Reading Help Relativity: The Special and General Theory
`
` Thus all our previous conclusions based on general relativity would `
` appear to be called in question. In reality we must make a subtle `
` detour in order to be able to apply the postulate of general `
` relativity exactly. I shall prepare the reader for this in the `
` following paragraphs. `
` `
` `
` Notes `
` `
` *) The field disappears at the centre of the disc and increases `
` proportionally to the distance from the centre as we proceed outwards. `
` `
` **) Throughout this consideration we have to use the Galileian `
` (non-rotating) system K as reference-body, since we may only assume `
` the validity of the results of the special theory of relativity `
` relative to K (relative to K1 a gravitational field prevails). `
` `
` `
` `
` EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM `
` `
` `
` The surface of a marble table is spread out in front of me. I can get `
` from any one point on this table to any other point by passing `
` continuously from one point to a " neighbouring " one, and repeating `
` this process a (large) number of times, or, in other words, by going `
` from point to point without executing "jumps." I am sure the reader `
` will appreciate with sufficient clearness what I mean here by " `
` neighbouring " and by " jumps " (if he is not too pedantic). We `
` express this property of the surface by describing the latter as a `
` continuum. `
` `
` Let us now imagine that a large number of little rods of equal length `
` have been made, their lengths being small compared with the dimensions `
` of the marble slab. When I say they are of equal length, I mean that `
` one can be laid on any other without the ends overlapping. We next lay `
` four of these little rods on the marble slab so that they constitute a `
` quadrilateral figure (a square), the diagonals of which are equally `
` long. To ensure the equality of the diagonals, we make use of a little `
` testing-rod. To this square we add similar ones, each of which has one `
` rod in common with the first. We proceed in like manner with each of `
` these squares until finally the whole marble slab is laid out with `
` squares. The arrangement is such, that each side of a square belongs `
` to two squares and each corner to four squares. `
` `
` It is a veritable wonder that we can carry out this business without `
` getting into the greatest difficulties. We only need to think of the `
` following. If at any moment three squares meet at a corner, then two `
` sides of the fourth square are already laid, and, as a consequence, `
` the arrangement of the remaining two sides of the square is already `
` completely determined. But I am now no longer able to adjust the `
` quadrilateral so that its diagonals may be equal. If they are equal of `
` their own accord, then this is an especial favour of the marble slab `
` and of the little rods, about which I can only be thankfully `
` surprised. We must experience many such surprises if the construction `
` is to be successful. `
` `
` If everything has really gone smoothly, then I say that the points of `
` the marble slab constitute a Euclidean continuum with respect to the `
` little rod, which has been used as a " distance " (line-interval). By `
` choosing one corner of a square as " origin" I can characterise every `
` other corner of a square with reference to this origin by means of two `
` numbers. I only need state how many rods I must pass over when, `
` starting from the origin, I proceed towards the " right " and then " `
` upwards," in order to arrive at the corner of the square under `
` consideration. These two numbers are then the " Cartesian co-ordinates `
` " of this corner with reference to the " Cartesian co-ordinate system" `
` which is determined by the arrangement of little rods. `
` `
` By making use of the following modification of this abstract `
` experiment, we recognise that there must also be cases in which the `
` experiment would be unsuccessful. We shall suppose that the rods " `
` expand " by in amount proportional to the increase of temperature. We `
` heat the central part of the marble slab, but not the periphery, in `
` which case two of our little rods can still be brought into `
` coincidence at every position on the table. But our construction of `
` squares must necessarily come into disorder during the heating, `
` because the little rods on the central region of the table expand, `
` whereas those on the outer part do not. `
` `
` With reference to our little rods -- defined as unit lengths -- the `
` marble slab is no longer a Euclidean continuum, and we are also no `
` longer in the position of defining Cartesian co-ordinates directly `
` with their aid, since the above construction can no longer be carried `
` out. But since there are other things which are not influenced in a `
` similar manner to the little rods (or perhaps not at all) by the `
` temperature of the table, it is possible quite naturally to maintain `
` the point of view that the marble slab is a " Euclidean continuum." `
` This can be done in a satisfactory manner by making a more subtle `
` stipulation about the measurement or the comparison of lengths. `
` `
` But if rods of every kind (i.e. of every material) were to behave in `
` the same way as regards the influence of temperature when they are on `
` the variably heated marble slab, and if we had no other means of `
` detecting the effect of temperature than the geometrical behaviour of `
` our rods in experiments analogous to the one described above, then our `
` best plan would be to assign the distance one to two points on the `
