Reading Help Relativity: The Special and General Theory
`
` We imagine a large portion of empty space, so far removed from stars `
` and other appreciable masses, that we have before us approximately the `
` conditions required by the fundamental law of Galilei. It is then `
` possible to choose a Galileian reference-body for this part of space `
` (world), relative to which points at rest remain at rest and points in `
` motion continue permanently in uniform rectilinear motion. As `
` reference-body let us imagine a spacious chest resembling a room with `
` an observer inside who is equipped with apparatus. Gravitation `
` naturally does not exist for this observer. He must fasten himself `
` with strings to the floor, otherwise the slightest impact against the `
` floor will cause him to rise slowly towards the ceiling of the room. `
` `
` To the middle of the lid of the chest is fixed externally a hook with `
` rope attached, and now a " being " (what kind of a being is immaterial `
` to us) begins pulling at this with a constant force. The chest `
` together with the observer then begin to move "upwards" with a `
` uniformly accelerated motion. In course of time their velocity will `
` reach unheard-of values -- provided that we are viewing all this from `
` another reference-body which is not being pulled with a rope. `
` `
` But how does the man in the chest regard the Process ? The `
` acceleration of the chest will be transmitted to him by the reaction `
` of the floor of the chest. He must therefore take up this pressure by `
` means of his legs if he does not wish to be laid out full length on `
` the floor. He is then standing in the chest in exactly the same way as `
` anyone stands in a room of a home on our earth. If he releases a body `
` which he previously had in his land, the accelertion of the chest will `
` no longer be transmitted to this body, and for this reason the body `
` will approach the floor of the chest with an accelerated relative `
` motion. The observer will further convince himself that the `
` acceleration of the body towards the floor of the chest is always of `
` the same magnitude, whatever kind of body he may happen to use for the `
` experiment. `
` `
` Relying on his knowledge of the gravitational field (as it was `
` discussed in the preceding section), the man in the chest will thus `
` come to the conclusion that he and the chest are in a gravitational `
` field which is constant with regard to time. Of course he will be `
` puzzled for a moment as to why the chest does not fall in this `
` gravitational field. just then, however, he discovers the hook in the `
` middle of the lid of the chest and the rope which is attached to it, `
` and he consequently comes to the conclusion that the chest is `
` suspended at rest in the gravitational field. `
` `
` Ought we to smile at the man and say that he errs in his conclusion ? `
` I do not believe we ought to if we wish to remain consistent ; we must `
` rather admit that his mode of grasping the situation violates neither `
` reason nor known mechanical laws. Even though it is being accelerated `
` with respect to the "Galileian space" first considered, we can `
` nevertheless regard the chest as being at rest. We have thus good `
` grounds for extending the principle of relativity to include bodies of `
` reference which are accelerated with respect to each other, and as a `
` result we have gained a powerful argument for a generalised postulate `
` of relativity. `
` `
` We must note carefully that the possibility of this mode of `
` interpretation rests on the fundamental property of the gravitational `
` field of giving all bodies the same acceleration, or, what comes to `
` the same thing, on the law of the equality of inertial and `
` gravitational mass. If this natural law did not exist, the man in the `
` accelerated chest would not be able to interpret the behaviour of the `
` bodies around him on the supposition of a gravitational field, and he `
` would not be justified on the grounds of experience in supposing his `
` reference-body to be " at rest." `
` `
` Suppose that the man in the chest fixes a rope to the inner side of `
` the lid, and that he attaches a body to the free end of the rope. The `
` result of this will be to strech the rope so that it will hang " `
` vertically " downwards. If we ask for an opinion of the cause of `
` tension in the rope, the man in the chest will say: "The suspended `
` body experiences a downward force in the gravitational field, and this `
` is neutralised by the tension of the rope ; what determines the `
` magnitude of the tension of the rope is the gravitational mass of the `
` suspended body." On the other hand, an observer who is poised freely `
` in space will interpret the condition of things thus : " The rope must `
` perforce take part in the accelerated motion of the chest, and it `
` transmits this motion to the body attached to it. The tension of the `
` rope is just large enough to effect the acceleration of the body. That `
` which determines the magnitude of the tension of the rope is the `
` inertial mass of the body." Guided by this example, we see that our `
` extension of the principle of relativity implies the necessity of the `
` law of the equality of inertial and gravitational mass. Thus we have `
` obtained a physical interpretation of this law. `
` `
` From our consideration of the accelerated chest we see that a general `
` theory of relativity must yield important results on the laws of `
` gravitation. In point of fact, the systematic pursuit of the general `
` idea of relativity has supplied the laws satisfied by the `
` gravitational field. Before proceeding farther, however, I must warn `
` the reader against a misconception suggested by these considerations. `
` A gravitational field exists for the man in the chest, despite the `
