# Reading Help Relativity: The Special and General Theory

`

` eq. 39: file eq39.gif `

` `

` In this way we satisfy the postulate of the constancy of the velocity `

` of light in vacuo for rays of light of arbitrary direction, both for `

` the system K and for the system K'. This may be shown in the following `

` manner. `

` `

` We suppose a light-signal sent out from the origin of K at the time t `

` = 0. It will be propagated according to the equation `

` `

` eq. 40: file eq40.gif `

` `

` or, if we square this equation, according to the equation `

` `

` x2 + y2 + z2 = c^2t2 = 0 . . . (10). `

` `

` It is required by the law of propagation of light, in conjunction with `

` the postulate of relativity, that the transmission of the signal in `

` question should take place -- as judged from K1 -- in accordance with `

` the corresponding formula `

` `

` r' = ct' `

` `

` or, `

` `

` x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a). `

` `

` In order that equation (10a) may be a consequence of equation (10), we `

` must have `

` `

` x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11). `

` `

` Since equation (8a) must hold for points on the x-axis, we thus have s `

` = I. It is easily seen that the Lorentz transformation really `

` satisfies equation (11) for s = I; for (11) is a consequence of (8a) `

` and (9), and hence also of (8) and (9). We have thus derived the `

` Lorentz transformation. `

` `

` The Lorentz transformation represented by (8) and (9) still requires `

` to be generalised. Obviously it is immaterial whether the axes of K1 `

` be chosen so that they are spatially parallel to those of K. It is `

` also not essential that the velocity of translation of K1 with respect `

` to K should be in the direction of the x-axis. A simple consideration `

` shows that we are able to construct the Lorentz transformation in this `

` general sense from two kinds of transformations, viz. from Lorentz `

` transformations in the special sense and from purely spatial `

` transformations. which corresponds to the replacement of the `

` rectangular co-ordinate system by a new system with its axes pointing `

` in other directions. `

` `

` Mathematically, we can characterise the generalised Lorentz `

` transformation thus : `

` `

` It expresses x', y', x', t', in terms of linear homogeneous functions `

` of x, y, x, t, of such a kind that the relation `

` `

` x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a). `

` `

` is satisficd identically. That is to say: If we substitute their `

` expressions in x, y, x, t, in place of x', y', x', t', on the `

` left-hand side, then the left-hand side of (11a) agrees with the `

` right-hand side. `

` `

` `

` `

` APPENDIX II `

` `

` MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD") `

` (SUPPLEMENTARY TO SECTION 17) `

` `

` `

` We can characterise the Lorentz transformation still more simply if we `

` introduce the imaginary eq. 25 in place of t, as time-variable. If, in `

` accordance with this, we insert `

` `

` x[1] = x `

` x[2] = y `

` x[3] = z `

` x[4] = eq. 25 `

` `

` and similarly for the accented system K1, then the condition which is `

` identically satisfied by the transformation can be expressed thus : `

` `

` x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 `

` (12). `

` `

` That is, by the afore-mentioned choice of " coordinates," (11a) [see `

` the end of Appendix II] is transformed into this equation. `

` `

` We see from (12) that the imaginary time co-ordinate x[4], enters into `

` the condition of transformation in exactly the same way as the space `

` co-ordinates x[1], x[2], x[3]. It is due to this fact that, according `

` to the theory of relativity, the " time "x[4], enters into natural `

` laws in the same form as the space co ordinates x[1], x[2], x[3]. `

` `

` A four-dimensional continuum described by the "co-ordinates" x[1], `

` x[2], x[3], x[4], was called "world" by Minkowski, who also termed a `

` point-event a " world-point." From a "happening" in three-dimensional `

` space, physics becomes, as it were, an " existence " in the `

` four-dimensional " world." `

` `

` This four-dimensional " world " bears a close similarity to the `

` three-dimensional " space " of (Euclidean) analytical geometry. If we `

