special theory of relativity the equations which express the general laws of nature pass over into equations of the same form when by making use of the Lorentz transformation we replace the space-time variables X, Y, Z and T of a Galilean reference body K by the space-time variables X prime, Y prime, Z prime and T prime of a new reference body K prime according to the general theory of relativity on the other hand by application of arbitrary substitutions of the Gauss variables X sub 1 X sub 2 X sub 3 and X sub 4 the equations must pass over into equations of the same form for every transformation not only the Lorentz transformation corresponds to the transition of one Gauss coordinate system into another if we desire to adhere to our old-time three-dimensional view of things then we can characterise the development which is being undergone by the fundamental idea of the general theory of relativity as follows the special theory of relativity has reference to Galilean domains i.e. to those in which no gravitational field exists in this connection a Galilean reference body serves as body of reference i.e. a rigid body the state of motion of which is so chosen by the Galilean law of the uniform rectilinear motion of isolated material points holds relatively to it certain considerations suggest that we should refer the same Galilean domains to non-Galilean reference bodies also a gravitational field of a special kind is then present with respect to these bodies cf. sections 20 and 23 in gravitational fields there are no such things as rigid bodies with Euclidean properties thus the fictitious rigid body of reference is of no avail in the general theory of relativity the motion of clocks is also influenced by gravitational fields and in such a way that a physical definition of time which is made directly with the aid of clocks has by no means the same degree of plausibility as in the special theory of relativity for this reason non-rigid reference bodies are used which are as a whole not only moving in any way whatsoever but which also suffer alterations in form and lib during their motion clocks for which the law of motion is of any kind however irregular serve for the definition of time we have to imagine these clocks fixed at a point on the non-rigid reference body these clocks satisfy only the one condition that the readings which are observed simultaneously on adjacent clocks in space differ from each other by an indefinitely small amount this non-rigid reference body which might appropriately be termed a reference mollusk is in the main equivalent to a Gaussian four-dimensional coordinate system chosen arbitrarily that which gives the mollusk a certain comprehensibility as compared with the Gauss coordinate system is the really unjustified formal retention of the separate existence of the space coordinates as opposed to the time coordinate every point on the mollusk is treated as a space point and every material point which is at rest relatively to it so long as the mollusk is considered as reference body the general principle of relativity requires that all these mollusks can be used as reference bodies with equal right and equal success in the formulation of the general laws of nature the laws themselves must be quite independent of the choice of mollusk the great power possessed by the general principle of relativity lies in comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above section 29 the solution of the problem of gravitation on the basis of the general principle of relativity if the reader has followed all our previous considerations he will have no further difficulty in understanding the methods leading to the solution of the problem of gravitation we start off on a consideration of a Galilean domain i.e. a domain in which there is no gravitational field relative to the Galilean reference body k the behaviour of measuring rods and clocks with reference to k is known from the special theory of relativity likewise the behaviour of isolated material points the latter move uniformly and in straight lines now let us refer this domain to a random Gauss coordinate system or to a mollusk as reference body k prime then with respect to k prime there is a gravitational field g of a particular kind we learn the behaviour of measuring rods and clocks and also of freely moving material points with reference to k prime simply by mathematical transformation we interpret this behaviour as the behaviour of measuring rods clocks and material points under the influence of the gravitational field g here upon we introduce a hypothesis that the influence of the gravitational field on measuring rods clocks and freely moving material points continues to take place according to the same laws even in the case where the prevailing gravitational field is not derivable from the Galilean special case simply by means of a transformation of coordinates the next step is to investigate the spacetime behaviour of the gravitational field g which was derived from the Galilean special case simply by transformation of the coordinates this behaviour is formulated in a law which is always valid to find out how the reference body or mollusk used in the description may be chosen this law is not yet the general law of the gravitational field since the gravitational field under consideration is of a special kind in order to find out the general law of field of gravitation we still require to obtain a generalization of the law as found above however by taking into consideration the following demands a. the required generalization must likewise satisfy the general postulate of relativity b. if there is any matter in the domain under consideration