Reading Help Relativity: The Special and General Theory
two " counter-points " are identical (indistinguishable from each `
` other). An elliptical universe can thus be considered to some extent `
` as a curved universe possessing central symmetry. `
` `
` It follows from what has been said, that closed spaces without limits `
` are conceivable. From amongst these, the spherical space (and the `
` elliptical) excels in its simplicity, since all points on it are `
` equivalent. As a result of this discussion, a most interesting `
` question arises for astronomers and physicists, and that is whether `
` the universe in which we live is infinite, or whether it is finite in `
` the manner of the spherical universe. Our experience is far from being `
` sufficient to enable us to answer this question. But the general `
` theory of relativity permits of our answering it with a moduate degree `
` of certainty, and in this connection the difficulty mentioned in `
` Section 30 finds its solution. `
` `
` `
` `
` THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY `
` `
` `
` According to the general theory of relativity, the geometrical `
` properties of space are not independent, but they are determined by `
` matter. Thus we can draw conclusions about the geometrical structure `
` of the universe only if we base our considerations on the state of the `
` matter as being something that is known. We know from experience that, `
` for a suitably chosen co-ordinate system, the velocities of the stars `
` are small as compared with the velocity of transmission of light. We `
` can thus as a rough approximation arrive at a conclusion as to the `
` nature of the universe as a whole, if we treat the matter as being at `
` rest. `
` `
` We already know from our previous discussion that the behaviour of `
` measuring-rods and clocks is influenced by gravitational fields, i.e. `
` by the distribution of matter. This in itself is sufficient to exclude `
` the possibility of the exact validity of Euclidean geometry in our `
` universe. But it is conceivable that our universe differs only `
` slightly from a Euclidean one, and this notion seems all the more `
` probable, since calculations show that the metrics of surrounding `
` space is influenced only to an exceedingly small extent by masses even `
` of the magnitude of our sun. We might imagine that, as regards `
` geometry, our universe behaves analogously to a surface which is `
` irregularly curved in its individual parts, but which nowhere departs `
` appreciably from a plane: something like the rippled surface of a `
` lake. Such a universe might fittingly be called a quasi-Euclidean `
` universe. As regards its space it would be infinite. But calculation `
` shows that in a quasi-Euclidean universe the average density of matter `
` would necessarily be nil. Thus such a universe could not be inhabited `
` by matter everywhere ; it would present to us that unsatisfactory `
` picture which we portrayed in Section 30. `
` `
` If we are to have in the universe an average density of matter which `
` differs from zero, however small may be that difference, then the `
` universe cannot be quasi-Euclidean. On the contrary, the results of `
` calculation indicate that if matter be distributed uniformly, the `
` universe would necessarily be spherical (or elliptical). Since in `
` reality the detailed distribution of matter is not uniform, the real `
` universe will deviate in individual parts from the spherical, i.e. the `
` universe will be quasi-spherical. But it will be necessarily finite. `
` In fact, the theory supplies us with a simple connection * between `
` the space-expanse of the universe and the average density of matter in `
` it. `
` `
` `
` Notes `
` `
` *) For the radius R of the universe we obtain the equation `
` `
` eq. 28: file eq28.gif `
` `
` The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27; `
` p is the average density of the matter and k is a constant connected `
` with the Newtonian constant of gravitation. `
` `
` `
` `
` APPENDIX I `
` `
` SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION `
` (SUPPLEMENTARY TO SECTION 11) `
` `
` `
` For the relative orientation of the co-ordinate systems indicated in `
` Fig. 2, the x-axes of both systems pernumently coincide. In the `
` present case we can divide the problem into parts by considering first `
` only events which are localised on the x-axis. Any such event is `
` represented with respect to the co-ordinate system K by the abscissa x `
` and the time t, and with respect to the system K1 by the abscissa x' `
` and the time t'. We require to find x' and t' when x and t are given. `
` `
` A light-signal, which is proceeding along the positive axis of x, is `
` transmitted according to the equation `
` `
` x = ct `
` `
` or `
` `
` x - ct = 0 . . . (1). `
` `
` Since the same light-signal has to be transmitted relative to K1 with `
` the velocity c, the propagation relative to the system K1 will be `
` represented by the analogous formula `
` `
` x' - ct' = O . . . (2) `
` `
` Those space-time points (events) which satisfy (x) must also satisfy `
` (2). Obviously this will be the case when the relation `
` `
` (x' - ct') = l (x - ct) . . . (3). `
` `
` is fulfilled in general, where l indicates a constant ; for, according `
` to (3), the disappearance of (x - ct) involves the disappearance of `
` (x' - ct'). `
` `
` If we apply quite similar considerations to light rays which are being `
` transmitted along the negative x-axis, we obtain the condition `
` `
` (x' + ct') = �(x + ct) . . . (4). `
` `
` By adding (or subtracting) equations (3) and (4), and introducing for `