` introduce into the latter a new Cartesian co-ordinate system (x'[1], `

` x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are `

` linear homogeneous functions of x[1], x[2], x[3] which identically `

` satisfy the equation `

` `

` x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2 `

` `

` The analogy with (12) is a complete one. We can regard Minkowski's " `

` world " in a formal manner as a four-dimensional Euclidean space (with `

` an imaginary time coordinate) ; the Lorentz transformation corresponds `

` to a " rotation " of the co-ordinate system in the fourdimensional " `

` world." `

` `

` `

` `

` APPENDIX III `

` `

` THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY `

` `

` `

` From a systematic theoretical point of view, we may imagine the `

` process of evolution of an empirical science to be a continuous `

` process of induction. Theories are evolved and are expressed in short `

` compass as statements of a large number of individual observations in `

` the form of empirical laws, from which the general laws can be `

` ascertained by comparison. Regarded in this way, the development of a `

` science bears some resemblance to the compilation of a classified `

` catalogue. It is, as it were, a purely empirical enterprise. `

` `

` But this point of view by no means embraces the whole of the actual `

` process ; for it slurs over the important part played by intuition and `

` deductive thought in the development of an exact science. As soon as a `

` science has emerged from its initial stages, theoretical advances are `

` no longer achieved merely by a process of arrangement. Guided by `

` empirical data, the investigator rather develops a system of thought `

` which, in general, is built up logically from a small number of `

` fundamental assumptions, the so-called axioms. We call such a system `

` of thought a theory. The theory finds the justification for its `

` existence in the fact that it correlates a large number of single `

` observations, and it is just here that the " truth " of the theory `

` lies. `

` `

` Corresponding to the same complex of empirical data, there may be `

` several theories, which differ from one another to a considerable `

` extent. But as regards the deductions from the theories which are `

` capable of being tested, the agreement between the theories may be so `

` complete that it becomes difficult to find any deductions in which the `

` two theories differ from each other. As an example, a case of general `

` interest is available in the province of biology, in the Darwinian `

` theory of the development of species by selection in the struggle for `

` existence, and in the theory of development which is based on the `

` hypothesis of the hereditary transmission of acquired characters. `

` `

` We have another instance of far-reaching agreement between the `

` deductions from two theories in Newtonian mechanics on the one hand, `

` and the general theory of relativity on the other. This agreement goes `

` so far, that up to the preseat we have been able to find only a few `

` deductions from the general theory of relativity which are capable of `

` investigation, and to which the physics of pre-relativity days does `

` not also lead, and this despite the profound difference in the `

` fundamental assumptions of the two theories. In what follows, we shall `

` again consider these important deductions, and we shall also discuss `

` the empirical evidence appertaining to them which has hitherto been `

` obtained. `

` `

` (a) Motion of the Perihelion of Mercury `

` `

` According to Newtonian mechanics and Newton's law of gravitation, a `

` planet which is revolving round the sun would describe an ellipse `

` round the latter, or, more correctly, round the common centre of `

` gravity of the sun and the planet. In such a system, the sun, or the `

` common centre of gravity, lies in one of the foci of the orbital `

` ellipse in such a manner that, in the course of a planet-year, the `

` distance sun-planet grows from a minimum to a maximum, and then `

` decreases again to a minimum. If instead of Newton's law we insert a `

` somewhat different law of attraction into the calculation, we find `

` that, according to this new law, the motion would still take place in `

` such a manner that the distance sun-planet exhibits periodic `

