Reading Help Relativity: The Special and General Theory
reference-bodies, in the sense of the method followed in the special `
` theory of relativity, is in general not possible in space-time `
` description. The Gauss co-ordinate system has to take the place of the `
` body of reference. The following statement corresponds to the `
` fundamental idea of the general principle of relativity: "All Gaussian `
` co-ordinate systems are essentially equivalent for the formulation of `
` the general laws of nature." `
` `
` We can state this general principle of relativity in still another `
` form, which renders it yet more clearly intelligible than it is when `
` in the form of the natural extension of the special principle of `
` relativity. According to the special theory of relativity, the `
` equations which express the general laws of nature pass over into `
` equations of the same form when, by making use of the Lorentz `
` transformation, we replace the space-time variables x, y, z, t, of a `
` (Galileian) reference-body K by the space-time variables x1, y1, z1, `
` t1, of a new reference-body K1. According to the general theory of `
` relativity, on the other hand, by application of arbitrary `
` substitutions of the Gauss variables x[1], x[2], x[3], x[4], the `
` equations must pass over into equations of the same form; for every `
` transformation (not only the Lorentz transformation) corresponds to `
` the transition of one Gauss co-ordinate system into another. `
` `
` If we desire to adhere to our "old-time" three-dimensional view of `
` things, then we can characterise the development which is being `
` undergone by the fundamental idea of the general theory of relativity `
` as follows : The special theory of relativity has reference to `
` Galileian domains, i.e. to those in which no gravitational field `
` exists. In this connection a Galileian reference-body serves as body `
` of reference, i.e. a rigid body the state of motion of which is so `
` chosen that the Galileian law of the uniform rectilinear motion of `
` "isolated" material points holds relatively to it. `
` `
` Certain considerations suggest that we should refer the same Galileian `
` domains to non-Galileian reference-bodies also. A gravitational field `
` of a special kind is then present with respect to these bodies (cf. `
` Sections 20 and 23). `
` `
` In gravitational fields there are no such things as rigid bodies with `
` Euclidean properties; thus the fictitious rigid body of reference is `
` of no avail in the general theory of relativity. The motion of clocks `
` is also influenced by gravitational fields, and in such a way that a `
` physical definition of time which is made directly with the aid of `
` clocks has by no means the same degree of plausibility as in the `
` special theory of relativity. `
` `
` For this reason non-rigid reference-bodies are used, which are as a `
` whole not only moving in any way whatsoever, but which also suffer `
` alterations in form ad lib. during their motion. Clocks, for which the `
` law of motion is of any kind, however irregular, serve for the `
` definition of time. We have to imagine each of these clocks fixed at a `
` point on the non-rigid reference-body. These clocks satisfy only the `
` one condition, that the "readings" which are observed simultaneously `
` on adjacent clocks (in space) differ from each other by an `
` indefinitely small amount. This non-rigid reference-body, which might `
` appropriately be termed a "reference-mollusc", is in the main `
` equivalent to a Gaussian four-dimensional co-ordinate system chosen `
` arbitrarily. That which gives the "mollusc" a certain `
` comprehensibility as compared with the Gauss co-ordinate system is the `
` (really unjustified) formal retention of the separate existence of the `
` space co-ordinates as opposed to the time co-ordinate. Every point on `
` the mollusc is treated as a space-point, and every material point `
` which is at rest relatively to it as at rest, so long as the mollusc `
` is considered as reference-body. The general principle of relativity `
` requires that all these molluscs can be used as reference-bodies with `
` equal right and equal success in the formulation of the general laws `
` of nature; the laws themselves must be quite independent of the choice `
` of mollusc. `
` `
` The great power possessed by the general principle of relativity lies `
` in the comprehensive limitation which is imposed on the laws of nature `
` in consequence of what we have seen above. `
` `
` `
` `
` THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL `
` PRINCIPLE OF RELATIVITY `
` `
` `
` If the reader has followed all our previous considerations, he will `
` have no further difficulty in understanding the methods leading to the `
` solution of the problem of gravitation. `
` `
` We start off on a consideration of a Galileian domain, i.e. a domain `
` in which there is no gravitational field relative to the Galileian `
` reference-body K. The behaviour of measuring-rods and clocks with `
` reference to K is known from the special theory of relativity, `
` likewise the behaviour of "isolated" material points; the latter move `
` uniformly and in straight lines. `
` `
` Now let us refer this domain to a random Gauss coordinate system or to `
` a "mollusc" as reference-body K1. Then with respect to K1 there is a `
` gravitational field G (of a particular kind). We learn the behaviour `
` of measuring-rods and clocks and also of freely-moving material points `
` with reference to K1 simply by mathematical transformation. We `
` interpret this behaviour as the behaviour of measuring-rods, docks and `
` material points tinder the influence of the gravitational field G. `
` Hereupon we introduce a hypothesis: that the influence of the `
` gravitational field on measuringrods, clocks and freely-moving `
