Grad-Shafranov equation in noncircular stationary axisymmetric spacetimes

TL;DR

This paper formulates a general relativistic magnetohydrodynamics framework in stationary axisymmetric spacetimes, deriving a versatile Grad-Shafranov equation applicable even in noncircular spacetimes relevant for magnetars and similar astrophysical objects.

Contribution

It introduces a comprehensive Grad-Shafranov equation for noncircular spacetimes, expanding the analysis of magnetized relativistic stars beyond previous circular assumptions.

Findings

Derived the general GS equation for noncircular spacetimes

Established the applicability to magnetars with toroidal fields

Discussed limits and special cases of the GS equation

Abstract

A formulation is developed for general relativistic ideal magnetohydrodynamics in stationary axisymmetric spacetimes. We reduce basic equations to a single second-order partial differential equation, the so-called Grad-Shafranov (GS) equation. Our formulation is most general in the sense that it is applicable even when a stationary axisymmetric spacetime is noncircular, that is, even when it is impossible to foliate a spacetime with two orthogonal families of two-surfaces. The GS equation for noncircular spacetimes is crucial for the study of relativistic stars with a toroidal magnetic field or meridional flow, such as magnetars, since the existence of a toroidal field or meridional flow violates the circularity of a spacetime. We also derive the wind equation in noncircular spacetimes, and discuss various limits of the GS equation.