Theories of the Universe

Milton K. Munitz

• Publisher: Free Press
• Year: 2008
• ISBN:
  9781439119280
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

The theoretical physicist shares his latest thoughts on the nature of space and time in this anthology of selections from Princeton University Press. Along with eminent colleagues, Hawking extends theoretical frontiers by speculating on the big questions of modern cosmology.

Modern Theories of the Universe

Introduction

THE INCREASE IN ATTENTION to cosmological questions since the second decade of the present century is due primarily to the promise held out for a successful prosecution of studies in this area due to the use of new instruments, both optical and conceptual, that made their appearance during this period. It is true that the use of new instruments, however radical their innovations, are generally but improvements in one form or another upon older models. And this we find to be the case with the more powerful telescopes that the astronomer uses to gather his data. It is similarly true of the various mathematical tools that the physicist uses to construct his theoretical models. Yet the difference from the old may be so great, that they warrant an altogether fresh outlook. When for example, Hubble began his studies with the use of the 100-inch telescope, or Einstein incorporated into his relativity theory the basic ideas of non-Euclidean geometry and the tensor calculus, these more refined instruments brought such a wealth of information or insight, that they helped to raise the subject of cosmology to a new level altogether. Questions that had been endlessly debated in earlier periods, either now found definite answers or were seen in an altogether different light because of the rich field of possibilities opened up.

We must recognize, at the same time, a certain parallel in the problems confronting cosmology today and those which engaged the attention of cosmologists at the beginnings of modern astronomy following upon the Copernican revolution. In that earlier period, as we have seen, the assimilation of the earth as a member of the planetary family of the sun, and the sun as a member of the system of stars, raised the fundamental question as to the extent and possible structure of the universe, where the latter is taken as made up of the stars as its basic astronomical units. Today the scale has shifted from stars to galaxies. We know now what was only surmised in the eighteenth century by Wright, Kant, Herschel and others. Our Galaxy is a vast but finite system of stars that is located among an enormously extensive population of galactic systems of different types, shapes and structures. The universe, as far as present knowledge indicates, is made up of such galaxies as its basic units. The problem with which cosmology is now primarily concerned is then: How extensive is the universe of galaxies? What spatial structure does it have? Is the “space” or “geometry” to be used in describing it of a finite or an infinite sort? Is the universe, when regarded from a temporal point of view, one that had a finite origin in the past or has it always been in existence? If the latter, has it always possessed roughly the same structure as it does now? These alternatives and the several refinements and combinations that result from their analyses yield a variety of theories actively discussed in the present phase of our subject.

The general theory of relativity, which Einstein proposed in 1915, undertook to widen and generalize the ideas contained in the special theory of relativity of 1905. It also provided through its new interpretation of gravitational phenomena, a schematic base for encompassing the whole range of mechanical phenomena dealt with in classical physics. Its superiority over the older physics consisted in its greater systematic simplicity and the capacity it had both for making predictions and for offering explanations over a wider range and with a greater precision than had been possible formerly. The first major success of the general theory of relativity was obtained in dealing with the dynamics of the solar system. It was here that the famous three “crucial” tests of the theory were obtained. It was Einstein himself, in a paper published in 1917, who went on to show that the general theory of relativity had another useful line of theoretical application. This time its basic equations were designed to assist in understanding the structure of the physical universe as a whole, as distinguished from the relatively restricted problems encountered in the dynamics of the solar system. In this first cosmological application of general relativity (briefly and simply expounded by Einstein in his discussion of “Considerations on the Universe as a Whole”), the way was pointed out for making use of non-Euclidean geometry in framing a world-view. The universe was pictured as being at once both finite and unbounded. Since then, not only Einstein, but others as well have worked out numerous other possibilities within the framework of “relativistic cosmology” on the basis of the leads furnished by this path-breaking investigation.

What Hubble established by 1924 in his epoch-making observational studies of “the realm of the nebulae” was the general answer to the much-debated question as to the status of the nebulae. The two possibilities, that they were independent stellar systems lying beyond the confines of our own Galaxy, or on the other hand, were proper parts of our own Galaxy, was settled in favor of their existence as independent systems. The preliminary “reconnaissance” of the observable region of the nebulae, as recounted by Hubble, established two main empirical results. One was that the extra-galactic nebulae (or, as some writers prefer to call them, simply, “the galaxies”) are distributed throughout space in a uniform and homogeneous manner. The other was that the spectra of these galaxies display a shift toward the red—the greater the shift the more distant they are. This latter feature is commonly interpreted as signifying that the galaxies are receding from us and from one another in such a fashion that the more distant the galaxy, the faster is its velocity of recession. It is this fact which lends support to the belief that the universe is “expanding.” A further selection by Hubble, given below, sketches the observational program to be undertaken by the 200-inch Palomar telescope. It indicates both the progress made since the initial exploratory studies and some of the typical problems still facing the astronomer.

