Relativistic magnetized star with poloidal and toroidal fields

Abstract

We study the inner magnetohydrodynamic structure of a general relativistic magnetized star, with poloidal and toroidal fields. The star is taken to be differentially rotating, stationary, axisymmetric, and made from perfect, infinitely conducting fluid. Strong toroidal fields of up to ${10}^{17} \mathrm{G}$ can be created from the initial poloidal field by a variation of the mechanism proposed by Meier et al. and Klu\ifmmode \acute{z}\else \'{z}\fi{}niak and Ruderman. It is also found that the redshifted toroidal field and the redshifted chemical potential are constants along a magnetic surface. We prove that a spacetime containing an ideal magnetohydrodynamic fluid which flows only azimuthally is circular in the sense of Carter if, and only if, the magnetic field has only poloidal or only toroidal components. Further, we show through post Newtonian analysis that, even when this criterion is breached, spacetime inside astrophysical compact objects where the magnetic field is less than ${10}^{19} \mathrm{G}$ can be considered circular. In both cases the metric inside the star assumes a simple form, with only one nonvanishing off diagonal term. It is shown that imposing chemical equilibrium forces the magnetic field to assume a force-free configuration. We derive the form of the electric 4-current in force-free relativistic magnetohydrodynamics. The connection between both field components is then given through the vector potential and is used to rule out some field configurations. We derive a new separability condition on the metric which shows that not every pure fluid metric can be dressed with a frozen in, force-free field.