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 $f_{,rr} / (1-\mu^{2}) + f_{,\mu\mu} / \Delta + K(f(r,\mu),r,\mu) = 0$, 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.


Introduction
gave the equation for a force-free magnetosphere (FFM) of the curved Kerr spacetime, which can describe the energy extraction process. The rotational energy of the Kerr black hole (KBH) can be converted into the thermal and kinetic energy of the surrounding plasma, and there could exist an outgoing electromagnetically driven wind. In this physical picture the electron would emit many photons, which can produce a plentiful supply of the electron-positron pairs. The above energy source is a part of the central engines for gamma-ray bursts and active galactic nuclei. Exact analytic solutions of this highly nonlinear equation can provide a well understanding of the energy extraction process and can be helpful for numerical simulation. But there exists few analytic works to deal with this highly nonlinear equation. Based on a new mathematical form of this equation, we discuss about an interesting phenomenon called Meissner effect, which is the expulsion of magnetic field lines out of the event horizon and the quenching of jet power for the KBH with high spin [7,12] . King et al. (1975) [10] found this effect from the Wald vacuum solution [11] . Other types of black holes, such as Kerr-Newman black hole and string theory black holes, also show this behavior [13] . But this effect was never seen in the general relativistic magnetohydrodynamic simulations [6,7] .   [7] try to explain the absence of Meissner effect. They think it is a geometry effect and the steady axisymmetric fields only become radial near the event horizon to evade Meissner effect. but this explanation is not very satisfactory and convinced [6] . Pan & Yu (2016) [6] also try to answer this question in the steady axisymmetic force-free magnetosphere, but they need many special condition, such as, H φ = ωA φ , where A φ is the φ component of the vector potential A, ω is the angular velocity of magnetic fields and H φ corresponds to twice the poloidal electric current. In this paper, we try to answer this question using more general conditions. The paper is organized as follows. In Section 2, basic equation derived by Blandford & Znajek (1977) [1] and its Lagrangian are showed. Then we discuss about the physical meaning of its Lagrangian in Section 3. In section 4, we prove that this equation can be reduced to and D(s, y, z) are source functions. In Section 5, we prove that the Meissner effect does not appear in a steady axisymmetric and magnetically dominated KBH-FFM with the condition dω The summary is in Section 6.

Physical meaning of the Lagrangian
The Lorentz invariant 1 where the Carter field components given by Znajek (1977) [8] are Then we can get 1 c is the velocity of an observer rotating around the KBH with angular velocity ω with respect to the Carter observers (ν Here, g is the determinant of the metric tensor g µν . . This relation can be found in MacDonald & Thorne (1982) [3] . The inner and the outer light surface will never intersect each other [2] when 0 < ω < a 2Mr + , there exists a region between the inner light surface and the outer light surface such that L < 0. According to the equation: [3] . Inside the inner light surface we have Komissarov (2004) [2] analyzed this poloidal electric field in detail at this region.

Equation forms
Let p ≡ A φ and f (r, µ) = F(p, r, µ). Here, F(p, r, µ) and L(p, r, µ) are two functions of p, r and µ. f (r, µ) is a function of the variables r and µ. Then we have the following results: f ,rr = p ,rr F ,p + p 2 ,r F ,pp + 2p ,r F ,pr + F ,rr , f ,µµ = p ,µµ F ,p + p 2 ,µ F ,pp + 2p ,r F ,pµ + F ,µµ . We apply the above results to compare Eq.(5) with equation then we have F ,p = √ |L|. If we know the relation between ω and A φ , we can also get the relation between f and F ,rr and F ,µµ are not partial derivatives with respect to p, so F ,rr and F ,µµ have no terms of p ,r , p ,rr , p ,µ , p ,µµ , then C = −( Using coordinate transformation, we can simplify Now we transform the equation where, a 11 = Solving the equations a 11 = a 22 and a 12 = 0, we can obtain the following equations: The relation y ,Rµ = y ,µR leads to The substitution z = R(R)U(µ) will separate Eq.(12) into where n is a non-negative integer. The solutions of Eq.(13) are the Chebyshev polynomials of the first kind. The simplest solution of Eq.(13) is z = µR so that y = (a)  (1 − µ 2 )(R 2 − 1).

