Teukolsky Equation Research Articles

Mode analysis for the linearized Einstein equations on the Kerr metric: the large $\mathfrak{a}$ case

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1 -form wave operator on the Kerr metric in the large \mathfrak{a} case. By mode solutions we mean solutions of the form e^{-it_*\sigma}\tilde{h}(r,\theta,\varphi) where t_{*} is a suitable time variable. The corresponding Fourier-transformed 1 -form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and \tilde{h} has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of Häfner et al. (2021). No mode solutions exist for \Im \sigma\ge 0,\, \sigma\neq 0 . For \sigma=0 mode solutions are Coulomb solutions for the 1 -form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and Bäckdahl (2018) and Aksteiner et al. (2021).

Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes

Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes

A New Gauge for Gravitational Perturbations of Kerr Spacetimes II: The Linear Stability of Schwarzschild Revisited

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of Benomio (A new gauge for gravitational perturbations of Kerr spacetimes I: the linearised theory, 2022, https://arxiv.org/abs/2211.00602), specialised to the |a|=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|a|=0$$\\end{document} case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin ±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pm 2$$\\end{document} Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of Dafermos et al. (Acta Math 222:1–214, 2019), which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.

Teukolsky equation for near-extremal black holes beyond general relativity: Near-horizon analysis

Teukolsky equation for near-extremal black holes beyond general relativity: Near-horizon analysis

Uniform Boundedness for Solutions to the Teukolsky Equation on Schwarzschild from Conservation Laws of Linearised Gravity

We consider the equations of linearised gravity on the Schwarzschild spacetime in a double null gauge. Applying suitably commuted versions of the conservation laws derived in earlier work of the second author we establish control on the gauge invariant Teukolsky quantities α[±2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upalpha ^{[\\pm 2]}$$\\end{document} without any reference to the decoupled Teukolsky wave equation satisfied by these quantities. More specifically, we uniformly bound the energy flux of all first derivatives of α[±2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\upalpha ^{[\\pm 2]}$$\\end{document} along any outgoing cone from an initial data quantity at the level of first derivatives of the linearised curvature and second derivatives of the linearised connection components. Analogous control on the energy fluxes along any ingoing cone is established a posteriori directly from the Teukolsky equation using the outgoing bounds.

Analytically separating the source of the Teukolsky equation

Recent gravitational wave detections from black hole mergers have underscored the critical role black hole perturbation theory and the Teukolsky equation play in understanding the behavior of black holes. The separable nature of the Teukolsky equation has long been leveraged to study the vacuum linear Teukolsky equation; however, as theory and measurements advance, solving the sourced Teukolsky equation is becoming a frontier of research. In particular, second-order calculations, such as in quasinormal mode and self-force problems, have extended sources. This paper presents a novel method for analytically separating the Teukolsky equation’s source, aimed to improve efficiency. Separating the source is a nontrivial problem due to the angular and radial mixing of generic quantities in Kerr spacetime. We provide a proof-of-concept demonstration of our method and show that it is accurate, separating the Teukolsky source produced by the stress-energy tensor of an ideal gas cloud surrounding a Kerr black hole. The detailed application of our method is provided in an accompanying notebook. Our approach opens up a new avenue for accurate black hole perturbation theory calculations with sources in both the time and frequency domain. Published by the American Physical Society 2024

Distinguishability of binary extreme-mass-ratio inspirals in low frequency band

The inspiral of compact stellar objects into massive black holes are one of the main astrophysical sources for the Laser Interferometer Space Antenna (LISA) and Taiji. These extreme-mass-ratio inspirals (EMRIs) have great potential for cosmology and fundamental physics. A binary extreme-mass-ratio inspiral (b-EMRI) describes the case where binary black holes (BBHs) are captured by a supermassive black hole. The b-EMRIs serve as multi-band gravitational wave sources and provide insights into the dynamics of nuclei and tests of general relativity. However, if the b-EMRIs can be distinguished from the normal EMRIs or not is still not clear. In this work, with a few of assumptions, and using the Teukolsky equation, we calculate the approximate gravitational waves of b-EMRIs and assess their detectability by space-based detectors. We also decouple the secondary object information from the Teukolsky equation, enabling us to calculate the energy fluxes and generate the waveforms more conveniently. Variations in the quadrupole of the binary result in small but non-negligible changes in energy fluxes and waveforms, making it possible to distinguish b-EMRI signals with data analysis. This opens up the potential of using b-EMRIs to test gravity theories and for further astrophysical studies.