only its inertial mass and thus according to section 15 only its energy is of importance for its effect in exciting a field c. gravitational field and matter together must satisfy the law of the conservation of energy and of impulse finally the general principle of relativity permits us to determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent i.e. which have already been fitted into the frame of the special theory of relativity in this connection we proceed in principle according to the method which has already been explained for measuring rods, clocks and freely moving material points the theory of gravitation derived in this way from the general postulate of relativity excels not only in its beauty nor in removing the defect attaching to classical mechanics which was brought to light in section 21 nor in interpreting the empirical law of the equality of inertial and gravitational mass but it has also already explained a result of observation in astronomy against which classical mechanics is powerless if we can find the application of the theory to the case where the gravitational fields can be regarded as being weak and in which all masses move with respect to the coordinate system with velocities which are small compared with the velocity of light we then obtain as a first approximation the Newtonian theory thus the latter theory is obtained here without any particular assumption whereas Newton had to introduce the hypothesis that the force of attraction between mutually attracting material points is inversely proportional to the square of the distance between them if we increase the accuracy of the calculation deviations from the theory of Newton make their appearance practically all of which must nevertheless escape the test of observation owing to their smallness we must draw attention here to one of these deviations according to Newton's theory a planet moves around the sun in an ellipse which would permanently maintain its position with respect to the fixed stars if we could disregard the motion of the fixed stars themselves and the action of the other planets under consideration thus if we correct the observed motion of the planets for these two influences and if Newton's theory be strictly correct we ought to obtain for the orbit of the planet an ellipse which is fixed with reference to the fixed stars this deduction which can be tested with great accuracy has been confirmed for all the planets save one with the precision that is capable of being obtained by the delicacy of observation attainable at the present time the sole exception is Mercury the planet which lies nearest the sun since the time of Laverier it has been known that the eclipse corresponding to the orbit of Mercury after it has been corrected as mentioned above is not stationary with respect to the fixed stars but that it rotates exceedingly slowly in the plane of the orbit and in the sense of the orbital motion the value obtained for this rotary movement of the orbital ellipse was 43 seconds of arc per century an amount ensured to be correct to within a few seconds of arc this effect can be explained by the means of classical mechanics only on the assumption of hypotheses which have little probability and which were devised solely for this purpose on the basis of the general theory of relativity it is found that the ellipse of every planet around the sun must necessarily rotate in the manner indicated above that for all the planets with the exception of Mercury this rotation is too small to be detected with the delicacy of observation possible at the present time but that in the case of Mercury it must amount to 43 seconds of arc per century a result which is strictly in agreement with observation apart from this one it has hitherto been possible to make only two deductions from the theory which admit of being tested by observation to wit the curvature of light rays the gravitational field of the sun first observed by Eddington and others in 1919 and the displacement of the spectral lines of light reaching us from large stars as compared with the corresponding lines for light produced in an analogous manner terrestrally i.e. by the same kind of atom established by Adams in 1924 these two deductions from the theory have both been confirmed this is the LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org recording by Linda Lu Relativity the special and general theory by Albert Einstein considerations on the universe as a whole sections 30, 31 and 32 section 30 cosmological difficulties of Newton's theory apart from the difficulty discussed in section 21 there is a second fundamental difficulty attending classical celestial mechanics which to the best of my knowledge was first discussed in detail by the astronomer Seliger if we pondered over the question as to how the universe considered as a whole is to be regarded the first answer that suggests itself to us is surely this as we guard space and time universe is infinite there are stars everywhere so that the density of matter although very variable in detail is nevertheless on the average everywhere the same in other words however far we might travel through space we should find everywhere in attenuated swarm of fixed stars of approximately the same kind and density this view is not in harmony with the theory of Newton the latter theory rather requires that the universe should have a kind of center in which the density of the stars is a maximum and that as we proceed towards from the center the group density of the stars should diminish until finally at great distances it is succeeded by an infinite region of emptiness the stellar universe ought to be a finite island in