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. `
` `
`