Sommer Fold-Lenz method in general relativity and its generalization

We consider the applicability and justification of the Sommerfeld-Lenz method of obtaining exact solutions of Einstein equations as well as some of its applications. The method consists in introducing a parabolic reference frame specified by the set of freely falling particles in the given Newtonian space. Further on, using the equivalence principle, we construct the corresponding analog of such reference frame in a stationary space-time of General Relativity. The procedure for finding a metric is reduced to the construction of the corresponding «local Galileo deformation» or «Lorentz deformation» of the Minkowski metric. We suggest a number of modifications and generalizations of the Sommerfeld-Lenz method and consider its applications. Generalizing this method to spherically symmetric configurations of General Relativity with matter, we find the law of transformation of energy-momentum tensor owing to «Lorentz deformation». It is shown that the metrics of Schwarzschild, Reissner-Nordstrom, de Sitter, and their compositions can be obtained by the Sommerfeld-Lenz method. On the other hand, Tolman, Fisher solutions, and internal Schwarzschild solution can not be obtained by this method. In this way the metric of Reissner-Nordstrom-de Sitter is built with using a modified Sommerfeld-Lenz method. Then on the basis of set of free test particles of the Special Relativity the non-inertial reference frame with differential rotation is built. This reference frame can be interpreted as a model of configuration with spin. The «physical-geometric» derivation of Kerr metric is proposed through generalizing the Sommerfeld-Lenz method to rotating reference frame. We give a natural kinematic interpretation for the ellipsoidal shape of the surfaces of constant rotation and for the reference frame with the differential rotation is obtained. Also the connection of the «flattening» parameter of ellipsoids with a momentum of test particles (or impact parameter). Thus, the Sommerfeld-Lenz method allows to make a simple and clear procedure of obtaining the Kerr metric.