Teukolsky Amplitudes Overview

• Teukolsky amplitudes are coefficients in black hole perturbation theory that vary with context, representing seed functions, scattering parameters, QNM eigenfunctions, or gauge-invariant fields.
• They serve as connection coefficients linking horizon/infinity solutions and QNM eigenfunctions, with normalization conventions impacting physical interpretations.
• Empirical and numerical studies demonstrate that amplitude values depend on initial data, analytic continuations, and asymptotic matching, requiring careful specification of the framework.

Searching arXiv for recent and relevant papers on Teukolsky amplitudes across initial-data, scattering, QNM, and physical-space interpretations. arXiv search query: "Teukolsky amplitudes Kerr scattering QNM Teukolsky Starobinsky constants" Teukolsky amplitudes are not a single universally fixed object. In the literature, the term appears in at least five distinct but related senses: the overall prefactor of Teukolsky-wave initial data; the coefficients multiplying canonical horizon and infinity solutions of the separated radial Teukolsky equation; the complex coefficients of quasinormal-mode eigenfunctions; gauge-invariant spin $\pm2$ or coupled electromagnetic-gravitational radiative fields in physical-space analyses; and the coefficients governing late-time tails or Cauchy-horizon blow-up (Fernández et al., 2021, Alcoforado et al., 2024, Shlapentokh-Rothman et al., 2020, Zhu et al., 2023, Collingbourne et al., 2023, Ma et al., 2021, Gurriaran, 2024). The multiplicity is not merely terminological: each notion of amplitude is tied to a different formulation of the Teukolsky problem, a different normalization convention, and a different physical question.

1. Terminological scope and basic distinctions

In initial-data studies, “amplitude” usually means a seed-function prefactor. In the comparison of linear Brill and Teukolsky waves, Teukolsky waves are the standard axisymmetric quadrupolar vacuum solutions of the linearized Einstein equations on flat spacetime, in transverse-traceless gauge, restricted to even-parity, axisymmetric $\ell=2$, $m=0$ data at a moment of time symmetry. There the operative viewpoint is explicit: the amplitude is the dimensionless overall prefactor $\epsilon$ entering the generating function $F(t,r)$, and all perturbative metric quantities scale linearly with $\epsilon$ (Fernández et al., 2021).

In frequency-domain Kerr perturbation theory, by contrast, amplitudes are coefficients of basis solutions. The radial Teukolsky equation admits distinguished solutions characterized by ingoing or outgoing behavior at the horizon and at infinity, and a general homogeneous solution is expanded as a linear combination of those basis elements. In that setting the relevant amplitudes are the connection coefficients $a_{\mathcal H^\pm}^{[s]}$, $a_{\mathcal I^\pm}^{[s]}$, or their transformed-system analogues $A_{\mathcal H^\pm}^{[s]}$, $A_{\mathcal I^\pm}^{[s]}$ (Shlapentokh-Rothman et al., 2020). Closely related are the standard transmission, reflection, and incidence amplitudes $\ell=2$0, $\ell=2$1, and $\ell=2$2 for the $\ell=2$3 radial solutions (Nasipak, 2024).

In quasinormal-mode analyses, amplitude can mean the coefficient of a full eigenfunction rather than the coefficient of a single asymptotic wave. The cleanest formulation in the supplied literature defines $\ell=2$4 as the complex coefficient multiplying a quasinormal mode with known radial eigenfunction, known angular structure, and known time dependence $\ell=2$5 (Zhu et al., 2023). This is distinct from a coefficient extracted at one radius.

In physical-space stability papers, the term is used for gauge-invariant radiative fields themselves. On Schwarzschild, the tensorial Teukolsky variables are $\ell=2$6 and $\ell=2$7, the spin-$\ell=2$8 and spin-$\ell=2$9 amplitudes built from the extreme linearized Weyl curvature components (Collingbourne et al., 2023). On Reissner–Nordström, the spin $m=0$0 Teukolsky-type amplitudes are the gauge-invariant one-forms

$m=0$1

(Giorgi, 2018). In Kerr–Newman, the relevant radiative amplitudes form a coupled system $m=0$2 and their negative-sign counterparts, rather than a single decoupled scalar (Giorgi, 2023).

A plausible implication is that “Teukolsky amplitudes” should always be read relative to the analytical framework in use: Cauchy data, scattering, QNM decomposition, physical-space radiation fields, or asymptotic tail coefficients.

