$d$-Dimensional Black Hole Entropy Spectrum from Quasinormal Modes

Starting from recent observations about quasinormal modes, we use semiclassical arguments to derive the Bekenstein-Hawking entropy spectrum for $d$-dimensional spherically symmetric black holes. We find that, as first suggested by Bekenstein, the entropy spectrum is equally spaced: $S_{\mathrm{B}\mathrm{H}}=k\ln (m_0)n$, where $m_0$ is a fixed integer that must be derived from the microscopic theory. As shown in O. Dreyer, gr-qc/0211076, 4D loop quantum gravity yields precisely such a spectrum with $m_0=3$ providing the Immirzi parameter is chosen appropriately. For $d$-dimensional black holes of radius $R_H(M)$, our analysis predicts the existence of a unique quasinormal mode frequency in the large damping limit ${\omega }^{(d)}(M)={\alpha }^{(d)}c/R_H(M)$ with coefficient ${\alpha }^{(d)}=\frac{(d-3)}{4\pi }\ln (m_0)$, where $m_0$ is an integer.