A WKB formula for echoes

Abstract

We present a unified approach for the study of idealized gravitational compact objects like wormholes (WHs) and horizonless stars, here simulated by the presence of boundary conditions at a deeply inner wall. At the classical level, namely neglecting quantum effects, the presence of the wall leads to characteristic echoes following the usual ringdown phase, such that it can discriminate black holes from other horizonless, and probably exotic, compact objects. With regard to this issue, an analytical though approximated expression for the complex frequencies of the quasinormal echoes is found and discussed. At the quantum level, we show that static WHs do not radiate.