On the gravitational instability in an expanding universe.Reformulation of Lifshitz's theory

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  • On the gravitational instability in an expanding universe.Reformulation of Lifshitz s theory

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In view of a new interest motivated by the discovery of a cosmic black-body radiation for the problem of galaxy formation in an evolving universe, an attempt is made to refor­ mulate Lifshitz's relativistic theory of the gravitational instability i:1 such a way that the rewritten formalism permits us to make a full comparison with Bonnar's approach based on the Newtonian cosmology. The essence of our reformulation lies in deriving a differential equation for the density contrast oE/E which is a 4-scalar independent of Lifshitz's coor­ dinate conditions. The differential equation is solved, except for such a time interval that non-adiabatic processes are effective. l\t the later stage, our equation is the same as Bonnor's, but this does not necessarily n1ean that both approaches are equivalent to each other at the stage. The latter situation is illustrated by considering a mass concentration in a spherical region, which depends not only on the density contrast, but also on the metric perturbation. 1 l proposed a new approach to the formation of galaxies m close connection with the presence 2 l of a cosmic black-body radiation in an evolving universe. 3 l His approach is similar to Bonnor's 4 ) in the sense that the Newtonian gravitational theory was applied to the density fluctuation, while the effect of radiation drag was taken into consideration and he did not neces­ sarily postulate that the initial fluctuations were of a statistical origin. It must be noticed, however, that the Newtonian theory would not be ap­ plicable at the early stage of the cosmic evolution where the radiation drag plays a significant role. In this circumstance, we must pay attention to Lifshitz's relativistic theory 5 l of the gravitational instability. In principle, we can apply his theory to any stage of the cosmic evolution, by taking into account non­ adiabatic processes, if necessary. Nevertheless, most workers keep his theory at a respectful distance, simply because of the situation that the relativistic treatment is less transparent than its Newtonian counterpart. The above situation is partly due to the fact that the perturbed part of the metric tensor includes in general the mode of gravitational waves (in the ex

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