Second order gauge invariant gravitational perturbations of a Kerr black hole

C.O.Lousto (AEI; M.Campanelli; Potsdam)

REVIEW

Second order gauge invariant gravitational perturbations of a Kerr black hole

M.Campanelli, C.O.Lousto (AEI, Potsdam)

General Relativity and Quantum Cosmology · Submitted 5 Nov 1998 · 45 references

Pith's verdict: A real construction — the tetrad-invariant piece checks out and the second-order Kerr wave equation is genuinely new — but the gauge-vector integrability is assumed, not shown, and the paper says so.

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arXiv IDgr-qc/9811019 (version 2)

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CitationC.O.Lousto (AEI, M.Campanelli, Potsdam). "Second order gauge invariant gravitational perturbations of a Kerr black hole." arXiv:gr-qc/9811019 (1998). https://pith.science/paper/KDMKTUMP

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Bottom line: the tetrad-invariant object is genuine and the algebra checks out; the gauge-invariant waveform is a well-motivated recipe with one load-bearing step assumed rather than proved, and the authors are upfront about that. Worth a serious read, not a desk reject and not a rubber stamp either. What is actually new: extending Teukolsky decoupling to arbitrary perturbative order (Eqs. 7, 11) is straightforward but nobody had written it down cleanly, and the explicit second-order tetrad invariant (Eq. 32) is a real construction — the class III factor and the -(2/3)(psi3)^2/psi2 correction do cancel the variations in Eqs. (28) and (31). The claim that psi_I satisfies the same operator with a corrected source (Eqs. 47-49) is the useful result, and Appendices A and B give the Newman-Penrose-to-metric dictionary a numerical group would actually need. Appendix C's Schwarzschild limit is a sensible cross-check against the Gleiser-Nicasio-Price-Pullin results. Where it is soft: Sec. IV.C says outright that the existence of xi^t and xi^phi is 'assumed a priori', tied to Chrzanowski's outgoing-radiation-gauge existence. That is an honest acknowledgement, but the integrability conditions for Eq. (50) are never checked, and that is the step the whole gauge-invariant construction rests on. I would want at least a paragraph showing the mixed partials commute for the difference xi^ORG - xi. On the stress-test's sign worry: I do not think it lands. The paper defines the invariant combination as the difference of transformed and original gauge vectors, which is the standard object and is invariant by the lemma in Sec. IV.B; the ORG vector plays the role of the transformed one. That is bookkeeping, not a sign error. Two smaller things: the a posteriori constants c_t, c_phi (Eqs. 51-52) make psi_I non-unique, which the paper admits; and there are no numerical results — the evolution is deferred to [42], in preparation. Neither undermines the existence claim. Who should read it: anyone doing second-order perturbation theory around rotating holes, and numerical groups wanting a gauge-clean extraction target. Send it out.

Referee report

Major objections (3)