2. Separated Kerr theory: horizon, infinity, and scattering amplitudes

For the homogeneous radial Teukolsky equation on Kerr, the most standard amplitude notion is asymptotic. In the monodromy-based analysis of the radial equation, the physical radial solutions are defined by their behavior near the outer horizon and spatial infinity: $m=0$3

$m=0$4

together with the mixed asymptotics defining reflection and incidence amplitudes (Nasipak, 2024). These $m=0$5-coefficients are the conventional scattering amplitudes.

The frequency-space analysis of the full subextremal Kerr problem formulates the same structure in a basis adapted to the original Teukolsky variable and to the Dafermos–Holzegel–Rodnianski transformed system. A general homogeneous mode can be written as

$m=0$6

or equivalently

$m=0$7

These coefficients are the mode amplitudes whose boundedness the paper controls uniformly across real frequencies (Shlapentokh-Rothman et al., 2020).

The central denominator in the connection problem is the Wronskian

$m=0$8

Its nonvanishing excludes nontrivial outgoing real modes, and quantitative bounds on $m=0$9 yield quantitative control of the amplitudes on bounded frequency sets (Shlapentokh-Rothman et al., 2020).

The same paper also gives exact separated flux identities relating horizon and infinity amplitudes. For the transformed variable, one form is

$\epsilon$0

where $\epsilon$1, $\epsilon$2 is the radial Teukolsky–Starobinsky constant, and $\epsilon$3 are boundary conversion constants (Shlapentokh-Rothman et al., 2020). In this formulation, amplitudes are not merely labels of asymptotic expansions; they are linked by conserved or almost-conserved identities.

3. Spin reversal, Teukolsky–Starobinsky constants, and amplitude normalization

Spin reversal is the main algebraic mechanism relating different amplitude conventions. In separated Kerr theory, the Teukolsky–Starobinsky identities map spin $\epsilon$4 and spin $\epsilon$5 solutions into one another and introduce the radial Teukolsky–Starobinsky constant $\epsilon$6, which enters directly into mode-amplitude and scattering relations (Costa et al., 2021).

For the asymptotic basis modes, the radial Teukolsky–Starobinsky operators act diagonally up to multiplicative constants. The paper on “facts and fictions” records explicit horizon and infinity factors such as

$\epsilon$7

and a horizon factor

$\epsilon$8

with $\epsilon$9 (Costa et al., 2021). These formulas control how horizon and infinity amplitudes transform under spin reversal.

That same paper emphasizes a major sign distinction. For $F(t,r)$0, the radial Teukolsky–Starobinsky constant is positive on real separated modes; for $F(t,r)$1, $F(t,r)$2 can be negative (Costa et al., 2021). The paper interprets this as permitting a non-superradiant amplification channel for higher spins. This is directly relevant to amplitude interpretation, because $F(t,r)$3 appears in the flux formulas for reflected and transmitted amplitudes.

Canonical quantization on Kerr uses the same constants in a different way. The mode-by-mode Hertz/curvature relation is imposed by

$F(t,r)$4

and the separated normalization constants satisfy

$F(t,r)$5

$F(t,r)$6

where $F(t,r)$7 is the radial Teukolsky–Starobinsky constant in that notation and $F(t,r)$8 (Iuliano et al., 2023). The paper states that the canonical commutation relations can be implemented if and only if the radial Teukolsky–Starobinsky constant is strictly positive, and identifies this positivity as valid for Maxwell and gravitational perturbations.

The monodromy-based study of the MST parameter $F(t,r)$9 places amplitude normalization into a different analytic framework. It identifies $\epsilon$0 as the monodromy eigenvalue at infinity and then feeds $\epsilon$1 into the standard MST amplitude formulas for transmission, incidence, and reflection (Nasipak, 2024). The paper also derives the monodromy matrix at infinity in the physical $\epsilon$2 basis and expresses the transmission-amplitude ratio

$\epsilon$3

in terms of the Stokes multipliers $\epsilon$4 (Nasipak, 2024). This suggests that amplitude normalization can be viewed equally as a problem in analytic continuation and monodromy data.

4. Initial-data amplitudes for Teukolsky waves

A different usage arises in numerical-relativity initial data based on Teukolsky waves. In the linear comparison of Brill and Teukolsky waves, the metric perturbation is generated from the seed function

$\epsilon$5

where $\epsilon$6 is the dimensionless overall amplitude and $\epsilon$7 is the wavelength parameter (Fernández et al., 2021). At $\epsilon$8, a moment of time symmetry, the radial coefficients are

$\epsilon$9

$a_{\mathcal H^\pm}^{[s]}$0

The paper emphasizes the transverse-traceless conditions

$a_{\mathcal H^\pm}^{[s]}$1

and defines the spatial perturbation by $a_{\mathcal H^\pm}^{[s]}$2 (Fernández et al., 2021). In that setting, “Teukolsky amplitude” means precisely the linear prefactor $a_{\mathcal H^\pm}^{[s]}$3.