` material points continues to take place according to the same laws, `
` even in the case where the prevailing gravitational field is not `
` derivable from the Galfleian special care, simply by means of a `
` transformation of co-ordinates. `
` `
` The next step is to investigate the space-time behaviour of the `
` gravitational field G, which was derived from the Galileian special `
` case simply by transformation of the coordinates. This behaviour is `
` formulated in a law, which is always valid, no matter how the `
` reference-body (mollusc) used in the description may be chosen. `
` `
` This law is not yet the general law of the gravitational field, since `
` the gravitational field under consideration is of a special kind. In `
` order to find out the general law-of-field of gravitation we still `
` require to obtain a generalisation of the law as found above. This can `
` be obtained without caprice, however, by taking into consideration the `
` following demands: `
` `
` (a) The required generalisation must likewise satisfy the general `
` postulate of relativity. `
` `
` (b) If there is any matter in the domain under consideration, only its `
` inertial mass, and thus according to Section 15 only its energy is `
` of importance for its etfect in exciting a field. `
` `
` (c) Gravitational field and matter together must satisfy the law of `
` the conservation of energy (and of impulse). `
` `
` Finally, the general principle of relativity permits us to determine `
` the influence of the gravitational field on the course of all those `
` processes which take place according to known laws when a `
` gravitational field is absent i.e. which have already been fitted into `
` the frame of the special theory of relativity. In this connection we `
` proceed in principle according to the method which has already been `
` explained for measuring-rods, clocks and freely moving material `
` points. `
` `
` The theory of gravitation derived in this way from the general `
` postulate of relativity excels not only in its beauty ; nor in `
` removing the defect attaching to classical mechanics which was brought `
` to light in Section 21; nor in interpreting the empirical law of `
` the equality of inertial and gravitational mass ; but it has also `
` already explained a result of observation in astronomy, against which `
` classical mechanics is powerless. `
` `
` If we confine the application of the theory to the case where the `
` gravitational fields can be regarded as being weak, and in which all `
` masses move with respect to the coordinate system with velocities `
` which are small compared with the velocity of light, we then obtain as `
` a first approximation the Newtonian theory. Thus the latter theory is `
` obtained here without any particular assumption, whereas Newton had to `
` introduce the hypothesis that the force of attraction between mutually `
` attracting material points is inversely proportional to the square of `
` the distance between them. If we increase the accuracy of the `
` calculation, deviations from the theory of Newton make their `
` appearance, practically all of which must nevertheless escape the test `
` of observation owing to their smallness. `
` `
` We must draw attention here to one of these deviations. According to `
` Newton's theory, a planet moves round the sun in an ellipse, which `
` would permanently maintain its position with respect to the fixed `
` stars, if we could disregard the motion of the fixed stars themselves `
` and the action of the other planets under consideration. Thus, if we `
` correct the observed motion of the planets for these two influences, `
` and if Newton's theory be strictly correct, we ought to obtain for the `
` orbit of the planet an ellipse, which is fixed with reference to the `
` fixed stars. This deduction, which can be tested with great accuracy, `
` has been confirmed for all the planets save one, with the precision `
` that is capable of being obtained by the delicacy of observation `
` attainable at the present time. The sole exception is Mercury, the `
` planet which lies nearest the sun. Since the time of Leverrier, it has `
` been known that the ellipse corresponding to the orbit of Mercury, `
` after it has been corrected for the influences mentioned above, is not `
` stationary with respect to the fixed stars, but that it rotates `
` exceedingly slowly in the plane of the orbit and in the sense of the `
` orbital motion. The value obtained for this rotary movement of the `
` orbital ellipse was 43 seconds of arc per century, an amount ensured `
` to be correct to within a few seconds of arc. This effect can be `
` explained by means of classical mechanics only on the assumption of `
` hypotheses which have little probability, and which were devised `
` solely for this purponse. `
` `
` On the basis of the general theory of relativity, it is found that the `
` ellipse of every planet round the sun must necessarily rotate in the `
` manner indicated above ; that for all the planets, with the exception `
` of Mercury, this rotation is too small to be detected with the `
` delicacy of observation possible at the present time ; but that in the `
` case of Mercury it must amount to 43 seconds of arc per century, a `
` result which is strictly in agreement with observation. `
` `
` Apart from this one, it has hitherto been possible to make only two `
` deductions from the theory which admit of being tested by observation, `
` to wit, the curvature of light rays by the gravitational field of the `
` sun,*x and a displacement of the spectral lines of light reaching `
` us from large stars, as compared with the corresponding lines for `
` light produced in an analogous manner terrestrially (i.e. by the same `
` kind of atom).** These two deductions from the theory have both `
