Magnetic tidal Love numbers clarified

In this brief paper, we clarify certain aspects related to the magnetic (i.e., odd parity or axial) tidal Love numbers of a star in general relativity. Magnetic tidal deformations of a compact star had been computed in 2009 independently by Damour and Nagar [1] and by Binnington and Poisson [2]. More recently, Landry and Poisson [3] showed that the magnetic tidal Love numbers depend on the assumptions made on the fluid, in particular they are different (and of opposite sign) if the fluid is assumed to be in static equilibrium or if it is irrotational. We show that the zero-frequency limit of the Regge-Wheeler equation forces the fluid to be irrotational. For this reason, the results of Damour and Nagar are equivalent to those of Landry and Poisson for an irrotational fluid, and are expected to be the most appropriate to describe realistic configurations.