the infinite ocean of space begin with note proof according to the theory of Newton the number of lines of force which come from infinity and terminate in a mass M is proportional to the mass M if on the average the mass density rho sub zero is constant throughout the universe then a sphere of volume v will enclose the average man rho sub zero v thus the number of lines of force passing through the surface f of the sphere into its interior rho sub zero v for unit area of the surface of the sphere the number of lines of force which enters the sphere is thus proportional to rho sub zero v over f or 2 rho sub zero r hence the intensity of the field at the surface would ultimately become infinite with increasing radius r of the sphere which is impossible end footnote this conception isn't itself not very satisfactory it is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space never to return and without ever again coming into interaction with other objects of nature such a finite material universe would be destined to become gradually but systematically impoverished in order to escape this dilemma Seliger suggested a modification of Newton's law in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly then would result from the inverse square law in this way it is possible for the mean density of matter to be constant everywhere even to infinity without infinitely large gravitational fields being produced we thus free ourselves from the distasteful conception that the material universe ought to possess something of the nature of a center of course we purchase our emancipation from the fundamental difficulties mentioned at the cost of a modification and complication of Newton's law which is neither empirical nor theoretical foundation we can imagine innumerable laws which would serve the same purpose without our being able to state a reason why one of them is to be preferred to the others for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton and a section 30 section 31 the possibility of a finite and yet unbounded universe but speculations on the structure of the universe also move in quite another direction the development of non-euclidean geometry led to the recognition of the fact that we can cast out on the infiniteness of our space without coming into conflict with the laws of thought or with experience Reimann, Helmholtz these questions have already been treated in detail and with unsurpassable acidity by Helmholtz and Poincare whereas I can only touch on them briefly here in the first place we imagine an existence in two-dimensional space flat beings with flat implements and in particular flat rigid measuring rods are free to move in a plane for them nothing exists outside of this plane that which they observe to happen to themselves and to their flat things is the all-inclusive reality of their plane in particular the constructions of plane euclidean geometry can be carried out by means of the rods for example the lattice construction considered in section 24 in contrast to ours the universe of these beings is two-dimensional but like ours it extends to infinity in their universe there is room for an infinite number of identical squares made up of rods i.e. its volume surface is infinite if these beings say their universe is quote, plane there is sense in this statement because they mean that they can perform the constructions of plane euclidean geometry with their rods in this connection the individual rods always represent the same distance independently of their position let us consider now a second two-dimensional existence but this time on a spherical surface instead of on a plane the flat beings with their measuring rods and other objects fit exactly on the surface they are unable to leave it their whole universe of observation extends exclusively over the surface of the sphere are these beings able to regard the geometry of their universe as being plane geometry and their rods with all as a realization of distance they cannot do this for if they attempt to realize a straight line they will obtain a curve which we three-dimensional beings designate as a great circle i.e. a self-contained line of definite finite length which can be measured up by means of a measuring rod similarly this universe has a finite area that can be compared with the area of a square constructed with rods great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits but the spherical surface beings do not need to go on a world tour in order to perceive that they are not living in a Euclidean universe they can convince themselves of this on every part of their world provided they do not use too small a piece of it starting from a point they draw straight lines arcs of circles are stretched in three-dimensional space of equal length in all directions they will call the line joining the free ends of these lines a circle for a plane surface the ratio of the circumference of a circle to its diameter both lengths being measured with the same rod is, according to Euclidean geometry of the plane equal to a constant value which is independent of the diameter of the circle on their spherical surface our flat beings would find for this ratio the value equation 27 pi times sine parentheses little r over big r and parentheses divided by parentheses little r over big r ie smaller value than pi the difference being more considerable greater is the radius of the circle in comparison with the radius r of the world sphere by means of this relation spherical beings can determine the radius of their universe quote world unquote even when only a relatively small part of their world sphere for their measurements but if this part is very small