# Reading Help Relativity: The Special and General Theory

`

` *) First observed by Eddington and others in 1919. (Cf. Appendix `

` III, pp. 126-129). `

` `

` **) Established by Adams in 1924. (Cf. p. 132) `

` `

` `

` `

` `

` PART III `

` `

` CONSIDERATIONS ON THE UNIVERSE AS A WHOLE `

` `

` `

` COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY `

` `

` `

` Part from the difficulty discussed in Section 21, there is a second `

` fundamental difficulty attending classical celestial mechanics, which, `

` to the best of my knowledge, was first discussed in detail by the `

` astronomer Seeliger. If we ponder over the question as to how the `

` universe, considered as a whole, is to be regarded, the first answer `

` that suggests itself to us is surely this: As regards space (and time) `

` the universe is infinite. There are stars everywhere, so that the `

` density of matter, although very variable in detail, is nevertheless `

` on the average everywhere the same. In other words: However far we `

` might travel through space, we should find everywhere an attenuated `

` swarm of fixed stars of approrimately the same kind and density. `

` `

` This view is not in harmony with the theory of Newton. The latter `

` theory rather requires that the universe should have a kind of centre `

` in which the density of the stars is a maximum, and that as we proceed `

` outwards from this centre the group-density of the stars should `

` diminish, until finally, at great distances, it is succeeded by an `

` infinite region of emptiness. The stellar universe ought to be a `

` finite island in the infinite ocean of space.* `

` `

` This conception is in itself not very satisfactory. It is still less `

` satisfactory because it leads to the result that the light emitted by `

` the stars and also individual stars of the stellar system are `

` perpetually passing out into infinite space, never to return, and `

` without ever again coming into interaction with other objects of `

` nature. Such a finite material universe would be destined to become `

` gradually but systematically impoverished. `

` `

` In order to escape this dilemma, Seeliger suggested a modification of `

` Newton's law, in which he assumes that for great distances the force `

` of attraction between two masses diminishes more rapidly than would `

` result from the inverse square law. In this way it is possible for the `

` mean density of matter to be constant everywhere, even to infinity, `

` without infinitely large gravitational fields being produced. We thus `

` free ourselves from the distasteful conception that the material `

` universe ought to possess something of the nature of a centre. Of `

` course we purchase our emancipation from the fundamental difficulties `

` mentioned, at the cost of a modification and complication of Newton's `

` law which has neither empirical nor theoretical foundation. We can `

` imagine innumerable laws which would serve the same purpose, without `

` our being able to state a reason why one of them is to be preferred to `

` the others ; for any one of these laws would be founded just as little `

` on more general theoretical principles as is the law of Newton. `

` `

` `

` Notes `

` `

` *) Proof -- According to the theory of Newton, the number of "lines `

` of force" which come from infinity and terminate in a mass m is `

` proportional to the mass m. If, on the average, the Mass density p[0] `

` is constant throughout tithe universe, then a sphere of volume V will `

` enclose the average man p[0]V. Thus the number of lines of force `

` passing through the surface F of the sphere into its interior is `

` proportional to p[0] V. For unit area of the surface of the sphere the `

` number of lines of force which enters the sphere is thus proportional `

` to p[0] V/F or to p[0]R. Hence the intensity of the field at the `

` surface would ultimately become infinite with increasing radius R of `

` the sphere, which is impossible. `

` `

` `

` `

` THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE `

` `

` `

` But speculations on the structure of the universe also move in quite `

` another direction. The development of non-Euclidean geometry led to `

` the recognition of the fact, that we can cast doubt on the `

` infiniteness of our space without coming into conflict with the laws `

` of thought or with experience (Riemann, Helmholtz). These questions `

` have already been treated in detail and with unsurpassable lucidity by `

` Helmholtz and Poincar�, whereas I can only touch on them briefly here. `

` `

` In the first place, we imagine an existence in two dimensional space. `

` Flat beings with flat implements, and in particular flat rigid `

` measuring-rods, are free to move in a plane. For them nothing exists `

` outside of this plane: that which they observe to happen to themselves `

` and to their flat " things " is the all-inclusive reality of their `

` plane. In particular, the constructions of plane Euclidean geometry `

` can be carried out by means of the rods e.g. the lattice construction, `

` considered in Section 24. In contrast to ours, the universe of `

` these beings is two-dimensional; but, like ours, it extends to `

` infinity. In their universe there is room for an infinite number of `

` identical squares made up of rods, i.e. its volume (surface) is `

` infinite. If these beings say their universe is " plane," there is `

` sense in the statement, because they mean that they can perform the `

` constructions of plane Euclidean geometry with their rods. In this `

` connection the individual rods always represent the same distance, `

` independently of their position. `

` `

` Let us consider now a second two-dimensional existence, but this time `

` on a spherical surface instead of on a plane. The flat beings with `

