Additional symmetry of the spherically symmetric configurations, the conservation laws and their application

We construct a 2 +2 decomposition of the Einstein equations and the energy-momentum tensor for a spherically symmetric configurations, induced by the spherical symmetry. In this case, the canonical expansion of the energy-momentum tensor is reduced to the expansion of the radial-time component of the tensor in the radialtime or isotropic bases. We confine ourselves to the case of a radial-time basis. Radial-time part of the Einstein equations is presented in terms of proper radial-time vectors of energy-momentum tensor. We introduce the dual variables, and it is shown that spherically symmetric configurations in general relativity have additional hidden unimodular symmetry. The consequence of this symmetry is the existence of local and integral conservation laws, and in particular, the mass function. Thus, the existence of this additional symmetry indicates the origins of the existence of the mass function. This allows us to construct general formal representations of the metrics for spherically symmetric configurations in terms of the mass function and the corresponding adapted basis. We introduce some simple canonical bases consistent with spherical symmetry. It turns out that to obtain exact solutions it is convenient to use a variety of non-orthogonal coordinate systems. Among these bases we discuss the basis of spatial coordinates (r,R), where r is the radial Lagrangian coordinate, R is the radial coordinate of the curvatures coordinate system and the coordinate basis (t,R), where r is the proper time. As an application we consider the spherically symmetric configurations of the charged dust. We construct local and integral conservation laws, as a consequences of the unimodular symmetry. We obtain the complete set of integrals of motion and exact solutions are found. Note that in the particular case of electrovacuum spaces unimodular symmetry vector is reduced to the Killing vector, and unimodular symmetry itself - to the isometry, which leads to a generalized Birkhoff theorem. We obtain general internal solution for the charged dust configurations. In the special case of spatial coordinates (r,R), with appropriate simplifications, it is reduced to the Ori solution. In the case of the coordinate basis (t,R), the constructed solution coincides with the Pavlov solution up to notation.