Topology of Some Spheroidal Metrics

The solutions of Einstein's vacuum field equations, Rμν = 0, are found when quasi-oblate and prolate spheroidal coordinates are used. The solutions for the ``Newtonian'' potential can be written as a linear combination of Legendre polynomides of integral order l. For oblate coordinates the solutions for each l have a ring singularity and have a double sheeted topology; one can get from one sheet to the other by going through the ring. When the l = 0 and l = 1 solutions are combined an infinite-sheeted topology results from the nonlinear character of the field equations. In general the geometry is asymptotically flat on only one sheet; on the others it is highly distorted. In some cases the region near the ``ring singularity'' opens out into a multisheeted infinite space. For the prolate coordinates the solutions contain a line singularity of finite length. In general, the prolate coordinate solutions are much less rich in varied topologies than are the oblate solutions.