[Sec. IV.C, Eqs. (45)-(46), (50)] The paper states that 'the existence of the ξ^t and ξ^ϕ components are assumed a priori and they are related to the existence of the outgoing radiation gauge proved in Ref. [32].' This is the load-bearing step for Eq. (41). The reconstruction in Eq. (50) requires the mixed partials of ξ^t and ξ^ϕ to commute, and the integrability conditions are neither written down nor checked. Please either display the integrability conditions and show that they hold (or that they follow from Chrzanowski's construction), or state explicitly that the existence of ξ^{ORG} - ξ is established only up to this assumption. As written, the abstract's 'we prove' (abstract:s2) rests on an assumed step.
[Sec. IV.B, Eq. (41)] The text says 'Making the replacement ξ^(1) → -ξ^(1) into Eq. (40) above generates a totally invariant object,' which would give a correction term with a minus sign, but Eq. (41) writes +ψ^(1)_{4,μ}(ξ^{ORG μ} - ξ^μ) + .... With the definition of ξ in Eq. (34) (x → x + εξ), the combination ξ^{ORG} - ξ transforms as ξ^{ORG} - ξ → ξ^{ORG} - ξ - 2δξ under a first-order gauge change, which is not invariant. The construction is invariant only if ξ^{ORG} is defined with the opposite sign (ORG → actual) or if the correction term enters with the opposite sign. Please state the convention explicitly and reconcile Eq. (41) with the text.
[Sec. IV.A, Eq. (41)] The correction term involves l^r(0)_{,μ}ξ^μ/l^r(0) and m^ϑ(0)_{,μ}ξ^μ/m^ϑ(0). Under a tetrad rotation l → A l, m → e^{iθ} m, these ratios pick up A_{,μ}ξ^μ/A and iθ_{,μ}ξ^μ, so they are not manifestly tetrad invariant. Since ψ^(2)_4TI is already tetrad invariant, the correction term must be tetrad invariant by itself for ψ_I to be. Please show the cancellation explicitly, or state that the correction term is evaluated in the Kinnersley tetrad and that the tetrad invariance of ψ_I is up to that choice (as footnote 3 does for Eq. (32)).

Minor objections (7)

[Eqs. (28)-(29)] The relative sign between the (A-1) and iθ terms in Eq. (28) and the corresponding combination in Eq. (29) should be double-checked; as written, the two do not cancel for the boost/rotation subgroup unless a convention is fixed.
[Eq. (32) and footnote 3] The generalization to a generic zeroth-order tetrad is stated but not derived; please give the derivation or a reference.
[Eq. (50)] The reconstruction formula for ξ^i from its derivatives is formal; the mixed-partial subtraction term is not clearly derived. Please spell out the inversion.
[Eqs. (51)-(52)] The gauge-fixing constants c^t and c^ϕ are defined by integrals over all time; the denominators could vanish for special configurations. Please comment on this.
[Appendix A, Eq. (A4); Ref. [33]] Eq. (A4) writes 'k^(1)' where 'κ^(1)' is meant; Ref. [33] gives the year as '178' where '1978' is meant; and the line for 2γ^(1) contains '3µ - 2µ + γ - γ', which looks like a typographical slip.
[Sec. IV.A] The claim that the quadratic term in (32) 'does not vanish relative to ψ^(2)_4 for large r' is stated but not shown; please give the asymptotic behavior.
[Ref. [42]] The numerical implementation is deferred to Ref. [42], which is in preparation; the abstract's claim of 'direct comparison with full numerical approaches' is therefore prospective. This is acceptable for a formalism paper, but the abstract could be more explicit.

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The paper

The paper claims

The paper builds a second-order gravitational waveform around a rotating black hole that does not change when coordinates or the null frame are changed, and shows it obeys the same Teukolsky wave equation as the first-order field with a new

The reading

First-order gravitational perturbations of a Kerr black hole are described by the Weyl scalar $psi_4$, which is already gauge invariant at that order, so a computed waveform does not depend on which coordinates were used. At second order that protection is gone: $psi_4^{(2)}$ shifts under both infinitesimal coordinate changes and rotations of the null tetrad, which makes a second-order perturbative waveform hard to compare with a full numerical simulation. The authors set out to repair this by explicitly constructing a corrected field $psi_I$ — the second-order piece of $psi_4$ plus quadratic corrections built from first-order quantities — that is invariant under both kinds of transformation and that reduces to the ordinary second-order $psi_4$ in an asymptotically flat gauge far from the source. They then argue that $psi_I$ satisfies a single wave equation with the same Teukolsky operator as at first order and a source term assembled entirely from first-order data. If that holds, second-order perturbation theory around a spinning hole becomes a practical, gauge-independent benchmark for numerical relativity in the regime where the merged remnant rotates rapidly.

