Echoes of massless scalar field induced from hairy Schwarzschild black hole

Abstract

A hairy Schwarzschild black hole describes the deformation of Schwarzschild black hole with an exponent correction due to the introducing of additional sources. Inspired by the novel feature that the hairy Schwarzschild black hole can have double photon spheres for certain parameters space, in this paper we study the echo signals of a massless scalar field from this hairy Schwarzschild black hole by calculating its time evolution. We mainly analyze how the hairy parameters and scalar field's angular momentum affect the echo waveform of the perturbing scalar field. Additionally, we roughly evaluate the time delay of echoes. We find that the hairy parameters have significant influences on the echo signals.