A. L. Zel'manov (GAISh) presented his theory of an anisotropic, non uniform universe. He criticized the theory of a uniform, isotropic universe, and pointed out that the postulate of homogeneity involves an unjustified extrapolation to the entire universe of the properties and characteristics of the observable portion of it. The formulation of a relativistic theory describing an anisotropic, nonuniform universe involves several other problems as well, and it leads to a reconsideration of the basic questions related to cosmology. One of the problems requiring such a re-evaluation is the problem of whether space and time are finite or infinite.
Zel'manov demonstrated mathematically that, when applied to the whole universe, the mutually exclusive, opposite concepts of "finite" and "infinite" no longer have the simple meanings which we are accustomed to give them on the basis of our experience of the macroscopic world. According to the anisotropic model of the universe, which takes into account the general theory of relativity, spatial finiteness or infinity is relative: with respect to single material frames of reference having a single motion, the universe is infinite, whereas with respect to other frames of reference it is finite. Thus it follows that our customary conceptions of "infinity" and "finiteness" are inapplicable to the universe as a whole.
It was also established by Zel'manov that in an anisotropic, nonuniform universe the spatial deformation may be so irregular that an expansion of the space in one region can be accompanied by contraction of the space in another region. Finally, he pointed out too that the anisotropy can affect
the behavior of a volume of space so much that a regular volume minimum becomes possible instead of a singularity.
M.F. Shirokov (Moscow Aeronautical Institute) and I.Z. Fisher presented their theory of a nonuniform, isotropic universe. They noted that Fridman's theory of a universe which is evolving in time does not formu late the problem of cosmology correctly, and they offered a formulation which is physically and mathematically more correct. The solutions obtained by Shirokov and Fisher do not involve the difficulties, related to the infinite density of matter at some initial moment and to the finite lifetime of the universe, which plagued Fridman's theory.
Ya. A. Kipper (Institute of Physics and Astronomy, Academy of Sciences of the Estonian SSR) discussed the gravitational paradox in Newton's theory of gravitation, and he indicated a possible means of eliminating this paradox.
In a second contribution, A. L. Zel'manov also considered the gravitational paradox in the quasi-Newtonian approximation, and he analyzed the reasons for it.
I. D. Novikov (GAISh) discussed some cosmological models in the quasi-Newtonian approximation.
A number of other interesting reports were also presented at the conference. It is to be hoped that this conference on extragalactic astronomy and cosmology will have a stimulating effect on the future development of these two fields of study.