Reading Help Relativity: The Special and General Theory
In order to attain the greatest possible clearness, let us return to `
` our example of the railway carriage supposed to be travelling `
` uniformly. We call its motion a uniform translation ("uniform" because `
` it is of constant velocity and direction, " translation " because `
` although the carriage changes its position relative to the embankment `
` yet it does not rotate in so doing). Let us imagine a raven flying `
` through the air in such a manner that its motion, as observed from the `
` embankment, is uniform and in a straight line. If we were to observe `
` the flying raven from the moving railway carriage. we should find that `
` the motion of the raven would be one of different velocity and `
` direction, but that it would still be uniform and in a straight line. `
` Expressed in an abstract manner we may say : If a mass m is moving `
` uniformly in a straight line with respect to a co-ordinate system K, `
` then it will also be moving uniformly and in a straight line relative `
` to a second co-ordinate system K1 provided that the latter is `
` executing a uniform translatory motion with respect to K. In `
` accordance with the discussion contained in the preceding section, it `
` follows that: `
` `
` If K is a Galileian co-ordinate system. then every other co-ordinate `
` system K' is a Galileian one, when, in relation to K, it is in a `
` condition of uniform motion of translation. Relative to K1 the `
` mechanical laws of Galilei-Newton hold good exactly as they do with `
` respect to K. `
` `
` We advance a step farther in our generalisation when we express the `
` tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate `
` system devoid of rotation, then natural phenomena run their course `
` with respect to K1 according to exactly the same general laws as with `
` respect to K. This statement is called the principle of relativity (in `
` the restricted sense). `
` `
` As long as one was convinced that all natural phenomena were capable `
` of representation with the help of classical mechanics, there was no `
` need to doubt the validity of this principle of relativity. But in `
` view of the more recent development of electrodynamics and optics it `
` became more and more evident that classical mechanics affords an `
` insufficient foundation for the physical description of all natural `
` phenomena. At this juncture the question of the validity of the `
` principle of relativity became ripe for discussion, and it did not `
` appear impossible that the answer to this question might be in the `
` negative. `
` `
` Nevertheless, there are two general facts which at the outset speak `
` very much in favour of the validity of the principle of relativity. `
` Even though classical mechanics does not supply us with a sufficiently `
` broad basis for the theoretical presentation of all physical `
` phenomena, still we must grant it a considerable measure of " truth," `
` since it supplies us with the actual motions of the heavenly bodies `
` with a delicacy of detail little short of wonderful. The principle of `
` relativity must therefore apply with great accuracy in the domain of `
` mechanics. But that a principle of such broad generality should hold `
` with such exactness in one domain of phenomena, and yet should be `
` invalid for another, is a priori not very probable. `
` `
` We now proceed to the second argument, to which, moreover, we shall `
` return later. If the principle of relativity (in the restricted sense) `
` does not hold, then the Galileian co-ordinate systems K, K1, K2, etc., `
` which are moving uniformly relative to each other, will not be `
` equivalent for the description of natural phenomena. In this case we `
` should be constrained to believe that natural laws are capable of `
` being formulated in a particularly simple manner, and of course only `
` on condition that, from amongst all possible Galileian co-ordinate `
` systems, we should have chosen one (K[0]) of a particular state of `
` motion as our body of reference. We should then be justified (because `
` of its merits for the description of natural phenomena) in calling `
` this system " absolutely at rest," and all other Galileian systems K " `
` in motion." If, for instance, our embankment were the system K[0] then `
` our railway carriage would be a system K, relative to which less `
` simple laws would hold than with respect to K[0]. This diminished `
` simplicity would be due to the fact that the carriage K would be in `
` motion (i.e."really")with respect to K[0]. In the general laws of `
` nature which have been formulated with reference to K, the magnitude `
` and direction of the velocity of the carriage would necessarily play a `
` part. We should expect, for instance, that the note emitted by an `