The same paper compares these linear Teukolsky data to linear Brill data through the gauge-invariant Moncrief master function. For Teukolsky data, the only nonzero mode is

$a_{\mathcal H^\pm}^{[s]}$4

reflecting that the Teukolsky construction is purely quadrupolar by construction (Fernández et al., 2021). After transforming the linearized Brill wave to TT gauge, the authors use an empirical comparison normalization

$a_{\mathcal H^\pm}^{[s]}$5

with $a_{\mathcal H^\pm}^{[s]}$6, for which the coefficient functions $a_{\mathcal H^\pm}^{[s]}$7 look qualitatively similar (Fernández et al., 2021).

In cylindrical-coordinate BSSN evolutions, the amplitude parameter is instead denoted $a_{\mathcal H^\pm}^{[s]}$8 and inserted directly into the seed function

$a_{\mathcal H^\pm}^{[s]}$9

with $a_{\mathcal I^\pm}^{[s]}$0 (Alcoforado et al., 2024). The paper uses $a_{\mathcal I^\pm}^{[s]}$1 for the weak-wave regime and amplitudes

$a_{\mathcal I^\pm}^{[s]}$2

for nonlinear runs, reporting apparent-horizon formation for $a_{\mathcal I^\pm}^{[s]}$3 (Alcoforado et al., 2024). It also states that $a_{\mathcal I^\pm}^{[s]}$4 in its normalization corresponds to $a_{\mathcal I^\pm}^{[s]}$5 in Hilditch et al., because their amplitude is twice of theirs (Alcoforado et al., 2024). The explicit comparison shows that even the simplest initial-data amplitude parameter is convention-dependent across papers.

5. Quasinormal-mode amplitudes and spatially resolved fitting

In QNM analysis, the amplitude problem is not limited to asymptotic coefficients. The Kerr study based on hyperboloidal, horizon-penetrating coordinates defines the decomposition

$a_{\mathcal I^\pm}^{[s]}$6

so that $a_{\mathcal I^\pm}^{[s]}$7 is the complex amplitude of a full QNM eigenfunction (Zhu et al., 2023). The paper then solves for these amplitudes by linear regression, either on a single time slice using both $a_{\mathcal I^\pm}^{[s]}$8 and $a_{\mathcal I^\pm}^{[s]}$9, or over a spacetime slab.

A key conceptual distinction made there is between amplitudes of full eigenfunctions and amplitudes inferred at one radius. If one performs a time-only fit at fixed radius $A_{\mathcal H^\pm}^{[s]}$0, the extracted coefficient is

$A_{\mathcal H^\pm}^{[s]}$1

not the intrinsic QNM amplitude $A_{\mathcal H^\pm}^{[s]}$2 itself (Zhu et al., 2023). The paper uses the full radial structure to improve amplitude recovery and argues that this materially enlarges the fitting window over which amplitudes plateau.

The same work also stresses that amplitude extraction is limited by incompleteness of the QNM set and by transient contamination. In scattering experiments based on an approximately ingoing Gaussian pulse, the authors report that the fundamental mode and the first two overtones can be stably extracted for a period of at least $A_{\mathcal H^\pm}^{[s]}$3 for Kerr black holes not too close to extremality, but that fitting for higher overtones is generically unstable (Zhu et al., 2023). The abstract sharpens this as large uncertainties in the amplitudes of higher overtones $A_{\mathcal H^\pm}^{[s]}$4. The amplitude ambiguity is therefore not purely numerical; it is tied to prompt non-QNM content.

A plausible implication is that QNM amplitudes are inseparable from basis choice. In this literature, the coefficient of a damped sinusoid at null infinity, the coefficient of a full radial-angular eigenfunction, and the coefficient obtained after a least-squares fit over a spacetime region are related, but they are not identical objects.