indeed they will no longer be able to demonstrate that they are on a spherical world and not on a Euclidean plane for a small part of a spherical surface differs only slightly from a piece of a plane of the same size thus if the spherical surface beings are living on a planet which the solar system occupies only a negligibly small part of the spherical universe they have no means of determining whether they are living in a finite or an infinite universe because a piece of universe to which they have access is in both cases practically plain or Euclidean it follows directly from this discussion that for our sphere beings circumference of a circle first increases with radius until the circumference of the universe is reached and that it then forward gradually decreases to 0 for still further increasing values of the radius during this process the area of the circle continues to increase more and more until finally becomes equal to the total area of the whole world sphere perhaps the reader will wonder why we have placed our beings in this sphere rather than on another closed surface but this choice has its justifications in the fact that of all closed surfaces the sphere is unique in possessing the property that all points on it are equivalent I admit that the ratio of the circumference C of a circle to its radius R depends on R but for a given value of R it is the same for all points of the world sphere in other words the world sphere is a surface of constant curvature to this two-dimensional sphere universe there is a three-dimensional analogy namely the three-dimensional spherical space which was discovered by Riemann its points are likewise all equivalent it possesses a finite volume which is determined by its radius 2 pi squared R cubed is it possible to imagine a spherical space? to imagine a space means nothing else than that we imagine an epitome of our space experience i.e. of experience that we can have in the movement of rigid bodies in this sense we can imagine a spherical space suppose we draw lines or stretch strings in all directions from a point and mark off from each of these or the measuring rod all the free ends of these lengths lie on a spherical surface we can specially measure up the area f of the surface by means of a square made up of measuring rods if the universe is Euclidean then f equals 4 pi R squared if it is spherical then f is always less than 4 pi R squared with increasing values of R f increases from 0 up to a maximum value which is determined by the world radius but for still further increasing values of R the area gradually diminishes to 0 at first the straight lines which radiate from the starting point diverge farther and farther from one another but later they approach each other and finally they run together again at a counterpoint to the starting point under such conditions they have traversed the whole spherical space it is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface it is finite i.e. a finite volume and has no bounds it may be mentioned that there is yet another kind of curved space quote elliptical space it can be regarded as a curved space in which the two counterpoints are identical indistinguishable from each other an elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry it follows from what has been said that closed spaces without limits are conceivable from among these spherical space and the elliptical excels in its simplicity since all points on it are equivalent as a result of this discussion the most interesting question arises for astronomers and physicists and that is whether the universe in which we live is infinite or whether it is finite in the manner of the spherical universe our experience is far from being sufficient for us to answer this question but the general theory of relativity permits of our answering it with a moderate degree of certainty and in this connection the difficulty mentioned in section 30 finds its solution end of section 31 section 32 structure of space according to the general theory of relativity according to the general theory of relativity the geometrical properties of space are not independent but they are determined by matter thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known we know from experience that for a suitably chosen coordinate system the velocities of the stars are small as compared with the velocity of transmission of light we can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole if we treat the matter as being at rest we already know from our previous discussion that the behavior of measuring rods and clocks is influenced by gravitational fields i.e. by the distribution of matter this in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe but it is conceivable that our universe differs only slightly from a Euclidean one and this notion seems all the more probable since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun we might imagine that as regards geometry the universe behaves analogously to a surface which is irregularly curved in its individual parts but which nowhere departs appreciably from a plane something like the rippled surface of a lake such a universe might fittingly be called a quasi Euclidean universe as regards its space it would be infinite but calculation shows that in a quasi Euclidean universe the average density of matter would necessarily be nil thus such a universe could not be inhabited by matter everywhere it would present to us that unsatisfactory picture which we portrayed in section 30 if we are to have in the universe an average density of matter which differs from zero however small may be that difference then the universe cannot be quasi Euclidean