` variations; but in this case the angle described by the line joining `

` sun and planet during such a period (from perihelion--closest `

` proximity to the sun--to perihelion) would differ from 360^0. The line `

` of the orbit would not then be a closed one but in the course of time `

` it would fill up an annular part of the orbital plane, viz. between `

` the circle of least and the circle of greatest distance of the planet `

` from the sun. `

` `

` According also to the general theory of relativity, which differs of `

` course from the theory of Newton, a small variation from the `

` Newton-Kepler motion of a planet in its orbit should take place, and `

` in such away, that the angle described by the radius sun-planet `

` between one perhelion and the next should exceed that corresponding to `

` one complete revolution by an amount given by `

` `

` eq. 41: file eq41.gif `

` `

` (N.B. -- One complete revolution corresponds to the angle 2p in the `

` absolute angular measure customary in physics, and the above `

`

` eq. 39: file eq39.gif `

` `

` In this way we satisfy the postulate of the constancy of the velocity `

` of light in vacuo for rays of light of arbitrary direction, both for `

` the system K and for the system K'. This may be shown in the following `

` manner. `

` `

` We suppose a light-signal sent out from the origin of K at the time t `

` = 0. It will be propagated according to the equation `

` `

` eq. 40: file eq40.gif `

` `

` or, if we square this equation, according to the equation `

` `

` x2 + y2 + z2 = c^2t2 = 0 . . . (10). `

` `

` It is required by the law of propagation of light, in conjunction with `

` the postulate of relativity, that the transmission of the signal in `

` question should take place -- as judged from K1 -- in accordance with `

` the corresponding formula `

` `

` r' = ct' `

` `

` or, `

` `

` x'2 + y'2 + z'2 - c^2t'2 = 0 . . . (10a). `

` `

` In order that equation (10a) may be a consequence of equation (10), we `

` must have `

` `

` x'2 + y'2 + z'2 - c^2t'2 = s (x2 + y2 + z2 - c^2t2) (11). `

` `

` Since equation (8a) must hold for points on the x-axis, we thus have s `

` = I. It is easily seen that the Lorentz transformation really `

` satisfies equation (11) for s = I; for (11) is a consequence of (8a) `

` and (9), and hence also of (8) and (9). We have thus derived the `

` Lorentz transformation. `

` `

` The Lorentz transformation represented by (8) and (9) still requires `

` to be generalised. Obviously it is immaterial whether the axes of K1 `

` be chosen so that they are spatially parallel to those of K. It is `

` also not essential that the velocity of translation of K1 with respect `

` to K should be in the direction of the x-axis. A simple consideration `

` shows that we are able to construct the Lorentz transformation in this `

` general sense from two kinds of transformations, viz. from Lorentz `

` transformations in the special sense and from purely spatial `

` transformations. which corresponds to the replacement of the `

` rectangular co-ordinate system by a new system with its axes pointing `

` in other directions. `

` `

` Mathematically, we can characterise the generalised Lorentz `

` transformation thus : `

` `

` It expresses x', y', x', t', in terms of linear homogeneous functions `

` of x, y, x, t, of such a kind that the relation `

` `

` x'2 + y'2 + z'2 - c^2t'2 = x2 + y2 + z2 - c^2t2 (11a). `

` `

` is satisficd identically. That is to say: If we substitute their `

` expressions in x, y, x, t, in place of x', y', x', t', on the `

` left-hand side, then the left-hand side of (11a) agrees with the `

` right-hand side. `

` `

` `

` `

` APPENDIX II `

` `

` MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD") `

` (SUPPLEMENTARY TO SECTION 17) `

` `

` `

` We can characterise the Lorentz transformation still more simply if we `

` introduce the imaginary eq. 25 in place of t, as time-variable. If, in `

` accordance with this, we insert `

` `

` x[1] = x `

` x[2] = y `

` x[3] = z `

` x[4] = eq. 25 `

` `

` and similarly for the accented system K1, then the condition which is `

` identically satisfied by the transformation can be expressed thus : `

` `

` x[1]'2 + x[2]'2 + x[3]'2 + x[4]'2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 `