` been confirmed. `
` `
` `
` Notes `
` `
`
` theory of relativity, is in general not possible in space-time `
` description. The Gauss co-ordinate system has to take the place of the `
` body of reference. The following statement corresponds to the `
` fundamental idea of the general principle of relativity: "All Gaussian `
` co-ordinate systems are essentially equivalent for the formulation of `
` the general laws of nature." `
` `
` We can state this general principle of relativity in still another `
` form, which renders it yet more clearly intelligible than it is when `
` in the form of the natural extension of the special principle of `
` relativity. According to the special theory of relativity, the `
` equations which express the general laws of nature pass over into `
` equations of the same form when, by making use of the Lorentz `
` transformation, we replace the space-time variables x, y, z, t, of a `
` (Galileian) reference-body K by the space-time variables x1, y1, z1, `
` t1, of a new reference-body K1. According to the general theory of `
` relativity, on the other hand, by application of arbitrary `
` substitutions of the Gauss variables x[1], x[2], x[3], x[4], the `
` equations must pass over into equations of the same form; for every `
` transformation (not only the Lorentz transformation) corresponds to `
` the transition of one Gauss co-ordinate system into another. `
` `
` If we desire to adhere to our "old-time" three-dimensional view of `
` things, then we can characterise the development which is being `
` undergone by the fundamental idea of the general theory of relativity `
` as follows : The special theory of relativity has reference to `
` Galileian domains, i.e. to those in which no gravitational field `
` exists. In this connection a Galileian reference-body serves as body `
` of reference, i.e. a rigid body the state of motion of which is so `
` chosen that the Galileian law of the uniform rectilinear motion of `
` "isolated" material points holds relatively to it. `
` `
` Certain considerations suggest that we should refer the same Galileian `
` domains to non-Galileian reference-bodies also. A gravitational field `
` of a special kind is then present with respect to these bodies (cf. `
` Sections 20 and 23). `
` `
` In gravitational fields there are no such things as rigid bodies with `
` Euclidean properties; thus the fictitious rigid body of reference is `
` of no avail in the general theory of relativity. The motion of clocks `
` is also influenced by gravitational fields, and in such a way that a `
` physical definition of time which is made directly with the aid of `
` clocks has by no means the same degree of plausibility as in the `
` special theory of relativity. `
` `
` For this reason non-rigid reference-bodies are used, which are as a `
` whole not only moving in any way whatsoever, but which also suffer `
` alterations in form ad lib. during their motion. Clocks, for which the `
` law of motion is of any kind, however irregular, serve for the `
` definition of time. We have to imagine each of these clocks fixed at a `
` point on the non-rigid reference-body. These clocks satisfy only the `
` one condition, that the "readings" which are observed simultaneously `
` on adjacent clocks (in space) differ from each other by an `
` indefinitely small amount. This non-rigid reference-body, which might `
` appropriately be termed a "reference-mollusc", is in the main `
` equivalent to a Gaussian four-dimensional co-ordinate system chosen `
` arbitrarily. That which gives the "mollusc" a certain `
` comprehensibility as compared with the Gauss co-ordinate system is the `
` (really unjustified) formal retention of the separate existence of the `
` space co-ordinates as opposed to the time co-ordinate. Every point on `
` the mollusc is treated as a space-point, and every material point `
` which is at rest relatively to it as at rest, so long as the mollusc `
` is considered as reference-body. The general principle of relativity `
` requires that all these molluscs can be used as reference-bodies with `
` equal right and equal success in the formulation of the general laws `
` of nature; the laws themselves must be quite independent of the choice `
` of mollusc. `
` `
` The great power possessed by the general principle of relativity lies `
` in the comprehensive limitation which is imposed on the laws of nature `
` in consequence of what we have seen above. `
` `
` `
` `
` THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL `
` PRINCIPLE OF RELATIVITY `
` `
` `
` If the reader has followed all our previous considerations, he will `
` have no further difficulty in understanding the methods leading to the `
` solution of the problem of gravitation. `
` `
` We start off on a consideration of a Galileian domain, i.e. a domain `
` in which there is no gravitational field relative to the Galileian `
` reference-body K. The behaviour of measuring-rods and clocks with `
` reference to K is known from the special theory of relativity, `
` likewise the behaviour of "isolated" material points; the latter move `
` uniformly and in straight lines. `
` `
` Now let us refer this domain to a random Gauss coordinate system or to `
` a "mollusc" as reference-body K1. Then with respect to K1 there is a `
` gravitational field G (of a particular kind). We learn the behaviour `
` of measuring-rods and clocks and also of freely-moving material points `
` with reference to K1 simply by mathematical transformation. We `
` interpret this behaviour as the behaviour of measuring-rods, docks and `
` material points tinder the influence of the gravitational field G. `
` Hereupon we introduce a hypothesis: that the influence of the `
` gravitational field on measuringrods, clocks and freely-moving `
