Decoupling and non-decoupling dynamics of large D black holes

Abstract

The limit of large number of dimensions localizes the gravitational field of a black hole in a well-defined region near the horizon. The perturbative dynamics of the black hole can then be characterized in terms of states in the near-horizon geometry. We investigate this by computing the spectrum of quasinormal modes of the Schwarzschild black hole in the 1/D expansion, which we find splits into two classes. Most modes are non-decoupled modes: non-normalizable states of the near-horizon geometry that straddle between the near-horizon zone and the asymptotic zone. They have frequency of order D/r_0 (with r_0 the horizon radius), and are also present in a large class of other black holes. There also exist a much smaller number of decoupled modes: normalizable states of the near-horizon geometry that are strongly suppressed in the asymptotic region. They have frequency of order 1/r_0, and are specific of each black hole. Our results for their frequencies are in excellent agreement with numerical calculations, in some cases even in D=4.

Figures (9)

• Figure 1: Radial potential $V_s(r_*)$ for perturbations of the Schwarzschild black hole for $n=7$ and $\ell=2$. The horizon is at $r_*\to-\infty$. We use the coding solid/dashed/dot-dashed $=$ tensor/vector/scalar in this and in the next two figures. Units are $r_0=1$.
• Figure 2: Radial potential $V_s(r_*)$ for $n=1000$ and $\ell=2$. On the right is a blow-up of the potential near the peak at $r_*\simeq 1$.
• Figure 3: Radial potential $V_s(r_*)$ for $n=1000$ and $\ell=1000$. On the right is a blow-up of the potential near the peak at $r_*\simeq 1$.
• Figure 4: Solution of \ref{['hiover']} determining quasinormal frequencies for $0<|\hat{\omega}|^2<\omega_c^2$. The continuous line of frequencies should become a discrete spectrum when higher order terms at large $n$ are included. Near $\hat{\omega}=\omega_c$ the curve connects smoothly to the spectrum \ref{['onomega']}.
• Figure 5: Frequency of $\ell=2$, $k=0$ (fundamental) tensor quasinormal mode as a function of $D$. Solid lines: analytical results eq. \ref{['Reonomega']}, \ref{['Imonomega']}; dashed line: leading order result $\omega=(D-3)\omega_c$. Gray lines: numerical results Dias:2014eua. For $\text{Re}~\omega$ we only include data up to $D=30$ for greater clarity.