We use analytic perturbation theory to present a new approximate metric for a rigidly rotating perfect fluid source with equation of state (EOS) ǫ+(1−n)p = ǫ 0 . This EOS includes the interesting cases of strange matter, constant density and the fluid of the Wahlquist metric. It is fully matched to its approximate asymptotically flat exterior using Lichnerowicz junction conditions and it is shown to be a totally general matching using Darmois-Israel conditions and properties of the harmonic coordinates. Then we analyse the Petrov type of the interior metric and show first that, in accordance with previous results, in the case corresponding to Wahlquist's metric it can not be matched to the asymptotically flat exterior. Next, that this kind of interior can only be of Petrov types I, D or (in the static case) O and also that the non-static constant density case can only be of type I. Finally, we check that it can not be a source of Kerr's metric.

We compare an approximation of the singularity-free Wahlquist exact solution with a stationary and axisymmetric metric for a rigidly rotating perfect fluid with the equation of state µ + 3p = µ0, a sub-case of a global approximate metric obtained recently by some of us. We see that to have a fluid with vanishing twist vector everywhere in Wahlquist's metric the only option is to let its parameter r0 → 0 and using this in the comparison allows us in particular to determine the approximate relation between the angular velocity of the fluid in a set of harmonic coordinates and r0. Through some coordinate changes we manage to make every component of both approximate metrics equal. In this situation, the free constants of our metric take values that happen to be those needed for it to be of Petrov type D, the last condition that this fluid must verify to give rise to the Wahlquist solution.

We compare the results obtained from analytical perturbation theory and the AKM numerical code for an axistationary spacetime built from matching a rotating perfect fluid interior with the equation of state ε − 3p = 4B of the simple MIT bag model and an asymptotically flat exterior. We discuss the behaviour of the error in the metric components of the analytical approximation going to higher orders. Additionally, we check and comment the errors in multipole moments, central pressure and some other physical properties of the spacetime.

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Abstract. We compare metrics obtained through analytic perturbation theory with their numerical counterparts. The analytic solutions are computed with the CMMR post-Minkowskian and slow rotation approximation due to Cabezas et al. [1] for an asymptotically flat stationary spacetime containing a rotating perfect fluid compact source. The same spacetime is studied with the AKM numerical multi-domain spectral code [2,3] . We then study their differences inside the source, near the infinity and in the matching surface, or equivalently, the global character of the analytic perturbation scheme.