# 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 `

`