` slab, provided that the ends of one of our rods could be made to `
` coincide with these two points ; for how else should we define the `
` distance without our proceeding being in the highest measure grossly `
` arbitrary ? The method of Cartesian coordinates must then be `
` discarded, and replaced by another which does not assume the validity `
` of Euclidean geometry for rigid bodies.* The reader will notice `
` that the situation depicted here corresponds to the one brought about `
` by the general postitlate of relativity (Section 23). `
` `
` `
` Notes `
` `
` *) Mathematicians have been confronted with our problem in the `
` following form. If we are given a surface (e.g. an ellipsoid) in `
` Euclidean three-dimensional space, then there exists for this surface `
` a two-dimensional geometry, just as much as for a plane surface. Gauss `
` undertook the task of treating this two-dimensional geometry from `
` first principles, without making use of the fact that the surface `
` belongs to a Euclidean continuum of three dimensions. If we imagine `
` constructions to be made with rigid rods in the surface (similar to `
` that above with the marble slab), we should find that different laws `
` hold for these from those resulting on the basis of Euclidean plane `
` geometry. The surface is not a Euclidean continuum with respect to the `
` rods, and we cannot define Cartesian co-ordinates in the surface. `
` Gauss indicated the principles according to which we can treat the `
` geometrical relationships in the surface, and thus pointed out the way `
` to the method of Riemman of treating multi-dimensional, non-Euclidean `
` continuum. Thus it is that mathematicians long ago solved the formal `
` problems to which we are led by the general postulate of relativity. `
` `
` `
` `
` GAUSSIAN CO-ORDINATES `
` `
` `
` According to Gauss, this combined analytical and geometrical mode of `
` handling the problem can be arrived at in the following way. We `
` imagine a system of arbitrary curves (see Fig. 4) drawn on the surface `
` of the table. These we designate as u-curves, and we indicate each of `
` them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in `
` the diagram. Between the curves u= 1 and u= 2 we must imagine an `
` infinitely large number to be drawn, all of which correspond to real `
` numbers lying between 1 and 2. fig. 04 We have then a system of `
` u-curves, and this "infinitely dense" system covers the whole surface `
` of the table. These u-curves must not intersect each other, and `
` through each point of the surface one and only one curve must pass. `
` Thus a perfectly definite value of u belongs to every point on the `
` surface of the marble slab. In like manner we imagine a system of `
` v-curves drawn on the surface. These satisfy the same conditions as `
` the u-curves, they are provided with numbers in a corresponding `
` manner, and they may likewise be of arbitrary shape. It follows that a `
` value of u and a value of v belong to every point on the surface of `
` the table. We call these two numbers the co-ordinates of the surface `
` of the table (Gaussian co-ordinates). For example, the point P in the `
` diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring `
` points P and P1 on the surface then correspond to the co-ordinates `
` `
` P: u,v `
` `
` P1: u + du, v + dv, `
` `
` where du and dv signify very small numbers. In a similar manner we may `
` indicate the distance (line-interval) between P and P1, as measured `
` with a little rod, by means of the very small number ds. Then `
` according to Gauss we have `
` `
` ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2 `
` `
` where g[11], g[12], g[22], are magnitudes which depend in a perfectly `
` definite way on u and v. The magnitudes g[11], g[12] and g[22], `
` determine the behaviour of the rods relative to the u-curves and `
` v-curves, and thus also relative to the surface of the table. For the `
` case in which the points of the surface considered form a Euclidean `
` continuum with reference to the measuring-rods, but only in this case, `
` it is possible to draw the u-curves and v-curves and to attach numbers `
` to them, in such a manner, that we simply have : `
` `
` ds2 = du2 + dv2 `
` `
` Under these conditions, the u-curves and v-curves are straight lines `
` in the sense of Euclidean geometry, and they are perpendicular to each `
` other. Here the Gaussian coordinates are samply Cartesian ones. It is `
` clear that Gauss co-ordinates are nothing more than an association of `
` two sets of numbers with the points of the surface considered, of such `
` a nature that numerical values differing very slightly from each other `
` are associated with neighbouring points " in space." `
` `
` So far, these considerations hold for a continuum of two dimensions. `
` But the Gaussian method can be applied also to a continuum of three, `
` four or more dimensions. If, for instance, a continuum of four `
` dimensions be supposed available, we may represent it in the following `
` way. With every point of the continuum, we associate arbitrarily four `
` numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates." `
` Adjacent points correspond to adjacent values of the coordinates. If a `
` distance ds is associated with the adjacent points P and P1, this `
` distance being measurable and well defined from a physical point of `
` view, then the following formula holds: `
` `
` ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2, `
` `
` where the magnitudes g[11], etc., have values which vary with the `
` position in the continuum. Only when the continuum is a Euclidean one `
` is it possible to associate the co-ordinates x[1] . . x[4]. with the `
