Brent Meeker: It's probably pretty hard because you can't just have a rigid rotating hunk of matter (no rigid bodies in relativity). So you'd probably choose a perfect fluid (no viscosity) to model the matter with an appropriate equation of state to model the compressibility. No doubt it can be done numerically. Brent Meeker WDS: Just because rigid bodies do not exist, does not stop there from being a rigidly rotating hunk of matter (all interpoint-distances preserved as measured by the metric). BM: But then the problem is what stress-energy tensor is implied by this hunk - nothing realistic.
--it could well be realistic. Consider a rigidly rotating baseball. Or asteroid Ceres. Nothing unrealistic about it.
WDS: Further, if you produced a solution with inviscid fluid and not preserving interpoint distances, then I would dispute your solution because viscosity presumably would cause some kind of energy loss. BM: Loss to where? The energy can't get out. The effect of viscosity would just be to make the stress-energy tensor more complicated and possibly even turbulent.
----the energy can and will get out. Viscous losses ==> heat ==> radiated photons ==> out. Almost all real fluids have viscosity, so therefore an inviscid fluid solution would not be realistic as a steady-state solution. A genuinely steady-state solution should exist with a Kerr exterior and realistic interior. One possible class of "realistic interiors" -- perhaps even the only class -- would be "a rigidly rotating mass distribution." Prove or disprove: there exists an interior in this class, and a way to glue it onto a Kerr metric exterior. If does exist, then start finding examples. If it doesn't exist, that sounds like a great theorem. You've now almost gotten me to talk myself into trying to do exactly this...