# 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 `

` `

`