` convenience the constants a and b in place of the constants l and �, `
` where `
` `
` eq. 29: file eq29.gif `
` `
` and `
` `
` eq. 30: file eq30.gif `
` `
` we obtain the equations `
` `
` eq. 31: file eq31.gif `
` `
` We should thus have the solution of our problem, if the constants a `
` and b were known. These result from the following discussion. `
` `
` For the origin of K1 we have permanently x' = 0, and hence according `
` to the first of the equations (5) `
` `
` eq. 32: file eq32.gif `
` `
` If we call v the velocity with which the origin of K1 is moving `
` relative to K, we then have `
` `
` eq. 33: file eq33.gif `
` `
` The same value v can be obtained from equations (5), if we calculate `
` the velocity of another point of K1 relative to K, or the velocity `
` (directed towards the negative x-axis) of a point of K with respect to `
` K'. In short, we can designate v as the relative velocity of the two `
` systems. `
` `
` Furthermore, the principle of relativity teaches us that, as judged `
` from K, the length of a unit measuring-rod which is at rest with `
` reference to K1 must be exactly the same as the length, as judged from `
` K', of a unit measuring-rod which is at rest relative to K. In order `
` to see how the points of the x-axis appear as viewed from K, we only `
` require to take a " snapshot " of K1 from K; this means that we have `
` to insert a particular value of t (time of K), e.g. t = 0. For this `
` value of t we then obtain from the first of the equations (5) `
` `
` x' = ax `
` `
` Two points of the x'-axis which are separated by the distance Dx' = I `
` when measured in the K1 system are thus separated in our instantaneous `
` photograph by the distance `
` `
` eq. 34: file eq34.gif `
` `
` But if the snapshot be taken from K'(t' = 0), and if we eliminate t `
` from the equations (5), taking into account the expression (6), we `
` obtain `
` `
` eq. 35: file eq35.gif `
` `
` From this we conclude that two points on the x-axis separated by the `
` distance I (relative to K) will be represented on our snapshot by the `
` distance `
` `
` eq. 36: file eq36.gif `
` `
` But from what has been said, the two snapshots must be identical; `
` hence Dx in (7) must be equal to Dx' in (7a), so that we obtain `
` `
` eq. 37: file eq37.gif `
` `
` The equations (6) and (7b) determine the constants a and b. By `
` inserting the values of these constants in (5), we obtain the first `
` and the fourth of the equations given in Section 11. `
` `
` eq. 38: file eq38.gif `
` `
` Thus we have obtained the Lorentz transformation for events on the `
` x-axis. It satisfies the condition `
` `
` x'2 - c^2t'2 = x2 - c^2t2 . . . (8a). `
` `
` The extension of this result, to include events which take place `
` outside the x-axis, is obtained by retaining equations (8) and `
` supplementing them by the relations `
` `
`
` other). An elliptical universe can thus be considered to some extent `
` as a curved universe possessing central symmetry. `
` `
` It follows from what has been said, that closed spaces without limits `
` are conceivable. From amongst these, the spherical space (and the `
` elliptical) excels in its simplicity, since all points on it are `
` equivalent. As a result of this discussion, a most interesting `
` question arises for astronomers and physicists, and that is whether `
` the universe in which we live is infinite, or whether it is finite in `
` the manner of the spherical universe. Our experience is far from being `
` sufficient to enable us to answer this question. But the general `
` theory of relativity permits of our answering it with a moduate degree `
` of certainty, and in this connection the difficulty mentioned in `
` Section 30 finds its solution. `
` `
` `
` `
` THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY `
` `
` `
` According to the general theory of relativity, the geometrical `
` properties of space are not independent, but they are determined by `
` matter. Thus we can draw conclusions about the geometrical structure `
` of the universe only if we base our considerations on the state of the `
` matter as being something that is known. We know from experience that, `
` for a suitably chosen co-ordinate system, the velocities of the stars `
` are small as compared with the velocity of transmission of light. We `
` can thus as a rough approximation arrive at a conclusion as to the `
` nature of the universe as a whole, if we treat the matter as being at `
` rest. `
` `
` We already know from our previous discussion that the behaviour of `
` measuring-rods and clocks is influenced by gravitational fields, i.e. `
` by the distribution of matter. This in itself is sufficient to exclude `
` the possibility of the exact validity of Euclidean geometry in our `
` universe. But it is conceivable that our universe differs only `
` slightly from a Euclidean one, and this notion seems all the more `
` probable, since calculations show that the metrics of surrounding `
` space is influenced only to an exceedingly small extent by masses even `
` of the magnitude of our sun. We might imagine that, as regards `
` geometry, our universe behaves analogously to a surface which is `
` irregularly curved in its individual parts, but which nowhere departs `
` appreciably from a plane: something like the rippled surface of a `
` lake. Such a universe might fittingly be called a quasi-Euclidean `
` universe. As regards its space it would be infinite. But calculation `
` shows that in a quasi-Euclidean universe the average density of matter `
` would necessarily be nil. Thus such a universe could not be inhabited `
