Conformally Galilean 4-metric and kinemetric reference frames

In cosmology the synchronous reference frames are used for description of the different cosmological models. But there is the Fock approach which is connected with the conformally Galilean metric. This metric can describe the open cosmological models such as the open Universe of Friedman. Such metric has a number of advantages compared to the metric in the synchronous coordinates. For example, solutions in these coordinates are written down in a parametric form. The conformally Galilean metrics are connected to the normal reference frames, i.e. with reference frames without a rotation. In such frames of reference it is possible to introduce 4-velocity (the Zelmanov monad) which is proportionate to a normal vector to 3-hypersurface. It is possible to relate a system of coordinates with a direction of a normal vector (of the monad) which is orthogonal to 3-hypersurface. Then the set of 3-hypersurfaces can be numbered by the timelike coordinate. Such system of coordinates is termed a kinemetric system of coordinates. If to combine the coordinate line of time with the direction of normal vector (of the monad) to 3-hypersurface, we will get the kinemetric frame of reference. To unite a direction of the chosen monad with a line of time, it is necessary to effect corresponding rotational displacements of the monad in the tangent 4-space-time. After such procedure the reference frame will be the kinemetric frame of reference. Introduction of the proper (physical) time lengthways of a timelike direction leads to a metric in the synchronous frame of reference. Thus in this frame of reference the conformally Galilean metric can be written down in an explicit form for the open isotropic cosmological model. Hence, a transition into the kinemetric frame of reference and an introduction of the proper time it is equivalent to transition into the synchronous frame of reference.