` material points continues to take place according to the same laws, `
` even in the case where the prevailing gravitational field is not `
` derivable from the Galfleian special care, simply by means of a `
` transformation of co-ordinates. `
` `
` The next step is to investigate the space-time behaviour of the `
` gravitational field G, which was derived from the Galileian special `
` case simply by transformation of the coordinates. This behaviour is `
` formulated in a law, which is always valid, no matter how the `
` reference-body (mollusc) used in the description may be chosen. `
` `
` This law is not yet the general law of the gravitational field, since `
` the gravitational field under consideration is of a special kind. In `
` order to find out the general law-of-field of gravitation we still `
` require to obtain a generalisation of the law as found above. This can `
` be obtained without caprice, however, by taking into consideration the `
` following demands: `
` `
` (a) The required generalisation must likewise satisfy the general `
` postulate of relativity. `
` `
` (b) If there is any matter in the domain under consideration, only its `
` inertial mass, and thus according to Section 15 only its energy is `
` of importance for its etfect in exciting a field. `
` `
` (c) Gravitational field and matter together must satisfy the law of `
` the conservation of energy (and of impulse). `
` `
` Finally, the general principle of relativity permits us to determine `
` the influence of the gravitational field on the course of all those `
` processes which take place according to known laws when a `
` gravitational field is absent i.e. which have already been fitted into `
` the frame of the special theory of relativity. In this connection we `
` proceed in principle according to the method which has already been `
` explained for measuring-rods, clocks and freely moving material `
` points. `
` `
` The theory of gravitation derived in this way from the general `
` postulate of relativity excels not only in its beauty ; nor in `
` removing the defect attaching to classical mechanics which was brought `
` to light in Section 21; nor in interpreting the empirical law of `
` the equality of inertial and gravitational mass ; but it has also `
` already explained a result of observation in astronomy, against which `
` classical mechanics is powerless. `
` `
` If we confine the application of the theory to the case where the `
` gravitational fields can be regarded as being weak, and in which all `
` masses move with respect to the coordinate system with velocities `
` which are small compared with the velocity of light, we then obtain as `
` a first approximation the Newtonian theory. Thus the latter theory is `
` obtained here without any particular assumption, whereas Newton had to `
` introduce the hypothesis that the force of attraction between mutually `
` attracting material points is inversely proportional to the square of `
` the distance between them. If we increase the accuracy of the `
` calculation, deviations from the theory of Newton make their `
` appearance, practically all of which must nevertheless escape the test `
` of observation owing to their smallness. `
` `
` We must draw attention here to one of these deviations. According to `
` Newton's theory, a planet moves round the sun in an ellipse, which `
` would permanently maintain its position with respect to the fixed `
` stars, if we could disregard the motion of the fixed stars themselves `
` and the action of the other planets under consideration. Thus, if we `
` correct the observed motion of the planets for these two influences, `
` and if Newton's theory be strictly correct, we ought to obtain for the `
` orbit of the planet an ellipse, which is fixed with reference to the `
` fixed stars. This deduction, which can be tested with great accuracy, `
` has been confirmed for all the planets save one, with the precision `
` that is capable of being obtained by the delicacy of observation `
` attainable at the present time. The sole exception is Mercury, the `
` planet which lies nearest the sun. Since the time of Leverrier, it has `
` been known that the ellipse corresponding to the orbit of Mercury, `
` after it has been corrected for the influences mentioned above, is not `
` stationary with respect to the fixed stars, but that it rotates `
` exceedingly slowly in the plane of the orbit and in the sense of the `
` orbital motion. The value obtained for this rotary movement of the `
` orbital ellipse was 43 seconds of arc per century, an amount ensured `
` to be correct to within a few seconds of arc. This effect can be `
` explained by means of classical mechanics only on the assumption of `
` hypotheses which have little probability, and which were devised `
` solely for this purponse. `
` `
` On the basis of the general theory of relativity, it is found that the `
` ellipse of every planet round the sun must necessarily rotate in the `
` manner indicated above ; that for all the planets, with the exception `
` of Mercury, this rotation is too small to be detected with the `
` delicacy of observation possible at the present time ; but that in the `
` case of Mercury it must amount to 43 seconds of arc per century, a `
` result which is strictly in agreement with observation. `
` `
` Apart from this one, it has hitherto been possible to make only two `
` deductions from the theory which admit of being tested by observation, `
` to wit, the curvature of light rays by the gravitational field of the `
` sun,*x and a displacement of the spectral lines of light reaching `
` us from large stars, as compared with the corresponding lines for `
` light produced in an analogous manner terrestrially (i.e. by the same `
` kind of atom).** These two deductions from the theory have both `
` been confirmed. `
` `
` `
` Notes `
` `
`