Numerical solution of the 2 + 1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

Harms, Enno; Bernuzzi, Sebastiano; Bruegmann, Bernd

doi:10.1088/0264-9381/30/11/115013

Cited by 60 publications

(138 citation statements)

References 82 publications

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“…The peak emission is associated with the plunge phase, which occurs in the near-zone with some amplitude h max , which also sets the initial amplitude of the ringdown. In the near-zone, the ZDM wavefunctions depend on the overtone n, and so it is unlikely that they are collectively excited (note this goes against expectations of a power-law ringdown from [19,29,45] who studied initial data mostly supported away from the horizon). This means that the individual ZDMs receive characteristic amplitudes ∼ h max .…”

Section: Supplemental Materialsmentioning

confidence: 96%

Turbulent Black Holes

2015

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We show that rapidly-spinning black holes can display turbulent gravitational behavior which is mediated by a new type of parametric instability. This instability transfers energy from higher temporal and azimuthal spatial frequencies to lower frequencies-a phenomenon reminiscent of the inverse energy cascade displayed by 2 + 1-dimensional turbulent fluids. Our finding reveals a path towards gravitational turbulence for rapidly-spinning black holes, and provides the first evidence for gravitational turbulence in an asymptotically flat spacetime. Interestingly, this finding predicts observable gravitational wave signatures from such phenomena in black hole binaries with high spins and gives a gravitational description of turbulence relevant to the fluid-gravity duality.Black holes are fascinating objects. They play a fundamental role in a plethora of energetic phenomena in our universe, for example as the engines of active galactic nuclei, X-ray binaries, and possibly even as regulators of galactic structure. In addition, they have become central tools in the study of field theories through the framework of holography [1]. This includes attempts to understand superfluidity, superconductivity and quark-gluon plasmas obtained in energetic collisions (see e.g. [2][3][4]). One particularly exciting connection inspired by holography is the "fluid-gravity" duality, which indicates the dynamics of black holes in asymptotically anti-deSitter (AAdS) spacetimes in d + 1 dimensions can be mapped to the physics described by conformal fluids governed by viscous, relativistic hydrodynamics in d dimensions [5,6]. This opens the door to search for particular behavior known to exist on one side of the duality on the other. For instance, this duality has motivated studies showing that particular gravitational scenarios can become turbulent when their fluid counterparts have high Reynolds numbers [7][8][9]. Additionally, concepts in hydrydonamics, such as ensthropy, have geometric counterparts related to curvature quantities [7]. This duality can also shed light on poorly understood phenomena from a new perspective. Analyzing turbulence from an intrinsically gravitational point of view is thus an exciting prospect.In this work we develop a method to do precisely this and consider realistic, asymptotically flat black holes. Our analysis describes how gravitational turbulence is mediated by a parametric instability in the gravitational field -which does not require the "confining properties" of asymptotically AdS spacetimes-and motivates the definition of a gravitational Reynolds number. We first review general properties of turbulent flows, salient features of the fluid-gravity duality, and parametric instability.Hydrodynamic turbulence. Turbulence is a ubiquitous property of fluid flows with sufficiently high Reynolds number (Re ≡ ρ/ηvλ >> 1) [10,11]. Here v and λ refer to the typical velocity and wavelength of characteristic modes of the solution, and ρ, η the fluid density and viscosity. At high Re, nonlinear interactions prevail...

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Section: Supplemental Materialsmentioning

confidence: 96%

Turbulent Black Holes

2015

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“…It can be slowlyvarying and adiabatic (spin aligned with particle's angular momentum, the last-stable-orbit (LSO) moves towards the horizon) or quickly-varying and nonadiabatic (spin anti-aligned with particle's angular momentum, the 1 Note that by "challenging" we refer here to the limits of the radiation reaction model and not to the solution of the Teukolsky equation using the methods of Ref. [6,18]. The inclusion of the higher-order post-Newtonian information of Ref.…”

Section: Dynamics: Measuring Nonadiabaticitymentioning

confidence: 99%

The antikick strikes back: Recoil velocities for nearly extremal binary black hole mergers in the test-mass limit

Nagar

Harms²,

Bernuzzi³

et al. 2014

Phys. Rev. D

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Gravitational waves emitted from a generic binary black-hole merger carry away linear momentum anisotropically, resulting in a gravitational recoil, or "kick", of the center of mass. For certain merger configurations the time evolution of the magnitude of the kick velocity has a local maximum followed by a sudden drop. Perturbative studies of this "antikick" in a limited range of black hole spins have found that the antikick decreases for retrograde orbits as a function of negative spin. We analyze this problem using a recently developed code to evolve gravitational perturbations from a point-particle in Kerr spacetime driven by an effective-one-body resummed radiation reaction force at linear order in the mass ratio ν 1. Extending previous studies to nearly-extremal negative spins, we find that the well-known decrease of the antikick is overturned and, instead of approaching zero, the antikick increases again to reach ∆v/(cν 2 ) = 3.37 × 10 −3 for dimensionless spinâ = −0.9999. The corresponding final kick velocity is v end /(cν 2 ) = 0.076. This result is connected to the nonadiabatic character of the emission of linear momentum during the plunge. We interpret it analytically by means of the quality factor of the flux to capture quantitatively the main properties of the kick velocity. The use of such quality factor does not require trajectories nor horizon curvature distributions and should therefore be useful both in perturbation theory and numerical relativity.

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“…This set-up provides a good starting point for the study of the fully spectral method in (2+1)-dimensions. In particular, we can compare our numerical findings with some results available in the literature [27,28].…”

Section: Applicationsmentioning

confidence: 73%

“…treat here on hyperboloidal slices. Recently, studies of this kind were much-noticed, in particular when considering the behavior of the fields' decay at late times [27,28].…”

Section: The Teukolsky Equation On Hyperboloidal Slicesmentioning

confidence: 99%

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Axisymmetric fully spectral code for hyperbolic equations

Macedo

Ansorg

2014

Journal of Computational Physics

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We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for the time direction) collocation points. The method solves two issues of previous algorithms which were restricted to one spatial dimension, namely, (i) the inversion of a dense matrix and (ii) the acquisition of a sufficiently good initial-guess for non-linear systems of equations. For the first issue, we use the iterative bi-conjugate gradient stabilized method, which we equip with a pre-conditioner based on a singly diagonally implicit Runge-Kutta ("SDIRK"-) method. In this paper, the SDIRK-method is also used to solve issue (ii). The numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution. As an application, we solve general-relativistic wave equations on a black-hole space-time in so-called hyperboloidal slices and reproduce some recent results available in the literature.

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Numerical solution of the 2 + 1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

Abstract: In this work we present a formulation of the Teukolsky equation forNumerical solution of the 2+1 Teukolsky equation, application to late-time decays

Cited by 60 publications

References 82 publications

Turbulent Black Holes

Turbulent Black Holes

The antikick strikes back: Recoil velocities for nearly extremal binary black hole mergers in the test-mass limit

Axisymmetric fully spectral code for hyperbolic equations

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