Global two-light-surface force-free eigenvalue problem

Determine the first-order corrections to the poloidal current function I_1(\Psi_0) and field-line angular velocity \Omega_F^{(1)}(\Psi_0) by constructing a global rotating force-free magnetosphere that crosses both the inner and outer light surfaces in the spin–torsion-perturbed Kerr geometry.

Background

The paper derives the local generalized Znajek relation and explicitly computes the static anisotropic magnetostatic flux redistribution. However, the rotating Grad–Shafranov equation contains first-order current and field-line-rotation corrections that cannot be fixed locally. They must instead be selected by imposing regularity at both light surfaces together with the horizon, axis, asymptotic, and fixed-flux conditions.

Resolving this problem would provide the global force-free response of the Einstein–Cartan-induced metric perturbation, including the corrections to the load relation and the rotating magnetospheric fields. The paper’s leading Blandford–Znajek coefficient avoids requiring this full solution because the load factor is stationary at the imposed optimal point x=1/2, but higher-order and fully global corrections remain dependent on it.

References

After the metric perturbation has been specified, the first term is known, but I_1(\Psi_0) and \Omega_F{(1)}(\Psi_0) are fixed only by a global solution that crosses both light surfaces. We do not solve that global eigenvalue problem in this work.

— Model dependent analytic spin torsion corrections to Blandford Znajek energy extraction in Einstein Cartan gravity  (2608.23174 - Wu et al., 24 Aug 2026) in Section 5, subsection “Force-free magnetosphere and the generalized Znajek condition”