Matching the Kerr solution on the surface of a rotating perfect fluid

Abstract

We investigate the possible shapes of the surface of a rigidly rotating perfect fluid on which is matched the Kerr metric, using the Boyer (1965) surface condition. The solution, given in Figures 1 to 5, depends on three parameters,β = qK, q = a/m, η- (a/gwc), wherem denotes the mass of the source, a its angular momentum per unit mass, ω the angular velocity of rotation, andK is an integration constant appearing in Boyer's surface condition. When β < 1, as in Figures 1 to 3, there are, for givenq andη, two possible surfaces, of which the smaller one touches the ring-singularity atλ = a, z = 0. Whenβ > 1, as in Figures 4 and 5, there is only one possible surface of kidney-shaped tori, which also touch the ring singularity. In the case of a differentially rotating perfect fluid, we find a variety of possible strictly spheroidal surfaces, depending on the choice of an arbitrary integration functionτ(Ω) of the angular velocity Ω. If we chooseτ(Ω) so that, at each point on the surface, Ω is single-valued, then the resulting Ω distribution exhibits an equatorial acceleration, similar to what is observed on the surface of the sun. This angular velocity distribution turns out to be identical with Thorne's (1971) “angular velocity of cumulative dragging”.