# 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 `

` `

`