` organpipe placed with its axis parallel to the direction of travel `
` would be different from that emitted if the axis of the pipe were `
` placed perpendicular to this direction. `
` `
` Now in virtue of its motion in an orbit round the sun, our earth is `
` comparable with a railway carriage travelling with a velocity of about `
` 30 kilometres per second. If the principle of relativity were not `
` valid we should therefore expect that the direction of motion of the `
` earth at any moment would enter into the laws of nature, and also that `
` physical systems in their behaviour would be dependent on the `
` orientation in space with respect to the earth. For owing to the `
` alteration in direction of the velocity of revolution of the earth in `
` the course of a year, the earth cannot be at rest relative to the `
` hypothetical system K[0] throughout the whole year. However, the most `
` careful observations have never revealed such anisotropic properties `
` in terrestrial physical space, i.e. a physical non-equivalence of `
` different directions. This is very powerful argument in favour of the `
` principle of relativity. `
` `
` `
` `
` THE THEOREM OF THE `
` ADDITION OF VELOCITIES `
` EMPLOYED IN CLASSICAL MECHANICS `
` `
` `
` Let us suppose our old friend the railway carriage to be travelling `
` along the rails with a constant velocity v, and that a man traverses `
` the length of the carriage in the direction of travel with a velocity `
` w. How quickly or, in other words, with what velocity W does the man `
` advance relative to the embankment during the process ? The only `
` possible answer seems to result from the following consideration: If `
` the man were to stand still for a second, he would advance relative to `
` the embankment through a distance v equal numerically to the velocity `
` of the carriage. As a consequence of his walking, however, he `
` traverses an additional distance w relative to the carriage, and hence `
` also relative to the embankment, in this second, the distance w being `
` numerically equal to the velocity with which he is walking. Thus in `
` total be covers the distance W=v+w relative to the embankment in the `
` second considered. We shall see later that this result, which `
` expresses the theorem of the addition of velocities employed in `
` classical mechanics, cannot be maintained ; in other words, the law `
` that we have just written down does not hold in reality. For the time `
` being, however, we shall assume its correctness. `
` `
` `
` `
` THE APPARENT INCOMPATIBILITY OF THE `
` LAW OF PROPAGATION OF LIGHT WITH THE `
` PRINCIPLE OF RELATIVITY `
` `
` `
` There is hardly a simpler law in physics than that according to which `
` light is propagated in empty space. Every child at school knows, or `
` believes he knows, that this propagation takes place in straight lines `
` with a velocity c= 300,000 km./sec. At all events we know with great `
` exactness that this velocity is the same for all colours, because if `
` this were not the case, the minimum of emission would not be observed `
` simultaneously for different colours during the eclipse of a fixed `
` star by its dark neighbour. By means of similar considerations based `
` on observa- tions of double stars, the Dutch astronomer De Sitter was `
` also able to show that the velocity of propagation of light cannot `
` depend on the velocity of motion of the body emitting the light. The `
` assumption that this velocity of propagation is dependent on the `
` direction "in space" is in itself improbable. `
` `
` In short, let us assume that the simple law of the constancy of the `
` velocity of light c (in vacuum) is justifiably believed by the child `
` at school. Who would imagine that this simple law has plunged the `
` conscientiously thoughtful physicist into the greatest intellectual `
` difficulties? Let us consider how these difficulties arise. `
` `
` Of course we must refer the process of the propagation of light (and `
` indeed every other process) to a rigid reference-body (co-ordinate `
` system). As such a system let us again choose our embankment. We shall `
` imagine the air above it to have been removed. If a ray of light be `
` sent along the embankment, we see from the above that the tip of the `
` ray will be transmitted with the velocity c relative to the `
` embankment. Now let us suppose that our railway carriage is again `
` travelling along the railway lines with the velocity v, and that its `
` direction is the same as that of the ray of light, but its velocity of `
` course much less. Let us inquire about the velocity of propagation of `
` the ray of light relative to the carriage. It is obvious that we can `
` here apply the consideration of the previous section, since the ray of `
