Equilibrium configuration of magnetar in general relativity

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It is well known that basic equations for the axisymmetric stationary ideal magnetohydrodynamic (MHD) system are reduced to a single second-order, nonlinear partial differential equation, so called the Grad-Shafranov (GS) equation. However, the GS equation for the most general case, when the stress-energy tensor is not circular so that one cannot make the usual foliation of spacetime into two orthogonal families of two-surfaces, has not been given explicitly in the framework of the general relativity. We derive the GS equation for the noncircular (the most general) axisymmetric stationary spacetimes. Such a study is needed to solve the equilibrium configuration of relativistic stars with strong toroidal magnetic field or convection such as magnetars.

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