What carries the argument

The load-bearing object is the corrected waveform of Eq. (41), $psi_I = psi_4^{(2)} + Q - Q_{AF} + psi_4^{(1)},_{mu}(xi^mu_{ORG} - xi^mu)$, where $Q$ is the quadratic tetrad-correcting term and $Q_{AF}$ is its counterpart evaluated in the outgoing radiation gauge. The tetrad part of the correction works because $psi_3^{(1)}$ shifts by $3a,psi_2^{(0)}$ under a null rotation of type I, which lets the rotation parameter be solved for and replaced; the coordinate part works because the first-order gauge vector can be recovered from variations of the tetrad-invariant $psi_2^{(1)}$ (giving $xi^r$ and $xi^vartheta$) and from variations of the metric and extrinsic curvature (giving the derivatives of $xi^t$ and $xi^phi$). The payoff is that the same Teukolsky operator $hat{T}$, written explicitly in Boyer-Lindquist coordinates and the Kinnersley tetrad, acts on the second-order invariant field, so an existing first-order evolution code can be reused with an added source term.

Extended reading notes

Core claim

The central claim is an existence proof by construction: for second-order perturbations about a Kerr black hole there exists a waveform $psi_I$ that is invariant under first-order coordinate (gauge) transformations and under tetrad rotations, and that coincides with the second-order $psi_4$ in an asymptotically flat gauge at the radiation zone. The construction proceeds in two stages. First, $psi_4^{(2)}$ is corrected by a quadratic term involving the first-order tetrad components and the first-order Weyl scalars, $2psi_4^{(1)}(l^r_{(1)}/l^r_{(0)} + m^vartheta_{(1)}/m^vartheta_{(0)}) - (2/3)(psi_3^{(1)})^2/psi_2^{(0)}$, which cancels the tetrad dependence, with a further subtraction in the outgoing radiation gauge so that the added piece does not survive at large $r$. Second, the residual coordinate dependence, which is linear in the first-order gauge vector $xi^mu$, is cancelled by adding $psi_4^{(1)}$ times the difference between the gauge vector in the outgoing radiation gauge and the actual one. The resulting $psi_I$ obeys $hat{T}psi_I = S_I$, where $hat{T}$ is exactly the operator governing first-order perturbations and $S_I$ is a source assembled from first-order perturbations and their derivatives. The authors also lay out how to impose initial data for $psi_I$, how to extract radiated energy, linear momentum and angular momentum from it, and how to fix the two residual integration constants by an a posteriori gauge choice.

Load-bearing premise

The construction assumes that the time and azimuthal components of the first-order gauge vector can be recovered by integrating their derivatives in Eqs. (45)-(46) with consistent integrability conditions, and that the outgoing-radiation-gauge reference vectors exist so that the difference between the two is well defined — the paper states in Sec. IV.C that these components are 'assumed a priori' and tied to the existence of the outgoing radiation gauge proved in Ref. [32].

Editorial extensions

If this is right

A second-order radiation field can be computed without committing to a coordinate system, so its waveform and energy flux can be compared directly with the covariant $psi_4$ produced by full numerical integrations of Einstein's equations.
Because the wave operator is unchanged from first order, the (2+1)-dimensional time-domain Teukolsky evolution code already used for first-order perturbations can be extended to second order by adding a source term built from the first-order solution.
The ratio of second-order to first-order radiated energy supplies the otherwise missing error estimate for the close-limit approximation, giving a systematic measure of how small the perturbative parameter must be for the approximation to hold.
The formalism reduces to the nonrotating case and is intended to reproduce the known second-order Schwarzschild close-limit results, while also giving access to higher multipoles such as $ell = 4$.
Radiated energy, linear momentum and angular momentum at future null infinity can all be computed from $psi_I$ using the standard Bondi and Landau-Lifshitz expressions, since $psi_I$ equals $psi_4^{(2)}$ in an asymptotically flat gauge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction amounts to a machine that converts any first-order gauge into an asymptotically flat one, which suggests that the outgoing radiation gauge may be the most economical setting for the actual numerical evolution, rather than evolving in an arbitrary gauge and correcting afterwards.
The method is presented for Kerr but the authors note it should carry over to any Petrov type D (or even type II) background, so the same template might be reused for second-order perturbations of nonrotating neutron stars or of cosmological backgrounds.
The non-uniqueness of the quadratic correction — one can always add products of first-order invariants — means that different implementations will produce waveforms differing at high order in $1/r$; a useful cross-check would be to verify that two such choices give the same radiated energy at infinity.
Because the correction requires integrating $xi^t$ and $xi^phi$ from their derivatives, the resulting waveform is nonlocal in the first-order data even though the evolution equation is local; this asymmetry between the equation and the observable is worth watching in any implementation.