` light plays the part of the man walking along relatively to the `
` carriage. The velocity w of the man relative to the embankment is here `
` replaced by the velocity of light relative to the embankment. w is the `
` required velocity of light with respect to the carriage, and we have `
` `
` w = c-v. `
` `
` The velocity of propagation ot a ray of light relative to the carriage `
` thus comes cut smaller than c. `
` `
` But this result comes into conflict with the principle of relativity `
` set forth in Section V. For, like every other general law of `
` nature, the law of the transmission of light in vacuo [in vacuum] `
` must, according to the principle of relativity, be the same for the `
` railway carriage as reference-body as when the rails are the body of `
` reference. But, from our above consideration, this would appear to be `
` impossible. If every ray of light is propagated relative to the `
` embankment with the velocity c, then for this reason it would appear `
` that another law of propagation of light must necessarily hold with `
` respect to the carriage -- a result contradictory to the principle of `
` relativity. `
` `
` In view of this dilemma there appears to be nothing else for it than `
` to abandon either the principle of relativity or the simple law of the `
` propagation of light in vacuo. Those of you who have carefully `
` followed the preceding discussion are almost sure to expect that we `
` should retain the principle of relativity, which appeals so `
` convincingly to the intellect because it is so natural and simple. The `
` law of the propagation of light in vacuo would then have to be `
` replaced by a more complicated law conformable to the principle of `
` relativity. The development of theoretical physics shows, however, `
` that we cannot pursue this course. The epoch-making theoretical `
` investigations of H. A. Lorentz on the electrodynamical and optical `
` phenomena connected with moving bodies show that experience in this `
` domain leads conclusively to a theory of electromagnetic phenomena, of `
` which the law of the constancy of the velocity of light in vacuo is a `
` necessary consequence. Prominent theoretical physicists were theref `
` ore more inclined to reject the principle of relativity, in spite of `
` the fact that no empirical data had been found which were `
` contradictory to this principle. `
` `
` At this juncture the theory of relativity entered the arena. As a `
`
` our example of the railway carriage supposed to be travelling `
` uniformly. We call its motion a uniform translation ("uniform" because `
` it is of constant velocity and direction, " translation " because `
` although the carriage changes its position relative to the embankment `
` yet it does not rotate in so doing). Let us imagine a raven flying `
` through the air in such a manner that its motion, as observed from the `
` embankment, is uniform and in a straight line. If we were to observe `
` the flying raven from the moving railway carriage. we should find that `
` the motion of the raven would be one of different velocity and `
` direction, but that it would still be uniform and in a straight line. `
` Expressed in an abstract manner we may say : If a mass m is moving `
` uniformly in a straight line with respect to a co-ordinate system K, `
` then it will also be moving uniformly and in a straight line relative `
` to a second co-ordinate system K1 provided that the latter is `
` executing a uniform translatory motion with respect to K. In `
` accordance with the discussion contained in the preceding section, it `
` follows that: `
` `
` If K is a Galileian co-ordinate system. then every other co-ordinate `
` system K' is a Galileian one, when, in relation to K, it is in a `
` condition of uniform motion of translation. Relative to K1 the `
` mechanical laws of Galilei-Newton hold good exactly as they do with `
` respect to K. `
` `
` We advance a step farther in our generalisation when we express the `
` tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate `
` system devoid of rotation, then natural phenomena run their course `
` with respect to K1 according to exactly the same general laws as with `
` respect to K. This statement is called the principle of relativity (in `
` the restricted sense). `
` `
` As long as one was convinced that all natural phenomena were capable `
` of representation with the help of classical mechanics, there was no `
` need to doubt the validity of this principle of relativity. But in `
` view of the more recent development of electrodynamics and optics it `
` became more and more evident that classical mechanics affords an `
` insufficient foundation for the physical description of all natural `
` phenomena. At this juncture the question of the validity of the `