` fact that there was no such field for the co-ordinate system first `
` chosen. Now we might easily suppose that the existence of a `
` gravitational field is always only an apparent one. We might also `
` think that, regardless of the kind of gravitational field which may be `
` present, we could always choose another reference-body such that no `
` gravitational field exists with reference to it. This is by no means `
` true for all gravitational fields, but only for those of quite special `
` form. It is, for instance, impossible to choose a body of reference `
` such that, as judged from it, the gravitational field of the earth (in `
` its entirety) vanishes. `
` `
` We can now appreciate why that argument is not convincing, which we `
` brought forward against the general principle of relativity at theend `
` of Section 18. It is certainly true that the observer in the `
` railway carriage experiences a jerk forwards as a result of the `
` application of the brake, and that he recognises, in this the `
` non-uniformity of motion (retardation) of the carriage. But he is `
` compelled by nobody to refer this jerk to a " real " acceleration `
` (retardation) of the carriage. He might also interpret his experience `
` thus: " My body of reference (the carriage) remains permanently at `
` rest. With reference to it, however, there exists (during the period `
` of application of the brakes) a gravitational field which is directed `
` forwards and which is variable with respect to time. Under the `
` influence of this field, the embankment together with the earth moves `
` non-uniformly in such a manner that their original velocity in the `
` backwards direction is continuously reduced." `
` `
` `
` `
` 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 `
` emphasised 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 `
` recognised " 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 the 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 `
` colour 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 behaviour 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 behaviour of bodies considered with respect to `
` the reference systems K and K1.* Newton saw this objection and `
` attempted to invalidate it, but without success. But E. Mach recognsed `
` 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 every body `
` of reference, whatever may be its state of motion. `
` `
` `
` Notes `
` `
` *) 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, e.g. in the case when `
` the reference-body is rotating uniformly. `
` `
` `
` `
` 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 Galileian domain relative to a Galileian 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 K1 which is accelerated relatively to K. But `
` since a gravitational field exists with respect to this new body of `
` reference K1, 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 `
`
` We imagine a large portion of empty space, so far removed from stars `
` and other appreciable masses, that we have before us approximately the `
` conditions required by the fundamental law of Galilei. It is then `
` possible to choose a Galileian reference-body for this part of space `
` (world), relative to which points at rest remain at rest and points in `
` motion continue permanently in uniform rectilinear motion. As `
` reference-body let us imagine a spacious chest resembling a room with `
` an observer inside who is equipped with apparatus. Gravitation `
` naturally does not exist for this observer. He must fasten himself `
` with strings to the floor, otherwise the slightest impact against the `
` floor will cause him to rise slowly towards the ceiling of the room. `
` `
` To the middle of the lid of the chest is fixed externally a hook with `
` rope attached, and now a " being " (what kind of a being is immaterial `
` to us) begins pulling at this with a constant force. The chest `
` together with the observer then begin to move "upwards" with a `
` uniformly accelerated motion. In course of time their velocity will `
` reach unheard-of values -- provided that we are viewing all this from `
` another reference-body which is not being pulled with a rope. `
` `
` But how does the man in the chest regard the Process ? The `
` acceleration of the chest will be transmitted to him by the reaction `
` of the floor of the chest. He must therefore take up this pressure by `
` means of his legs if he does not wish to be laid out full length on `
` the floor. He is then standing in the chest in exactly the same way as `
` anyone stands in a room of a home on our earth. If he releases a body `
` which he previously had in his land, the accelertion of the chest will `
` no longer be transmitted to this body, and for this reason the body `
` will approach the floor of the chest with an accelerated relative `
` motion. The observer will further convince himself that the `
` acceleration of the body towards the floor of the chest is always of `
` the same magnitude, whatever kind of body he may happen to use for the `
` experiment. `
` `
` Relying on his knowledge of the gravitational field (as it was `
` discussed in the preceding section), the man in the chest will thus `
` come to the conclusion that he and the chest are in a gravitational `
` field which is constant with regard to time. Of course he will be `
` puzzled for a moment as to why the chest does not fall in this `
` gravitational field. just then, however, he discovers the hook in the `
` middle of the lid of the chest and the rope which is attached to it, `
` and he consequently comes to the conclusion that the chest is `
` suspended at rest in the gravitational field. `
` `
` Ought we to smile at the man and say that he errs in his conclusion ? `
` I do not believe we ought to if we wish to remain consistent ; we must `