Original abstract

We investigate higher than the first order gravitational perturbations in the Newman-Penrose formalism. Equations for the Weyl scalar $\psi_4,$ representing outgoing gravitational radiation, can be uncoupled into a single wave equation to any perturbative order. For second order perturbations about a Kerr black hole, we prove the existence of a first and second order gauge (coordinates) and tetrad invariant waveform, $\psi_I$, by explicit construction. This waveform is formed by the second order piece of $\psi_4$ plus a term, quadratic in first order perturbations, chosen to make $\psi_I$ totally invariant and to have the appropriate behavior in an asymptotically flat gauge. $\psi_I$ fulfills a single wave equation of the form ${\cal T}\psi_I=S,$ where ${\cal T}$ is the same wave operator as for first order perturbations and $S$ is a source term build up out of (known to this level) first order perturbations. We discuss the issues of imposition of initial data to this equation, computation of the energy and momentum radiated and wave extraction for direct comparison with full numerical approaches to solve Einstein equations.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We investigate higher than the first order gravitational perturbations in the Newman-Penrose formalism. Equations for the Weyl scalar $\psi_4,$ representing outgoing gravitational radiation, can be uncoupled into a single wave equation to any perturbative order. For second order perturbations about a Kerr black hole, we prove the existence of a first and second order gauge (coordinates) and tetrad invariant waveform, $\psi_I$, by explicit construction. This waveform is formed by the second order piece of $\psi_4$ plus a term, quadratic in first order perturbations, chosen to make $\psi_I$ totally invariant and to have the appropriate behavior in an asymptotically flat gauge. $\psi_I$ fulfills a single wave equation of the form ${\cal T}\psi_I=S,$ where ${\cal T}$ is the same wave operator as for first order perturbations and $S$ is a source term build up out of (known to this level) first order perturbations. We discuss the issues of imposition of initial data to this equation, computation of the energy and momentum radiated and wave extraction for direct comparison with full numerical approaches to solve Einstein equations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate higher than the first order gravitational perturbations in the Newman-Penrose formalism.	contribution 0.82	0.90	0.03	asserted 0.88
s1	Equations for the Weyl scalar $\psi_4,$ representing outgoing gravitational radiation, can be uncoupled into a single wave equation to any perturbative order.	background 0.80	0.38	0.09	asserted 0.91
s2	For second order perturbations about a Kerr black hole, we prove the existence of a first and second order gauge (coordinates) and tetrad invariant waveform, $\psi_I$, by explicit construction.	result 0.60	0.96	0.73	asserted 1.00
s3	This waveform is formed by the second order piece of $\psi_4$ plus a term, quadratic in first order perturbations, chosen to make $\psi_I$ totally invariant and to have the appropriate behavior in an asymptotically flat gauge.	method 0.67	0.91	0.24	asserted 1.00
s4	$\psi_I$ fulfills a single wave equation of the form ${\cal T}\psi_I=S,$ where ${\cal T}$ is the same wave operator as for first order perturbations and $S$ is a source term build up out of (known to this level) first order perturbations.	result 0.83	0.88	0.32	asserted 1.00
s5	We discuss the issues of imposition of initial data to this equation, computation of the energy and momentum radiated and wave extraction for direct comparison with full numerical approaches to solve Einstein equations.	contribution 0.63	0.59	0.06	unclear 0.52

