The classical inverse scattering problem for a spherically symmetric gravitational field
Abstract
The inverse problem of the classical scattering of a particle by the gravitational field is formulated and solved in terms of the general theory of relativity. The solution generalizes the inversion algorithms in classical mechanics for the central potential in twodimensional space for the case of curved spacetime. A hydrodynamic model of a static fluid medium is considered. The radial distributions of the density and pressure of the gravitating matter are reconstructed on the basis of the known relationship between the classical scattering cross section of ultrarelativistic particles and the angle of deflection. The solution is reduced to integrating a secondorder nonlinear differential equation whose explicit form is determined from the scattering data. The method demonstrated here makes it possible to determine distributions of matter that correspond to ideally focusing gravitational systems.
 Publication:

Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki
 Pub Date:
 June 1983
 Bibcode:
 1983ZhETF..84.1921B
 Keywords:

 Gravitational Fields;
 Inverse Scattering;
 Relativistic Particles;
 Relativity;
 Classical Mechanics;
 Density Distribution;
 Hydrodynamic Equations;
 Radial Distribution;
 SpaceTime Functions;
 Astrophysics