Second Order Quasi-Normal Mode of the Schwarzschild Black Hole

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.

arXiv comment23 pages, no figure

References in corpus (37)

• Quasi-Normal Modes of Stars and Black Holes
• Evolution of Binary Black Hole Spacetimes
• Accurate Evolutions of Orbiting Black-Hole Binaries Without Excision
• Gravitational wave extraction from an inspiraling configuration of merging black holes
• Possibility of Direct Measurement of the Acceleration of the Universe Using 0.1 Hz Band Laser Interferometer Gravitational Wave Antenna in Space
• On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA
• Second-Order Cosmological Perturbations from Inflation
• Measuring gravitational waves from binary black hole coalescences: I. Signal to noise for inspiral, merger, and ringdown
• Relativistic second-order perturbations of the Einstein-de Sitter Universe
• Perturbations of spacetime: gauge transformations and gauge invariance at second order and beyond
• On the Back Reaction Problem for Gravitational Perturbations
• Second order gauge invariant gravitational perturbations of a Kerr black hole
• Evolving test-fields in a black-hole geometry
• Spin Flips and Precession in Black-Hole-Binary Mergers
• Second-order Gauge Invariant Cosmological Perturbation Theory: -- Einstein equations in terms of gauge invariant variables --
• Gravitational Waves from Supermassive Black Hole Coalescence in a Hierarchical Galaxy Formation Model
• Colliding black holes: how far can the close approximation go?
• Late-time dynamics of rapidly rotating black holes
• Second order perturbations of a Schwarzschild black hole
• Gravitational radiation from Schwarzschild black holes: the second order perturbation formalism
• Mode coupling in the nonlinear response of black holes
• Quick and dirty methods for studying black-hole resonances
• Relativistic second-order perturbations of nonzero-Λflat cosmological models and CMB anisotropies
• Gauge Invariant Variables in Two-Parameter Nonlinear Perturbations
• A superradiance resonance cavity outside rapidly rotating black holes
• Gravitational waves, inflation and the cosmic microwave background: towards testing the slow-roll paradigm
• Gauge Problem in the Gravitational Self-Force I. Harmonic Gauge Approach in the Schwarzschild Background
• High-order gauge-invariant perturbations of a spherical spacetime
• Nonlinear and Perturbative Evolution of Distorted Black Holes. II. Odd-parity Modes
• Gauge invariant formalism for second order perturbations of Schwarzschild spacetimes
• A time-domain fourth-order-convergent numerical algorithm to integrate black hole perturbations in the extreme-mass-ratio limit
• An Effective Search Method for Gravitational Ringing of Black Holes
• Reconstruction of Black Hole Metric Perturbations from Weyl Curvature II: The Regge-Wheeler gauge
• An Improved Search Method for Gravitational Ringing of Black Holes
• On Detection of Black Hole Quasi-Normal Ringdowns: Detection Efficiency and Waveform Parameter Determination in Matched Filtering
• Second order perturbations of a Schwarzschild black hole: inclusion of odd parity perturbations
• Spatial Correlation of Massive Black Hole Mergers: Probing the Formation Mechanism and Reionization

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