Second order perturbations of a Schwarzschild black hole

Jorge Pullin; Oscar Nicasio; Reinaldo Gleiser; Richard Price

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Second order perturbations of a Schwarzschild black hole

Reinaldo Gleiser, Oscar Nicasio, Richard Price, Jorge Pullin

General Relativity and Quantum Cosmology · Submitted 24 Oct 1995 · 10 references

Pith's verdict: A genuine foundational step: second-order Zerilli equation with a quadratic source, but the load-bearing asymptotic falloff of S_ren is asserted and deferred, not shown.

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arXiv IDgr-qc/9510049 (version 2)

Pith ID66XEBRPL, Pith's permanent record of this paper and its review

CitationJorge Pullin, Oscar Nicasio, Reinaldo Gleiser, Richard Price. "Second order perturbations of a Schwarzschild black hole." arXiv:gr-qc/9510049 (1995). https://pith.science/paper/66XEBRPL

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Editor's note

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This is the paper that set up second-order even-parity quadrupole perturbation theory for Schwarzschild, and it's worth your time if you work anywhere near black hole perturbation theory or numerical relativity benchmarking. The core structural point is right and, once you see it, almost inevitable: expanding the Einstein equations to second order gives you the same linear operator L as at first order plus a source quadratic in the first-order perturbations. The authors carry that through concretely — seven second-order equations, elimination down to a single Zerilli equation (11), an explicit renormalized source S_ren in (18), and a second-order power formula (20). That's real work, first-principles, no fitted parameters. The motivation is also honest: linearized theory has no internal error bar, and the only reliable one is to compute the next order. They say so plainly and then do it, at least in outline. Credit where due: the renormalization trick in (17), which absorbs the badly-growing pieces of chi into a redefinition so that the new source decays, is the clever move, and it's the kind of thing that makes the formalism usable rather than merely correct on paper. Now the soft spots, in proportion. The stress-test's central worry is fair: the chain (17) -> (18) -> (19) -> (20) rests on S_ren actually falling off at large r and chi_ren propagating as f(t - r*). That is stated, not proved — 'We will postpone a detailed proof to a lengthier publication.' It's a letter, so deferring algebra is normal, but this is the load-bearing step, and the naive large-r scaling of several terms in (18) is not manifestly decaying; the cancellation has to be exact. The typeset equation is hard to check as printed, which doesn't help. Second, the (1/7) d/dt(psi psidot) term in (19) is a total time derivative, and whether the combination in (20) is the physical gauge-invariant second-order power is not argued; the paper's own discussion of source non-uniqueness (eq. 12) makes that question live. Both are acknowledged gaps rather than hidden ones, which matters. The restriction to first-order l=2 source contributions is a real limitation for the general 'error measure' claim, but they justify it for the head-on case and flag it. The consistency check covering only time derivatives (footnote 7) is minor given the initial-value constraints. Who should read it: anyone building second-order perturbation codes, and anyone who wants to know what the linearized error bar actually is. It deserves a serious referee — the deferred proofs are the referee's job to demand, and the companion paper is the place to check them.

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Major objections (3)

[Eqs. (18)–(19) and the paragraph following (18)] The claim that S_ren decays asymptotically and that χ_ren propagates as a function of t − r* only is asserted, not proved; the authors acknowledge this with 'We will postpone a detailed proof to a lengthier publication.' The claim is load-bearing for (19), for the power formula (20), and for steps (iv)–(vi) of the algorithm. As typeset, several terms in (18) do not have manifestly decaying large-r behavior: the terms printed as '13r2 ψ,t ψ,rrr' and '13r2 ψ,r ψ,trr', and the term '−2r2 − M2 r3μλ ψ,t ψ,rr', need to be checked, and with ψ = O(1) at infinity (the normalization required by (16) for finite power) the cancellation must be exact. The letter should at least sketch the leading-order cancellation, or state explicitly that (19)–(20) are conditional on the deferred proof.
[Eq. (20) and the discussion around Eq. (12)] The paper acknowledges that the second-order source is gauge-dependent ('not unique'), but it does not argue that the combination appearing in (20) is invariant under the first-order gauge freedom. Since a first-order gauge transformation changes g⁽²⁾ and hence χ_ren, the physicality of the second-order correction in (20) needs an argument or an explicit check that changing the first-order gauge leaves (20) unchanged. Without such an argument, a reader cannot tell whether the ε² term in (20) is a physical correction or an artifact of the Regge-Wheeler first-order gauge choice.
[Paragraph beginning 'Our second, and more subtle, restriction'] The restriction to first-order ℓ=2 contributions only in the source means that for first-order data with ℓ≠2 components, (20) is not the complete second-order power, so the 'error measure' is a partial one. The paper acknowledges this and gives reasons (i)–(iii), but the abstract's claim to 'address the validity of many first order calculations' should be qualified accordingly, and the paper should state explicitly when the result is complete (pure ℓ=2 first-order data, as in the head-on collision from small separation).

