38 v 3 7 M ay 1 99 7 The Tolman - Bondi Model in the Ruban - Chernin Coordinates . 1 . Equations and Solutions . ∗

The Tolman-Bondi (TB) model is defined up to some transformation of a co-moving coordinate but the transformation is not fixed. The use of an arbitrary co-moving system of coordinates leads to the solution dependent on three functions f, F, F which are chosen independently in applications. The article studies the transformation rule which is given by the definition of an invariant mass. It is shown that the addition of the TB model by the definition of the transformation rule leads to the separation of the couples of functions (f , F) into nonin-tersecting classes. It is shown that every class is characterized only by the dependence of F on f and connected with unique system of co-moving coordinates. It is shown that the Ruban-Chernin system of coordinates corresponds to identical transformation. The dependence of Bonnor's solution on the Ruban-Chernin coordinate M by means of initial density and energy distribution is studied. It is shown that the simplest flat solution is reduced to an explicit dependence on the coordinate M. Several examples of initial conditions and transformation rules are studied. 1 The Introduction The observations show that in the large scale the Universe is not homogeneous. At the same time it is also supposed that the secreted centre is missing. These two properties separately are presented in the Friedmann-Robertson-Walker (FRW) and Tolman-Bondi (TB) models (Tolman 1934; Bondi 1947): the FRW model is homogeneous and does not include the secreted centre; the TB is nonhomogeneous and includes one. As the simplest nonhomogeneous model the TB model is used for interpretation of the observation data. The TB model is used to calculate the redshift A place of the TB model among the models consistent with the modern observation data is shown in (Baryshev et al. 1994). The TB model (Bondi 1947; Tolman 1934) describes the spherical symmetry dust motion with zero pressure in a co-moving system of coordinates. The solution of the Tolman's * gr-qc/9612038