d-Dimensional Black Hole Entropy Spectrum from Quasi-Normal Modes

Starting from recent observations\cite{hod,dreyer1} about quasi-normal modes, we use semi-classical arguments to derive the Bekenstein-Hawking entropy spectrum for $d$-dimensional spherically symmetric black holes. We find that the entropy spectrum is equally spaced: $S_{BH}=k\ln (m_0)n$, where $m_0$ is a fixed integer that must be derived from the microscopic theory. As shown in \cite{dreyer1},4-$d$ 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 requires the existence of a unique quasinormal mode frequency in the large damping limit ${\omega }^{(d)}(M)={\alpha }^{(d)}c\mathrm{/}{R}_H(M)$ with coefficient $\alpha^{(d)} = {(d-3)/over 4\pi} \ln(m_0)$,where$m_0$isanintegerand$\Gamma^{(d-2)}$ is the volume of the unit $d-2$ sphere.