` by matter everywhere ; it would present to us that unsatisfactory `
` picture which we portrayed in Section 30. `
` `
` If we are to have in the universe an average density of matter which `
` differs from zero, however small may be that difference, then the `
` universe cannot be quasi-Euclidean. On the contrary, the results of `
` calculation indicate that if matter be distributed uniformly, the `
` universe would necessarily be spherical (or elliptical). Since in `
` reality the detailed distribution of matter is not uniform, the real `
` universe will deviate in individual parts from the spherical, i.e. the `
` universe will be quasi-spherical. But it will be necessarily finite. `
` In fact, the theory supplies us with a simple connection * between `
` the space-expanse of the universe and the average density of matter in `
` it. `
` `
` `
` Notes `
` `
` *) For the radius R of the universe we obtain the equation `
` `
` eq. 28: file eq28.gif `
` `
` The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27; `
` p is the average density of the matter and k is a constant connected `
` with the Newtonian constant of gravitation. `
` `
` `
` `
` APPENDIX I `
` `
` SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION `
` (SUPPLEMENTARY TO SECTION 11) `
` `
` `
` For the relative orientation of the co-ordinate systems indicated in `
` Fig. 2, the x-axes of both systems pernumently coincide. In the `
` present case we can divide the problem into parts by considering first `
` only events which are localised on the x-axis. Any such event is `
` represented with respect to the co-ordinate system K by the abscissa x `
` and the time t, and with respect to the system K1 by the abscissa x' `
` and the time t'. We require to find x' and t' when x and t are given. `
` `
` A light-signal, which is proceeding along the positive axis of x, is `
` transmitted according to the equation `
` `
` x = ct `
` `
` or `
` `
` x - ct = 0 . . . (1). `
` `
` Since the same light-signal has to be transmitted relative to K1 with `
` the velocity c, the propagation relative to the system K1 will be `
` represented by the analogous formula `
` `
` x' - ct' = O . . . (2) `
` `
` Those space-time points (events) which satisfy (x) must also satisfy `
` (2). Obviously this will be the case when the relation `
` `
` (x' - ct') = l (x - ct) . . . (3). `
` `
` is fulfilled in general, where l indicates a constant ; for, according `
` to (3), the disappearance of (x - ct) involves the disappearance of `
` (x' - ct'). `
` `
` If we apply quite similar considerations to light rays which are being `
` transmitted along the negative x-axis, we obtain the condition `
` `
` (x' + ct') = �(x + ct) . . . (4). `
` `
` By adding (or subtracting) equations (3) and (4), and introducing for `
` convenience the constants a and b in place of the constants l and �, `
` where `
` `
` eq. 29: file eq29.gif `
` `
` and `
` `
` eq. 30: file eq30.gif `
` `
` we obtain the equations `
` `
` eq. 31: file eq31.gif `
` `
` We should thus have the solution of our problem, if the constants a `
` and b were known. These result from the following discussion. `
` `
` For the origin of K1 we have permanently x' = 0, and hence according `
` to the first of the equations (5) `
` `
` eq. 32: file eq32.gif `
` `
` If we call v the velocity with which the origin of K1 is moving `
` relative to K, we then have `
` `
` eq. 33: file eq33.gif `
` `
` The same value v can be obtained from equations (5), if we calculate `
` the velocity of another point of K1 relative to K, or the velocity `
` (directed towards the negative x-axis) of a point of K with respect to `
` K'. In short, we can designate v as the relative velocity of the two `
` systems. `
` `
` Furthermore, the principle of relativity teaches us that, as judged `
` from K, the length of a unit measuring-rod which is at rest with `
` reference to K1 must be exactly the same as the length, as judged from `
` K', of a unit measuring-rod which is at rest relative to K. In order `
` to see how the points of the x-axis appear as viewed from K, we only `
` require to take a " snapshot " of K1 from K; this means that we have `
` to insert a particular value of t (time of K), e.g. t = 0. For this `
` value of t we then obtain from the first of the equations (5) `
` `
` x' = ax `
` `
` Two points of the x'-axis which are separated by the distance Dx' = I `
` when measured in the K1 system are thus separated in our instantaneous `
` photograph by the distance `
` `
` eq. 34: file eq34.gif `
` `
` But if the snapshot be taken from K'(t' = 0), and if we eliminate t `
` from the equations (5), taking into account the expression (6), we `
` obtain `
` `
` eq. 35: file eq35.gif `
` `
` From this we conclude that two points on the x-axis separated by the `
` distance I (relative to K) will be represented on our snapshot by the `
` distance `
` `
` eq. 36: file eq36.gif `
` `
` But from what has been said, the two snapshots must be identical; `
` hence Dx in (7) must be equal to Dx' in (7a), so that we obtain `
` `
` eq. 37: file eq37.gif `
` `
` The equations (6) and (7b) determine the constants a and b. By `
` inserting the values of these constants in (5), we obtain the first `
` and the fourth of the equations given in Section 11. `
` `
` eq. 38: file eq38.gif `
` `
` Thus we have obtained the Lorentz transformation for events on the `
` x-axis. It satisfies the condition `
` `
` x'2 - c^2t'2 = x2 - c^2t2 . . . (8a). `
` `
` The extension of this result, to include events which take place `
` outside the x-axis, is obtained by retaining equations (8) and `
` supplementing them by the relations `
` `
`