# 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 `

`