# Reading Help Relativity: The Special and General Theory

system K1. In this connection the relation between the ordinary and `

` the accented magnitudes is given by the Lorentz transformation. Or in `

` brief : General laws of nature are co-variant with respect to Lorentz `

` transformations. `

` `

` This is a definite mathematical condition that the theory of `

` relativity demands of a natural law, and in virtue of this, the theory `

` becomes a valuable heuristic aid in the search for general laws of `

` nature. If a general law of nature were to be found which did not `

` satisfy this condition, then at least one of the two fundamental `

` assumptions of the theory would have been disproved. Let us now `

` examine what general results the latter theory has hitherto evinced. `

` `

` `

` `

` GENERAL RESULTS OF THE THEORY `

` `

` `

` It is clear from our previous considerations that the (special) theory `

` of relativity has grown out of electrodynamics and optics. In these `

` fields it has not appreciably altered the predictions of theory, but `

` it has considerably simplified the theoretical structure, i.e. the `

` derivation of laws, and -- what is incomparably more important -- it `

` has considerably reduced the number of independent hypothese forming `

` the basis of theory. The special theory of relativity has rendered the `

` Maxwell-Lorentz theory so plausible, that the latter would have been `

` generally accepted by physicists even if experiment had decided less `

` unequivocally in its favour. `

` `

` Classical mechanics required to be modified before it could come into `

` line with the demands of the special theory of relativity. For the `

` main part, however, this modification affects only the laws for rapid `

` motions, in which the velocities of matter v are not very small as `

` compared with the velocity of light. We have experience of such rapid `

` motions only in the case of electrons and ions; for other motions the `

` variations from the laws of classical mechanics are too small to make `

` themselves evident in practice. We shall not consider the motion of `

` stars until we come to speak of the general theory of relativity. In `

` accordance with the theory of relativity the kinetic energy of a `

` material point of mass m is no longer given by the well-known `

` expression `

` `

` eq. 15: file eq15.gif `

` `

` but by the expression `

` `

` eq. 16: file eq16.gif `

` `

` `

` This expression approaches infinity as the velocity v approaches the `

` velocity of light c. The velocity must therefore always remain less `

` than c, however great may be the energies used to produce the `

` acceleration. If we develop the expression for the kinetic energy in `

` the form of a series, we obtain `

` `

` eq. 17: file eq17.gif `

` `

` `

` When eq. 18 is small compared with unity, the third of these terms is `

` always small in comparison with the second, `

` `

` which last is alone considered in classical mechanics. The first term `

` mc^2 does not contain the velocity, and requires no consideration if `

` we are only dealing with the question as to how the energy of a `

` point-mass; depends on the velocity. We shall speak of its essential `

` significance later. `

` `

` The most important result of a general character to which the special `

` theory of relativity has led is concerned with the conception of mass. `

` Before the advent of relativity, physics recognised two conservation `

` laws of fundamental importance, namely, the law of the canservation of `

` energy and the law of the conservation of mass these two fundamental `

` laws appeared to be quite independent of each other. By means of the `

` theory of relativity they have been united into one law. We shall now `

` briefly consider how this unification came about, and what meaning is `

` to be attached to it. `

` `

` The principle of relativity requires that the law of the concervation `

` of energy should hold not only with reference to a co-ordinate system `

` K, but also with respect to every co-ordinate system K1 which is in a `

` state of uniform motion of translation relative to K, or, briefly, `

` relative to every " Galileian " system of co-ordinates. In contrast to `

` classical mechanics; the Lorentz transformation is the deciding factor `

` in the transition from one such system to another. `

` `

` By means of comparatively simple considerations we are led to draw the `

` following conclusion from these premises, in conjunction with the `

` fundamental equations of the electrodynamics of Maxwell: A body moving `

` with the velocity v, which absorbs * an amount of energy E[0] in `

` the form of radiation without suffering an alteration in velocity in `

` the process, has, as a consequence, its energy increased by an amount `

` `

` eq. 19: file eq19.gif `

` `

` In consideration of the expression given above for the kinetic energy `

` of the body, the required energy of the body comes out to be `

` `

` eq. 20: file eq20.gif `

