Stability of force-free magnetospheres

We analyze the dynamical evolution of a perturbed force-free magnetosphere of a rotating black hole, which is described by the Blandford-Znajek solution in the stationary limit. We find that the electromagnetic field perturbations can be classified into two categories: “trapped modes” and “traveling waves.” The trapped modes are analogous to the vacuum (without plasma) electromagnetic quasinormal modes in rotating black hole spacetimes, but with different eigenfrequencies and wave functions, due to their coupling with the background electromagnetic field and current. The traveling waves propagate freely to infinity or the black hole horizon along specific null directions, and they are closely related to the no-scattering Poynting flux solutions discovered by Brennan, Gralla and Jacobson. Our results suggest that the Blandford-Znajek solution is mode stable, and more importantly we expect this study to illuminate the dynamical behavior of force-free magnetospheres as well as to shed light on the path to new exact solutions.

Figure 1

A schematic depiction of the two classes of perturbations. The green and blue waves propagating away from and towards the black hole belong to the traveling wave class of solutions, while the red wave packet represents the trapped modes that are imperfectly confined by an effective radial potential. The black sphere in this figure represents the black hole, and the grey dashed circle represents a spherical photon orbit (for the relationship between such orbits and the trapped perturbations such as quasinormal modes, see e.g. [39, 40]).

Figure 2

An illustration of the contour (the red curve) in the complex $r$ plane with which we compute the inner product. The Schwarzschild wave function is analytical everywhere except at the branch cut (shown as a zigzag line), which starts at $r=2M$ and shoots upward to $+i\infty$.

Figure 3

The black dots represent $\delta \omega /a$ for the frequency shifts of the force-free trapped modes in the complex plane, where we have labeled the lines of points sharing the same $l$ (on straight lines through the origin), while the $m$ values are placed near the data points. For comparison, we also show as red circles, the corresponding $\delta \omega /a$ values for the vacuum quasinormal modes. Their trends are the same as the force-free modes and so we have not labeled them explicitly.