Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes

When a classical black hole is perturbed, its relaxation is governed by a set of quasinormal modes with complex frequencies $\omega ={\omega }_R+i{\omega }_I$. We show that this behavior is the same as that of damped harmonic oscillators whose real frequencies are $({\omega }_R^2+{\omega }_I^2{)}^{1/2}$, rather than simply ${\omega }_R$. Since, for highly excited modes, ${\omega }_I\gg {\omega }_R$, this observation changes drastically the physical understanding of the black hole spectrum and forces a reexamination of various results in the literature. In particular, adapting a derivation by Hod, we find that the area of the horizon of a Schwarzschild black hole is quantized in units $\Delta A=8\pi l_{\mathrm{Pl}}^2$, in contrast with the original result $\Delta A=4\log (3)l_{\mathrm{Pl}}^2$.

Figure 1

$\mathrm{Re}(2M{\omega }_{nl})$ against $\mathrm{Im}(2M{\omega }_{nl})$ for $l=2$ and $n=1,2,\dots ,12$, and for $n=20$, 30, 40. Data taken from 1.

Figure 2

$\mathrm{Re}[2M({\omega }_0{)}_{nl}]$ for $l=2$, against $n$.