Perturbative calculation of quasinormal modes of AdS Schwarzschild black holes

We calculate analytically quasinormal modes of AdS Schwarzschild black holes including first-order corrections. We consider massive scalar, gravitational and electromagnetic perturbations. Our results are in good agreement with numerical calculations. In the case of electromagnetic perturbations, ours is the first calculation to provide an analytic expression for quasinormal frequencies, because the effective potential vanishes at zeroth order. We show that the first-order correction is logarithmic.

Figure 1

The frequency gap (113) for scalar perturbations in $d=4$ for $r_H=1$ and $\ell =2$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (111)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 2

The frequency gap (113) for tensor perturbations in $d=4$ for $r_H=1$ and $\ell =0$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (119)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 3

The frequency gap (113) for tensor perturbations in $d=4$ for $r_H=0.2$ and $\ell =0$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (121)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 4

The frequency gap (113) for vector perturbations in $d=4$ for $r_H=1$ and $\ell =2$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (126)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 5

The frequency gap (113) for vector perturbations in $d=4$ for $r_H=0.2$ and $\ell =2$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (128)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 6

The frequency gap (113) for electromagnetic perturbations in $d=4$ for $r_H=100$ and $\ell =1$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (139)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 7

The frequency gap (113) for electromagnetic perturbations in $d=4$ for $r_H=1$ and $\ell =1$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (140)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.

Figure 8

The frequency gap (113) for electromagnetic perturbations in $d=4$ for $r_H=0.2$ and $\ell =1$: zeroth (horizontal lines) and first order (curved lines) analytical [Eq. (141)] compared with numerical data (diamonds) [7]. (a) and (b) are plots of $\mathrm{Re}\Delta {\omega }_n$ and $\mathrm{Im}\Delta {\omega }_n$, respectively, as functions of $n$.