Classical inversion algorithms for spherically symmetric gravitational fields

In the framework of general relativity inverse problems are formulated and solved for the classical scattering of a particle by a spherically symmetric gravitational field for fixed angular momentum, energy, and impact parameter. A hydrodynamic model of a static fluid sphere is considered. In the problems, the radial distributions of the density and pressure of the gravitating matter are recovered from classical scattering data for relativistic particles. The solution of the problems reduces to the integration of nonlinear second-order differential equations, the explicit form of which is determined by the scattering data. The constructed methods of solution can be used to obtain the distributions of matter corresponding to perfectly focusing gravitational systems. The algorithms give a unique possibility of direct determination of the internal structure of stars in astrophysics on the basis of scattering data.