6. Physical-space radiative amplitudes, late-time tail coefficients, and interior blow-up

In physical-space stability theory, the term “Teukolsky amplitude” often denotes the gauge-invariant radiative field itself. On Schwarzschild, the spin-$A_{\mathcal H^\pm}^{[s]}$5 tensorial amplitudes are

$A_{\mathcal H^\pm}^{[s]}$6

and the paper proves uniform boundedness of outgoing and ingoing fluxes for these quantities (Collingbourne et al., 2023). On Reissner–Nordström, the physically relevant spin-$A_{\mathcal H^\pm}^{[s]}$7 amplitudes are the gauge-invariant mixtures

$A_{\mathcal H^\pm}^{[s]}$8

which satisfy generalized spin-$A_{\mathcal H^\pm}^{[s]}$9 Teukolsky-type equations and, after a Chandrasekhar-type transformation, a Fackerell–Ipser-type equation (Giorgi, 2018). In Kerr–Newman, the radiative content is carried by a coupled gauge-invariant system $A_{\mathcal I^\pm}^{[s]}$0, and the paper treats boundedness and decay for that entire Teukolsky system rather than for a single scalar amplitude (Giorgi, 2023).

Late-time asymptotics introduce yet another amplitude concept: the coefficient of the Price tail. In the sharp decay analysis on Kerr, the leading coefficient is $A_{\mathcal I^\pm}^{[s]}$1. The paper states that the value of $A_{\mathcal I^\pm}^{[s]}$2 is characterized by an integral of the radiation field along future null infinity and proves a global conservation law relating that null-infinity integral, a horizon integral, and initial data (Ma et al., 2021). In the far region,

$A_{\mathcal I^\pm}^{[s]}$3

while in the interior region $A_{\mathcal I^\pm}^{[s]}$4,

$A_{\mathcal I^\pm}^{[s]}$5

For $A_{\mathcal I^\pm}^{[s]}$6, the horizon amplitude is $A_{\mathcal I^\pm}^{[s]}$7, with a faster $A_{\mathcal I^\pm}^{[s]}$8 law when $A_{\mathcal I^\pm}^{[s]}$9 (Ma et al., 2021).

Inside Kerr, the spin $\ell=2$00 field develops a different amplitude structure. The precise interior asymptotic theorem gives

$\ell=2$01

where $\ell=2$02 are the event-horizon Price-law amplitudes and

$\ell=2$03

for $\ell=2$04, while $\ell=2$05 (Gurriaran, 2024). The leading interior blow-up amplitude is therefore $\ell=2$06, modulated by the oscillatory factor $\ell=2$07. This is neither a scattering amplitude nor a QNM coefficient; it is a pointwise asymptotic coefficient near the Cauchy horizon.

The broad picture is therefore stratified. In some papers amplitudes are basis coefficients; in others they are gauge-invariant radiative fields; in others they are asymptotic tail constants. The data show that these usages are connected by transformation theory, conservation laws, and asymptotic matching, but they remain distinct notions rather than interchangeable ones.

7. Normalization conventions, ambiguities, and non-amplitude uses of related structures

Normalization dependence is a recurring issue. The QNM-fitting paper explicitly states that $\ell=2$08 depends on the normalization convention for the radial and angular eigenfunctions used to build the design matrix (Zhu et al., 2023). The Teukolsky-wave initial-data paper emphasizes that $\ell=2$09 is an empirical comparison choice, not a universal analytic equivalence (Fernández et al., 2021). The BSSN paper likewise notes a factor-of-two convention change relative to another amplitude parameter (Alcoforado et al., 2024). No amplitude statement is basis-free unless the normalization is declared.

A second caution is that not every Teukolsky-related paper is an amplitudes paper. The alternative radial ODE proposed as isospectral to the homogeneous radial Teukolsky equation is explicitly said not to derive ingoing or outgoing amplitudes, transmission or reflection coefficients, excitation factors, or source normalization constants (Hatsuda, 2020). Its relevance to amplitudes is indirect, through a simpler second-order ODE, asymptotic wave behaviors, and a Leaver-type continued fraction. Similarly, the twistor-manifold interpretation of Teukolsky operators explains the geometric origin of the differential operators but does not provide separated mode amplitudes, asymptotic matching data, or scattering coefficients (Araneda, 2019).

A final interpretive issue concerns the phrase “amplitude” itself. In the supplied literature, it may denote a numerical prefactor in initial data, a gauge-invariant curvature quantity, a connection coefficient in an ODE basis, a coefficient in a QNM expansion, or a late-time asymptotic constant. This suggests that the mathematically safest usage is always local to the formalism: one should specify whether the amplitude under discussion is a seed amplitude, a scattering amplitude, a QNM amplitude, a physical-space radiative field, or an asymptotic tail coefficient.