The gravitational condensation of galaxies and globular clusters of stars was discussed by L. M. Ozernoi (GAISh). He considers the structural development of the Metagalaxy to be a consequence of the gravitational instability of a nonuniform, essentially gaseous, medium with a tempera ture of the order of degrees centigrade and an average density of Calculations show that in such a medium large condensa tions will form at first, and the cooling of these will make them suscept ible to partition into individual protogalaxies. The latter will be stable with respect to disruptive tidal action. In addition, Ozernoi considered the formation of globular star clusters via a gravitational condensation of part of the protogalactic medium, which makes it possible to explain qualitatively certain observational data. In connection with this, he also concluded that the size of a globular cluster increases with the galacto centric distance, and this is in good qualitative agreement with observations.
G. I. Naan (Member of the Academy of Sciences of the Estonian SSR) reviewed certain philosophical works which excessively simplified or distorted the concept of cosmological infinity, and he pointed out how this concept should be considered from the viewpoint of modern physics.
The idea of an infinite cosmic expanse filled with a countless number of stars goes all the way back to the time of Democritus. This theory of an infinite universe first encountered difficulties in the middle of the 19th century, in connection with the so-called photometric paradox. The German astronomer Olbers demonstrated that if there were an infinite number of stars distributed uniformly throughout an infinite Euclidean universe, the sky would have to shine with a dazzling light, against the background of which our Sun would show up as a dark spot. The situation would be analogous for a uniform distribution of galaxies throughout an infinite space. Then, at the end of the 19th century, the gravitational paradox was discovered. Seeliger, another German astronomer, showed that on the basis of Newton's theory of universal gravitation the total attraction which all the particles of an infinite universe exert upon each individual particle in the universe (including the Earth) would be infinitely great.
The existence of the photometric and gravitational paradoxes induced many scientists to construct geometric models of a finite unbounded universe which closes upon itself as a result of an inward curvature. Long ago, Riemann pointed out that it is possible for the world to be finite and at the same time unbounded. The surface of a sphere constitutes
a two-dimensional analog of such a world. The area of a spherical surface is finite, but an imaginary two-dimensional being moving over such a surface would nowhere be able to detect the limits of his world, regardless of the length of the path it traverses. If this being maintained a constant direction of motion, then after some finite distance had been covered it would return to the starting point, having thereby completed a "round-the-world" journey. In other words, a spherical surface closes upon itself by virtue of its curvature.
From the point of view of relativistic cosmology, which is based on Einstein's theory of gravitation, a finite, unbounded universe closed in three dimensions is in principle quite permissible.
In his report Naan also stressed the fact that, whereas unboundedness in space and infinity in space amount to the same thing for Euclidean space, this is no longer the case when a more general, non-Euclidean geometry is used to describe the universe. He also gave the correct formulation of the problem by pointing out that at the present level of scientific development the most probable conclusion is the following: the universe has unbounded extension in space and time, and at the same time it is infinite in space and time. With respect to a purely spatial projection, the theoretical possibility exists that the universe is finite, but this possibility is not realized in practice. It is Naan's opinion that the problem of cosmological infinity should be solved by the combined efforts of astronomers, physicists, and philosophers.
On the basis of a study of cosmological solutions of Einstein's gravita tion equations, E. M. Lifshits, I. M. Khalatnikov, and V. V. Sudakov (Institute for Physical Problems, USSR Academy of Sciences) demonstrated in their joint contribution that the existence of a physical singularity in time, at which the density of matter becomes infinite, is not a necessary consequence of cosmological models in the general theory of relativity.
In the general case of an arbitrary distribution of matter and a gravi tational field, there is no physical singularity. The findings of Lifshits, Khalatnikov, and Sudakov rule out the possibility of the appearance of such a singularity in the future. This means that a contraction of our part of the universe (if it takes place) must inevitably turn into an expansion.