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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References

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gr-qc 2026-08 conditional novelty 6.0 of 10

A new sufficiency theorem ties black hole isospectrality to complex-linear perturbation systems, and the authors use it to prove parity isospectrality of subextremal Kerr-de Sitter black holes.
The Bondi--Sachs gauge, BMS frames, and memory in black hole perturbation theory
gr-qc 2026-06 unverdicted novelty 6.0 of 10

Introduces a gauge transformation framework for BMS frames in multiscale black hole perturbation theory on Kerr that incorporates memory effects and avoids infrared divergences.
Kerr Black Hole Ringdown in Effective Field Theory
gr-qc 2026-03 unverdicted novelty 6.0 of 10

Effective field theory yields model-independent corrections to Kerr black hole quasinormal modes that oscillate logarithmically near extremality, indicating discrete scale invariance.
Spin-up and mass-gain in hyperbolic encounters of spinning black holes
gr-qc 2025-10 unverdicted novelty 6.0 of 10

Scattering black holes gain spin and mass by absorbing emitted gravitational radiation, with spin-up up to 0.3 and mass gain up to 15% in near-threshold encounters.
Nonlinearities of Schwarzschild Black Hole Head-on Collisions
gr-qc 2025-08 unverdicted novelty 6.0 of 10

Quadratic quasi-normal mode amplitudes in Schwarzschild head-on collisions can be derived analytically via second-order perturbation theory bootstrapping.
Extracting linear and nonlinear quasinormal modes from black hole merger simulations
gr-qc 2023-10 conditional novelty 6.0 of 10

A public fitting pipeline recovers linear, retrograde and quadratic quasinormal modes from hundreds of binary black hole simulations and yields polynomial formulas for their amplitudes and phases in terms of mass rati...
Spin dependence of black hole ringdown nonlinearities
gr-qc 2023-08 accept novelty 6.0 of 10

In second-order Kerr perturbation theory the quadratic-to-linear ringdown amplitude ratio R_{220x220} is spin-dependent, dropping from about 0.16 at a/M=0.2 to about 0.08 at a/M=0.98, so it is not a universal constant.
The universal Teukolsky equations and black hole perturbations in higher-derivative gravity
gr-qc 2023-04 conditional novelty 6.0 of 10

A general Teukolsky-style framework reduces black hole perturbations in higher-derivative gravity to decoupled radial equations, applied to six-derivative gravity to compute spin-dependent quasinormal mode shifts.
Agnostic black hole spectroscopy: Quasinormal mode content of numerical relativity waveforms and limits of validity of linear perturbation theory
gr-qc 2023-02 conditional novelty 6.0 of 10

Higher quasinormal-mode overtones in binary black hole ringdown fits are overfitting artifacts, not physical modes; only the fundamental mode, the first overtone well after the peak, and spherical-spheroidal mode mixi...
Nonlinearities in Black Hole Ringdowns
gr-qc 2022-08 accept novelty 6.0 of 10

Second-order quasinormal modes are present in binary black hole ringdowns across mass ratios, with the quadratic (4,4) mode's amplitude scaling as the square of the (2,2) mode and rivaling the linear (4,4) mode.
Quasinormal-mode filters: a new approach to analyze the gravitational-wave ringdown of binary black-hole mergers
gr-qc 2022-07 conditional novelty 6.0 of 10

Two frequency-domain filters remove selected quasinormal modes from binary black-hole ringdowns, making subdominant effects visible without time-domain overfitting.
Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation
gr-qc 2022-06 conditional novelty 6.0 of 10

A decoupled master equation for gravitational perturbations of non-Kerr, non-Ricci-flat spinning black holes is derived, extending Teukolsky's formalism to modified gravity at arbitrary spin.
Stability of the fundamental quasinormal mode in time-domain observations against small perturbations
gr-qc 2022-05 accept novelty 6.0 of 10

