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 `
`