Investigations in relativistic cosmology

Abstract (by translator) A detailed report is given here of the general investigations carried out by the authors in the field of relativistic cosmology during the past years. The paper consists of two parts. The first part is devoted to a study of the singularities of the cosmological solutions of the gravitational equations. An attempt is made to provide an answer to one of the principal questions of modern cosmology: ‘does the general solution of the gravitational equations have a singularity?’ The authors give a negative answer to this question. The study carried out leads, in fact, to the general conclusion that the presence of a singularity with respect to time is not a necessary property of cosmological models of the general theory of relativity, and that the general case of an arbitrary distribution of matter and gravitational field does not lead to the appearance of a singularity. This result, however, does not exclude the possibility of the existence of more restricted classes of cosmological solutions of the gravitational equations which possess a true singularity. The search for such solutions, carried out by the authors, gives them reasons to believe that the possible types of such solutions are exhausted by those to which one is naturally led by the arguments given in §§ 3, 4 and Appendices C, G. As far as the solution corresponding to the real universe is concerned, its choice is connected with some deep physical requirements, which cannot be established on the basis of only the existing theory of gravitation. Only after establishing these requirements could it be determined whether or not the solution which satisfies them possesses a singularity. The second part of the paper contains an investigation of the gravitational stability of the isotropic model. There are grounds to believe that this model gives an adequate description of the present-day state of the universe considered on a large scale. The behaviour in time of various kinds of small perturbations to the isotropic model is studied. It is shown that perturbations which do not disturb the uniformity of the distribution of matter are either damped with time or remain constant. Perturbations which involve changes in the density of matter behave differently in expanding and contracting universes. In an expanding universe the changes in the density of matter grow slowly with time for long wavelength perturbations and decrease, with time for short wavelength perturbations. The contracting universe, however, is essentially unstable against such perturbations. The entire study is carried out on the basis of the Einstein equations in their classical form, without the ‘cosmological term’, in which form they follow logically from the general foundations of the theory of relativity.