Second Order Quasi-Normal Mode of the Schwarzschild Black Hole

TL;DR

This paper formulates and computes the second order quasi-normal modes of Schwarzschild black holes, revealing their potential for testing general relativity and improving gravitational wave data analysis.

Contribution

It derives the second order Zerilli equation, solves it numerically, and demonstrates that second order QNM frequencies are twice the first order ones, with implications for gravitational wave detection.

Findings

Second order QNM frequencies are twice the first order frequencies.

Second order GW amplitude can be up to 10% of the first order.

Detection of second order QNMs can test nonlinearity in general relativity.

Abstract

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