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 `
`