` their measuring-rods and other objects fit exactly on this surface and `

` they are unable to leave it. Their whole universe of observation `

` extends exclusively over the surface of the sphere. Are these beings `

` able to regard the geometry of their universe as being plane geometry `

` and their rods withal as the realisation of " distance " ? They cannot `

` do this. For if they attempt to realise a straight line, they will `

` obtain a curve, which we " three-dimensional beings " designate as a `

` great circle, i.e. a self-contained line of definite finite length, `

` which can be measured up by means of a measuring-rod. Similarly, this `

` universe has a finite area that can be compared with the area, of a `

` square constructed with rods. The great charm resulting from this `

` consideration lies in the recognition of the fact that the universe of `

` these beings is finite and yet has no limits. `

` `

` But the spherical-surface beings do not need to go on a world-tour in `

` order to perceive that they are not living in a Euclidean universe. `

` They can convince themselves of this on every part of their " world," `

` provided they do not use too small a piece of it. Starting from a `

` point, they draw " straight lines " (arcs of circles as judged in `

` three dimensional space) of equal length in all directions. They will `

` call the line joining the free ends of these lines a " circle." For a `

` plane surface, the ratio of the circumference of a circle to its `

` diameter, both lengths being measured with the same rod, is, according `

` to Euclidean geometry of the plane, equal to a constant value p, which `

` is independent of the diameter of the circle. On their spherical `

` surface our flat beings would find for this ratio the value `

` `

` eq. 27: file eq27.gif `

` `

` i.e. a smaller value than p, the difference being the more `

` considerable, the greater is the radius of the circle in comparison `

` with the radius R of the " world-sphere." By means of this relation `

` the spherical beings can determine the radius of their universe (" `

` world "), even when only a relatively small part of their worldsphere `

` is available for their measurements. But if this part is very small `

` indeed, they will no longer be able to demonstrate that they are on a `

` spherical " world " and not on a Euclidean plane, for a small part of `

` a spherical surface differs only slightly from a piece of a plane of `

` the same size. `

` `

` Thus if the spherical surface beings are living on a planet of which `

` the solar system occupies only a negligibly small part of the `

` spherical universe, they have no means of determining whether they are `

` living in a finite or in an infinite universe, because the " piece of `

` universe " to which they have access is in both cases practically `

` plane, or Euclidean. It follows directly from this discussion, that `

` for our sphere-beings the circumference of a circle first increases `

` with the radius until the " circumference of the universe " is `

` reached, and that it thenceforward gradually decreases to zero for `

` still further increasing values of the radius. During this process the `

` area of the circle continues to increase more and more, until finally `

` it becomes equal to the total area of the whole " world-sphere." `

` `

` Perhaps the reader will wonder why we have placed our " beings " on a `

` sphere rather than on another closed surface. But this choice has its `

` justification in the fact that, of all closed surfaces, the sphere is `

` unique in possessing the property that all points on it are `

` equivalent. I admit that the ratio of the circumference c of a circle `

` to its radius r depends on r, but for a given value of r it is the `

` same for all points of the " worldsphere "; in other words, the " `

` world-sphere " is a " surface of constant curvature." `

` `

` To this two-dimensional sphere-universe there is a three-dimensional `

` analogy, namely, the three-dimensional spherical space which was `

` discovered by Riemann. its points are likewise all equivalent. It `

` possesses a finite volume, which is determined by its "radius" `

` (2p2R3). Is it possible to imagine a spherical space? To imagine a `

` space means nothing else than that we imagine an epitome of our " `

` space " experience, i.e. of experience that we can have in the `

` movement of " rigid " bodies. In this sense we can imagine a spherical `

` space. `

` `

` Suppose we draw lines or stretch strings in all directions from a `

` point, and mark off from each of these the distance r with a `

` measuring-rod. All the free end-points of these lengths lie on a `

` spherical surface. We can specially measure up the area (F) of this `

` surface by means of a square made up of measuring-rods. If the `

` universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is `

` always less than 4pR2. With increasing values of r, F increases from `

` zero up to a maximum value which is determined by the " world-radius," `

` but for still further increasing values of r, the area gradually `

` diminishes to zero. At first, the straight lines which radiate from `

` the starting point diverge farther and farther from one another, but `

` later they approach each other, and finally they run together again at `

` a "counter-point" to the starting point. Under such conditions they `

` have traversed the whole spherical space. It is easily seen that the `

` three-dimensional spherical space is quite analogous to the `

` two-dimensional spherical surface. It is finite (i.e. of finite `

` volume), and has no bounds. `

` `

` It may be mentioned that there is yet another kind of curved space: " `

` elliptical space." It can be regarded as a curved space in which the `

` two " counter-points " are identical (indistinguishable from each `

