Matching stationary and axisymmetric spacetimes in perturbation theory: Lichnerowicz vs. Darmois junction conditions in harmonic gauge

We further analyse a stationary and axisymmetric global metric we have recently built, resulting from the matching between a Wahlquist perfect fluid interior and an asymptotically flat exterior. It involves the use of Lichnerowicz junction conditions and harmonic gauge, thus imposing additional restrictions to the matching that could derive into a loss of generality if one is to obtain consequences from them. We show that this is not the case since the Lichnerowicz matching, besides fixing the gauge part of the metrics, is completely equivalent to the Darmois one. This holds for the polytropic source case.