`
` Thus all our previous conclusions based on general relativity would `
` appear to be called in question. In reality we must make a subtle `
` detour in order to be able to apply the postulate of general `
` relativity exactly. I shall prepare the reader for this in the `
` following paragraphs. `
` `
` `
` Notes `
` `
` *) The field disappears at the centre of the disc and increases `
` proportionally to the distance from the centre as we proceed outwards. `
` `
` **) Throughout this consideration we have to use the Galileian `
` (non-rotating) system K as reference-body, since we may only assume `
` the validity of the results of the special theory of relativity `
` relative to K (relative to K1 a gravitational field prevails). `
` `
` `
` `
` EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM `
` `
` `
` The surface of a marble table is spread out in front of me. I can get `
` from any one point on this table to any other point by passing `
` continuously from one point to a " neighbouring " one, and repeating `
` this process a (large) number of times, or, in other words, by going `
` from point to point without executing "jumps." I am sure the reader `
` will appreciate with sufficient clearness what I mean here by " `
` neighbouring " and by " jumps " (if he is not too pedantic). We `
` express this property of the surface by describing the latter as a `
` continuum. `
` `
` Let us now imagine that a large number of little rods of equal length `
` have been made, their lengths being small compared with the dimensions `
` of the marble slab. When I say they are of equal length, I mean that `
` one can be laid on any other without the ends overlapping. We next lay `
` four of these little rods on the marble slab so that they constitute a `
` quadrilateral figure (a square), the diagonals of which are equally `
` long. To ensure the equality of the diagonals, we make use of a little `
` testing-rod. To this square we add similar ones, each of which has one `
` rod in common with the first. We proceed in like manner with each of `
` these squares until finally the whole marble slab is laid out with `
` squares. The arrangement is such, that each side of a square belongs `
` to two squares and each corner to four squares. `
` `
` It is a veritable wonder that we can carry out this business without `
` getting into the greatest difficulties. We only need to think of the `
` following. If at any moment three squares meet at a corner, then two `
` sides of the fourth square are already laid, and, as a consequence, `
` the arrangement of the remaining two sides of the square is already `
` completely determined. But I am now no longer able to adjust the `
` quadrilateral so that its diagonals may be equal. If they are equal of `
` their own accord, then this is an especial favour of the marble slab `
` and of the little rods, about which I can only be thankfully `
` surprised. We must experience many such surprises if the construction `
` is to be successful. `
` `
` If everything has really gone smoothly, then I say that the points of `
` the marble slab constitute a Euclidean continuum with respect to the `
` little rod, which has been used as a " distance " (line-interval). By `
` choosing one corner of a square as " origin" I can characterise every `
` other corner of a square with reference to this origin by means of two `
` numbers. I only need state how many rods I must pass over when, `
` starting from the origin, I proceed towards the " right " and then " `
` upwards," in order to arrive at the corner of the square under `
` consideration. These two numbers are then the " Cartesian co-ordinates `
` " of this corner with reference to the " Cartesian co-ordinate system" `
` which is determined by the arrangement of little rods. `
` `
` By making use of the following modification of this abstract `
` experiment, we recognise that there must also be cases in which the `
` experiment would be unsuccessful. We shall suppose that the rods " `
` expand " by in amount proportional to the increase of temperature. We `
` heat the central part of the marble slab, but not the periphery, in `
` which case two of our little rods can still be brought into `
` coincidence at every position on the table. But our construction of `
` squares must necessarily come into disorder during the heating, `
` because the little rods on the central region of the table expand, `
` whereas those on the outer part do not. `
` `
` With reference to our little rods -- defined as unit lengths -- the `
` marble slab is no longer a Euclidean continuum, and we are also no `
` longer in the position of defining Cartesian co-ordinates directly `
` with their aid, since the above construction can no longer be carried `
` out. But since there are other things which are not influenced in a `
` similar manner to the little rods (or perhaps not at all) by the `
` temperature of the table, it is possible quite naturally to maintain `
` the point of view that the marble slab is a " Euclidean continuum." `
` This can be done in a satisfactory manner by making a more subtle `
` stipulation about the measurement or the comparison of lengths. `
` `
` But if rods of every kind (i.e. of every material) were to behave in `
` the same way as regards the influence of temperature when they are on `
` the variably heated marble slab, and if we had no other means of `
` detecting the effect of temperature than the geometrical behaviour of `
` our rods in experiments analogous to the one described above, then our `
` best plan would be to assign the distance one to two points on the `
` slab, provided that the ends of one of our rods could be made to `
