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We formulate and calculate the second order quasi-normal modes (QNMs) of aSchwarzschild black hole (BH). Gravitational wave (GW) from a distorted BH, socalled ringdown, is well understood as QNMs in general relativity. Since QNMsfrom binary BH mergers will be detected with high signal-to-noise ratio by GWdetectors, it is also possible to detect the second perturbative order of QNMs,generated by nonlinear gravitational interaction near the BH. In the BHperturbation approach, we derive the master Zerilli equation for the metricperturbation to second order and explicitly regularize it at the horizon andspatial infinity. We numerically solve the second order Zerilli equation byimplementing the modified Leaver's continued fraction method. The second orderQNM frequencies are found to be twice the first order ones, and the GWamplitude is up to $\sim 10%$ that of the first order for the binary BHmergers. Since the second order QNMs always exist, we can use their detections(i) to test the nonlinearity of general relativity, in particular the no-hairtheorem, (ii) to remove fake events in the data analysis of QNM GWs and (iii)to measure the distance to the BH.

Second Order Quasi-Normal Mode of the Schwarzschild Black Hole