  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.
