Reading Help Relativity: The Special and General Theory
radioactive substances consist of negatively electrified particles `
` (electrons) of very small inertia and large velocity. By examining the `
` deflection of these rays under the influence of electric and magnetic `
` fields, we can study the law of motion of these particles very `
` exactly. `
` `
` In the theoretical treatment of these electrons, we are faced with the `
` difficulty that electrodynamic theory of itself is unable to give an `
` account of their nature. For since electrical masses of one sign repel `
` each other, the negative electrical masses constituting the electron `
` would necessarily be scattered under the influence of their mutual `
` repulsions, unless there are forces of another kind operating between `
` them, the nature of which has hitherto remained obscure to us.* If `
` we now assume that the relative distances between the electrical `
` masses constituting the electron remain unchanged during the motion of `
` the electron (rigid connection in the sense of classical mechanics), `
` we arrive at a law of motion of the electron which does not agree with `
` experience. Guided by purely formal points of view, H. A. Lorentz was `
` the first to introduce the hypothesis that the form of the electron `
` experiences a contraction in the direction of motion in consequence of `
` that motion. the contracted length being proportional to the `
` expression `
` `
` eq. 05: file eq05.gif `
` `
` This, hypothesis, which is not justifiable by any electrodynamical `
` facts, supplies us then with that particular law of motion which has `
` been confirmed with great precision in recent years. `
` `
` The theory of relativity leads to the same law of motion, without `
` requiring any special hypothesis whatsoever as to the structure and `
` the behaviour of the electron. We arrived at a similar conclusion in `
` Section 13 in connection with the experiment of Fizeau, the result `
` of which is foretold by the theory of relativity without the necessity `
` of drawing on hypotheses as to the physical nature of the liquid. `
` `
` The second class of facts to which we have alluded has reference to `
` the question whether or not the motion of the earth in space can be `
` made perceptible in terrestrial experiments. We have already remarked `
` in Section 5 that all attempts of this nature led to a negative `
` result. Before the theory of relativity was put forward, it was `
` difficult to become reconciled to this negative result, for reasons `
` now to be discussed. The inherited prejudices about time and space did `
` not allow any doubt to arise as to the prime importance of the `
` Galileian transformation for changing over from one body of reference `
` to another. Now assuming that the Maxwell-Lorentz equations hold for a `
` reference-body K, we then find that they do not hold for a `
` reference-body K1 moving uniformly with respect to K, if we assume `
` that the relations of the Galileian transformstion exist between the `
` co-ordinates of K and K1. It thus appears that, of all Galileian `
` co-ordinate systems, one (K) corresponding to a particular state of `
` motion is physically unique. This result was interpreted physically by `
` regarding K as at rest with respect to a hypothetical �ther of space. `
` On the other hand, all coordinate systems K1 moving relatively to K `
` were to be regarded as in motion with respect to the �ther. To this `
` motion of K1 against the �ther ("�ther-drift " relative to K1) were `
` attributed the more complicated laws which were supposed to hold `
` relative to K1. Strictly speaking, such an �ther-drift ought also to `
` be assumed relative to the earth, and for a long time the efforts of `
` physicists were devoted to attempts to detect the existence of an `
` �ther-drift at the earth's surface. `
` `
` In one of the most notable of these attempts Michelson devised a `
` method which appears as though it must be decisive. Imagine two `
` mirrors so arranged on a rigid body that the reflecting surfaces face `
` each other. A ray of light requires a perfectly definite time T to `
` pass from one mirror to the other and back again, if the whole system `
` be at rest with respect to the �ther. It is found by calculation, `
` however, that a slightly different time T1 is required for this `
` process, if the body, together with the mirrors, be moving relatively `
` to the �ther. And yet another point: it is shown by calculation that `
` for a given velocity v with reference to the �ther, this time T1 is `