Minor objections (8)

[Eq. (5)] The definition P2(θ) = 3(cos 2θ − 1)/2 does not equal the Legendre polynomial P2(cos θ) = (3cos²θ − 1)/2; at θ = 0 it gives 0 rather than 1. This is likely a typo, but it should be corrected because the multipole decomposition depends on it.
[Paragraph following Eq. (11)] The consistency check is described as verifying that the functions satisfy the second-order Einstein equations 'or rather, their time derivatives [7]'. Please state explicitly whether the second-order constraint equations are also satisfied, or whether footnote [7] is intended to cover this; a reader cannot tell from the present wording how complete the check is.
[Paragraph following Eq. (5)] The sentence 'through these expressions view the Regge-Wheeler perturbations as gauge invariant' is too strong. At second order the Regge-Wheeler functions are invariant under second-order gauge transformations but not under first-order ones, and the phrase should be qualified in light of the paper's own classification of 'second order gauge invariant' versus 'first and second order invariant'.
[Eq. (21) and the step list] In Eq. (21) the denominator is printed as '2r + 3m' rather than '2r + 3M'. In the step list, '(vi)' appears twice; the final item should be '(vii)'.
[Paragraph before Eq. (16)] The sentence 'radiation information is carried by the perturbations in g_θθ, g_φφ and g_θθ' lists g_θθ twice; the third component is presumably g_θφ and should be corrected.
[Introduction] The claim that there is 'only one generally reliable index of the accuracy of linearized perturbation theory' is too strong; comparison with full numerical solutions, as in [2], is also a reliable index where available. Please soften the wording.
[Paragraph following Eq. (11)] The explicit expression for S_RW is deferred ('somewhat lengthy and will be given elsewhere'). Since S_RW is the source for the paper's core equation (11), a reader would benefit from at least its leading large-r behavior or a pointer to where it appears.
[Throughout] There are several typographical errors: 'graviational' (before Eq. 16), 'funtion' (before Eq. 14), 'It seem appropriate' (after Eq. 19), and 'W e also acknowledge' (Acknowledgments).

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The paper

The paper claims

The equations for a black hole's gravitational waves, carried one order beyond the linear approximation, reduce to the same wave equation with a quadratic source.

The reading

Linear perturbation theory around a Schwarzschild black hole turns the full Einstein equations into a single wave equation for a small distortion, and that equation has been the workhorse of black hole astrophysics for decades. Its blind spot is that nothing inside linear theory says how small the distortion has to be for the linear answer to be trustworthy. This paper takes the next step in the expansion, writing the metric as a background plus $O(1)$ plus $O(2)$ terms, and shows that the second-order even-parity quadrupole equations collapse to a wave equation of exactly the same shape as the first-order one, with the only new ingredient a source term built from products of first-order quantities. The authors then argue that by redefining the second-order wave function in a suitable way, the source can be made to die off at large radius, so that the second-order radiation amplitude can be read off at infinity. If the construction holds, one can compute the leading correction to any linear black hole radiation calculation and thereby quote a real error bar instead of a guess.

What carries the argument

The load-bearing device is the renormalized second-order Zerilli function: the second-order wave function minus an explicit quadratic expression in the first-order perturbations, chosen so that the leftover source term decays at large radius instead of growing. The authors show that the second-order Zerilli equation is not unique, because the first-order gauge choice only shifts the quadratic terms, and they exploit that freedom to move the offending large-radius growth out of the source and into a redefinition of the wave function. The renormalized function is then the object whose asymptotic behaviour encodes the second-order gravitational wave amplitude, and the whole computational recipe, from initial data through numerical evolution to radiated power, is organized around it.

