# Reading Help Relativity: The Special and General Theory

is it possible to associate the co-ordinates x[1] . . x[4]. with the `

` points of the continuum so that we have simply `

` `

` ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2. `

` `

` In this case relations hold in the four-dimensional continuum which `

` are analogous to those holding in our three-dimensional measurements. `

` `

` However, the Gauss treatment for ds2 which we have given above is not `

` always possible. It is only possible when sufficiently small regions `

` of the continuum under consideration may be regarded as Euclidean `

` continua. For example, this obviously holds in the case of the marble `

` slab of the table and local variation of temperature. The temperature `

` is practically constant for a small part of the slab, and thus the `

` geometrical behaviour of the rods is almost as it ought to be `

` according to the rules of Euclidean geometry. Hence the imperfections `

` of the construction of squares in the previous section do not show `

` themselves clearly until this construction is extended over a `

` considerable portion of the surface of the table. `

` `

` We can sum this up as follows: Gauss invented a method for the `

` mathematical treatment of continua in general, in which " `

` size-relations " (" distances " between neighbouring points) are `

` defined. To every point of a continuum are assigned as many numbers `

` (Gaussian coordinates) as the continuum has dimensions. This is done `

` in such a way, that only one meaning can be attached to the `

` assignment, and that numbers (Gaussian coordinates) which differ by an `

` indefinitely small amount are assigned to adjacent points. The `

` Gaussian coordinate system is a logical generalisation of the `

` Cartesian co-ordinate system. It is also applicable to non-Euclidean `

` continua, but only when, with respect to the defined "size" or `

` "distance," small parts of the continuum under consideration behave `

` more nearly like a Euclidean system, the smaller the part of the `

` continuum under our notice. `

` `

` `

` `

` THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A `

` EUCLIDEAN CONTINUUM `

` `

` `

` We are now in a position to formulate more exactly the idea of `

` Minkowski, which was only vaguely indicated in Section 17. In `

` accordance with the special theory of relativity, certain co-ordinate `

` systems are given preference for the description of the `

` four-dimensional, space-time continuum. We called these " Galileian `

` co-ordinate systems." For these systems, the four co-ordinates x, y, `

` z, t, which determine an event or -- in other words, a point of the `

` four-dimensional continuum -- are defined physically in a simple `

` manner, as set forth in detail in the first part of this book. For the `

` transition from one Galileian system to another, which is moving `

` uniformly with reference to the first, the equations of the Lorentz `

` transformation are valid. These last form the basis for the derivation `

` of deductions from the special theory of relativity, and in themselves `

` they are nothing more than the expression of the universal validity of `

` the law of transmission of light for all Galileian systems of `

` reference. `

` `

` Minkowski found that the Lorentz transformations satisfy the following `

` simple conditions. Let us consider two neighbouring events, the `

` relative position of which in the four-dimensional continuum is given `

` with respect to a Galileian reference-body K by the space co-ordinate `

` differences dx, dy, dz and the time-difference dt. With reference to a `

` second Galileian system we shall suppose that the corresponding `

` differences for these two events are dx1, dy1, dz1, dt1. Then these `

` magnitudes always fulfil the condition* `

` `

` dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2. `

` `

` The validity of the Lorentz transformation follows from this `

` condition. We can express this as follows: The magnitude `

` `

` ds2 = dx2 + dy2 + dz2 - c^2dt2, `

` `

` which belongs to two adjacent points of the four-dimensional `

` space-time continuum, has the same value for all selected (Galileian) `

` reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1], `

` x[2], x[3], x[4], we also obtaill the result that `

` `

` ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2. `

` `

` is independent of the choice of the body of reference. We call the `

` magnitude ds the " distance " apart of the two events or `

` four-dimensional points. `

` `

` Thus, if we choose as time-variable the imaginary variable sq. rt. -I `

` . ct instead of the real quantity t, we can regard the space-time `

` contintium -- accordance with the special theory of relativity -- as a `

` ", Euclidean " four-dimensional continuum, a result which follows from `

` the considerations of the preceding section. `

` `

` `

` Notes `

` `

` *) Cf. Appendixes I and 2. The relations which are derived `

` there for the co-ordlnates themselves are valid also for co-ordinate `

` differences, and thus also for co-ordinate differentials (indefinitely `

` small differences). `

` `

` `

` `

` THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A `

` ECULIDEAN CONTINUUM `

` `

` `

` In the first part of this book we were able to make use of space-time `

` co-ordinates which allowed of a simple and direct physical `

` interpretation, and which, according to Section 26, can be regarded `

` as four-dimensional Cartesian co-ordinates. This was possible on the `