` `

` `

` Thus the body has the same energy as a body of mass `

` `

` eq.21: file eq21.gif `

` `

` moving with the velocity v. Hence we can say: If a body takes up an `

` amount of energy E[0], then its inertial mass increases by an amount `

` `

` eq. 22: file eq22.gif `

` `

` `

` the inertial mass of a body is not a constant but varies according to `

` the change in the energy of the body. The inertial mass of a system of `

` bodies can even be regarded as a measure of its energy. The law of the `

` conservation of the mass of a system becomes identical with the law of `

` the conservation of energy, and is only valid provided that the system `

` neither takes up nor sends out energy. Writing the expression for the `

` energy in the form `

` `

` eq. 23: file eq23.gif `

` `

` we see that the term mc^2, which has hitherto attracted our attention, `

` is nothing else than the energy possessed by the body ** before it `

` absorbed the energy E[0]. `

` `

` A direct comparison of this relation with experiment is not possible `

` at the present time (1920; see *** Note, p. 48), owing to the fact that `

` the changes in energy E[0] to which we can Subject a system are not `

` large enough to make themselves perceptible as a change in the `

` inertial mass of the system. `

` `

` eq. 22: file eq22.gif `

` `

` `

` is too small in comparison with the mass m, which was present before `

` the alteration of the energy. It is owing to this circumstance that `

` classical mechanics was able to establish successfully the `

` conservation of mass as a law of independent validity. `

` `

` Let me add a final remark of a fundamental nature. The success of the `

` Faraday-Maxwell interpretation of electromagnetic action at a distance `

` resulted in physicists becoming convinced that there are no such `

` things as instantaneous actions at a distance (not involving an `

` intermediary medium) of the type of Newton's law of gravitation. `

` According to the theory of relativity, action at a distance with the `

` velocity of light always takes the place of instantaneous action at a `

` distance or of action at a distance with an infinite velocity of `

` transmission. This is connected with the fact that the velocity c `

` plays a fundamental role in this theory. In Part II we shall see in `

` what way this result becomes modified in the general theory of `

` relativity. `

` `

` `

` Notes `

` `

` *) E[0] is the energy taken up, as judged from a co-ordinate system `

` moving with the body. `

` `

` **) As judged from a co-ordinate system moving with the body. `

` `

` ***[Note] The equation E = mc^2 has been thoroughly proved time and `

` again since this time. `

` `

` `

` `

` EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY `

` `

` `

` To what extent is the special theory of relativity supported by `

` experience? This question is not easily answered for the reason `

` already mentioned in connection with the fundamental experiment of `

` Fizeau. The special theory of relativity has crystallised out from the `

` Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of `

` experience which support the electromagnetic theory also support the `

` theory of relativity. As being of particular importance, I mention `

` here the fact that the theory of relativity enables us to predict the `

` effects produced on the light reaching us from the fixed stars. These `

` results are obtained in an exceedingly simple manner, and the effects `

` indicated, which are due to the relative motion of the earth with `

` reference to those fixed stars are found to be in accord with `

` experience. We refer to the yearly movement of the apparent position `

` of the fixed stars resulting from the motion of the earth round the `

` sun (aberration), and to the influence of the radial components of the `

` relative motions of the fixed stars with respect to the earth on the `

` colour of the light reaching us from them. The latter effect manifests `

` itself in a slight displacement of the spectral lines of the light `

` transmitted to us from a fixed star, as compared with the position of `

` the same spectral lines when they are produced by a terrestrial source `

` of light (Doppler principle). The experimental arguments in favour of `

` the Maxwell-Lorentz theory, which are at the same time arguments in `

` favour of the theory of relativity, are too numerous to be set forth `

` here. In reality they limit the theoretical possibilities to such an `

` extent, that no other theory than that of Maxwell and Lorentz has been `

` able to hold its own when tested by experience. `

` `

` But there are two classes of experimental facts hitherto obtained `

` which can be represented in the Maxwell-Lorentz theory only by the `

` introduction of an auxiliary hypothesis, which in itself -- i.e. `

` without making use of the theory of relativity -- appears extraneous. `

` `

` It is known that cathode rays and the so-called b-rays emitted by `

