Spectroscopy of the Schwarzschild Black Hole at Arbitrary Frequencies

Linear field perturbations of a black hole are described by the Green function of the wave equation that they obey. After Fourier decomposing the Green function, its two natural contributions are given by poles (quasinormal modes) and a largely unexplored branch cut in the complex frequency plane. We present new analytic methods for calculating the branch cut on a Schwarzschild black hole for arbitrary values of the frequency. The branch cut yields a power-law tail decay for late times in the response of a black hole to an initial perturbation. We determine explicitly the first three orders in the power-law and show that the branch cut also yields a new logarithmic behavior $T^{-2\ell -5}\ln T$ for late times. Before the tail sets in, the quasinormal modes dominate the black hole response. For electromagnetic perturbations, the quasinormal mode frequencies approach the branch cut at large overtone index $n$. We determine these frequencies up to $n^{-5/2}$ and, formally, to arbitrary order. Highly damped quasinormal modes are of particular interest in that they have been linked to quantum properties of black holes.

Figure 1

Full perturbation response $u_{\ell }$ to the Gaussian described above Eq. (10) compared to the late-time asymptotics. Solid (red): numerical solution; dashed (black): Eq. (10); lower curves: numerical solution minus the first (green), first 2 (blue) and first 4 (cyan) terms in Eq. (10).

Figure 2

$\Delta G_{\ell }$ as a function of $\nu$ for $r_{*}=0.1$ and $r_{*}^{\prime }=0.2$. (a) Using Eq. (5); dashed (green): $s=\ell =2$, continuous (blue): $s=0$, $\ell =1$, dot-dashed (orange): $s=\ell =1$. Note, the interesting behavior near the algebraically special frequency [22] at $\nu =4$ for $s=2$. (b) $s=0$, $\ell =1$ for small $\nu$; continuous (blue) using Eq. (5); dashed (red) using Eq. (8) to $O({\nu }^{15})$—see Ref. 30. (c) $s=0$, $\ell =1$ for large $\nu$; continuous (blue) using Eq. (5); dashed (red) using the asymptotics of Eq. (11).

Figure 3

Log-log plot of QNM frequencies for $s=\ell =1$ from the asymptotics Eq. (17) in dashed (green) and the numerical data in Ref. 41 in dotted (red). The two upper curves correspond to $400|\mathrm{Im}({\omega }_{\ell n})+\frac{n}{2}|$ and the two lower curves to $\mathrm{Re}({\omega }_{\ell n})$.