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

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 $f_{,rr} / (1-μ^{2}) + f_{,μμ} / Δ+ K(f(r,μ),r,μ) = 0$, where $μ= -\cosθ$. 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 $dω/d A_ϕ \leqslant 0$ and $H_ϕ(dH_ϕ/dA_ϕ) \geqslant 0$, here $A_ϕ$ is the $ϕ$ component of the vector potential $\vec{A}$, $ω$ is the angular velocity of magnetic fields and ${H_ϕ}$ corresponds to twice the poloidal electric current.