X iv : g r - qc / 0 10 80 24 v 1 8 A ug 2 00 1 Comment on ‘ An extreme critical space - time : echoing and black - hole perturbations ’ Gérard

We show that the black hole perturbations of the Hayward static solution to the massless Einstein-Klein-Gordon equations are actually gauge artifacts resulting from the linearization of a coordinate transformation. In a recent paper [1], Hayward discussed a homothetic, static solution to the massless Einstein-Klein-Gordon equations. With the spherically symmetric ansatz ds = rdΩ − 2edxdx (1) (where dΩ is the line-element of the unit sphere and (r, γ) are functions of the null coordinates x), this very simple solution reads r = (−xx) γ = 0 φ = 1 2 ln(−x/x) (2) (where φ is the scalar field). Another form of the solution (2), showing explicitly its static and homothetic character, is ds = e(dΩ + 2dρ − 2dτ). (3) Email: [email protected] Email: [email protected]