Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole

Abstract

In the analysis of a binary black hole coalescence, it is necessary to include gravitational self-interactions in order to describe the transition of the gravitational wave signal from the merger to the ringdown stage. In this paper we study the phenomenology of the generation and propagation of nonlinearities in the ringdown of a Schwarzschild black hole, using second-order perturbation theory. Following earlier work, we show that the Green's function and its causal structure determines how both first-order and second-order perturbations are generated, and hence highlight that both of these solutions share some physical properties. In particular, we discuss the sense in which both linear and quadratic quasi-normal modes (QNMs) are generated in the vicinity of the peak of the gravitational potential barrier (loosely referred to as the light ring). Among the second-order perturbations, there are solutions with linear QNM frequencies (whose amplitudes are thus renormalized from their linear values), as well as quadratic QNM frequencies with a distinct spectrum. Moreover, we show using a WKB analysis that, in the eikonal limit, waves generated inside the light ring propagate towards the black hole horizon, and only waves generated outside propagate towards an asymptotic observer. These results might be relevant for recent discussions on the validity of perturbation theory close to the merger. Finally, we argue that even if nonlinearities are small, quadratic QNMs may be detectable and would likely be useful for improving ringdown models of higher angular harmonics and future tests of gravity.

The quantum optics of gravitational waves

“…Strong nonlinear astronomical sources, perhaps those already known to emit GWs, seem therefore reasonable candidates to investigate. A recent step in this direction has been proposed in [20] where nonlinear effects present in black hole's ringdown [40][41][42][43][44][45] have been considered.…”

Nonlinearities in Black Hole Ringdowns

“…The proportionality coefficient between (A (2,2,0) ) 2 and A (2,2,0)×(2,2,0) (4,4) (which we expect to be order unity [36,41]) comes from the spacetime dependence of the full quadratic source as well as the Green's function. While in principle this can be computed, we use the fact that it should only depend on the dimensionless spin χ f of the remnant BH.…”

“…Quadratic QNMs.-Second-order perturbation theory has been studied for both Schwarzschild and Kerr BHs [31][32][33][34][35][36][37][38][39][40][41]. This involves the same Teukolsky operator as in Eq.…”

“…(1) acting on the second-order curvature correction, and a complicated source S that depends quadratically on the linear perturbations [39,40,42]. The second-order solution results from a rather involved integral of this source against the Green's function G [36,41]. We only need to know that it is quadratic in the linear perturbation and that, after enough time, it is well-approximated by the quadratic QNMs.…”

“…As the linear (2, ±2, 0) modes are most important, it is promising to investigate the quadratic QNMs they generate, which primarily appear in the ( , m) = (4, ±4) modes [34,35,41]. The quadratic QNM coming from the (2, 2) mode would have frequency ω (2,2,0)×(2,2,0) ≡ 2ω (2,2,0) and would decay faster than the linear fundamental mode (4, 4, 0), but slower than the first linear overtone (4, 4, 1), regardless of the BH spin [43].…”