A mathematical form of force-free magnetosphere equation around Kerr black holes and its application to Meissner effect

TL;DR

This paper reformulates the force-free magnetosphere equation around Kerr black holes into a new mathematical form and demonstrates that the Meissner effect does not occur under certain conditions.

Contribution

It introduces a novel mathematical formulation of the force-free magnetosphere equation around Kerr black holes and applies it to analyze the Meissner effect.

Findings

The magnetosphere equation can be rewritten in a new, more tractable form.

Under specific conditions, Kerr black holes do not exhibit the Meissner effect.

The new formulation aids in understanding magnetic field behavior near black holes.

Abstract

Based on the Lagrangian of the steady axisymmetric force-free magnetosphere (FFM) equation around Kerr black holes(KBHs), we find that the FFM equation can be rewritten in a new form as $f_{,rr} / (1-\mu^{2}) + f_{,\mu\mu} / \Delta + K(f(r,\mu),r,\mu) = 0$, where $\mu = -\cos\theta$. By coordinate transformation, the form of the above equation can be given by $s_{,yy} + s_{,zz} + D(s(y,z),y,z) = 0$. Based on the form, we prove finally that the Meissner effect is not possessed by a KBH-FFM with the condition where $d\omega/d A_{\phi} \leqslant 0$ and $H_{\phi}(dH_{\phi}/dA_{\phi}) \geqslant 0$, here $A_{\phi}$ is the $\phi$ component of the vector potential $\vec{A}$, $\omega$ is the angular velocity of magnetic fields and ${H_{\phi}}$ corresponds to twice the poloidal electric current.