The Wahlquist exterior: an approach to relativistic stationary axisymmetric perturbations

Rotating bodies of finite size in the context of general relativity remain very poorly understood; one of the issues is in establishing the precise nature of the conditions that must be satisfied in order to match with a suitable vacuum solution. Several well-known fluid solutions exist, but so far only one of them describes a bounded matter distribution. This is the Wahlquist solution, which happens to possess an unusual shape to its boundary, and because of this many consider it not to describe an isolated rotating body. So far, this claim is yet to be decisively proved. Recent work has suggested that this may well be the case, but it did not consider the issue of the exterior appearance of the boundary. An attempt is made to follow up the investigations regarding the apparent non-asymptotic flatness of the Wahlquist solution to second order, and to eventually arrive at a physical interpretation for the shape of the fluid. The slow rotation matching conditions are developed from first principles, and we demonstrate that by perturbing the boundary of the Wahlquist solution, it is possible to generate invariant Cauchy boundary data as viewed in the exterior Weyl coordinates. The exterior metric is then obtained to first and second order in the rotation speed using the Ernst potential method, where we show that it is possible to perform up to second order Cauchy matching of the interior and exterior fields. It is shown that while the first order solution is asymptotically flat, the second order solution is not so, and we show that the non-asymptotic flatness is due to the interior multipole expansion of a field originating from two-point masses present outside the fluid.