on the contrary the results of calculation indicate that if matter be distributed uniformly the universe would necessarily be spherical or elliptical since in reality the detailed distribution of matter is not uniform the real universe will deviate in individual parts from the spherical i.e. the universe will be quasi spherical but it will be necessarily for night in fact the theory supplies us with a simple connection begin footnote for the radius r of the universe we obtain the equation r squared equals 2 over kappa rho the use of the cgs system in this equation gives 2 over k equals 1 to the 8th power times 10 to the 27th power p is the average density of the matter and k is a constant connected with the Newtonian constant of gravitation n footnote between the space expanse of the universe and the average density of matter in it end section 32 end of part 3 end of relativity special in general theory by Albert Einstein the recordings are in the public domain for more information or to volunteer please visit LibriVox.org recording by ML Cohen Cleveland Ohio March 2007 relativity the special in general theory by Albert Einstein appendix 3 the experimental confirmation of the general theory of relativity from a systematic theoretical point of view we may imagine the process of evolution of an empirical science to be a continuous process of induction theories are evolved and are expressed in short compass as statements of a large number of individual observation in the form of empirical laws from which the general laws can be ascertained by comparison regarded in this way the development of a science bears some resemblance to the compilation of a classified catalog it is, as it were, a purely empirical enterprise but this point of view by no means embraces the whole of the actual process for it slurs over the important part played by intuition and deductive thought in the development of an exact science as soon as a science has emerged from its initial stages theoretical advances are no longer achieved merely by a process of arrangement guided by empirical data the investigator rather develops a system of thought which, in general is built up logically from a small number of fundamental assumptions the so-called axioms we call such a system of thought a theory the theory finds the justification for its existence in the fact that it correlates a large number of single observations and it is just here that the truth of the theory lies corresponding to the same complex of empirical data there may be several theories which differ from one another to a considerable extent but as regards the deduction from the theories which are capable of being tested the agreement between the theories may be so complete that it becomes difficult to find such deductions in which the two theories differ from each other as an example a case of general interest is available in the province of biology in the Darwinian theory of the development of species by selection in the struggle for existence and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characters we have another instance of far-reaching agreement between the deductions from the two theories and Newtonian mechanics on the one hand and the general theory of relativity on the other this agreement goes so far that up to the present we have been able to find only a few deductions to the theory of relativity which are capable of investigation and to which the physics of pre-relativity days does not also lead and this despite the profound difference in the fundamental assumption of the two theories in what follows we shall again consider these important deductions and we shall also discuss the empirical evidence appertaining to them which has hitherto been obtained a prediction of the perihelion of Mercury according to Newtonian mechanics and Newton's law of gravitation a planet which is revolving around the sun would describe an ellipse around the ladder or more correctly around the common center of gravity of the sun and the planet in such a system the sun or the common center of gravity lies in one of the foci of the orbital ellipse in such a manner that in the course of a planet year a distant sun-planet grows from a minimum to a maximum and then decreases again to a minimum if instead of Newton's law we insert a somewhat different law of attraction into the calculation we find that according to this new law the motion would still take place in such a manner that the distant sun-planet exhibits periodic variations but in this case the angle described by the line joining sun and planet during such a period prints these parallelian closest proximity to the sun to parallelian and prints these would differ from 360 degrees the line of the orbit would not then be a closed one but in the course of time it would fill up an annular part of the orbital plane that is between the circle of least and the circle of greatest distance of the planet from the sun according also to the general theory of relativity which differs of course from the theory of Newton a small variation from the Newton Kepler motion of a planet in its orbit should take place and in such a way that the angle described by the radius sun-planet between one parallelian and the next should exceed that corresponding to one complete revolution by an amount given by the formula plus 24 pi cubed a squared divided by t squared c squared times the quantity 1 minus e squared nb one complete revolution corresponds to the angle 2 to the pi power in the absolute angle measure customary in physics and the above expression gives the amount by which the radius sun-planet exceeds this angle during the interval between one parallelian