`

` *) First observed by Eddington and others in 1919. (Cf. Appendix `

` III, pp. 126-129). `

` `

` **) Established by Adams in 1924. (Cf. p. 132) `

` `

` `

` `

` `

` PART III `

` `

` CONSIDERATIONS ON THE UNIVERSE AS A WHOLE `

` `

` `

` COSMOLOGICAL DIFFICULTIES OF NEWTON'S THEORY `

` `

` `

` Part from the difficulty discussed in Section 21, there is a second `

` fundamental difficulty attending classical celestial mechanics, which, `

` to the best of my knowledge, was first discussed in detail by the `

` astronomer Seeliger. If we ponder over the question as to how the `

` universe, considered as a whole, is to be regarded, the first answer `

` that suggests itself to us is surely this: As regards space (and time) `

` the universe is infinite. There are stars everywhere, so that the `

` density of matter, although very variable in detail, is nevertheless `

` on the average everywhere the same. In other words: However far we `

` might travel through space, we should find everywhere an attenuated `

` swarm of fixed stars of approrimately the same kind and density. `

` `

` This view is not in harmony with the theory of Newton. The latter `

` theory rather requires that the universe should have a kind of centre `

` in which the density of the stars is a maximum, and that as we proceed `

` outwards from this centre the group-density of the stars should `

` diminish, until finally, at great distances, it is succeeded by an `

` infinite region of emptiness. The stellar universe ought to be a `

` finite island in the infinite ocean of space.* `

` `

` This conception is in itself not very satisfactory. It is still less `

` satisfactory because it leads to the result that the light emitted by `

` the stars and also individual stars of the stellar system are `

` perpetually passing out into infinite space, never to return, and `

` without ever again coming into interaction with other objects of `

` nature. Such a finite material universe would be destined to become `

` gradually but systematically impoverished. `

` `

` In order to escape this dilemma, Seeliger suggested a modification of `

` Newton's law, in which he assumes that for great distances the force `

` of attraction between two masses diminishes more rapidly than would `

` result from the inverse square law. In this way it is possible for the `

` mean density of matter to be constant everywhere, even to infinity, `

` without infinitely large gravitational fields being produced. We thus `

` free ourselves from the distasteful conception that the material `

` universe ought to possess something of the nature of a centre. Of `

` course we purchase our emancipation from the fundamental difficulties `

` mentioned, at the cost of a modification and complication of Newton's `

` law which has neither empirical nor theoretical foundation. We can `

` imagine innumerable laws which would serve the same purpose, without `

` our being able to state a reason why one of them is to be preferred to `

` the others ; for any one of these laws would be founded just as little `

` on more general theoretical principles as is the law of Newton. `

` `

` `

` Notes `

` `

` *) Proof -- According to the theory of Newton, the number of "lines `

` of force" which come from infinity and terminate in a mass m is `

` proportional to the mass m. If, on the average, the Mass density p[0] `

` is constant throughout tithe universe, then a sphere of volume V will `

` enclose the average man p[0]V. Thus the number of lines of force `

` passing through the surface F of the sphere into its interior is `

` proportional to p[0] V. For unit area of the surface of the sphere the `

` number of lines of force which enters the sphere is thus proportional `

` to p[0] V/F or to p[0]R. Hence the intensity of the field at the `

` surface would ultimately become infinite with increasing radius R of `

` the sphere, which is impossible. `

` `

` `

` `

` THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE `

` `

` `

` But speculations on the structure of the universe also move in quite `

` another direction. The development of non-Euclidean geometry led to `

` the recognition of the fact, that we can cast doubt on the `

` infiniteness of our space without coming into conflict with the laws `

` of thought or with experience (Riemann, Helmholtz). These questions `

` have already been treated in detail and with unsurpassable lucidity by `

` Helmholtz and Poincar�, whereas I can only touch on them briefly here. `

` `

` In the first place, we imagine an existence in two dimensional space. `

` Flat beings with flat implements, and in particular flat rigid `

` measuring-rods, are free to move in a plane. For them nothing exists `

` outside of this plane: that which they observe to happen to themselves `

` and to their flat " things " is the all-inclusive reality of their `

` plane. In particular, the constructions of plane Euclidean geometry `

` can be carried out by means of the rods e.g. the lattice construction, `

` considered in Section 24. In contrast to ours, the universe of `

` these beings is two-dimensional; but, like ours, it extends to `

` infinity. In their universe there is room for an infinite number of `

` identical squares made up of rods, i.e. its volume (surface) is `

` infinite. If these beings say their universe is " plane," there is `

` sense in the statement, because they mean that they can perform the `

` constructions of plane Euclidean geometry with their rods. In this `

` connection the individual rods always represent the same distance, `

` independently of their position. `

` `

` Let us consider now a second two-dimensional existence, but this time `

` on a spherical surface instead of on a plane. The flat beings with `

` their measuring-rods and other objects fit exactly on this surface and `