` different when the body is moving perpendicularly to the planes of the `
` mirrors from that resulting when the motion is parallel to these `
` planes. Although the estimated difference between these two times is `
` exceedingly small, Michelson and Morley performed an experiment `
` involving interference in which this difference should have been `
` clearly detectable. But the experiment gave a negative result -- a `
` fact very perplexing to physicists. Lorentz and FitzGerald rescued the `
` theory from this difficulty by assuming that the motion of the body `
` relative to the �ther produces a contraction of the body in the `
` direction of motion, the amount of contraction being just sufficient `
` to compensate for the differeace in time mentioned above. Comparison `
` with the discussion in Section 11 shows that also from the `
` standpoint of the theory of relativity this solution of the difficulty `
` was the right one. But on the basis of the theory of relativity the `
` method of interpretation is incomparably more satisfactory. According `
` to this theory there is no such thing as a " specially favoured " `
` (unique) co-ordinate system to occasion the introduction of the `
` �ther-idea, and hence there can be no �ther-drift, nor any experiment `
` with which to demonstrate it. Here the contraction of moving bodies `
` follows from the two fundamental principles of the theory, without the `
` introduction of particular hypotheses ; and as the prime factor `
` involved in this contraction we find, not the motion in itself, to `
` which we cannot attach any meaning, but the motion with respect to the `
` body of reference chosen in the particular case in point. Thus for a `
` co-ordinate system moving with the earth the mirror system of `
` Michelson and Morley is not shortened, but it is shortened for a `
` co-ordinate system which is at rest relatively to the sun. `
` `
` `
` Notes `
` `
` *) The general theory of relativity renders it likely that the `
` electrical masses of an electron are held together by gravitational `
` forces. `
` `
` `
` `
` MINKOWSKI'S FOUR-DIMENSIONAL SPACE `
` `
` `
` The non-mathematician is seized by a mysterious shuddering when he `
` hears of "four-dimensional" things, by a feeling not unlike that `
` awakened by thoughts of the occult. And yet there is no more `
` common-place statement than that the world in which we live is a `
` four-dimensional space-time continuum. `
` `
` Space is a three-dimensional continuum. By this we mean that it is `
` possible to describe the position of a point (at rest) by means of `
` three numbers (co-ordinales) x, y, z, and that there is an indefinite `
` number of points in the neighbourhood of this one, the position of `
` which can be described by co-ordinates such as x[1], y[1], z[1], which `
` may be as near as we choose to the respective values of the `
` co-ordinates x, y, z, of the first point. In virtue of the latter `
` property we speak of a " continuum," and owing to the fact that there `
` are three co-ordinates we speak of it as being " three-dimensional." `
` `
` Similarly, the world of physical phenomena which was briefly called " `
` world " by Minkowski is naturally four dimensional in the space-time `
` sense. For it is composed of individual events, each of which is `
` described by four numbers, namely, three space co-ordinates x, y, z, `
` and a time co-ordinate, the time value t. The" world" is in this sense `
` also a continuum; for to every event there are as many "neighbouring" `
` events (realised or at least thinkable) as we care to choose, the `
` co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely `
` small amount from those of the event x, y, z, t originally considered. `
` That we have not been accustomed to regard the world in this sense as `
` a four-dimensional continuum is due to the fact that in physics, `
` before the advent of the theory of relativity, time played a different `
` and more independent role, as compared with the space coordinates. It `
` is for this reason that we have been in the habit of treating time as `
` an independent continuum. As a matter of fact, according to classical `
` mechanics, time is absolute, i.e. it is independent of the position `
` and the condition of motion of the system of co-ordinates. We see this `
` expressed in the last equation of the Galileian transformation (t1 = `
` t) `
` `
` The four-dimensional mode of consideration of the "world" is natural `
` on the theory of relativity, since according to this theory time is `
` robbed of its independence. This is shown by the fourth equation of `