` (12). `

` `

` That is, by the afore-mentioned choice of " coordinates," (11a) [see `

` the end of Appendix II] is transformed into this equation. `

` `

` We see from (12) that the imaginary time co-ordinate x[4], enters into `

` the condition of transformation in exactly the same way as the space `

` co-ordinates x[1], x[2], x[3]. It is due to this fact that, according `

` to the theory of relativity, the " time "x[4], enters into natural `

` laws in the same form as the space co ordinates x[1], x[2], x[3]. `

` `

` A four-dimensional continuum described by the "co-ordinates" x[1], `

` x[2], x[3], x[4], was called "world" by Minkowski, who also termed a `

` point-event a " world-point." From a "happening" in three-dimensional `

` space, physics becomes, as it were, an " existence " in the `

` four-dimensional " world." `

` `

` This four-dimensional " world " bears a close similarity to the `

` three-dimensional " space " of (Euclidean) analytical geometry. If we `

` introduce into the latter a new Cartesian co-ordinate system (x'[1], `

` x'[2], x'[3]) with the same origin, then x'[1], x'[2], x'[3], are `

` linear homogeneous functions of x[1], x[2], x[3] which identically `

` satisfy the equation `

` `

` x'[1]^2 + x'[2]^2 + x'[3]^2 = x[1]^2 + x[2]^2 + x[3]^2 `

` `

` The analogy with (12) is a complete one. We can regard Minkowski's " `

` world " in a formal manner as a four-dimensional Euclidean space (with `

` an imaginary time coordinate) ; the Lorentz transformation corresponds `

` to a " rotation " of the co-ordinate system in the fourdimensional " `

` world." `

` `

` `

` `

` APPENDIX III `

` `

` THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY `

` `

` `

` From a systematic theoretical point of view, we may imagine the `

` process of evolution of an empirical science to be a continuous `

` process of induction. Theories are evolved and are expressed in short `

` compass as statements of a large number of individual observations in `

` the form of empirical laws, from which the general laws can be `

` ascertained by comparison. Regarded in this way, the development of a `

` science bears some resemblance to the compilation of a classified `

` catalogue. It is, as it were, a purely empirical enterprise. `

` `

` But this point of view by no means embraces the whole of the actual `

` process ; for it slurs over the important part played by intuition and `

` deductive thought in the development of an exact science. As soon as a `

` science has emerged from its initial stages, theoretical advances are `

` no longer achieved merely by a process of arrangement. Guided by `

` empirical data, the investigator rather develops a system of thought `

` which, in general, is built up logically from a small number of `

` fundamental assumptions, the so-called axioms. We call such a system `

` of thought a theory. The theory finds the justification for its `

` existence in the fact that it correlates a large number of single `

` observations, and it is just here that the " truth " of the theory `

` lies. `

` `

` Corresponding to the same complex of empirical data, there may be `

` several theories, which differ from one another to a considerable `

` extent. But as regards the deductions from the theories which are `

` capable of being tested, the agreement between the theories may be so `

` complete that it becomes difficult to find any deductions in which the `

` two theories differ from each other. As an example, a case of general `

` interest is available in the province of biology, in the Darwinian `

` theory of the development of species by selection in the struggle for `

` existence, and in the theory of development which is based on the `

` hypothesis of the hereditary transmission of acquired characters. `

` `

` We have another instance of far-reaching agreement between the `

` deductions from two theories in Newtonian mechanics on the one hand, `

` and the general theory of relativity on the other. This agreement goes `

` so far, that up to the preseat we have been able to find only a few `

` deductions from the general theory of relativity which are capable of `

` investigation, and to which the physics of pre-relativity days does `

` not also lead, and this despite the profound difference in the `

` fundamental assumptions of the two theories. In what follows, we shall `

` again consider these important deductions, and we shall also discuss `

` the empirical evidence appertaining to them which has hitherto been `

` obtained. `

` `

` (a) Motion of the Perihelion of Mercury `

` `

` According to Newtonian mechanics and Newton's law of gravitation, a `

` planet which is revolving round the sun would describe an ellipse `

` round the latter, or, more correctly, round the common centre of `

` gravity of the sun and the planet. In such a system, the sun, or the `

` common centre of gravity, lies in one of the foci of the orbital `

` ellipse in such a manner that, in the course of a planet-year, the `

` distance sun-planet grows from a minimum to a maximum, and then `

` decreases again to a minimum. If instead of Newton's law we insert a `

` somewhat different law of attraction into the calculation, we find `

` that, according to this new law, the motion would still take place in `

` such a manner that the distance sun-planet exhibits periodic `

` variations; but in this case the angle described by the line joining `

` sun and planet during such a period (from perihelion--closest `

` proximity to the sun--to perihelion) would differ from 360^0. The line `

` of the orbit would not then be a closed one but in the course of time `

` it would fill up an annular part of the orbital plane, viz. between `

` the circle of least and the circle of greatest distance of the planet `

` from the sun. `

` `

` According also to the general theory of relativity, which differs of `

` course from the theory of Newton, a small variation from the `

` Newton-Kepler motion of a planet in its orbit should take place, and `

` in such away, that the angle described by the radius sun-planet `

` between one perhelion and the next should exceed that corresponding to `

` one complete revolution by an amount given by `

` `

` eq. 41: file eq41.gif `

` `

` (N.B. -- One complete revolution corresponds to the angle 2p in the `

` absolute angular measure customary in physics, and the above `

`