## Cosmology Before Einstein

Discoveries in observational astronomy would provide the basis for the revolution in cosmology that took place after 1900. It should nevertheless be noted that the study of cosmology existed before this time, largely as the study of general questions about the universe as a whole. These investigations reduced, in some cases, to speculative intellectual exercises, but they prepared the way for new mathematical descriptions of the universe and would become important when the revolutionary observational findings of the 1920s were in place. It was out of this speculative, older tradition that analysis of the universe based on Einstein's general theory of relativity emerged, and the latter would eventually provide the theoretical framework for all of modern work in cosmology.

In the eighteenth century a general picture of the universe emerged in which space is populated by stars, self-luminous objects similar to the Sun. The notion of an extended, possibly infinite distribution of stars in space raised certain puzzles about what we should expect to observe in the sky. In particular, there seemed to be a question about the background level of brightness in the sky that would result from such a distribution of light sources. The fact that the sky is dark at night emerged as something that required explanation. The question was discussed by the Swiss astronomer Philippe Loys de Cheseaux (1718—1751) in 1744 and again independently in 1826 by Heinrich Wilhelm Olbers (1758-1840), a German physician and amateur astronomer. The puzzle is known today as Olber's paradox, a name introduced by the British cosmolo-gist Herman Bondi (1919-) in 1958.

Let us assume that stars are evenly distributed so that their average density in space is constant. Consider an imaginary, thin, spherical shell centered on the Earth. Our goal is to calculate the total intensity of the light reaching the Earth from all the stars contained within this shell. The number of stars will be proportional to the area of the surface of the shell multiplied by the small thickness of the shell. This area is proportional to the square of the radius. On the other hand, the intensity of light from each star will be proportional to one over the square of the radius. It follows that the intensity of light coming from all of the stars in the shell is equal to a constant times the thickness of the shell. As we go deeper into space, we encounter spheres of ever-increasing radius. If all of the starlight reached us, then the total intensity of the starlight from all of the stars in a given sphere would be proportional to its radius. However, some of the light from the more distant stars will be blocked by closer stars. The total amount of starlight from all the stars in a sphere will not increase indefinitely with the radius but will eventually reach a limiting value. Under any reasonable assumptions about the average size and brightness of stars and their density in space it turns out that this value is very high. The sky should be ablaze with light of great intensity. The paradox evident in the darkness of the night sky seems to follow from simple and plausible assumptions about the universe as a whole.

There have been two classes of solution to Olbers' paradox. The first, advanced by Cheseaux and Olbers, involves an explanation in terms of the physical process of light transmission. It is suggested that as light travels through space, some of it is absorbed by intervening matter present in the form of dust, fluid, or gas, leading to a reduction in its intensity. This explanation has been severely criticized on thermodynamical grounds since any energy absorbed by the intervening matter would result in heating and the reemission of radiation. The second explanation involves cosmological assumptions about the distribution of matter throughout the universe. One such argument involves what is called a hierarchic universe. It is supposed that matter is arranged in the form of a sequence of hierarchies so that the overall density of luminous sources of radiation decreases as one moves outward in such a way as to compensate for the increased number of sources. An explanation of this sort was presented by Carl Charlier (1862—1934) early in the twentieth century.

Explanations of the dark night sky using the concept of a hierarchic universe tended to be highly theoretical. Some astronomers suggested solutions that were simpler and supported by what was then known about the universe. Like many, if not most, of his scientific contemporaries, Harlow Shapley in 1915 believed that the whole universe consisted of the Milky Way galaxy as well as possibly a few satellite objects about the galaxy. In this conception the universe is like an island suspended in infinite empty space. Such a universe avoids the puzzle of the night sky since matter is not distributed indefinitely but is circumscribed by the boundaries of the galaxy. Of course, this explanation became unsatisfactory when it was recognized that the galaxy is just one of countless nebular stellar systems scattered throughout distant space.

A puzzle related to Olbers' paradox concerns the question of the gravitational field exerted at a given point by an indefinitely extended distribution of matter in space. Given a system of bodies, the mathematical function that specifies the strength of the gravitational field at each point in space resulting from this system is known as the gravitational potential function of the system. If we let the system of bodies be the whole universe, we are confronted with the problem of how to define a universal potential function. In the late nineteenth century Carl Neumann (1832-1925) pointed out that it was not clear, given standard Newtonian gravitational theory, how one would obtain a mathematical function that is well defined. Even the slightest variation in the density of matter could lead, on a cosmological scale, to singularities in the potential function.

Proposed resolutions of the gravity potential problem followed the same two lines of reasoning used to explain Olbers' paradox: modifying physical processes on the one hand, and on the other, making suitable assumptions about the large-scale distribution of matter in the universe. The solution given by Neumann, and later, by Hugo von Seeliger (1849-1924), was to modify the physical law of gravitation. The gravitational force acting between two bodies separated by a distance r is multiplied by a factor proportional to a quantity of the form exp (—r). For small values of r the law reduces to the usual Newtonian law, while for larger values of r the force drops off to a value close to zero. The operation of the modified law is equivalent to supposing that over very long distances a repulsive force acts, counteracting the force of gravity and leading to general equilibrium on a large scale.

A different resolution of the potential problem would be to suppose, as Shapley did, that all of the matter of the universe is located within circumscribed boundaries. A more abstract approach is to use the concept of hierarchical ordering, where the density of distant matter is supposed to decrease in such a way as to make negligible the contribution of this matter to the gravitational potential function.

### Non-Euclidean Geometry

In the early nineteenth century, mathematicians showed that there are geometries different from the time-honored geometry of Euclid. There is no single absolute and true geometry but many different and mutually inconsistent geometries. Pioneers in this revolutionary field of study were Nikolai Lobachevsky (1793-1856) in Russia and Janos Bolyai (1802-1860) in Hungary. Later in the century, important further work was done by the great German mathematician Georg Riemann (1826-1866), who adapted methods from analysis and calculus to the abstract study of geometrical spaces.

Consider a line and a point not on the line, both lying in a plane. In Euclidean geometry, there is a unique line through the point parallel to the given line. This fact can be shown to be equivalent to the assertion that the sum of the angles of a triangle is equal to two right angles. This property of Euclidean geometry defines its character at the most fundamental level. Lobachevsky and Bolyai considered geometrical systems in which it was possible, through a point not on a line, to draw an infinite number of lines parallel to the given line. This property is equivalent to supposing that the angles of a triangle sum to less than two right angles. A geometrical system with this property is known as hyperbolic geometry because the relations between angles and lengths in it are described using the hyperbolic trigonometric functions. Riemann showed that one could also obtain a geometry different from both Euclidean and hyperbolic geometry by supposing that there are no parallel lines: for any point not on a given line, every line through this point intersects the given line. In the resulting geometry, known as elliptical, or Riemannian, geometry, the angles of a triangle sum to a value greater than two right angles.

It was tacitly assumed throughout history that physical space and the universe itself are described by Euclidean geometry. In the late nineteenth and early twentieth centuries, astronomers questioned this assumption and considered the possibility of spatial relations based on hyperbolic or elliptical geometry. Observational tests were proposed to decide the geometry of our universe and, if this geometry was elliptical or hyperbolic, to determine the radius of curvature of space. Such tests, which would have had fundamental implications for cosmology, were inconclusive, the margin of error being too large to distinguish between Euclidean and non-Euclidean cases. Around 1900 Simon Newcomb (1832-1925) and Karl Schwarzschild (1873-1916) discussed these questions in quantitative terms. Speculation about the mathematician's "fairyland of geometry" and its applicability to the physical world were widespread during the period.

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