` the Lorentz transformation: `
` `
` eq. 24: file eq24.gif `
` `
` `
` Moreover, according to this equation the time difference Dt1 of two `
` events with respect to K1 does not in general vanish, even when the `
` time difference Dt1 of the same events with reference to K vanishes. `
` Pure " space-distance " of two events with respect to K results in " `
` time-distance " of the same events with respect to K. But the `
` discovery of Minkowski, which was of importance for the formal `
` development of the theory of relativity, does not lie here. It is to `
` be found rather in the fact of his recognition that the `
` four-dimensional space-time continuum of the theory of relativity, in `
` its most essential formal properties, shows a pronounced relationship `
` to the three-dimensional continuum of Euclidean geometrical `
` space.* In order to give due prominence to this relationship, `
` however, we must replace the usual time co-ordinate t by an imaginary `
` magnitude eq. 25 proportional to it. Under these conditions, the `
` natural laws satisfying the demands of the (special) theory of `
` relativity assume mathematical forms, in which the time co-ordinate `
` plays exactly the same role as the three space co-ordinates. Formally, `
` these four co-ordinates correspond exactly to the three space `
` co-ordinates in Euclidean geometry. It must be clear even to the `
` non-mathematician that, as a consequence of this purely formal `
` addition to our knowledge, the theory perforce gained clearness in no `
` mean measure. `
` `
` These inadequate remarks can give the reader only a vague notion of `
` the important idea contributed by Minkowski. Without it the general `
` theory of relativity, of which the fundamental ideas are developed in `
` the following pages, would perhaps have got no farther than its long `
` clothes. Minkowski's work is doubtless difficult of access to anyone `
` inexperienced in mathematics, but since it is not necessary to have a `
` very exact grasp of this work in order to understand the fundamental `
` ideas of either the special or the general theory of relativity, I `
` shall leave it here at present, and revert to it only towards the end `
` of Part 2. `
` `
` `
` Notes `
` `
` *) Cf. the somewhat more detailed discussion in Appendix II. `
` `
` `
` `
` `
` PART II `
` `
` THE GENERAL THEORY OF RELATIVITY `
`
` (electrons) of very small inertia and large velocity. By examining the `
` deflection of these rays under the influence of electric and magnetic `
` fields, we can study the law of motion of these particles very `
` exactly. `
` `
` In the theoretical treatment of these electrons, we are faced with the `
` difficulty that electrodynamic theory of itself is unable to give an `
` account of their nature. For since electrical masses of one sign repel `
` each other, the negative electrical masses constituting the electron `
` would necessarily be scattered under the influence of their mutual `
` repulsions, unless there are forces of another kind operating between `
` them, the nature of which has hitherto remained obscure to us.* If `
` we now assume that the relative distances between the electrical `
` masses constituting the electron remain unchanged during the motion of `
` the electron (rigid connection in the sense of classical mechanics), `
` we arrive at a law of motion of the electron which does not agree with `
` experience. Guided by purely formal points of view, H. A. Lorentz was `
` the first to introduce the hypothesis that the form of the electron `
` experiences a contraction in the direction of motion in consequence of `
` that motion. the contracted length being proportional to the `
` expression `
` `
` eq. 05: file eq05.gif `
` `
` This, hypothesis, which is not justifiable by any electrodynamical `
` facts, supplies us then with that particular law of motion which has `
` been confirmed with great precision in recent years. `
` `
` The theory of relativity leads to the same law of motion, without `
` requiring any special hypothesis whatsoever as to the structure and `
` the behaviour of the electron. We arrived at a similar conclusion in `
` Section 13 in connection with the experiment of Fizeau, the result `
` of which is foretold by the theory of relativity without the necessity `
` of drawing on hypotheses as to the physical nature of the liquid. `
` `
` The second class of facts to which we have alluded has reference to `
` the question whether or not the motion of the earth in space can be `
` made perceptible in terrestrial experiments. We have already remarked `