Time-domain simulations show that the prompt ringdown of a perturbed black hole is modified only in proportion to the perturbation size, so the fundamental quasinormal mode measured by gravitational-wave detectors is ...
Black hole perturbation theory and gravitational self-force
gr-qc 2021-01 conditional novelty 6.0 of 10

The paper provides a unified review and a new multiscale framework for generating post-adiabatic gravitational waveforms from generic nonresonant orbits around Kerr black holes.
Numerical computation of second order vacuum perturbations of Kerr black holes
gr-qc 2020-10 conditional novelty 6.0 of 10

A pseudo-spectral time-domain code evolves second-order vacuum perturbations of Kerr black holes, showing that a linear m=2 wave drives second-order m=0 and m=4 waves that ring down at the expected quasinormal frequencies.
Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric
gr-qc 2020-08 conditional novelty 6.0 of 10

The authors give an explicit chain of Newman-Penrose transport equations that reconstructs the first-order Kerr metric perturbation from the Teukolsky solution alone, and demonstrate it at leading order in a large-dis...
Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms
gr-qc 2020-07 conditional novelty 6.0 of 10

A new 'partially flat' coordinate gauge, combined with reformulated compactified characteristic equations, removes a spurious logarithmic term from Cauchy-characteristic waveform extraction and yields simple asymptoti...
Kicking gravitational wave detectors with recoiling black holes
gr-qc 2019-08 conditional novelty 6.0 of 10

Equal-mass black holes spinning at 97% of the maximum, in the hangup-kick configuration, recoil up to about 4,700 km/s, and the corresponding waveforms are distinguishable with LIGO at signal-to-noise ratios near 30.
Gravitational Recoil From Accretion-Aligned Black-Hole Binaries
gr-qc 2012-01 conditional novelty 6.0 of 10

Partially aligned black-hole spins can produce merger recoils near 4000 to 5000 km/s, and accretion-aligned populations give roughly a 1-in-4000 chance of a line-of-sight kick above 2000 km/s.
Total recoil: the maximum kick from nonspinning black-hole binary inspiral
gr-qc 2006-10 accept novelty 6.0 of 10

The largest numerical-relativity parameter study to date finds a maximum gravitational recoil of 175.2 plus or minus 11 km/s for nonspinning black hole binaries at mass ratio q of about 0.36.
Disentangling Spin-induced Precession vs. Eccentricity in GW190521 and GW231123
gr-qc 2026-10 conditional novelty 5.0 of 10

Using full numerical-relativity simulations as templates, the authors infer GW190521 was both eccentric and spin-precessing at 20 Hz (e_20 about 0.21), while GW231123 was precessing but nearly circular (e_20 about 0.06).
Parameter Estimation with Targeted Eccentric Numerical-Relativity Simulations for GW200208_22 and GW190620
gr-qc 2025-07 unverdicted novelty 5.0 of 10

Bayesian parameter estimation with targeted eccentric numerical-relativity waveforms yields eccentricity estimates of e20 ≈ 0.2 for GW200208_22 and e10 ≈ 0.19 for GW190620, reinforcing the eccentric hypothesis.
The AdS Perspective on the Nonlinear Tails in Black Hole Ringdown
gr-qc 2025-06 conditional novelty 5.0 of 10

The known t^{-(2L+2)} nonlinear ringdown tail is rederived via AdS2 x S2, with a proposed but incorrectly normalized Aretakis amplitude relation.
The Fourth RIT binary black hole simulations catalog: Extension to Eccentric Orbits
gr-qc 2022-01 conditional novelty 5.0 of 10

A public catalog of 1,881 numerical relativity black hole binary simulations, 824 of them eccentric, plus new empirical exponential fits connecting peak radiation and remnant black hole properties.
The second RIT binary black hole simulations catalog and its application to gravitational waves parameter estimation
gr-qc 2019-01 conditional novelty 5.0 of 10

