Black hole lasers, a mode analysis

We show that the black hole laser effect discovered by Corley and Jacobson should be described in terms of frequency eigenmodes that are spatially bound. The spectrum contains a discrete and finite set of complex frequency modes, which appear in pairs and which encode the laser effect. In addition, it contains real frequency modes that form a continuous set when space is infinite, and which are only elastically scattered, i.e., not subject to any Bogoliubov transformation. The quantization is straightforward, but the calculation of the asymptotic fluxes is rather involved. When the number of complex frequency modes is small, our expressions differ from those given earlier. In particular, when the region between the horizons shrinks, there is a minimal distance under which no complex frequency mode exists, and no radiation is emitted. Finally, we relate this effect to other dynamical instabilities found for rotating black holes and in electric fields, and we give the conditions to get this type of instability.

Figure 1

Velocity profile $v(x)$ as a function of $\kappa x$, for ${\kappa }_B={\kappa }_W$, $D=0.33$, and $\kappa L=8$. The horizons are located at $\kappa x=\pm 8$, where $v(x)+1=0$.

Figure 2

Graphical resolution of Eq. (16). On the left, the straight lines represent $\omega -k{v}_{+}$ in the supersonic regime $v_{+}<-1$ for three different values of $\omega$. The middle one, which is tangent to $\Omega (k)$, determines the critical frequency ${\omega }_{\max }$. For $\omega <{\omega }_{\max }$, the two extra negative roots correspond to right moving modes with respect to the fluid since ${\partial }_k\Omega >0$. On the right, $k^u$ is the positive root describing the right moving mode, see Eq. (18), $k^{(1)}$ the most negative root, $k^{(2)}$ that with the smallest norm, see Eq. (22).

Figure 3

By making use of numerical results borrowed from 20, we have represented the square norm of ${\mathcal{B}}_{\omega }^{(2)}$, the amplitude of the negative frequency trapped mode of Eq. (22), as a function of $\omega$ real. Near a complex frequency ${\lambda }_a={\omega }_a-i{\Gamma }_a$, which solves $S_{22}=1$, see Eq. (36), $|{\mathcal{B}}_{\omega }^{(2)}{|}^2$ behaves as a Lorentzian: $\sim |\omega -{\omega }_a-i{\Gamma }_a{|}^{-2}$, see Eqs. (33, 34). The dots are the numerical values, whereas the continuous line is a fitted sum of Lorentzians. The remarkable agreement establishes that the complex frequencies ${\lambda }_a$ can be deduced from the analysis of the real frequency modes. The frequency $\omega$ has been expressed in the units of ${\omega }_{\max }$, see Fig. 2, so that there is no resonance above $\omega /{\omega }_{\max }=1$. In the present case there are 13 resonances. The narrow peaks, ${\Gamma }_a\sim 0$, are due to the fact that the surface gravities are equal, ${\kappa }_B={\kappa }_W$, which leads to $z_{\omega }=w_{\omega }$ in Eq. (41).