From the time of the construction of the “static” Einstein model (“static” because it did not envisage the possibility of the recession of the galaxies or the expansion of the system as a whole) and the exploratory observational studies by Hubble and others of the properties of the galaxies, the field of scientific cosmology has developed rapidly. In De Sitter’s discussion of “Relativity and Modern Theories of the Universe,” the reader will find an expert summary of the development of relativistic cosmologies up to the period of the early nineteen-thirties. It is written by one who himself played an important role in that development. Relativistic cosmologies find their common base in the “field-equations” of the general theory of relativity. Since the field-equations of that theory, being its most fundamental part, are deliberately stated in a most general form, they leave unspecified the particular values to be given to certain key variables. The specification of these to suit, for example, the cosmologie field of their application, is made in accordance with reasons that seem compelling to a given author or at a given stage of research. The series of resulting equations, being in effect tools of calculation and mathematical devices of representation, constitute what we mean by “models of the universe as a whole.” It was in this fashion that the original “Einstein model” and the one proposed by De Sitter shortly afterward (known as the “De Sitter model”) were constructed. These “Einstein” and “De Sitter” models, however, were essentially “static” models. They were soon to be replaced by a group of models which took as their central feature the “expansion” of space. These are designated as “the expanding-universe models of general relativity.” Eddington and his pupil Lemaître shared a prominent role in the development and popularization of these ideas. Each of them offers below his own highly instructive account of this side of the subject.

In the mid-thirties, the English cosmologist E. A. Milne undertook to challenge the almost exclusive reliance hitherto put upon general relativity as a base from which to construct theoretical models of the universe. In a system of thought which he identified by the name “Kinematic Relativity,” Milne offered a most ingenious conceptual framework in terms of which to construct a model of the expanding universe. It is an approach in which the notion of time assumes a central importance as contrasted with the more “geometrizing” emphases of relativity theory. His approach illustrates a mode of thought that would seem, however, for all its suggestiveness, not to have won any considerable following. One of the reasons, perhaps, for the reluctance of scientists in general to accept Milne’s point of view, is its highly rationalistic character. It places its primary emphasis on deuctions made from allegedly “self-evident” premisses rather than on the leads and checks furnished by observational experience.

The more conventional, orthodox, and antirationalistic viewpoint of scientists is illustrated in the discussion by H. P. Robertson of “Geometry as a Branch of Physics.” Robertson writes primarily from the vantage point of “orthodox” relativistic cosmology, to which he himself made important contributions. He stresses the necessary role of observational experience in scientific cosmology. Such a role, he argues, arises particularly with respect to the choice of a “metric” or geometry to be used in constructing an adequate model of the universe as a whole.

Since the end of the Second World War the subject of cosmology has continued to receive important contributions of both an observational and theoretical sort. The use of more powerful instruments such as the 200-inch Hale telescope, the 48-inch Schmidt telescope, and others, has increased in a significant way the quantity and quality of observational data. Among the most important results obtained in this direction is the recent recalibration of the distance-scale for galaxies, effected by Baαade at Palomar Observatory. This has given grounds for a considerable revision upward in the calculated range to which our instruments may be taken as probing in space. By this change in the scale to be used for the calculation of cosmic distances, some of the difficulties faced by earlier models of the “expanding universe” no longer hold. These included particularly the problem of how to accept the so-called “age of the universe,” which came out on some views to be less than the calculated ages of some of the constituents of the universe, such as the earth. Another development which promises to hold important consequences for the whole subject of cosmology is the relatively recent use of radio astronomy as a means of gathering data, in addition to the conventional reliance upon optical telescopes.

Meanwhile on a conceptual level the current field of interest is divided among those who favor a “continuous creation” theory of the expanding universe as against those who uphold an “evolutionary” type of cosmology. In the arguments offered by Bondi, Sciama and Hoyle, who are among the chief protagonists of the “continuous creation” theory, we find the universe assigned an infinite time-scale. In its gross features, the universe is thought of as being ever the same, no matter at what point in time it is considered or from whatever station in space it is viewed. Matter, it is claimed, is created continuously in elementary form. It is out of such randomly produced particles that eventually agglomerations of matter of the size of galaxies and clusters of galaxies come to be formed. While the continuing expansion of the universe removes some galaxies from our range of observability, other newly-formed systems come to take their place and thus help to keep the universe in a “steady-state.”

In opposition to this view are those who favor an “evolutionary” world picture, like Lemaître or Gamow. They would account for the variety of observational facts, including the relative abundances of the various elements as found throughout space, in terms of a universe that “originated” at some finite epoch in the past. The event which marked such an origin is pictured as both catacylsmic in its proportions and unique in the thermonuclear and other physical conditions which it possessed. It is from such a beginning, conceived as a “Primeval Atom” by Lemaître and as an extremely compressed state of matter called “ylem” by Gamow, that the present constitution of the universe is taken as having evolved.

Considerations on the Universe as a Whole

ALBERT EINSTEIN:

Cosmological Difficulties of Newton’s Theory

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 approximately 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 improverished.

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From Albert Einstein, Relativity: The Special and General Theory, translated by R. W. Lawson, 1920, Part III, Chapters XXX, XXXI, XXXII. Reprinted by kind permission of the publisher, Peter Smith, New York.

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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.

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 Poincaré, 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, arc 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 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 π, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

i.e. a smaller value than π, 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 world-sphere 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 thence-forward gra...