Appendix A: Metric coefficients

The following table gives the metric coefficients in normalized units in three cases: the Minkowski (M), Schwarzschild (S) and Kerr (K) space-times.

$$\begin{array}{\vert c\vert c\vert c\vert c\vert} \hline & {... ...sqrt{1-2m+a^2m^2}}{1+a^2m^2+2 a^2 m^3}\\ \hline \end{array}$$ (A.1)

where $m=GM/r_{\rm a}c^2$ and a=Jc/GM² (where J is the total angular momentum of the black hole and $0\le a\le1$). The radii $x_{\rm h}$ and $x_{\rm e}$ are respectively the radius of the horizon and of the ergosphere. We consider only the case where $0\le m \le \frac{1}{2}$ (the Alfvén point is outside the ergosphere). The minimum value $\omega'_{\rm min}$ of $\omega '$ corresponds to the condition $K_{\rm A} \ge 0$ (positive total angular momentum L) and the maximum value $\omega'_{\rm max}$ corresponds to the condition $M_{\rm A}^2>0$ (the Alfvén point must be inside the light surface).