Rotating fluid masses in general relativity

In what follows we shall derive some properties of the gravitational field of an isolated, axially symmetric, uniformly rotating mass of perfect fluid in a steady state, according to the general theory of relativity. Several exact models describing rotating fluids are known in Newtonian mechanics, the Maclaurin and Jacobi ellipsoids ((6)) being perhaps the most interesting. In general relativity, no such exact solution is known in its entirety, although Kerr ((4)) has exhibited a certain vacuum solution possessing features that one might expect of a space-time exterior to some rotating body. Throughout this paper we shall have Kerr's solution in mind. The question that we shall keep before us is whether a perfect fluid interior can be matched to any given exterior field. Our main results exhibit the class of all possible fluid boundaries, given the exterior field, and some relations between the pressure, density, 4-velocity, and interior metric tensor.