On Axially Symmetric Solutions of Einstein's Field Equations in Matter and an Example of Static Gravitational Shielding

A model system, consisting of a thin spherical shell with radius R and mass M and a point mass m at a distance s> R from the center of the sphere, held fixed by an appropriate strut, is solved to order raM. The stresses in the shell are not of the canonical Weyl type, and it is argued that the same is true for more realistic situations, e.g., rotating matter. Owing to the nonlinearity of Einstein's field equations, the field of the point mass is shielded from the in-terior of the shell by a factor ~ lying between 1- 3M/R and 1- 2M/R, and the field outside the shell explicitly depends on R. In the last decade, very powerful methods have been worked out which allow one to construct a great many axially symmetric and stationary solutions of Ein-stein's vacuum field equations. (See, e.g., [ 1-3] for an overview.) However, much less is known about realistic material sources for these vacuum solutions, an