` in Section 5 that all attempts of this nature led to a negative `
` result. Before the theory of relativity was put forward, it was `
` difficult to become reconciled to this negative result, for reasons `
` now to be discussed. The inherited prejudices about time and space did `
` not allow any doubt to arise as to the prime importance of the `
` Galileian transformation for changing over from one body of reference `
` to another. Now assuming that the Maxwell-Lorentz equations hold for a `
` reference-body K, we then find that they do not hold for a `
` reference-body K1 moving uniformly with respect to K, if we assume `
` that the relations of the Galileian transformstion exist between the `
` co-ordinates of K and K1. It thus appears that, of all Galileian `
` co-ordinate systems, one (K) corresponding to a particular state of `
` motion is physically unique. This result was interpreted physically by `
` regarding K as at rest with respect to a hypothetical �ther of space. `
` On the other hand, all coordinate systems K1 moving relatively to K `
` were to be regarded as in motion with respect to the �ther. To this `
` motion of K1 against the �ther ("�ther-drift " relative to K1) were `
` attributed the more complicated laws which were supposed to hold `
` relative to K1. Strictly speaking, such an �ther-drift ought also to `
` be assumed relative to the earth, and for a long time the efforts of `
` physicists were devoted to attempts to detect the existence of an `
` �ther-drift at the earth's surface. `
` `
` In one of the most notable of these attempts Michelson devised a `
` method which appears as though it must be decisive. Imagine two `
` mirrors so arranged on a rigid body that the reflecting surfaces face `
` each other. A ray of light requires a perfectly definite time T to `
` pass from one mirror to the other and back again, if the whole system `
` be at rest with respect to the �ther. It is found by calculation, `
` however, that a slightly different time T1 is required for this `
` process, if the body, together with the mirrors, be moving relatively `
` to the �ther. And yet another point: it is shown by calculation that `
` for a given velocity v with reference to the �ther, this time T1 is `
` different when the body is moving perpendicularly to the planes of the `
` mirrors from that resulting when the motion is parallel to these `
` planes. Although the estimated difference between these two times is `
` exceedingly small, Michelson and Morley performed an experiment `
` involving interference in which this difference should have been `
` clearly detectable. But the experiment gave a negative result -- a `
` fact very perplexing to physicists. Lorentz and FitzGerald rescued the `
` theory from this difficulty by assuming that the motion of the body `
` relative to the �ther produces a contraction of the body in the `
` direction of motion, the amount of contraction being just sufficient `
` to compensate for the differeace in time mentioned above. Comparison `
` with the discussion in Section 11 shows that also from the `
` standpoint of the theory of relativity this solution of the difficulty `
` was the right one. But on the basis of the theory of relativity the `
` method of interpretation is incomparably more satisfactory. According `
` to this theory there is no such thing as a " specially favoured " `
` (unique) co-ordinate system to occasion the introduction of the `
` �ther-idea, and hence there can be no �ther-drift, nor any experiment `
` with which to demonstrate it. Here the contraction of moving bodies `
` follows from the two fundamental principles of the theory, without the `
` introduction of particular hypotheses ; and as the prime factor `
` involved in this contraction we find, not the motion in itself, to `
` which we cannot attach any meaning, but the motion with respect to the `
` body of reference chosen in the particular case in point. Thus for a `
` co-ordinate system moving with the earth the mirror system of `
` Michelson and Morley is not shortened, but it is shortened for a `
` co-ordinate system which is at rest relatively to the sun. `
` `
` `
` Notes `
` `
` *) The general theory of relativity renders it likely that the `
` electrical masses of an electron are held together by gravitational `
` forces. `
` `
` `
` `
` MINKOWSKI'S FOUR-DIMENSIONAL SPACE `
` `
` `
` The non-mathematician is seized by a mysterious shuddering when he `
` hears of "four-dimensional" things, by a feeling not unlike that `
` awakened by thoughts of the occult. And yet there is no more `
` common-place statement than that the world in which we live is a `