` coincide with these two points ; for how else should we define the `
` distance without our proceeding being in the highest measure grossly `
` arbitrary ? The method of Cartesian coordinates must then be `
` discarded, and replaced by another which does not assume the validity `
` of Euclidean geometry for rigid bodies.* The reader will notice `
` that the situation depicted here corresponds to the one brought about `
` by the general postitlate of relativity (Section 23). `
` `
` `
` Notes `
` `
` *) Mathematicians have been confronted with our problem in the `
` following form. If we are given a surface (e.g. an ellipsoid) in `
` Euclidean three-dimensional space, then there exists for this surface `
` a two-dimensional geometry, just as much as for a plane surface. Gauss `
` undertook the task of treating this two-dimensional geometry from `
` first principles, without making use of the fact that the surface `
` belongs to a Euclidean continuum of three dimensions. If we imagine `
` constructions to be made with rigid rods in the surface (similar to `
` that above with the marble slab), we should find that different laws `
` hold for these from those resulting on the basis of Euclidean plane `
` geometry. The surface is not a Euclidean continuum with respect to the `
` rods, and we cannot define Cartesian co-ordinates in the surface. `
` Gauss indicated the principles according to which we can treat the `
` geometrical relationships in the surface, and thus pointed out the way `
` to the method of Riemman of treating multi-dimensional, non-Euclidean `
` continuum. Thus it is that mathematicians long ago solved the formal `
` problems to which we are led by the general postulate of relativity. `
` `
` `
` `
` GAUSSIAN CO-ORDINATES `
` `
` `
` According to Gauss, this combined analytical and geometrical mode of `
` handling the problem can be arrived at in the following way. We `
` imagine a system of arbitrary curves (see Fig. 4) drawn on the surface `
` of the table. These we designate as u-curves, and we indicate each of `
` them by means of a number. The Curves u= 1, u= 2 and u= 3 are drawn in `
` the diagram. Between the curves u= 1 and u= 2 we must imagine an `
` infinitely large number to be drawn, all of which correspond to real `
` numbers lying between 1 and 2. fig. 04 We have then a system of `
` u-curves, and this "infinitely dense" system covers the whole surface `
` of the table. These u-curves must not intersect each other, and `
` through each point of the surface one and only one curve must pass. `
` Thus a perfectly definite value of u belongs to every point on the `
` surface of the marble slab. In like manner we imagine a system of `
` v-curves drawn on the surface. These satisfy the same conditions as `
` the u-curves, they are provided with numbers in a corresponding `
` manner, and they may likewise be of arbitrary shape. It follows that a `
` value of u and a value of v belong to every point on the surface of `
` the table. We call these two numbers the co-ordinates of the surface `
` of the table (Gaussian co-ordinates). For example, the point P in the `
` diagram has the Gaussian co-ordinates u= 3, v= 1. Two neighbouring `
` points P and P1 on the surface then correspond to the co-ordinates `
` `
` P: u,v `
` `
` P1: u + du, v + dv, `
` `
` where du and dv signify very small numbers. In a similar manner we may `
` indicate the distance (line-interval) between P and P1, as measured `
` with a little rod, by means of the very small number ds. Then `
` according to Gauss we have `
` `
` ds2 = g[11]du2 + 2g[12]dudv = g[22]dv2 `
` `
` where g[11], g[12], g[22], are magnitudes which depend in a perfectly `
` definite way on u and v. The magnitudes g[11], g[12] and g[22], `
` determine the behaviour of the rods relative to the u-curves and `
` v-curves, and thus also relative to the surface of the table. For the `
` case in which the points of the surface considered form a Euclidean `
` continuum with reference to the measuring-rods, but only in this case, `
` it is possible to draw the u-curves and v-curves and to attach numbers `
` to them, in such a manner, that we simply have : `
` `
` ds2 = du2 + dv2 `
` `
` Under these conditions, the u-curves and v-curves are straight lines `
` in the sense of Euclidean geometry, and they are perpendicular to each `
` other. Here the Gaussian coordinates are samply Cartesian ones. It is `
` clear that Gauss co-ordinates are nothing more than an association of `
` two sets of numbers with the points of the surface considered, of such `
` a nature that numerical values differing very slightly from each other `
` are associated with neighbouring points " in space." `
` `
` So far, these considerations hold for a continuum of two dimensions. `
` But the Gaussian method can be applied also to a continuum of three, `
` four or more dimensions. If, for instance, a continuum of four `
` dimensions be supposed available, we may represent it in the following `
` way. With every point of the continuum, we associate arbitrarily four `
` numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates." `
` Adjacent points correspond to adjacent values of the coordinates. If a `
` distance ds is associated with the adjacent points P and P1, this `
` distance being measurable and well defined from a physical point of `
` view, then the following formula holds: `
` `
` ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2, `
` `
` where the magnitudes g[11], etc., have values which vary with the `
` position in the continuum. Only when the continuum is a Euclidean one `
` is it possible to associate the co-ordinates x[1] . . x[4]. with the `
`