EXACT SOLUTIONS OF THE CONFORMALLY FLAT UNIVERSE. I. THE EVOLUTION OF MODEL AS THE PROBLEM ABOUT A PARTICLE MOVEMENT IN A FORCE FIELD
Baranov A.M., Saveljev E.V.
The problem reduction of an evolution modelling of the open Universe for conformally flat spacetime metric in Fock’s form to an equivalent problem of a particle movement with an unit mass in a force field is demonstrated. The exact cosmological models filled with a substance and radiation in an approximation of the perfect fluid are found since the Friedman solution by means an introduction of set "mechanical" potentials. In the article the possibility of deriving from the Einstein equations exact cosmological solutions for the open Universe by reduction to the equivalent problem of a mass particle motion in the force field is considered. The cosmological model is filled by substance in an approximation of the perfect fluid with nonzero pressure, generally speaking. The metric of 4D space-time is taken in the Fock form as the metric conformal to the Minkowski metric. This metric has the dependence on one variable. A square of the variable is product of advanced and retarded times. The using of mechanical interpretation of the gravitation equations leads to a possibility of consideration of various mechanics force fields with the subsequent physical interpretation of the found exact cosmological solutions. First of all a movement of a free particle with an unit mass (a mechanical force equals to zero) is considered, i.e. the particle moves on inertia. The fourth degree of discovered law of movement is a conformal factor of the cosmological metric which is conformally flat. This case corresponds to the exact cosmological solution without pressure, coinciding with known the Friedman solution for the open Universe. After that the force field leading to uniformly decelerated motion of a particle is considered. The force potential is taken in the form of linear function. The tangent of a slope angle of the function curve coincides with particle acceleration. Such research leads to the exact cosmological solution asymptotically describing both an incoherent dust, and the ultrarelativistic substance which may be interpreted as an equilibrium radiation. Further a square-law function without a linear term and a constant value is taken as a force potential. Such potential can be interpreted as potential of the free oscillator. The solution of corresponding equation of motion is written down in the form of a cosine function with some initial phase related to the ratio between parameters which define dust-like and ultrarelativistic substance. This conclusion becomes obvious after concidering asymptotic behaviour of pressure and energy density. Besides, the series expansion of a root of the fourth degree from a conformal factor asymptoticly coincides with the law of uniformly decelerated motion in previous case that indicates its particular character.
Keywords: open cosmological models, particle, Newton’s equation, function of state.
UDC: 530.12; 530.51
PACS: 04.20.-q; 98.80.Jk
Please cite this article in English as: Baranov A.M., Saveljev E.V. Exact solutions of the conformally flat Universe. I. The evolution of model as the problem about a particle movement in a force field. Space, Time and Fundamental Interactions, 2020, no. 3, pp. 27-36.