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

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−μ2)+f,μμ/Δ+K(f(r,μ),r,μ)=0f,rr/(1−μ2)+f,μμ/Δ+K(f(r,μ),r,μ)=0, where μ=−cos⁡θμ=−cos⁡θ. With coordinate transformation, the above equation can be given as s,yy+s,zz+D(s(y,z),y,z)=0s,yy+s,zz+D(s(y,z),y,z)=0. Using this form, we prove that the Meissner effect is not possessed by a KBH–FFM with the condition dω/dAϕ⩽0dω/dAϕ⩽0 and Hϕ(dHϕ/dAϕ)⩾0Hϕ(dHϕ/dAϕ)⩾0, here AϕAϕ is the ϕ   component of the vector potential A→, ω   is the angular velocity of magnetic fields and HϕHϕ corresponds to twice the poloidal electric current.