# Analytic properties of force-free jets in the Kerr spacetime -- III: uniform field solution

## TL;DR

This paper investigates the uniqueness of solutions for force-free magnetospheres around Kerr black holes, revealing a unique solution with a discontinuity and current sheet when proper boundary and constraint conditions are applied.

## Contribution

It clarifies the relationship between boundary conditions and solution uniqueness, providing a numerical method to find a unique force-free field configuration around Kerr black holes.

## Key findings

- Multiple solutions approach uniform field at infinity
- Numerical solution shows a unique configuration with a current sheet
- Constraint and boundary conditions are interconnected via radiation conditions

## 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 $A_\phi$ depending on two "free" functions $\Omega(A_\phi)$ and $I(A_\phi)$, where $A_\phi$ is the $\phi$ component of the vector potential of the electromagnetic field, $\Omega$ 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.