and the next close prints in this expression a represents the major semi-axis of the ellipse e its eccentricity c the velocity of light and t the period of revolution of the planet our result may also be stated as follows according to the general theory of relativity the major axis of the ellipse rotates around the sun in the same sense as the orbital motion of the planet theory requires that this rotation should amount the 43 seconds of arc per century for the planet Mercury but for the other planets of our solar system its magnitude should be so small that it would necessarily escape detection footnote especially since the next planet Venus has an orbit that is almost an exact circle which makes it more difficult to locate the perihelion with precision end footnote in point of fact astronomers have found that the theory of Newton does not suffice to calculate the observed motion of Mercury with an exactness corresponding to that of the delicacy of observation attainable at the present time after taking account of all the disturbing influences exerted on Mercury by the remaining planets it was found levier 1859 in newcom 1895 close that an unexplained perihelion movement of the orbit of Mercury remained over the amount of which does not differ sensibly from the above mentioned plus 43 seconds of arc per century the uncertainty of the empirical results amounts to a few seconds only b. deflection of light by a gravitational field in section 22 it has been already mentioned that according to the general theory of relativity a ray of light will experience a curvature of its path when passing through a gravitational field this curvature being similar to that experienced by the path of a body which is projected through a gravitational field as a result of this theory we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter for a ray of light which passes the Sun at a distance of delta Sun radii from its center the angle of deflection parentheses alpha close should amount to alpha equals 1.7 seconds of arc divided by delta it may be added that according to the theory half of this deflection is produced by the Newtonian field of attraction of the Sun and the other half by the geometrical modification parentheses quote curvature and parentheses of space caused by the Sun this result admits of an experimental test by means of the photographic registration of stars during a total eclipse of the Sun the only reason why we must wait for a total eclipse is because every other time the atmosphere is so strongly illuminated by the light from the Sun that the stars situated near the Sun's disc are invisible the predicted effect can be seen clearly from the accompanying diagram readers annotation figure five the Earth is shown as a dot at the bottom of the diagram a straight line proceeding from there labeled D sub 1 proceeds upward and slightly to the right passing the Sun at a tangent Sun being represented by a circle a second line labeled D sub 2 starts at the Earth proceeds at a relatively smaller angle which results in its passing the Sun at a greater distance than the initial line D1 which is signified by the symbol delta after passing the Sun the line becomes parallel to D sub 1 end of readers annotation if the Sun parentheses S and parentheses were not present a star which is practically infinitely distant would be seen in the direction D sub 1 as observed from the Earth but as a consequence of the deflection of light from the star by the Sun the star will be seen in the direction D sub 2 that is at a somewhat greater distance from the center of the Sun than corresponds to its real position in practice the question is tested in the following way the stars in the neighborhood of the Sun are photographed during a solar eclipse in addition a second photograph of the same stars is taken when the Sun is situated at another position in the sky that is a few months earlier later as compared with the standard photograph the positions of the stars on the eclipse photograph ought to appear displaced radially outwards parentheses away from the center of the Sun close brands by an amount corresponding to the angle A we are indebted to the royal society and to royal astronomical society for the investigation of this important deduction undaunted by the war the difficulties of both a material and a psychological nature aroused by the war these societies equip two expeditions to Sobral, Brazil and to the island of Príncipe, West Africa and sent several of Britain's most celebrated astronomers parentheses Eddington Cottingham, Cromelin Davidson and Prins in order to obtain photographs of the solar eclipse of 29th May 1919 the relative discrepancies to be expected between the stellar photographs obtained during the eclipse and a comparison photographs amounted to a few hundredths of a millimeter only thus great accuracy was necessary in making the adjustments required for taking of the photographs and in their subsequent measurement the results of the measurements confirm the theory in a thoroughly satisfactory manner the rectangular components of the observed and of the calculated deviation of the stars, parentheses and seconds of an arc and parentheses are set forth in the following table of results readers annotation the table consists of measurements on seven stars which are then tabulated in four additional columns which are entitled first coordinate and second coordinate and then for each of those the observed and calculated measurements are given and readers annotation number of the star 11 first coordinate, observed minus 0.19, calculated minus 0.22 second coordinate, observed plus 0.16, calculated plus 0.02 star numbered 5 first