` radioactive substances consist of negatively electrified particles `

`

` the accented magnitudes is given by the Lorentz transformation. Or in `

` brief : General laws of nature are co-variant with respect to Lorentz `

` transformations. `

` `

` This is a definite mathematical condition that the theory of `

` relativity demands of a natural law, and in virtue of this, the theory `

` becomes a valuable heuristic aid in the search for general laws of `

` nature. If a general law of nature were to be found which did not `

` satisfy this condition, then at least one of the two fundamental `

` assumptions of the theory would have been disproved. Let us now `

` examine what general results the latter theory has hitherto evinced. `

` `

` `

` `

` GENERAL RESULTS OF THE THEORY `

` `

` `

` It is clear from our previous considerations that the (special) theory `

` of relativity has grown out of electrodynamics and optics. In these `

` fields it has not appreciably altered the predictions of theory, but `

` it has considerably simplified the theoretical structure, i.e. the `

` derivation of laws, and -- what is incomparably more important -- it `

` has considerably reduced the number of independent hypothese forming `

` the basis of theory. The special theory of relativity has rendered the `

` Maxwell-Lorentz theory so plausible, that the latter would have been `

` generally accepted by physicists even if experiment had decided less `

` unequivocally in its favour. `

` `

` Classical mechanics required to be modified before it could come into `

` line with the demands of the special theory of relativity. For the `

` main part, however, this modification affects only the laws for rapid `

` motions, in which the velocities of matter v are not very small as `

` compared with the velocity of light. We have experience of such rapid `

` motions only in the case of electrons and ions; for other motions the `

` variations from the laws of classical mechanics are too small to make `

` themselves evident in practice. We shall not consider the motion of `

` stars until we come to speak of the general theory of relativity. In `

` accordance with the theory of relativity the kinetic energy of a `

` material point of mass m is no longer given by the well-known `

` expression `

` `

` eq. 15: file eq15.gif `

` `

` but by the expression `

` `

` eq. 16: file eq16.gif `

` `

` `

` This expression approaches infinity as the velocity v approaches the `

` velocity of light c. The velocity must therefore always remain less `

` than c, however great may be the energies used to produce the `

` acceleration. If we develop the expression for the kinetic energy in `

` the form of a series, we obtain `

` `

` eq. 17: file eq17.gif `

` `

` `

` When eq. 18 is small compared with unity, the third of these terms is `

` always small in comparison with the second, `

` `

` which last is alone considered in classical mechanics. The first term `

` mc^2 does not contain the velocity, and requires no consideration if `

` we are only dealing with the question as to how the energy of a `

` point-mass; depends on the velocity. We shall speak of its essential `

` significance later. `

` `

` The most important result of a general character to which the special `

` theory of relativity has led is concerned with the conception of mass. `

` Before the advent of relativity, physics recognised two conservation `

` laws of fundamental importance, namely, the law of the canservation of `

` energy and the law of the conservation of mass these two fundamental `

` laws appeared to be quite independent of each other. By means of the `

` theory of relativity they have been united into one law. We shall now `

` briefly consider how this unification came about, and what meaning is `

` to be attached to it. `

` `

` The principle of relativity requires that the law of the concervation `

` of energy should hold not only with reference to a co-ordinate system `

` K, but also with respect to every co-ordinate system K1 which is in a `

` state of uniform motion of translation relative to K, or, briefly, `

` relative to every " Galileian " system of co-ordinates. In contrast to `

` classical mechanics; the Lorentz transformation is the deciding factor `

` in the transition from one such system to another. `

` `

` By means of comparatively simple considerations we are led to draw the `

` following conclusion from these premises, in conjunction with the `

` fundamental equations of the electrodynamics of Maxwell: A body moving `

` with the velocity v, which absorbs * an amount of energy E[0] in `

` the form of radiation without suffering an alteration in velocity in `

` the process, has, as a consequence, its energy increased by an amount `

` `

` eq. 19: file eq19.gif `

` `

` In consideration of the expression given above for the kinetic energy `

` of the body, the required energy of the body comes out to be `

` `

` eq. 20: file eq20.gif `

` `

` `

` Thus the body has the same energy as a body of mass `