Perturbative and semi-analytical solutions to Teukolsky equations for massive fermions

In this work, we aim at solving the Teukolsky equations for a fermion with mass m e ≠ 0 in the presence of a rotating black hole with mass M. We consider two different regimes: m˜e=M−1me≪1 and a ω ≪ 1; m˜e≪1 and a ω ≳ 1. We treat each of these two regimes in different ways: we use a perturbative approach for the first, similar to the usual one employed for spin 0, 1, 2 and mass-less 1/2 fields, but with two small parameters (aω and m˜e ); as we shall see, the second can be treated with a semi-analytical approach. In a forthcoming paper we shall study the remaining two cases in which m˜e≳1 , while a ω ≪ 1 or a ω ≳ 1. The regime with m˜e≪1 , but a ω ≳ 1 is probably the most interesting from the astrophysics point of view, but this last two cases might be of some interest for the study of the interaction of fermions with very small black holes, which may be formed, for example, in the last stages of the Hawking evaporation.

Mode stability for gravitational instantons of type D

We study Ricci-flat perturbations of gravitational instantons of Petrov type D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme spin weight satisfy a Riemannian version of the separable Teukolsky equation. As a step toward infinitesimal rigidity of the type D Kerr and Taub-bolt families of instantons, we prove mode stability, i.e. that the Teukolsky equation admits no solutions compatible with regularity and asymptotic (local) flatness.

Scattering of spinning compact objects from a worldline EFT

We study the EFT of a spinning compact object and show that with appropriate gauge fixing, computations become amenable to worldline quantum field theory techniques. We use the resulting action to compute Compton and one-loop scattering amplitudes at fourth order in spin. By matching these amplitdes to solutions of the Teukolsky equations, we fix the values of Wilson coefficients appearing in the EFT such that it reproduces Kerr black hole scattering. We keep track of the spin supplementary condition throughout our computations and discuss alternative ways to ensure its preservation.

Challenges in quasinormal mode extraction: Perspectives from numerical solutions to the Teukolsky equation

Challenges in quasinormal mode extraction: Perspectives from numerical solutions to the Teukolsky equation

Teukolsky-like equations in a non-vacuum axisymmetric type D spacetime

We study an axisymmetric metric satisfying the Petrov type D property with some additional ansatze, but without assuming the vacuum condition. We find that our metric in turn becomes conformal to the Kerr metric deformed by one function of the radial coordinate. We then study the gravitational-wave equations on this background metric in the case that the conformal factor is unity. We find that under an appropriate gauge condition, the homogeneous wave equations admit the separation of the variables, which is also helpful for solving the nonhomogeneous equations. The resultant ordinary differential equation for the radial coordinate gives a natural extension of the Teukolsky equation.

Geometric background for the Teukolsky equation revisited

The aim of this review paper is to revisit the geometric framework of the Teukolsky equation in a form that is suitable for analysts working on this equation. We introduce spinor bundles, the Newman–Penrose formalism and the Geroch–Held–Penrose (GHP) formalism. In particular, we develop the case of Kerr spacetimes, for which we provide detailed computations.

Conformal scattering theories for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the constructions of conformal scattering theories for the tensorial wave equation such as the tensorial Fackerell-Ipser and the spin ±1 Teukolsky equations on Schwarzschild spacetime. In our strategy, we construct the conformal scattering for the tensorial Fackerell-Ipser equations which are obtained from the Maxwell equation and spin ±1 Teukolsky equations. Our method combines Penrose’s conformal compactification and the energy decay results of the tensorial fields satisfying the tensorial Fackerell-Ipser equation to prove the energy equality of the fields through the conformal boundary (resp. ) and the initial Cauchy hypersurface . We will prove the well-posedness of the Goursat problem by using a generalization of Hörmander’s results for the tensorial wave equations. By using the results for the tensorial Fackerell-Ipser equations we will establish the construction of conformal scattering for the spin ±1 Teukolsky equations.

In horizon penetrating coordinates: Kerr black hole metric perturbation, construction and completion

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington–Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.

Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-D black holes: an exact approach

We present a novel method that solves Teukolsky equations with the source to calculate radiation fluxes at infinity and event horizon for any perturbation fields of type-D black holes. For the first time, we use the confluent Heun function to obtain the exact solutions of ingoing and outgoing waves for the Teukolsky equation. This benefits from our derivation of the asymptotic analytic expression of the confluent Heun function at infinity. It is interesting to note that these exact solutions are not subject to any constraints, such as low-frequency and weak-field. To illustrate the correctness, we apply these exact solutions to calculate the gravitational, electromagnetic, and scalar radiations emitted by a particle in circular orbits around a Schwarzschild black hole. Numerical results show that the proposed exact solution appreciably improves the computational accuracy and efficiency compared with the 23rd post-Newtonian order expansion and the Mano-Suzuki-Takasugi method.

Peeling for tensorial wave equations on Schwarzschild spacetime

In this paper, we establish the asymptotic behavior along outgoing and incoming radial geodesics, i.e. the peeling property for the tensorial Fackerell–Ipser and spin [Formula: see text] Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity [Formula: see text] and the initial Cauchy hypersurface [Formula: see text] in a neighborhood of spacelike infinity [Formula: see text] far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

Universal Teukolsky equations and black hole perturbations in higher-derivative gravity

Universal Teukolsky equations and black hole perturbations in higher-derivative gravity

Scattering in black hole backgrounds and higher-spin amplitudes. Part II

We continue to investigate correspondences between, on the one hand, scattering amplitudes for massive higher-spin particles and gravitons in appropriate quantum-to-classical limits, and on the other hand, classical gravitational interactions of spinning black holes according to general relativity. We first construct an ansatz for a gravitational Compton amplitude, at tree level, constrained only by locality, crossing symmetry, unitarity and consistency with the linearized-Kerr 3-point amplitude, to all orders in the black hole’s spin. We then explore the extent to which a unique classical Compton amplitude can be identified by comparing with the results of the classical process of scattering long-wavelength gravitational waves off an exact Kerr black hole, determined by appropriate solutions of the Teukolsky equation. Up to fourth order in spin, we find complete agreement with a previously conjectured exponential form of the tree-level Compton amplitude. At higher orders, we extract tree-level contributions from the Teukolsky amplitude by an analytic continuation from a physical (a/GM < 1) to a particle-like (a/GM > 1) regime. Up to the sixth order in spin, we identify a unique conservative part of the amplitude which is insensitive both to the choice of boundary conditions at the black hole horizon and to branch choices in the analytic continuation. The remainder of the amplitude is determined modulo an overall sign from a branch choice, with the sign flipping under exchanging purely ingoing and purely outgoing boundary conditions at the horizon. Along the way, we make contact with novel applications of massive spinor-helicity variables pertaining to their relation to EFT operators and (spinning) partial amplitudes.

Perturbations of Spinning Black Holes beyond General Relativity: Modified Teukolsky Equation

The detection of gravitational waves from compact binary mergers by the LIGO/Virgo collaboration has, for the first time, allowed for tests of relativistic gravity in the strong, dynamical and nonlinear regime. Outside Einstein's relativity, spinning black holes may be different from their general relativistic counterparts, and their merger may then lead to a modified ringdown. We study the latter and, for the first time, derive a modified Teukolsky equation, i.e., a set of linear, decoupled differential equations that describe dynamical perturbations of non-Kerr black holes for the radiative Newman-Penrose scalars $\Psi_0$ and $\Psi_4$. We first focus on non-Ricci-flat, Petrov type D black hole backgrounds in modified gravity, and derive the modified Teukolsky equation through direct decoupling and through a new approach, proposed by Chandrasekhar, that uses certain gauge conditions. We then extend this analysis to non-Ricci-flat, Petrov type I black hole backgrounds in modified gravity, assuming they can be treated as a linear perturbation of Petrov type D, black hole backgrounds in GR by generalizing Chandrasekhar's approach, and derive the decoupled modified Teukolsky equation. Our work lays the foundation to study the gravitational waves emitted in the ringdown phase of black hole coalescence in modified gravity for black holes of any spin. Our work can also be extended to compute gravitational waves emitted by extreme mass-ratio binary inspirals in modified gravity.