Asymptotic Quasinormal Frequencies for Black Holes in Non-Asymptotically Flat Spacetimes
Vitor Cardoso, Jose Natario, Ricardo Schiappa
TL;DR
The paper extends the monodromy technique used for asymptotically flat black holes to non-flat spacetimes and derives analytic expressions for the asymptotic quasinormal frequencies of Schwarzschild–de Sitter and large Schwarzschild–AdS black holes in four dimensions. By formulating perturbations as a Schrödinger-like problem $-d^2Φ/dx^2 + V(x)Φ = ω^2 Φ$ and performing monodromy matching around horizons with surface gravities $k_H$, $k_C$, and $k_F$, it yields concrete spectra: for dS, a cosh-based condition $cosh(π ω/k_H - π ω/k_C) + 3 cosh(π ω/k_H + π ω/k_C) = 0$ in the $j→0$ limit, with specific $j$-dependent cases; for AdS, large black holes give $ ext{offset}/R^2$ and $ ext{gap}/R^2$ with complex values that precisely match prior numerical results, e.g. $ ext{offset} = 0.572975 + 0.419193 i$ and $ ext{gap} = 1.29904 + 2.25 i$, all scaled by $1/R^2$. These findings reinforce the relevance of quasinormal spectra to quantum gravity questions, including greybody factors and holographic dualities, and set the stage for extending the analysis to $d$-dimensions and charged configurations.
Abstract
The exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrodinger-like equation to the complex plane and then performing a method of monodromy matching at the several poles in the plane. While this method was successfully used in asymptotically flat spacetime, as applied to both the Schwarzschild and Reissner-Nordstrom solutions, its extension to non-asymptotically flat spacetimes has not been achieved yet. In this work it is shown how to extend the method to this case, with the explicit analysis of Schwarzschild de Sitter and large Schwarzschild Anti-de Sitter black holes, both in four dimensions. We obtain, for the first time, analytic expressions for the asymptotic quasinormal frequencies of these black hole spacetimes, and our results match previous numerical calculations with great accuracy. We also list some results concerning the general classification of asymptotic quasinormal frequencies in d-dimensional spacetimes.