` principle of relativity became ripe for discussion, and it did not `
` appear impossible that the answer to this question might be in the `
` negative. `
` `
` Nevertheless, there are two general facts which at the outset speak `
` very much in favour of the validity of the principle of relativity. `
` Even though classical mechanics does not supply us with a sufficiently `
` broad basis for the theoretical presentation of all physical `
` phenomena, still we must grant it a considerable measure of " truth," `
` since it supplies us with the actual motions of the heavenly bodies `
` with a delicacy of detail little short of wonderful. The principle of `
` relativity must therefore apply with great accuracy in the domain of `
` mechanics. But that a principle of such broad generality should hold `
` with such exactness in one domain of phenomena, and yet should be `
` invalid for another, is a priori not very probable. `
` `
` We now proceed to the second argument, to which, moreover, we shall `
` return later. If the principle of relativity (in the restricted sense) `
` does not hold, then the Galileian co-ordinate systems K, K1, K2, etc., `
` which are moving uniformly relative to each other, will not be `
` equivalent for the description of natural phenomena. In this case we `
` should be constrained to believe that natural laws are capable of `
` being formulated in a particularly simple manner, and of course only `
` on condition that, from amongst all possible Galileian co-ordinate `
` systems, we should have chosen one (K[0]) of a particular state of `
` motion as our body of reference. We should then be justified (because `
` of its merits for the description of natural phenomena) in calling `
` this system " absolutely at rest," and all other Galileian systems K " `
` in motion." If, for instance, our embankment were the system K[0] then `
` our railway carriage would be a system K, relative to which less `
` simple laws would hold than with respect to K[0]. This diminished `
` simplicity would be due to the fact that the carriage K would be in `
` motion (i.e."really")with respect to K[0]. In the general laws of `
` nature which have been formulated with reference to K, the magnitude `
` and direction of the velocity of the carriage would necessarily play a `
` part. We should expect, for instance, that the note emitted by an `
` organpipe placed with its axis parallel to the direction of travel `
` would be different from that emitted if the axis of the pipe were `
` placed perpendicular to this direction. `
` `
` Now in virtue of its motion in an orbit round the sun, our earth is `
` comparable with a railway carriage travelling with a velocity of about `
` 30 kilometres per second. If the principle of relativity were not `
` valid we should therefore expect that the direction of motion of the `
` earth at any moment would enter into the laws of nature, and also that `
` physical systems in their behaviour would be dependent on the `
` orientation in space with respect to the earth. For owing to the `
` alteration in direction of the velocity of revolution of the earth in `
` the course of a year, the earth cannot be at rest relative to the `
` hypothetical system K[0] throughout the whole year. However, the most `
` careful observations have never revealed such anisotropic properties `
` in terrestrial physical space, i.e. a physical non-equivalence of `
` different directions. This is very powerful argument in favour of the `
` principle of relativity. `
` `
` `
` `
` THE THEOREM OF THE `
` ADDITION OF VELOCITIES `
` EMPLOYED IN CLASSICAL MECHANICS `
` `
` `
` Let us suppose our old friend the railway carriage to be travelling `
` along the rails with a constant velocity v, and that a man traverses `
` the length of the carriage in the direction of travel with a velocity `
` w. How quickly or, in other words, with what velocity W does the man `
` advance relative to the embankment during the process ? The only `
` possible answer seems to result from the following consideration: If `
` the man were to stand still for a second, he would advance relative to `
` the embankment through a distance v equal numerically to the velocity `
` of the carriage. As a consequence of his walking, however, he `
` traverses an additional distance w relative to the carriage, and hence `
` also relative to the embankment, in this second, the distance w being `
` numerically equal to the velocity with which he is walking. Thus in `
` total be covers the distance W=v+w relative to the embankment in the `
` second considered. We shall see later that this result, which `
` expresses the theorem of the addition of velocities employed in `
` classical mechanics, cannot be maintained ; in other words, the law `
` that we have just written down does not hold in reality. For the time `
