# Reading Help Relativity: The Special and General Theory

Galilei) is executing an accelerated and in general curvilinear motion `

` with respect to the accelerated reference-body K1 (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 Galileian 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 `

` K1). 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. `

` `

` In 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 neighbourhood 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.[2]* `

` `

` 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 unlinlited domain of validity ; `

` its results hold only so long as we are able to disregard the `

` influences of gravitational fields on the phenomena (e.g. 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 realised, in `

` which the electrical masses are quite at rest relatively to each `

` other, and to the co-ordinate system. Should we be justified in saying `

` that for this reason electrostatics is overthrown by the `

` field-equations of Maxwell in 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 Iaws 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 " Galileian " 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 K1 possessing any kind of motion, `

` then relative to K1 there exists a gravitational field which is `

` variable with respect to space and time.[3]** The character of this `

` field will of course depend on the motion chosen for K1. 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 gravitationial 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 realised in the most beautiful `

` manner. But between the clear vision of this goal and its actual `

` realisation 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. `

` `

` `

` Notes `

` `

` *) 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. (Cf. Appendix `

` III.) `

` `

` **) This follows from a generalisation of the discussion in `

` Section 20 `

` `

` `

` `

` BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE `

` `

` `

` Hitherto I have purposely refrained from speaking about the physical `

` interpretation of space- and time-data in the case of the 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 Galileian `

` reference-body as regards the domain considered, and the results of `

` the special theory of relativity hold relative to K. Let us supposse `

` the same domain referred to a second body of reference K1, which is `

` rotating uniformly with respect to K. In order to fix our ideas, we `

` shall imagine K1 to be in the form of a plane circular disc, which `

` rotates uniformly in its own plane about its centre. An observer who `

` is sitting eccentrically on the disc K1 is sensible of a force which `

` acts outwards 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.* 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 signification of time- and space-data with `

` reference to the circular disc K1, 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 centre 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 Galileian reference-body K. As judged from this body, the `

` clock at the centre 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 centre of the circular disc, i.e. as observed from K. `

` It is obvious that the same effect would be noted by an observer whom `

` we will imagine sitting alongside his clock at the centre 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 with the aid of clocks 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 co-ordinates 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 Galileian 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 `

` measaring-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 p = 3.14 . . ., but a larger number,[4]** whereas of course, `

` for a disc which is at rest with respect to K, this operation would `

` yield p 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 and 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 co-ordinates x, y, z relative to the `

` disc by means of the method used in discussing the special theory, and `

` as long as the co- ordinates and times of events have not been `

` defined, we cannot assign an exact meaning to the natural laws in `

` which these occur. `

` `

`

` with respect to the accelerated reference-body K1 (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 Galileian 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 `

` K1). 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. `

` `

` In 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 neighbourhood 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.[2]* `

` `

` 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 unlinlited domain of validity ; `

` its results hold only so long as we are able to disregard the `

` influences of gravitational fields on the phenomena (e.g. 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 realised, in `

` which the electrical masses are quite at rest relatively to each `

` other, and to the co-ordinate system. Should we be justified in saying `

` that for this reason electrostatics is overthrown by the `

` field-equations of Maxwell in 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 Iaws 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 " Galileian " 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 K1 possessing any kind of motion, `

` then relative to K1 there exists a gravitational field which is `

` variable with respect to space and time.[3]** The character of this `

` field will of course depend on the motion chosen for K1. 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 gravitationial 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 realised in the most beautiful `

` manner. But between the clear vision of this goal and its actual `

` realisation 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. `

` `

` `

` Notes `

` `

` *) 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. (Cf. Appendix `

` III.) `

` `

` **) This follows from a generalisation of the discussion in `

` Section 20 `

` `

` `

` `

` BEHAVIOUR OF CLOCKS AND MEASURING-RODS ON A ROTATING BODY OF REFERENCE `

` `

` `

` Hitherto I have purposely refrained from speaking about the physical `

` interpretation of space- and time-data in the case of the 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 Galileian `

` reference-body as regards the domain considered, and the results of `

` the special theory of relativity hold relative to K. Let us supposse `

` the same domain referred to a second body of reference K1, which is `

` rotating uniformly with respect to K. In order to fix our ideas, we `

` shall imagine K1 to be in the form of a plane circular disc, which `

` rotates uniformly in its own plane about its centre. An observer who `

` is sitting eccentrically on the disc K1 is sensible of a force which `

` acts outwards 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.* 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 signification of time- and space-data with `

` reference to the circular disc K1, 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 centre 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 Galileian reference-body K. As judged from this body, the `

` clock at the centre 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 centre of the circular disc, i.e. as observed from K. `

` It is obvious that the same effect would be noted by an observer whom `

` we will imagine sitting alongside his clock at the centre 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 with the aid of clocks 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 co-ordinates 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 Galileian 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 `

` measaring-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 p = 3.14 . . ., but a larger number,[4]** whereas of course, `

` for a disc which is at rest with respect to K, this operation would `

` yield p 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 and 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 co-ordinates x, y, z relative to the `

` disc by means of the method used in discussing the special theory, and `

` as long as the co- ordinates and times of events have not been `

` defined, we cannot assign an exact meaning to the natural laws in `

` which these occur. `

` `

`