Selection rules for black-hole quantum transitions

We suggest that quantum transitions of black holes comply with selection rules, analogous to those of atomic spectroscopy. In order to identify such rules, we apply Bohr’s correspondence principle to the quasinormal ringing frequencies of black holes. In this context, classical ringing frequencies with an asymptotically vanishing real part ${\omega }_R$ correspond to virtual quanta, and may thus be interpreted as forbidden quantum transitions. With this motivation, we calculate the quasinormal spectrum of neutrino fields in spherically symmetric black-hole spacetimes. It is shown that ${\omega }_R\to 0$ for these resonances, suggesting that the corresponding fermionic transitions are quantum mechanically forbidden.

Figure 1

Neutrino QNM frequencies of the RN black hole. Results are displayed for $l=\frac{1}{2}$, with $Q=0$ (triangles), $Q=0.3$ (circles) and $0.4$ (squares). Solid lines are a guide to the eye. The real part ${\omega }_R$ features a damped periodic dependence on the imaginary part ${\omega }_I$, with a period that becomes smaller with increasing black-hole charge. The spacing of the frequencies oscillates around $\Delta {\omega }_I=i2\pi T_{BH}$.

Figure 2

Neutrino QNM frequencies of the RN black hole for $l=\frac{1}{2},\frac{3}{2}$ and $\frac{5}{2}$ (from bottom to top), with $Q=0.4$. For clarity, we display only the envelope of the ${\omega }_R({\omega }_I)$ oscillatory dependence. The real part of the neutrino resonances asymptotically approaches zero, approximately as ${\omega }_R\propto {\omega }_I^{-1/2}$ (solid curves).