` being, however, we shall assume its correctness. `
` `
` `
` `
` THE APPARENT INCOMPATIBILITY OF THE `
` LAW OF PROPAGATION OF LIGHT WITH THE `
` PRINCIPLE OF RELATIVITY `
` `
` `
` There is hardly a simpler law in physics than that according to which `
` light is propagated in empty space. Every child at school knows, or `
` believes he knows, that this propagation takes place in straight lines `
` with a velocity c= 300,000 km./sec. At all events we know with great `
` exactness that this velocity is the same for all colours, because if `
` this were not the case, the minimum of emission would not be observed `
` simultaneously for different colours during the eclipse of a fixed `
` star by its dark neighbour. By means of similar considerations based `
` on observa- tions of double stars, the Dutch astronomer De Sitter was `
` also able to show that the velocity of propagation of light cannot `
` depend on the velocity of motion of the body emitting the light. The `
` assumption that this velocity of propagation is dependent on the `
` direction "in space" is in itself improbable. `
` `
` In short, let us assume that the simple law of the constancy of the `
` velocity of light c (in vacuum) is justifiably believed by the child `
` at school. Who would imagine that this simple law has plunged the `
` conscientiously thoughtful physicist into the greatest intellectual `
` difficulties? Let us consider how these difficulties arise. `
` `
` Of course we must refer the process of the propagation of light (and `
` indeed every other process) to a rigid reference-body (co-ordinate `
` system). As such a system let us again choose our embankment. We shall `
` imagine the air above it to have been removed. If a ray of light be `
` sent along the embankment, we see from the above that the tip of the `
` ray will be transmitted with the velocity c relative to the `
` embankment. Now let us suppose that our railway carriage is again `
` travelling along the railway lines with the velocity v, and that its `
` direction is the same as that of the ray of light, but its velocity of `
` course much less. Let us inquire about the velocity of propagation of `
` the ray of light relative to the carriage. It is obvious that we can `
` here apply the consideration of the previous section, since the ray of `
` light plays the part of the man walking along relatively to the `
` carriage. The velocity w of the man relative to the embankment is here `
` replaced by the velocity of light relative to the embankment. w is the `
` required velocity of light with respect to the carriage, and we have `
` `
` w = c-v. `
` `
` The velocity of propagation ot a ray of light relative to the carriage `
` thus comes cut smaller than c. `
` `
` But this result comes into conflict with the principle of relativity `
` set forth in Section V. For, like every other general law of `
` nature, the law of the transmission of light in vacuo [in vacuum] `
` must, according to the principle of relativity, be the same for the `
` railway carriage as reference-body as when the rails are the body of `
` reference. But, from our above consideration, this would appear to be `
` impossible. If every ray of light is propagated relative to the `
` embankment with the velocity c, then for this reason it would appear `
` that another law of propagation of light must necessarily hold with `
` respect to the carriage -- a result contradictory to the principle of `
` relativity. `
` `
` In view of this dilemma there appears to be nothing else for it than `
` to abandon either the principle of relativity or the simple law of the `
` propagation of light in vacuo. Those of you who have carefully `
` followed the preceding discussion are almost sure to expect that we `
` should retain the principle of relativity, which appeals so `
` convincingly to the intellect because it is so natural and simple. The `
` law of the propagation of light in vacuo would then have to be `
` replaced by a more complicated law conformable to the principle of `
` relativity. The development of theoretical physics shows, however, `
` that we cannot pursue this course. The epoch-making theoretical `
` investigations of H. A. Lorentz on the electrodynamical and optical `
` phenomena connected with moving bodies show that experience in this `
` domain leads conclusively to a theory of electromagnetic phenomena, of `
` which the law of the constancy of the velocity of light in vacuo is a `
` necessary consequence. Prominent theoretical physicists were theref `
` ore more inclined to reject the principle of relativity, in spite of `
` the fact that no empirical data had been found which were `
` contradictory to this principle. `
` `
` At this juncture the theory of relativity entered the arena. As a `
`