Can one hear the shape of a black hole singularity?

Abstract

We compute the large overtone quasinormal frequencies $ω_n$ of asymptotically AdS black holes with interior near-singularity Kasner epochs. We show that the Kasner exponents can be extracted cleanly from the large $n$ behaviour of the dispersion $\partial ω_n/\partial k^2$ at $k=0$, where $k$ is a boundary momentum. We show that Kasner transitions, as occur in BKL near-singularity chaos, are audible in the quasinormal frequencies and reveal a map between overtone number $n$ and proper time to the singularity. Our results are obtained using a WKB analysis that isolates non-analytic exponents, determined by the near-singularity spacetime, from non-universal data such as the optical distance from the boundary to the singularity. We check our formulae in several numerical models.