` they are unable to leave it. Their whole universe of observation `

` extends exclusively over the surface of the sphere. Are these beings `

` able to regard the geometry of their universe as being plane geometry `

` and their rods withal as the realisation of " distance " ? They cannot `

` do this. For if they attempt to realise a straight line, they will `

` obtain a curve, which we " three-dimensional beings " designate as a `

` great circle, i.e. a self-contained line of definite finite length, `

` which can be measured up by means of a measuring-rod. Similarly, this `

` universe has a finite area that can be compared with the area, of a `

` square constructed with rods. The great charm resulting from this `

` consideration lies in the recognition of the fact that the universe of `

` these beings is finite and yet has no limits. `

` `

` But the spherical-surface beings do not need to go on a world-tour in `

` order to perceive that they are not living in a Euclidean universe. `

` They can convince themselves of this on every part of their " world," `

` provided they do not use too small a piece of it. Starting from a `

` point, they draw " straight lines " (arcs of circles as judged in `

` three dimensional space) of equal length in all directions. They will `

` call the line joining the free ends of these lines a " circle." For a `

` plane surface, the ratio of the circumference of a circle to its `

` diameter, both lengths being measured with the same rod, is, according `

` to Euclidean geometry of the plane, equal to a constant value p, which `

` is independent of the diameter of the circle. On their spherical `

` surface our flat beings would find for this ratio the value `

` `

` eq. 27: file eq27.gif `

` `

` i.e. a smaller value than p, the difference being the more `

` considerable, the greater is the radius of the circle in comparison `

` with the radius R of the " world-sphere." By means of this relation `

` the spherical beings can determine the radius of their universe (" `

` world "), even when only a relatively small part of their worldsphere `

` is available for their measurements. But if this part is very small `

` indeed, they will no longer be able to demonstrate that they are on a `

` spherical " world " and not on a Euclidean plane, for a small part of `

` a spherical surface differs only slightly from a piece of a plane of `

` the same size. `

` `

` Thus if the spherical surface beings are living on a planet of which `

` the solar system occupies only a negligibly small part of the `

` spherical universe, they have no means of determining whether they are `

` living in a finite or in an infinite universe, because the " piece of `

` universe " to which they have access is in both cases practically `

` plane, or Euclidean. It follows directly from this discussion, that `

` for our sphere-beings the circumference of a circle first increases `

` with the radius until the " circumference of the universe " is `

` reached, and that it thenceforward gradually decreases to zero for `

` still further increasing values of the radius. During this process the `

` area of the circle continues to increase more and more, until finally `

` it becomes equal to the total area of the whole " world-sphere." `

` `

` Perhaps the reader will wonder why we have placed our " beings " on a `

` sphere rather than on another closed surface. But this choice has its `

` justification in the fact that, of all closed surfaces, the sphere is `

` unique in possessing the property that all points on it are `

` equivalent. I admit that the ratio of the circumference c of a circle `

` to its radius r depends on r, but for a given value of r it is the `

` same for all points of the " worldsphere "; in other words, the " `

` world-sphere " is a " surface of constant curvature." `

` `

` To this two-dimensional sphere-universe there is a three-dimensional `

` analogy, namely, the three-dimensional spherical space which was `

` discovered by Riemann. its points are likewise all equivalent. It `

` possesses a finite volume, which is determined by its "radius" `

` (2p2R3). Is it possible to imagine a spherical space? To imagine a `

` space means nothing else than that we imagine an epitome of our " `

` space " experience, i.e. of experience that we can have in the `

` movement of " rigid " bodies. In this sense we can imagine a spherical `

` space. `

` `

` Suppose we draw lines or stretch strings in all directions from a `

` point, and mark off from each of these the distance r with a `

` measuring-rod. All the free end-points of these lengths lie on a `

` spherical surface. We can specially measure up the area (F) of this `

` surface by means of a square made up of measuring-rods. If the `

` universe is Euclidean, then F = 4pR2 ; if it is spherical, then F is `

` always less than 4pR2. With increasing values of r, F increases from `

` zero up to a maximum value which is determined by the " world-radius," `

` but for still further increasing values of r, the area gradually `

` diminishes to zero. At first, the straight lines which radiate from `

` the starting point diverge farther and farther from one another, but `

` later they approach each other, and finally they run together again at `

` a "counter-point" to the starting point. Under such conditions they `

` have traversed the whole spherical space. It is easily seen that the `

` three-dimensional spherical space is quite analogous to the `

` two-dimensional spherical surface. It is finite (i.e. of finite `

` volume), and has no bounds. `

` `

` It may be mentioned that there is yet another kind of curved space: " `

` elliptical space." It can be regarded as a curved space in which the `

` two " counter-points " are identical (indistinguishable from each `

`