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Second Order Quasi-Normal Mode of the Schwarzschild Black Hole
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Second Order Quasi-Normal Mode of the Schwarzschild Black Hole
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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.
Forward citations
Cited by 8 Pith papers
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Early-Time Nonlinear Growth in an Unstable Q-Ball Hairy Black Hole
The early growth of the weakly responding scalar component in an unstable Q-ball hairy black hole is dominated by a second-order QNM sourced by the linear unstable mode, even while evolution remains perturbative.
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The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory
Introduces a gauge transformation framework for BMS frames in multiscale black hole perturbation theory on Kerr that incorporates memory effects and avoids infrared divergences.
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Black Hole Ringdown Nonlinearities in the Large-D Limit
In the large-D limit, analytic third-order nonlinear corrections to quasinormal modes improve ringdown modeling accuracy by several orders of magnitude for head-on black hole collisions.
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Early-Time Nonlinear Growth in an Unstable Q-Ball Hairy Black Hole
In an unstable Q-ball hairy black hole, the early growth of the weakly responding scalar component is dominated by a second-order QNM rather than its linear response.
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Can Oscillatory and Persistent Nonlinearities Be Bridged in Black Hole Ringdown?
Quadratic quasinormal modes and Christodoulou memory effect are related through bridge coefficients depending primarily on remnant black hole parameters.
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Excitation factors for horizonless compact objects: long-lived modes, echoes, and greybody factors
Excitation factors of long-lived quasinormal modes in horizonless compact objects scale with their small imaginary frequency, suppressing early contributions and producing a hierarchy where prompt ringdown uses ordina...
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Can Oscillatory and Persistent Nonlinearities Be Bridged in Black Hole Ringdown?
Quadratic quasinormal modes and the memory effect in black hole ringdown are related through bridge coefficients that depend primarily on remnant black hole parameters.
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Greybody factors, reflectionless scattering modes, and echoes of ultracompact horizonless objects
High-frequency quasi-reflectionless scattering modes in the greybody factors of ultracompact horizonless objects are responsible for echoes in the time-domain response.
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