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We investigate the nonlinear evolution of black hole ringdown in the framework of higher-order metric perturbation theory.
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At the horizon ( r ∗ → − ∞ r_{*}\to-\infty ), S ℓ m ( 2 ) ( t , r ∗ ) S^{(2)}_{\ell m}(t,r_{*}) actually behaves as ∝ e r ∗ / 2 M \propto e^{r*/2M} , not as ∝ r ∗ − 2 \propto r_{*}^{-2} . However, this difference does not affect ϕ F ( 2 ) \phi^{(2)}_{F} (i.e., second-order QNMs and tail), since G F ( t − t ′ , x , x ′ ) G_{F}(t-t^{\prime};x,x^{\prime}) vanishes for x ′ < 0 x^{\prime}<0 assuming an observer’s position at x > 0 x>0
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