A Static Generalization of the Schwarzschild Solution, that Gives not Asymptotically Dipole Term

In this paper we study the static axisymmetric solutions of the vacuum Einstein equations. Among static axisymmetric vacuum solutions of the most interest are the asymptotically flat solutions reducing to the Schwarzschild solution. The purpose of this paper is to obtain a static solution which turned out to be appropriate for describing the gravitational field around an axisymmetric mass distribution. In this paper the method of singular sources os considered and some new applications are presented. By mean of the method of singular sources it is possible to construct gravitational multipoles which generalize the Schwarzschild solution. The linearity of the gravistatic equations makes it possible to solve the problem of superposition of two or several known solutions. The obtained static vacuum axisymmetric generalization of the Schwarzschild solution near two points of horizon has coordinate singularities. In the obtained solution the dipole term is absent, and we have found the corresponding condition. If one considers axially symmetric solutions of gravistatics, then construction of gravitational multipoles becomes ambiguous. It means that different solutions can give asymptotically the same Newtonian limit.

gravistatic, Schwarzschild solution, Einstein equation, Weyl metric, asymptotically flat