SPACE, TIME AND FUNDAMENTAL INTERACTIONS

Exact solutions of the conformally flat Universe. II. The linear equation of state and multidimensional models

Baranov A.M., Savelev E.V.

The finding problem of conformally flat cosmological models as exact solutions of the equations of gravitation for different equations of state with linear connection between pressure and energy density is demonstrated within the limits of multidimensional space-time with one time-like direction. In this case the energy-momentum tensor (EMT) is taken as generalisation of EMT in an approach of the perfect Pascal fluid for space-time with four dimensions. The special case is EMT for an incoherent dust with zero pressure is related to the open model of Friedman's Universe. It is claimed that such approach leads to an identification of some discrete set of equations of state for which conformal factors are connected with the harmonic functions as solutions of the Laplace equations in multidimensional Euclidean spaces of an integer dimensionality. Dimensionality of these spaces, in turn, is defined by a concrete equation of state. For four-dimensional space-times the corresponding table is constructed. This table allows to trace connection between a discrete set of linear equations of state and dimensionality of the auxiliary Euclidean spaces and also the functional expression of conformal factors of the open cosmological models related to potential functions, which are solutions of the Laplace equations in these auxiliary Euclidean spaces. Thus it can be seen that three dimensional spatial-like manifold restricts a selection of discrete physically interpreted equations of state for the finding of exact solutions of the gravitation equations related to potential functions. Therefore, on the one hand, any linear equation of state can be approximated with any accuracy by any rational fraction. On the other hand, the exact solution of the many-dimensional equations of Einstein can be found only related via to potential functions when the spatial extension of space-time will be made up to necessary multidimension. Such possibility appears for any linear equation of state with a rational constant of proportionality at growth of the space dimensionality \(N\) (\(N> 3\)). For such space-times the similar table is constructed, but without fixing of dimensionality of a spatial hypersurface. Here each value of spatial dimensionality \(N\) corresponds to \(2N+1\) of linear equations of state. This table demonstrates the possibilities for each such equation of state with a rational constant of proportionality between pressure and density of energy under construction for any open cosmological model with the conformally flat metric, but in corresponding space-time with dimensionality more than four.