Relativity - The Special and General Theory by Albert Einstein (early reader books txt) 📕

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 measuringrods, 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 measuringrods 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