Meissner effect
Our proof is similar to that in Pan & Yu (2016) [6] . But theirs needs many hypotheses and only applies to a special case. If there is Meissner effect, then all magnetic field lines should not cross the event horizon.
If there exists a magnetic field line which only cross the inner light surface and does not cross the event horizon, then there must exists a curve A φ = constant which only cross the inner light surface and does not cross the event horizon. We will prove that magnetic field lines of this type do not exist, so that the curves likeÕA in the Fig.1 (b) , do not exist. And we do not require the FFM to be symmetric with respect to the equator. In (y, z) space, Eq.(5) becomes where κ is Σ ∆+(m 2 −a 2 )(1−µ 2 ) . Making use of Green's theorem and Eq.(17), we obtain In Fig.1 (b) the tangential vector and the normal vector ofÕA is − → tv = (dy, dz) and − → nv = (A φ,y , A φ,z ), respectively. They have the relation A φ,y dy + A φ,z dz = 0. The curvė ABCO is the inner light surface, and ĀBCO L y (A φ,y dz − A φ,z dy) = 0. Using boundary condition A φ,y (r = ∞) > 0, we can make sure that there must exists a region where the of A φ on the right equipotential line is higher than that on the left (.i.e. c2 > c1). This boundary condition is a necessary condition for the extraction of energy. Then in Fig.1 (b) the cross product − → tv × − → nv is always perpendicular to the paper plane and pointing outside so that −A φ,y dz + A φ,z dy > 0. Inside the inner light surface L y > 0 leads to ı OA L y (A φ,y dz − A φ,z dy) < 0, and g 33 ω + g 03 is less than 0 under the condition 0 < ω < a 2Mr + [2] . κ is always bigger than 0. If we assume that H φ H ′ φ 0 and ω ′ 0, then L ,A φ = 2(g 33 ω+g 03 )ω ′ 0, and (18)). The inner light surface is located between event horizon and the the ergosphere [2] . So the magnetic field lines which cross the inner light surface must also cross the event horizon. The Meissner effect expels magnetic fields out of the event horizon and it also expels magnetic fields out of the inner light surface. The magnetic fields outside the inner light surface could not go to the region between the event horizon and the inner light surface, and the Meissner effect expels magnetic fields out of the event horizon, so there can only exist closed poloidal field line or constant A φ , ω and H φ between the event horizon and the inner light surface. But the closed poloidal fields does not exist. The reason is as follows. One of the necessary conditions for the steady-state force-free approximation not break down is 1 2 F 2 > 0 [2,8] [see Eq. (71), Eq.(72) and Section 3 in Komissarov (2004) [2] ]. Because Gralla & Jacobson (2014) [9] have proved: a stationary, axisymmetric, force-free and magnetically dominated field ( 1 2 F 2 > 0) configuration cannot possess a closed loop of poloidal field line [see Eq. (98) in Gralla & Jacobson (2014) [9] ]. This leads to any closed poloidal field line not exist in the region between event horizon and the inner light surface. The final case left is that A φ , ω and H φ must be constant in this region . The boundary condition H φ = sinθ(ω(r 2 + +a 2 )−a) r 2 + +a 2 cos 2 θ A φ,θ [2] at the event horizon leads to H φ = 0 in the region between the event horizon and the inner light surface, so no magnetic field or electric field exists. This means 1 2 F 2 = 0 in this region. These configuration are unrealistic. So the Meissner effect does not appear in a stationary, axisymmetric and magnetically dominated KBH-FFM. If ω = constant, the relations between H φ and A φ in Pan & Yu (2016) [6] , which include the Blandfold-  [5] also shows that no Meissner effect occurs for some magnetospheres, which are consistent with the condition dω dA φ 0 and dH 2 φ dA φ 0. When ω = a 2Mr + , the inner light surface will meet the event horizon, there will be no rotational energy for the KBH to extract [1] , and our proof do not apply to this situation. We think the Meissner effect will occur in this case.

Conclusion
We give a new mathematical form of FFM equation. The simplest case of this equation have the form s ,yy + s ,zz = ρs, where ρ and s are independent. The stable Schrödinger equation has this form and it has some analytic solutions. However in our case, ρ is a very complicated function of y, z so it is hard to give an analytic solution. Although the Green's function of Laplace's equation can be easily obtained for the positive quarter-plane or half-plane, the integral equation is hard to solve. In other cases where FFM equations are high nonlinear, the solutions are more difficult to get. On the other hand, analyzing the light surface function L = g 33 ω 2 + 2g 03 ω + g 00 provides important insights on understanding this equation. We will focus on it in our future work.