Appendix C: Domain of definition of the Bernoulli function

Figure C.1: Domain of definition of the Bernoulli function: we consider a Schwarzschild black hole and draw this domain for different values of m and $\omega '$ (variations of $\Theta '$ correspond only to a contraction or a dilatation of the domain along the y axis. We adopt here $\Theta '=1$). Cases a) and b): m=0.2<m*. There are three possible configurations depending on the value of $\omega '$. Case a) corresponds to $\omega '=0.3<\omega '_*$ and case b to $\omega '=0.5>\omega '_*$. The case $\omega '=\omega '_*$ looks very similar to case b) and is not plotted. We added the fast/slow mode Mach curves (dotted line) and the gravitational throat curve (dashed line) with the corresponding slow (S) and fast (F) critical point. The thick line is the solution passing through S, F and the Alfvén point A. Case c) m=m*=1/3. This is the critical case where it is still possible to find a solution (here $\omega '=0.3$ and $\Theta '=1$).

The Bernoulli function $H\left(x,y\right)$ is defined for ${\cal D}(x,y)>0$ which gives the following condition:

 

A(x) Y² - 2 B(x) Y + C(x) > 0 (C.1)

where $Y=\tilde{h}(1) y / \tilde{h}(y)$ is a function of y and $\Theta '$ only and A, B and C are functions of x, $\omega '$, a and m only (notice that it is completely independent of the function $\tilde{s}(x)$). The function Y(y) is strictly increasing from Y(0)=0 to $Y(+\infty)=+\infty$ with Y(1)=1. So we can focus to the pressureless case $\Theta'=0$ and Y=y, all other cases $\Theta'>0$ corresponding only to a contraction of the domain along the y-axis. The coefficients A, B and C are given by

$$A(x) = -\left(\tilde{\varpi}^2x\right)^2\xi^2(x)$$ (C.2)

$$B(x) = M_{\rm A}^2\left(\tilde{\varpi}^2 x\right)^2$$ (C.3)

$$\tilde{\varpi}^2\left\lbrace \tilde{\varpi}^2(1) \xi^2(x)\right.$$ C(x) = (C.4)

$$\left. - M_{\rm A}^2 \left( \tilde g_{tt}(1)\tilde g_{\phi\phi} x... ...)\tilde g_{t\phi} x + \tilde g_{\phi\phi}(1)\tilde g_{tt} \right)\right\rbrace.$$

The radius x being fixed, the Eq. (C.1) has 1 or 2 positive roots delimiting the domain where H(x,y) is well defined. Many configurations are possible for a Kerr metric and we do not specify them in details here. We discuss only the case of the Schwarzschild metric. Four different configurations are possible depending on m (with a critical case at m_*=1/3) and $\omega '$(with a critical case at $\omega'=27 m^2 \left(1-2m\right)^2$). This is illustrated in Fig. C.1.