` four-dimensional space-time continuum. `
` `
` Space is a three-dimensional continuum. By this we mean that it is `
` possible to describe the position of a point (at rest) by means of `
` three numbers (co-ordinales) x, y, z, and that there is an indefinite `
` number of points in the neighbourhood of this one, the position of `
` which can be described by co-ordinates such as x[1], y[1], z[1], which `
` may be as near as we choose to the respective values of the `
` co-ordinates x, y, z, of the first point. In virtue of the latter `
` property we speak of a " continuum," and owing to the fact that there `
` are three co-ordinates we speak of it as being " three-dimensional." `
` `
` Similarly, the world of physical phenomena which was briefly called " `
` world " by Minkowski is naturally four dimensional in the space-time `
` sense. For it is composed of individual events, each of which is `
` described by four numbers, namely, three space co-ordinates x, y, z, `
` and a time co-ordinate, the time value t. The" world" is in this sense `
` also a continuum; for to every event there are as many "neighbouring" `
` events (realised or at least thinkable) as we care to choose, the `
` co-ordinates x[1], y[1], z[1], t[1] of which differ by an indefinitely `
` small amount from those of the event x, y, z, t originally considered. `
` That we have not been accustomed to regard the world in this sense as `
` a four-dimensional continuum is due to the fact that in physics, `
` before the advent of the theory of relativity, time played a different `
` and more independent role, as compared with the space coordinates. It `
` is for this reason that we have been in the habit of treating time as `
` an independent continuum. As a matter of fact, according to classical `
` mechanics, time is absolute, i.e. it is independent of the position `
` and the condition of motion of the system of co-ordinates. We see this `
` expressed in the last equation of the Galileian transformation (t1 = `
` t) `
` `
` The four-dimensional mode of consideration of the "world" is natural `
` on the theory of relativity, since according to this theory time is `
` robbed of its independence. This is shown by the fourth equation of `
` the Lorentz transformation: `
` `
` eq. 24: file eq24.gif `
` `
` `
` Moreover, according to this equation the time difference Dt1 of two `
` events with respect to K1 does not in general vanish, even when the `
` time difference Dt1 of the same events with reference to K vanishes. `
` Pure " space-distance " of two events with respect to K results in " `
` time-distance " of the same events with respect to K. But the `
` discovery of Minkowski, which was of importance for the formal `
` development of the theory of relativity, does not lie here. It is to `
` be found rather in the fact of his recognition that the `
` four-dimensional space-time continuum of the theory of relativity, in `
` its most essential formal properties, shows a pronounced relationship `
` to the three-dimensional continuum of Euclidean geometrical `
` space.* In order to give due prominence to this relationship, `
` however, we must replace the usual time co-ordinate t by an imaginary `
` magnitude eq. 25 proportional to it. Under these conditions, the `
` natural laws satisfying the demands of the (special) theory of `
` relativity assume mathematical forms, in which the time co-ordinate `
` plays exactly the same role as the three space co-ordinates. Formally, `
` these four co-ordinates correspond exactly to the three space `
` co-ordinates in Euclidean geometry. It must be clear even to the `
` non-mathematician that, as a consequence of this purely formal `
` addition to our knowledge, the theory perforce gained clearness in no `
` mean measure. `
` `
` These inadequate remarks can give the reader only a vague notion of `
` the important idea contributed by Minkowski. Without it the general `
` theory of relativity, of which the fundamental ideas are developed in `
` the following pages, would perhaps have got no farther than its long `
` clothes. Minkowski's work is doubtless difficult of access to anyone `
` inexperienced in mathematics, but since it is not necessary to have a `
` very exact grasp of this work in order to understand the fundamental `
` ideas of either the special or the general theory of relativity, I `
` shall leave it here at present, and revert to it only towards the end `
` of Part 2. `
` `
` `
` Notes `
` `
` *) Cf. the somewhat more detailed discussion in Appendix II. `
` `
` `
` `
` `
` PART II `
` `
` THE GENERAL THEORY OF RELATIVITY `
`