A public catalog of 320 black-hole-binary simulations, applied as the sole template bank to GW150914, reproduces LIGO's published mass and spin ranges for that event.
Remnant mass, spin, and recoil from spin aligned black-hole binaries
gr-qc 2014-06 conditional novelty 5.0 of 10

Aligned-spin black hole binary simulations give a maximum merger recoil of 526 plus or minus 23 km/s at mass ratio 0.62, roughly 80 km/s above earlier estimates, plus fourth-order fitting formulas for remnant mass, sp...
Calibration of Moving Puncture Simulations
gr-qc 2006-10 conditional novelty 5.0 of 10

A fourth-order moving-puncture binary black hole code is calibrated: it converges at fourth order at the resolutions used, agrees with an independent code, and its dominant error at small extraction radii is finite-di...
The Nonlinear Tails in Black Hole Ringdown: the Scattering Perspective
gr-qc 2025-07 conditional novelty 4.0 of 10

Nonlinear ringdown tails in the transverse-traceless gauge decay as t^{-(2ℓ+1)}, and this paper rederives that law from in-in scattering diagrams.
Generation and propagation of nonlinear quasi-normal modes of a Schwarzschild black hole
gr-qc 2022-08 conditional novelty 4.0 of 10

Second-order perturbation theory shows that quadratic quasi-normal modes of a Schwarzschild black hole are generated near the potential peak, and in the eikonal limit only waves generated outside the light ring propag...
The RIT binary black hole simulations catalog
gr-qc 2017-03 conditional novelty 4.0 of 10

A public catalog of 126 numerical-relativity black-hole-binary waveforms and remnant properties is released for independent gravitational-wave analysis.
Remnant of binary black-hole mergers: New simulations and peak luminosity studies
gr-qc 2016-10 accept novelty 4.0 of 10

Refitting remnant mass, spin and recoil formulas on 175 binary black hole simulations and adding a new peak luminosity formula, with residuals of about 0.1 to 0.2 percent for mass and spin and about 5 percent for reco...
Modeling the source of GW150914 with targeted numerical-relativity simulations
gr-qc 2016-07 accept novelty 4.0 of 10

Given the same black-hole masses and spins inferred for GW150914, two independent numerical-relativity codes produce waveforms that match to 99.8 percent and remnant masses and spins that agree to three significant figures.
Extraction of Gravitational Waves in Numerical Relativity
gr-qc 2016-06 accept novelty 2.0 of 10

A systematic review of the four methods used to extract gravitational waves from numerical relativity simulations, with first-principles derivations, implementation recipes, and an accuracy comparison.

How later papers used it

88 later papers cite it

Evidence: substantial. 59 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 11 of the 182 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool. 3 more papers by the same authors are listed: 3 used its method or tool.

Uses its method or tool 4 of 43 papers with this label · 3 by the same authors

Defined the derivatives following this paper.

Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric (2020) · 0.92 · paraphrased: the licence does not allow quoting

By the same authors

Performed the extraction of gravitational radiation from numerical relativity simulations using formulas (22), (23) and (24) from this paper for the energy and linear and angular momentum radiated in terms of the extracted Weyl scalar Ψ4 at the observer location sphere.

The maximum radiated energy and final spin of high speed collision of two black holes (2024) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Performed the extraction of gravitational radiation from numerical relativity simulations using formulas (22) and (23) from this paper for the energy and linear momentum radiated, and formulas from another work for angular momentum radiated, in terms of the extracted Weyl scalar Ψ4 at the observer location sphere.

Parameter Estimation with Targeted Eccentric Numerical-Relativity Simulations for GW200208_22 and GW190620 (2025) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Derived extrapolation formulas for the radiated energy and momenta calculated from the Weyl scalar ψ4 as in this paper.

