It is well known that in Newtonian gravity if the center of a spherical cavity inside a sphere of uniform density is not concentric with the sphere then the gravitational field inside the cavity will be non zero and uniform. Is there an exact solution of Einstein equations of general relativity which corresponds to such a distribution of matter?
I'm asking this question in connection with the discussion of subtleties of Einstein's interpretation of gravitational field in https://arxiv.org/abs/physics/0204044 and http://www.mathpages.com/home/kmath622/kmath622.htm
In a 1950 letter to Max von Laue Einstein, in contrast to modern views, directly identifies the presence of gravity with non zero Christoffel symbols, not with the non zero Riemann tensor (the fragment of this letter is reproduced in J.J. Stachel, How Einstein discovered general relativity: a historical tale with some contemporary morals). This view is of course a bit strange because, as Peter Havas noted in 1967 (https://arxiv.org/abs/1512.09253), Christoffel symbols can be made non zero in flat Minkowski space-time simply by introducing curvilinear coordinates, without any involvement of coordinate system's acceleration. Nevertheless identifying gravity with only space-time curvature misses some essential points as discussed in the sources mentioned above (for example, the Riemann tensor for the space-time of a cosmic string vanishes everywhere except at the string. Nevertheless there is a gravitational effect - deflection of light by the cosmic string).