` basis of the law of the constancy of the velocity of tight. But `

` according to Section 21 the general theory of relativity cannot `

` retain this law. On the contrary, we arrived at the result that `

` according to this latter theory the velocity of light must always `

` depend on the co-ordinates when a gravitational field is present. In `

` connection with a specific illustration in Section 23, we found `

` that the presence of a gravitational field invalidates the definition `

` of the coordinates and the ifine, which led us to our objective in the `

` special theory of relativity. `

` `

` In view of the resuIts of these considerations we are led to the `

` conviction that, according to the general principle of relativity, the `

` space-time continuum cannot be regarded as a Euclidean one, but that `

` here we have the general case, corresponding to the marble slab with `

` local variations of temperature, and with which we made acquaintance `

` as an example of a two-dimensional continuum. Just as it was there `

` impossible to construct a Cartesian co-ordinate system from equal `

` rods, so here it is impossible to build up a system (reference-body) `

` from rigid bodies and clocks, which shall be of such a nature that `

` measuring-rods and clocks, arranged rigidly with respect to one `

` another, shaIll indicate position and time directly. Such was the `

` essence of the difficulty with which we were confronted in Section `

` 23. `

` `

` But the considerations of Sections 25 and 26 show us the way to `

` surmount this difficulty. We refer the fourdimensional space-time `

` continuum in an arbitrary manner to Gauss co-ordinates. We assign to `

` every point of the continuum (event) four numbers, x[1], x[2], x[3], `

` x[4] (co-ordinates), which have not the least direct physical `

` significance, but only serve the purpose of numbering the points of `

` the continuum in a definite but arbitrary manner. This arrangement `

` does not even need to be of such a kind that we must regard x[1], `

` x[2], x[3], as "space" co-ordinates and x[4], as a " time " `

` co-ordinate. `

` `

` The reader may think that such a description of the world would be `

` quite inadequate. What does it mean to assign to an event the `

` particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these `

` co-ordinates have no significance ? More careful consideration shows, `

` however, that this anxiety is unfounded. Let us consider, for `

` instance, a material point with any kind of motion. If this point had `

` only a momentary existence without duration, then it would to `

` described in space-time by a single system of values x[1], x[2], x[3], `

` x[4]. Thus its permanent existence must be characterised by an `

` infinitely large number of such systems of values, the co-ordinate `

` values of which are so close together as to give continuity; `

` corresponding to the material point, we thus have a (uni-dimensional) `

` line in the four-dimensional continuum. In the same way, any such `

` lines in our continuum correspond to many points in motion. The only `

` statements having regard to these points which can claim a physical `

` existence are in reality the statements about their encounters. In our `

` mathematical treatment, such an encounter is expressed in the fact `

` that the two lines which represent the motions of the points in `

` question have a particular system of co-ordinate values, x[1], x[2], `

` x[3], x[4], in common. After mature consideration the reader will `

` doubtless admit that in reality such encounters constitute the only `

` actual evidence of a time-space nature with which we meet in physical `

` statements. `

` `

` When we were describing the motion of a material point relative to a `

` body of reference, we stated nothing more than the encounters of this `

` point with particular points of the reference-body. We can also `

` determine the corresponding values of the time by the observation of `

` encounters of the body with clocks, in conjunction with the `

` observation of the encounter of the hands of clocks with particular `

` points on the dials. It is just the same in the case of `

` space-measurements by means of measuring-rods, as a litttle `

` consideration will show. `

` `

` The following statements hold generally : Every physical description `

` resolves itself into a number of statements, each of which refers to `

` the space-time coincidence of two events A and B. In terms of Gaussian `

` co-ordinates, every such statement is expressed by the agreement of `

` their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the `

` description of the time-space continuum by means of Gauss co-ordinates `

` completely replaces the description with the aid of a body of `

` reference, without suffering from the defects of the latter mode of `

` description; it is not tied down to the Euclidean character of the `

` continuum which has to be represented. `

` `

` `

` `

` EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY `

` `

` `

` We are now in a position to replace the pro. visional formulation of `

` the general principle of relativity given in Section 18 by an exact `

` formulation. The form there used, "All bodies of reference K, K1, `

` etc., are equivalent for the description of natural phenomena `

` (formulation of the general laws of nature), whatever may be their `

` state of motion," cannot be maintained, because the use of rigid `

