Question about the Blandford–Znajek process

Question

I've been trying to understand the Blandford–Znajek process which explains how Gamma Ray Bursts form, but it's still not clear to me how it works. In this article, several theories about the formation of GRBs are mentioned, and in the 4th page, the author talks about the Blandford–Znajek process.

In that part he mentions: "These rotating field lines induce an electromagnetic force that accelerates charged plasma at relativistic speeds along the axis of rotation. Due to the radial component of the field, the particle spirals as it leaves.", and there is a figure taken from Kip Thorne's book Black Holes and Time Warps illustrating this. I'm having trouble understanding this figure and why the particles spiral as they leave. Does anyone know what he's talking about? If you know any equations involved in this I would appreciate it.

Answer 1

If there is a radial component along the jet axis, then as particles move away from the BH they experience a Lorentz force, $$\vec{F}=q\vec{v}\times\vec{B},$$ that is always perpendicular to the field lines.

Since $\vec{F}\cdot\vec{v}=0$, then no work is done on the particle and its speed and kinetic energy $(\gamma-1)mc^2$ would remain constant, hence its Lorentz factor $\gamma$ is constant. Also, since the force acts perpendicularly to the field lines, we can also deduce that velocity parallel to the field must be constant and therefore that the magnitudes of the components of velocity both perpendicular and parallel to the field are constant.

Hence the Lorentz force provides a centripetal acceleration and the particle executes a helical path along the field lines that results in the emission of doppler-beamed synchrotron radiation.