coordinate, observed plus 0.29, calculated plus 0.31 second coordinate, observed negative 0.46, calculated minus 0.43 star number 4, observed 0.11, calculated 0.10 second coordinate, observed 0.83, calculated plus 0.74 star number 3, observed plus 0.20 plus 0.20, calculated plus 0.12 second coordinate, observed plus 1.00 calculated plus 0.87 star number 6, observed at the first coordinate plus 0.10, calculated plus 0.04 second coordinate, observed plus 0.57 calculated plus 0.40 number the star 10, observed minus 0.08 calculated plus 0.09 second coordinate, observed plus 0.35 calculated plus 0.32 number the star 2 observed plus 0.95 calculated plus 0.85 and at the second coordinate observed minus 0.27 calculated minus 0.09 C displacement of the spectral line towards the red in section 23 it has been shown that in a system K prime which is in rotation with regard to a Galilean system K clocks of identical construction and which are considered at rest with respect to the rotating reference body go at rates which are dependent on the position of the clocks we shall now examine this dependence quantitatively a clock which is situated at a distance r from the center of the disk has a velocity relative to K which is given by v equals omega r where omega represents the angular velocity of rotation of the disk K prime with respect to K if v sub 0 represents the number unit time relative to K when the clock is at rest then the quote rate of the clock when it is moving relative to K with a velocity v but at rest with respect to the disk will in accordance with section 12 be given by v equals v sub 0 times the square root of 1 minus v squared over c squared or with sufficient accuracy by v equals v 0 times the quantity 1 minus 1 half v squared over c squared this expression may be also stated in the following form v equals v sub 0 times the quantity 1 minus 1 over c squared times omega squared r squared over 2 if we represent the difference of potential of the centrifugal force between the position of the clock and the center of the disk by phi that is the work considered negatively which must be performed on the unit of mass against the centrifugal force in order to transport it from the position of the clock on the rotating disk to the center of the disk then we have phi equals minus omega squared r squared divided by 2 from this it follows that v times the quantity 1 plus phi over c squared in the first place we see from this expression that the two clocks of identical construction will go at different rates when situated at different distances from the center of the disk this result is also valid from the standpoint of an observer who is rotating with the disk now as judged from the disk the latter is in the gravitational field phi hence the result we have obtained will hold quite generally for gravitational fields furthermore we can regard an atom which is emitting spectral lines as a clock so that the following statement will hold an atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated the frequency of an atom situated on the surface of a heavenly body will be somewhat less than the frequency of an atom of the same element which is situated in free space or on the surface of a smaller celestial body close parentheses period now phi equals minus k times m over r where k is Newton's constant of gravitation and m is the mass of the heavenly body thus the displacement towards the red ought to take place for spectral lines produced at the surface of stars as compared with the spectral lines of the same element produced at the surface of the earth the amount of this displacement being v sub zero minus v divided by v sub zero equals k over c squared times m over r for the sun the displacement towards the red predicted by theory amounts to about 2 millionths of the wavelength a trustworthy calculation is not possible in the case of the stars in general neither the mass m nor the radius r is known it is an open question whether or not this effect exists and at the present time astronomers are working with great zeal towards the solution owing to the smallness of the effect in the case of the sun it is difficult to form an opinion as to its existence whereas greb and bachem parentheses bond and prens as a result of their own measurements and schwarzschild on the cyanogen vans have placed the existence of the effect almost beyond doubt other investigators particularly saint john have been led to the opposite opinion in consequence of their measurements mean displacements of lines towards the less-refrangable end of the spectrum are certainly revealed by statistical investigation of the thick stars but up to the present the examination of the available data does not allow of any definite decision being arrived at as to whether or not these displacements are to be referred in reality to the effect of gravitation the results of observation have been collected together and discussed in detail from the standpoint of the question which has been engaging our attention here in a paper by E. Freundlich entitled Sir Profunder, Alamingan, Rebatat's Theory parentheses Dionysos and Schafton 1919 number 35 page 520 Julia Springer, Berlin close friends period at all events a definite decision will be reached during the next few years if the displacement of spectral lines towards the red by the gravitational potential does not exist then the general theory of relativity will be untenable on the other hand if the cause of the displacement of spectral lines be definitely traced to the gravitational potential then the study of this displacement will furnish us with important information as to the mass of the heavenly bodies end of appendix 3