` reference-bodies, in the sense of the method followed in the special `

`

` points of the continuum so that we have simply `

` `

` ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2. `

` `

` In this case relations hold in the four-dimensional continuum which `

` are analogous to those holding in our three-dimensional measurements. `

` `

` However, the Gauss treatment for ds2 which we have given above is not `

` always possible. It is only possible when sufficiently small regions `

` of the continuum under consideration may be regarded as Euclidean `

` continua. For example, this obviously holds in the case of the marble `

` slab of the table and local variation of temperature. The temperature `

` is practically constant for a small part of the slab, and thus the `

` geometrical behaviour of the rods is almost as it ought to be `

` according to the rules of Euclidean geometry. Hence the imperfections `

` of the construction of squares in the previous section do not show `

` themselves clearly until this construction is extended over a `

` considerable portion of the surface of the table. `

` `

` We can sum this up as follows: Gauss invented a method for the `

` mathematical treatment of continua in general, in which " `

` size-relations " (" distances " between neighbouring points) are `

` defined. To every point of a continuum are assigned as many numbers `

` (Gaussian coordinates) as the continuum has dimensions. This is done `

` in such a way, that only one meaning can be attached to the `

` assignment, and that numbers (Gaussian coordinates) which differ by an `

` indefinitely small amount are assigned to adjacent points. The `

` Gaussian coordinate system is a logical generalisation of the `

` Cartesian co-ordinate system. It is also applicable to non-Euclidean `

` continua, but only when, with respect to the defined "size" or `

` "distance," small parts of the continuum under consideration behave `

` more nearly like a Euclidean system, the smaller the part of the `

` continuum under our notice. `

` `

` `

` `

` THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A `

` EUCLIDEAN CONTINUUM `

` `

` `

` We are now in a position to formulate more exactly the idea of `

` Minkowski, which was only vaguely indicated in Section 17. In `

` accordance with the special theory of relativity, certain co-ordinate `

` systems are given preference for the description of the `

` four-dimensional, space-time continuum. We called these " Galileian `

` co-ordinate systems." For these systems, the four co-ordinates x, y, `

` z, t, which determine an event or -- in other words, a point of the `

` four-dimensional continuum -- are defined physically in a simple `

` manner, as set forth in detail in the first part of this book. For the `

` transition from one Galileian system to another, which is moving `

` uniformly with reference to the first, the equations of the Lorentz `

` transformation are valid. These last form the basis for the derivation `

` of deductions from the special theory of relativity, and in themselves `

` they are nothing more than the expression of the universal validity of `

` the law of transmission of light for all Galileian systems of `

` reference. `

` `

` Minkowski found that the Lorentz transformations satisfy the following `

` simple conditions. Let us consider two neighbouring events, the `

` relative position of which in the four-dimensional continuum is given `

` with respect to a Galileian reference-body K by the space co-ordinate `

` differences dx, dy, dz and the time-difference dt. With reference to a `

` second Galileian system we shall suppose that the corresponding `

` differences for these two events are dx1, dy1, dz1, dt1. Then these `

` magnitudes always fulfil the condition* `

` `

` dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2. `

` `

` The validity of the Lorentz transformation follows from this `

` condition. We can express this as follows: The magnitude `

` `

` ds2 = dx2 + dy2 + dz2 - c^2dt2, `

` `

` which belongs to two adjacent points of the four-dimensional `

` space-time continuum, has the same value for all selected (Galileian) `

` reference-bodies. If we replace x, y, z, sq. rt. -I . ct , by x[1], `

` x[2], x[3], x[4], we also obtaill the result that `

` `

` ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2. `

` `

` is independent of the choice of the body of reference. We call the `

` magnitude ds the " distance " apart of the two events or `

` four-dimensional points. `

` `

` Thus, if we choose as time-variable the imaginary variable sq. rt. -I `

` . ct instead of the real quantity t, we can regard the space-time `

` contintium -- accordance with the special theory of relativity -- as a `

` ", Euclidean " four-dimensional continuum, a result which follows from `

` the considerations of the preceding section. `

` `

` `

` Notes `

` `

` *) Cf. Appendixes I and 2. The relations which are derived `

` there for the co-ordlnates themselves are valid also for co-ordinate `

` differences, and thus also for co-ordinate differentials (indefinitely `

` small differences). `

` `

` `

` `

` THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF REALTIIVTY IS NOT A `

` ECULIDEAN CONTINUUM `

` `

` `

` In the first part of this book we were able to make use of space-time `

` co-ordinates which allowed of a simple and direct physical `

` interpretation, and which, according to Section 26, can be regarded `

` as four-dimensional Cartesian co-ordinates. This was possible on the `

