Analytic Properties of Force-free Jets in the Kerr Spacetime. III. Uniform Field Solution

Abstract

The structure of steady axisymmetric force-free magnetosphere of a Kerr black hole (BH) is governed by a second-order partial differential equation of A φ depending on two "free" functions Ω(A φ ) and I(A φ ), where A φ is the φ component of the vector potential of the electromagnetic field, Ω is the angular velocity of the magnetic field lines and I is the poloidal electric current. In this paper, we investigate the solution uniqueness. Taking asymptotically uniform field as an example, analytic studies imply that there are infinitely many solutions approaching uniform field at infinity, while only a unique one is found in general relativistic magnetohydrodynamic simulations. To settle down the disagreement, we reinvestigate the structure of the governing equation and numerically solve it with given constraint condition and boundary condition. We find that the constraint condition (field lines smoothly crossing the light surface (LS)) and boundary conditions at horizon and at infinity are connected via radiation conditions at horizon and at infinity, rather than being independent. With appropriate constraint condition and boundary condition, we numerically solve the governing equation and find a unique solution. Contrary to naive expectation, our numerical solution yields a discontinuity in the angular velocity of the field lines and a current sheet along the last field line crossing the event horizon. We also briefly discuss the applicability of the perturbation approach to solving the governing equation.

Toward a Full MHD Jet Model of Spinning Black Holes. I. Framework and a Split Monopole Example

“…The numerical techniques are similar to those in previous work (e.g. Contopoulos et al 2013;Nathanail & Contopoulos 2014;Mahlmann et al 2018;Pan et al 2017;Huang et al 2016Huang et al , 2018Huang et al , 2019. On can obtain converged {Ψ (l) , Ω (l) , L (l) } with field lines smoothly crossing the inner/outer Alfvén surface after sufficient evolution.…”

Magnetosphere structure of a Kerr black hole: Marginally force-free equatorial boundary condition

“…In this case, many solutions are expected [15,16], but the many-solutions scenario is in conflict with several previous FFE simulations [5,[17][18][19][20][21][22]. To explain the discrepancy on the uniqueness of uniform field solution, Pan et al [23,24] proposed that the two eigenfunctions are not independent; instead, they are related by the radiation condition at infinity, which is formulated asÊ θ =B φ , withÊ θ andB φ being the θ component of electric field and φ component of the magnetic field measured by zero-angularmomentum-observers, respectively. As for the uniform field solution, the radiation condition is explicitly expressed as…”

“…Consequently, the force-free magnetospheres is governed by energy conservation equation of electromagnetic field, or conventionally called as the GS equation. In the Kerr spacetime, the axisymmetric and steady GS equation can be written in a compact form [23] A φ,rr + sin 2 θ ∆ A φ,µµ K(r, θ; Ω)…”

“…To explain the discrepancy about the solution uniqueness from different approaches, Pan et al [23] proposed that the two eigenfunctions Ω(A φ ) and I(A φ ) are connected by the radiation condition at infinity instead of being independent, which was readily confirmed by recent high-accuracy FFE simulations done by East and Yang [1]. In addition, there are other interesting features in the structure of the BH magnetosphere showing up in the simulations, e.g., an equatorial current sheet naturally develops with the ergosphere, and the magnetic dominance marginally loses on the current sheet, i.e.…”

An improved algorithm for crossing curved light surfaces: rapidly rotating pulsar magnetospheres in curved space–time

“…The force-free, axisymmetric and steady magnetosphere outside the NS is governed by the curved-spacetime GS equation (e.g. Gralla & Jacobson 2014;Gralla et al 2016), which we write in the symmetric form proposed by Pan et al (2017), 0 = Ψ,rr + sin 2 θ α 2 r 2 Ψ,µµ K(r, θ; Ω)…”