` rather admit that his mode of grasping the situation violates neither `
` reason nor known mechanical laws. Even though it is being accelerated `
` with respect to the "Galileian space" first considered, we can `
` nevertheless regard the chest as being at rest. We have thus good `
` grounds for extending the principle of relativity to include bodies of `
` reference which are accelerated with respect to each other, and as a `
` result we have gained a powerful argument for a generalised postulate `
` of relativity. `
` `
` We must note carefully that the possibility of this mode of `
` interpretation rests on the fundamental property of the gravitational `
` field of giving all bodies the same acceleration, or, what comes to `
` the same thing, on the law of the equality of inertial and `
` gravitational mass. If this natural law did not exist, the man in the `
` accelerated chest would not be able to interpret the behaviour of the `
` bodies around him on the supposition of a gravitational field, and he `
` would not be justified on the grounds of experience in supposing his `
` reference-body to be " at rest." `
` `
` Suppose that the man in the chest fixes a rope to the inner side of `
` the lid, and that he attaches a body to the free end of the rope. The `
` result of this will be to strech the rope so that it will hang " `
` vertically " downwards. If we ask for an opinion of the cause of `
` tension in the rope, the man in the chest will say: "The suspended `
` body experiences a downward force in the gravitational field, and this `
` is neutralised by the tension of the rope ; what determines the `
` magnitude of the tension of the rope is the gravitational mass of the `
` suspended body." On the other hand, an observer who is poised freely `
` in space will interpret the condition of things thus : " The rope must `
` perforce take part in the accelerated motion of the chest, and it `
` transmits this motion to the body attached to it. The tension of the `
` rope is just large enough to effect the acceleration of the body. That `
` which determines the magnitude of the tension of the rope is the `
` inertial mass of the body." Guided by this example, we see that our `
` extension of the principle of relativity implies the necessity of the `
` law of the equality of inertial and gravitational mass. Thus we have `
` obtained a physical interpretation of this law. `
` `
` From our consideration of the accelerated chest we see that a general `
` theory of relativity must yield important results on the laws of `
` gravitation. In point of fact, the systematic pursuit of the general `
` idea of relativity has supplied the laws satisfied by the `
` gravitational field. Before proceeding farther, however, I must warn `
` the reader against a misconception suggested by these considerations. `
` A gravitational field exists for the man in the chest, despite the `
` fact that there was no such field for the co-ordinate system first `
` chosen. Now we might easily suppose that the existence of a `
` gravitational field is always only an apparent one. We might also `
` think that, regardless of the kind of gravitational field which may be `
` present, we could always choose another reference-body such that no `
` gravitational field exists with reference to it. This is by no means `
` true for all gravitational fields, but only for those of quite special `
` form. It is, for instance, impossible to choose a body of reference `
` such that, as judged from it, the gravitational field of the earth (in `
` its entirety) vanishes. `
` `
` We can now appreciate why that argument is not convincing, which we `
` brought forward against the general principle of relativity at theend `
` of Section 18. It is certainly true that the observer in the `
` railway carriage experiences a jerk forwards as a result of the `
` application of the brake, and that he recognises, in this the `
` non-uniformity of motion (retardation) of the carriage. But he is `
` compelled by nobody to refer this jerk to a " real " acceleration `
` (retardation) of the carriage. He might also interpret his experience `
` thus: " My body of reference (the carriage) remains permanently at `
` rest. With reference to it, however, there exists (during the period `
` of application of the brakes) a gravitational field which is directed `
` forwards and which is variable with respect to time. Under the `
` influence of this field, the embankment together with the earth moves `
` non-uniformly in such a manner that their original velocity in the `
` backwards direction is continuously reduced." `
` `
` `
` `
` 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 `
` emphasised 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 `
` recognised " 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 the 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 `
` colour 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 behaviour 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 behaviour of bodies considered with respect to `
` the reference systems K and K1.* Newton saw this objection and `
` attempted to invalidate it, but without success. But E. Mach recognsed `
` 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 every body `
` of reference, whatever may be its state of motion. `
` `
` `
` Notes `
` `
` *) 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, e.g. in the case when `
` the reference-body is rotating uniformly. `
` `
` `
` `
` 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 Galileian domain relative to a Galileian 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 K1 which is accelerated relatively to K. But `
` since a gravitational field exists with respect to this new body of `
` reference K1, 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 `
`