Static, Axially Symmetric, Interior Solution in General Relativity

A static, axially symmetric, interior solution of the Einstein field equations which matches on smoothly to one of the Weyl exterior solutions is exhibited. It is obtained essentially by guessing an interior metric and requiring that the calculated stress-energy tensor not be grossly unphysical. Boundary conditions at the surface of the matter are employed which guarantee the existence of, but do not exhibit explicitly, coordinates in which the metric components and their derivatives are continuous. Finally, it is shown that this method can easily be generalized to obtain interior solutions for any Weyl metric which indicates that it has a positive mass source.

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