I’m reading about the debate in the 1930s between Einstein and Silberstein about the (non-)existence of a static two-body vacuum solution of the Einstein field equations.
Silberstein claimed to have found just such a solution as a special case of Weyl’s metric. However, he then concluded that the existence of an unphysical solution implies that Einstein’s gravitational theory has to be modified.
Meanwhile, Einstein dismissed Silberstein’s solution on two grounds. First, he claimed that there are additional singularities; second, he claimed that a solution that yields singularities is in any case not a proper solution of a field theory, so it certainly cannot be used to discredit that theory.
I disagree with Silberstein… just because there exist solutions that are unphysical does not unmake a theory. The equations of ballistics also yield unphysical solutions, such as cannonballs going underground or flying backwards in time… but it simply means that we chose unphysical initial conditions, not that the theory is wrong.
I also disagree with Einstein’s second argument though… field theory or not, some singularities can be quite useful and physically meaningful, be it, say, the “point mass” in Newton’s theory, the “point source” in electromagnetism, or, well, singularities in general relativity representing compact (point) masses.
But both these issues are more philosophy than physics. I am more interested in Einstein’s first argument… is it really true that Silberstein’s solution yields more than two singularities?
That is because when I actually calculate with Silberstein’s metric, I find regular behavior everywhere except at the two singular points. I see no sign whatsoever of the supposed singular line between them. What am I missing?