` `

` eq.21: file eq21.gif `

` `

` moving with the velocity v. Hence we can say: If a body takes up an `

` amount of energy E[0], then its inertial mass increases by an amount `

` `

` eq. 22: file eq22.gif `

` `

` `

` the inertial mass of a body is not a constant but varies according to `

` the change in the energy of the body. The inertial mass of a system of `

` bodies can even be regarded as a measure of its energy. The law of the `

` conservation of the mass of a system becomes identical with the law of `

` the conservation of energy, and is only valid provided that the system `

` neither takes up nor sends out energy. Writing the expression for the `

` energy in the form `

` `

` eq. 23: file eq23.gif `

` `

` we see that the term mc^2, which has hitherto attracted our attention, `

` is nothing else than the energy possessed by the body ** before it `

` absorbed the energy E[0]. `

` `

` A direct comparison of this relation with experiment is not possible `

` at the present time (1920; see *** Note, p. 48), owing to the fact that `

` the changes in energy E[0] to which we can Subject a system are not `

` large enough to make themselves perceptible as a change in the `

` inertial mass of the system. `

` `

` eq. 22: file eq22.gif `

` `

` `

` is too small in comparison with the mass m, which was present before `

` the alteration of the energy. It is owing to this circumstance that `

` classical mechanics was able to establish successfully the `

` conservation of mass as a law of independent validity. `

` `

` Let me add a final remark of a fundamental nature. The success of the `

` Faraday-Maxwell interpretation of electromagnetic action at a distance `

` resulted in physicists becoming convinced that there are no such `

` things as instantaneous actions at a distance (not involving an `

` intermediary medium) of the type of Newton's law of gravitation. `

` According to the theory of relativity, action at a distance with the `

` velocity of light always takes the place of instantaneous action at a `

` distance or of action at a distance with an infinite velocity of `

` transmission. This is connected with the fact that the velocity c `

` plays a fundamental role in this theory. In Part II we shall see in `

` what way this result becomes modified in the general theory of `

` relativity. `

` `

` `

` Notes `

` `

` *) E[0] is the energy taken up, as judged from a co-ordinate system `

` moving with the body. `

` `

` **) As judged from a co-ordinate system moving with the body. `

` `

` ***[Note] The equation E = mc^2 has been thoroughly proved time and `

` again since this time. `

` `

` `

` `

` EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY `

` `

` `

` To what extent is the special theory of relativity supported by `

` experience? This question is not easily answered for the reason `

` already mentioned in connection with the fundamental experiment of `

` Fizeau. The special theory of relativity has crystallised out from the `

` Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of `

` experience which support the electromagnetic theory also support the `

` theory of relativity. As being of particular importance, I mention `

` here the fact that the theory of relativity enables us to predict the `

` effects produced on the light reaching us from the fixed stars. These `

` results are obtained in an exceedingly simple manner, and the effects `

` indicated, which are due to the relative motion of the earth with `

` reference to those fixed stars are found to be in accord with `

` experience. We refer to the yearly movement of the apparent position `

` of the fixed stars resulting from the motion of the earth round the `

` sun (aberration), and to the influence of the radial components of the `

` relative motions of the fixed stars with respect to the earth on the `

` colour of the light reaching us from them. The latter effect manifests `

` itself in a slight displacement of the spectral lines of the light `

` transmitted to us from a fixed star, as compared with the position of `

` the same spectral lines when they are produced by a terrestrial source `

` of light (Doppler principle). The experimental arguments in favour of `

` the Maxwell-Lorentz theory, which are at the same time arguments in `

` favour of the theory of relativity, are too numerous to be set forth `

` here. In reality they limit the theoretical possibilities to such an `

` extent, that no other theory than that of Maxwell and Lorentz has been `

` able to hold its own when tested by experience. `

` `

` But there are two classes of experimental facts hitherto obtained `

` which can be represented in the Maxwell-Lorentz theory only by the `

` introduction of an auxiliary hypothesis, which in itself -- i.e. `

` without making use of the theory of relativity -- appears extraneous. `

` `

` It is known that cathode rays and the so-called b-rays emitted by `

` radioactive substances consist of negatively electrified particles `

`