Perturbative extraction of gravitational waveforms generated with Numerical Relativity (2015) (same authors) · 0.81 · paraphrased: the licence does not allow quoting

How the literature uses it

83 arXiv papers cite this in 177 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 86 sentences without a confident label

Contests it 1 paper

Our result is consistent with [ref] up to the overall minus sign due to different signatures, which corrects some errors in [ref] .

Perturbations of spinning black holes in dynamical Chern-Simons gravity I. Slow rotation equations · Perturbed NP quantities · contest 0.82

Builds on it 1 paper

We further show that our formalism can be generalized to higher order in both $ζ$ and $ϵ$, which is thus a bGR extension of the higher-order Teukolsky formalism in GR developed in [ref] .

Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation · Executive summary · 0.99

Uses its method or tool 42 papers

From the waveforms we compute the radiated energy and radiated linear and angular momentum using the formulas given in [ref] .

Evolutions of unequal mass, highly spinning black hole binaries · Results · 1.00
Thus, [equation] where $S_i$ and $S_f$ are the initial and final angular momentum before and after scattering, respectively. $J_GW$ is the angular momentum radiated away by s, which we compute as [ref] , [equation] Here, $h^+$ and $h^×$ are polarizations of the strain, and a dot denotes derivatives with respect to time.

Spin-up and mass-gain in hyperbolic encounters of spinning black holes · Extraction of observables · 0.97
The extraction of gravitational radiation from the numerical relativity simulations is performed using the formulas (22),(23),(24) from [ref] for the energy and linear and angular momentum radiated, respectively, in terms of the extracted Weyl scalar $Ψ_4$ at the observer location sphere $R_obs=113M$.

The maximum radiated energy and final spin of high speed collision of two black holes · Techniques · 1.00

Compares with it 6 papers

Here we provide a complete listing of these relations (compare also to Eq. (A4) of [ref] ):

Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric · Linearized NP spin coefficients in terms of the linearized metric · 0.86
Similarly to <ref>, we expand $Ψ_4$ as follows: [equation] This is in contrast to other works on QQNMs [ref] , which define their perturbative expansion similarly to Campanelli and Lousto [ref] , as [equation] The background and first-order perturbations are identical in both expansions.

Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime · Perturbative expansion · 0.93
In [ref] , the Teukolsky equation was successfully extended to second- and higher-order, so let us show now that these higher-order equations are very similar to what we obtained in this paper.

Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation · Connection to the second-order Teukolsky formalism in GR · 0.84

Cites it as support 2 papers

Even if we do not specialize the perturbed tetrad or gauge, $Ψ^(2)_4L$ and $Ψ^(2)_4$ contain identical GW content [ref] .

Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime · General case · 0.99
On the one hand, the inability of the linear-order Teukolsky equation to estimate the errors due to the use of a perturbative expansion makes the study of higher-order Teukolsky equations necessary [ref] .

Perturbations of spinning black holes beyond General Relativity: Modified Teukolsky equation · Connection to the second-order Teukolsky formalism in GR · 0.97

Cites it as background 18 papers

The equation for first order perturbations was originally derived by Teukolsky [ref] , and was later generalized to $n$-th order perturbations by Campanelli & Lousto [ref] .

Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric · Master equations for perturbations of a Petrov Type D spacetime · 0.87
This involves the same Teukolsky operator as in Eq. [ref] acting on the second-order curvature correction, and a complicated source $𝒮$ that depends quadratically on the linear perturbations [ref] .

Nonlinearities in Black Hole Ringdowns · Introduction · 0.89
There have been a number of investigations on this topic, starting with analyses on Schwarzschild black holes [ref] , which characterized the QQNM frequency spectrum and the sources that drive these quadratic modes, followed by generalizations to Kerr black holes [ref] .

Generation and propagation of nonlinear quasi-normal modes of a Schwarzschild black hole · Introduction · 0.92

Discussion

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