` basis of the law of the constancy of the velocity of tight. But `

` according to Section 21 the general theory of relativity cannot `

` retain this law. On the contrary, we arrived at the result that `

` according to this latter theory the velocity of light must always `

` depend on the co-ordinates when a gravitational field is present. In `

` connection with a specific illustration in Section 23, we found `

` that the presence of a gravitational field invalidates the definition `

` of the coordinates and the ifine, which led us to our objective in the `

` special theory of relativity. `

` `

` In view of the resuIts of these considerations we are led to the `

` conviction that, according to the general principle of relativity, the `

` space-time continuum cannot be regarded as a Euclidean one, but that `

` here we have the general case, corresponding to the marble slab with `

` local variations of temperature, and with which we made acquaintance `

` as an example of a two-dimensional continuum. Just as it was there `

` impossible to construct a Cartesian co-ordinate system from equal `

` rods, so here it is impossible to build up a system (reference-body) `

` from rigid bodies and clocks, which shall be of such a nature that `

` measuring-rods and clocks, arranged rigidly with respect to one `

` another, shaIll indicate position and time directly. Such was the `

` essence of the difficulty with which we were confronted in Section `

` 23. `

` `

` But the considerations of Sections 25 and 26 show us the way to `

` surmount this difficulty. We refer the fourdimensional space-time `

` continuum in an arbitrary manner to Gauss co-ordinates. We assign to `

` every point of the continuum (event) four numbers, x[1], x[2], x[3], `

` x[4] (co-ordinates), which have not the least direct physical `

` significance, but only serve the purpose of numbering the points of `

` the continuum in a definite but arbitrary manner. This arrangement `

` does not even need to be of such a kind that we must regard x[1], `

` x[2], x[3], as "space" co-ordinates and x[4], as a " time " `

` co-ordinate. `

` `

` The reader may think that such a description of the world would be `

` quite inadequate. What does it mean to assign to an event the `

` particular co-ordinates x[1], x[2], x[3], x[4], if in themselves these `

` co-ordinates have no significance ? More careful consideration shows, `

` however, that this anxiety is unfounded. Let us consider, for `

` instance, a material point with any kind of motion. If this point had `

` only a momentary existence without duration, then it would to `

` described in space-time by a single system of values x[1], x[2], x[3], `

` x[4]. Thus its permanent existence must be characterised by an `

` infinitely large number of such systems of values, the co-ordinate `

` values of which are so close together as to give continuity; `

` corresponding to the material point, we thus have a (uni-dimensional) `

` line in the four-dimensional continuum. In the same way, any such `

` lines in our continuum correspond to many points in motion. The only `

` statements having regard to these points which can claim a physical `

` existence are in reality the statements about their encounters. In our `

` mathematical treatment, such an encounter is expressed in the fact `

` that the two lines which represent the motions of the points in `

` question have a particular system of co-ordinate values, x[1], x[2], `

` x[3], x[4], in common. After mature consideration the reader will `

` doubtless admit that in reality such encounters constitute the only `

` actual evidence of a time-space nature with which we meet in physical `

` statements. `

` `

` When we were describing the motion of a material point relative to a `

` body of reference, we stated nothing more than the encounters of this `

` point with particular points of the reference-body. We can also `

` determine the corresponding values of the time by the observation of `

` encounters of the body with clocks, in conjunction with the `

` observation of the encounter of the hands of clocks with particular `

` points on the dials. It is just the same in the case of `

` space-measurements by means of measuring-rods, as a litttle `

` consideration will show. `

` `

` The following statements hold generally : Every physical description `

` resolves itself into a number of statements, each of which refers to `

` the space-time coincidence of two events A and B. In terms of Gaussian `

` co-ordinates, every such statement is expressed by the agreement of `

` their four co-ordinates x[1], x[2], x[3], x[4]. Thus in reality, the `

` description of the time-space continuum by means of Gauss co-ordinates `

` completely replaces the description with the aid of a body of `

` reference, without suffering from the defects of the latter mode of `

` description; it is not tied down to the Euclidean character of the `

` continuum which has to be represented. `

` `

` `

` `

` EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY `

` `

` `

` We are now in a position to replace the pro. visional formulation of `

` the general principle of relativity given in Section 18 by an exact `

` formulation. The form there used, "All bodies of reference K, K1, `

` etc., are equivalent for the description of natural phenomena `

` (formulation of the general laws of nature), whatever may be their `

` state of motion," cannot be maintained, because the use of rigid `

` reference-bodies, in the sense of the method followed in the special `

`