Grad-Shafranov equation in noncircular stationary axisymmetric spacetimes

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.PACS numbers: 04.20.Cv,04.40. Dg,52.30.Cv,95.30.Qd,95.30.Sf,97.10.Cv

Equilibrium solutions of relativistic rotating stars with mixed poloidal and toroidal magnetic fields

“…1 In this paper, we present a formulation and numerical solutions for rapidly rotating relativistic stars with strong mixed poloidal and toroidal magnetic fields. Our solutions extend the above works [11][12][13][14] to highly deformed configurations as a result of rapid rotation and stronger magnetic fields, which can not be calculated from perturbative methods. We assume stationarity and axisymmetry of the gravitational, electromagnetic and matter fields, but we do not assume the spacetime to be circular, or spatially conformally flat.…”

“…Under the assumptions of stationarity and axisymmetry, however, one can find a set of integrability conditions and algebraic relations that the fluid variables must satisfy. Such a formulation for the system of relativistic ideal MHD equations is found in [10,11,13], and a fully covariant geometric formulation is derived in our previous work [15]. We rewrite the formulation in [10,11,13] as suited to our numerical method.…”

“…Such a formulation for the system of relativistic ideal MHD equations is found in [10,11,13], and a fully covariant geometric formulation is derived in our previous work [15]. We rewrite the formulation in [10,11,13] as suited to our numerical method. We introduce the basis, {t α , φ α , e α A }, associated with the coordinates t, φ and two other spatial coordinates x A , where t and φ are chosen along the symmetry vectors t α and φ α and normalized as t…”

“…Moreover our new method of solving Einstein's and Maxwell's equations are not restricted to axial symmetry. The first integrals of a system of relativistic magnetohydrodynamic (MHD) equations are derived assuming the above symmetries as well as perfect conductor conditions [10,11,15], and solved to obtain self-consistent, non-perturbative, equilibrium configurations for the first time.…”

Conformally flat noncircular spacetimes

“…Nevertheless, there is no room for the circular idealization when considering rotating neutron stars [1] surrounded by strong toroidal magnetic fields ranging ∼ 10 16 to 10 17 G, see also [2,3] and references therein. The circularity condition is a very severe restriction, which fails to hold in spacetimes allowing the existence of toroidal magnetic fields and meridional flows [3].…”