Extended reading notes

Core claim

The paper's central claim is that the second-order even-parity $m=0$, $l=2$ Einstein equations for a Schwarzschild background can be reduced, in the Regge-Wheeler gauge, to a single equation of the form $R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)$, and ultimately to a Zerilli equation $R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)$ of the same form as in linear theory, differing only by a source term quadratic in the first-order perturbations. Starting from seven nontrivial Einstein equations for four second-order functions, the authors use one equation to eliminate $H^{(2)}_2$, another to eliminate the time derivative of $H^{(2)}_0$, and Zerilli's substitution to reach one equation for a single function $R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)$, with $R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)$ playing the role of the Zerilli function. They report that the raw source term diverges at large radius, so they introduce a renormalized function $R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)$, whose source falls off asymptotically and which propagates as a function of $t-r_*$ alone; from it they obtain the radiated power accurate to second order, $P = (3/10)[R = (1-2M/r)K^{(2)}_{,r} + (1/7)[(1-2M/r)/(2r+3M)]K^{(1)}K^{(1)}_{,t} + O(1/r^2)]^2$ in the notation of the paper's eq. (20). The claimed payoff is a computable, order-by-order error measure on linear quadrupole radiation from black holes.

Load-bearing premise

The construction rests on the claim that the renormalized second-order function has a source that decays at large radius and travels purely as a function of $t-r_*$, and the authors explicitly defer the proof of this to a longer paper.

Editorial extensions

If this is right

Any linear black hole radiation calculation can in principle be followed by a second-order run whose output bounds the error of the linear answer, turning a guess about the expansion parameter into a computed number.
The radiated power to second order, given in eq. (20) as a square of the first-order time derivative plus a second-order correction, gives a concrete formula for the leading nonlinear correction to quadrupole emission.
The head-on collision of two black holes from small initial separation becomes a direct test case, since in that setting the first-order perturbations carry only $\ell=2$ and the restricted source used here is the complete source.
The same reduction is expected to go through for odd-parity perturbations, and the authors state that case is simpler than the even-parity one worked out here.
The formalism supplies a benchmark that full numerical relativity simulations of non-symmetric encounters could be checked against.

Reading between the lines

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

If the second-order correction is routinely computable, the practical boundary of perturbation theory could be mapped empirically: one would run the expansion until the second-order term reaches a fixed fraction of the first, rather than assuming a threshold on the expansion parameter.
The observation that strong nonlinearity near the horizon is absorbed by the hole while the outgoing wave stays nearly linear suggests the second-order correction will be largest in the near-zone and small in the radiation zone, which would explain why linear theory has outperformed naive expectations in head-on collisions.
The gauge dependence of the second-order wave equation implies that any second-order "correction" quoted without a stated gauge is ambiguous, so a useful error measure should be built from the asymptotically flat radiation amplitude rather than from intermediate metric functions.
A natural extension is to relax the restriction to first-order $\ell=2$ sources and include the full infinite sum of first-order multipole products, which would make the formalism applicable to equal-mass and off-axis encounters where higher multipoles are excited.

Original abstract

We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first order perturbations. This provides a formalism to address the validity of many first order calculations of interest in astrophysics.

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The abstract, sentence by sentence

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

We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order. The Einstein equations can be reduced to a single linear wave equation with a potential and a source term. The source term is quadratic in terms of the first order perturbations. This provides a formalism to address the validity of many first order calculations of interest in astrophysics.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the even-parity $\ell=2$ perturbations of a Schwarzschild black hole to second order.	contribution 1.00	0.93	0.07	asserted 0.94
s1	The Einstein equations can be reduced to a single linear wave equation with a potential and a source term.	method 0.83	0.59	0.09	asserted 0.98
s2	The source term is quadratic in terms of the first order perturbations.	method 0.69	0.59	0.08	asserted 1.00
s3	This provides a formalism to address the validity of many first order calculations of interest in astrophysics.	interpretation 0.96	0.40	0.32	asserted 0.87

What kind of paper the classifier reports this is: theoretical result 0.63 · evidence from proof or derivation 1.00.

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References

Works cited (10)

[1]

Davis M, Ruﬃni R , Press W H and Price R H 1971, Phys. Rev. Lett. 27 1466; Petrich L, Shapiro S, and Wasserman I 1985, Astrophys. J. Suppl. Ser. 58 297

View on Pith 1971
[2]

Price R H and Pullin J 1994, Phys. Rev. Lett. 72 3297

View on Pith 1994
[3]

Suen W M, private communication

View on Pith
[4]

Abrahams A and Price R 1996, Phys. Rev. D53 1963

View on Pith 1996
[5]

Regge T and Wheeler J A 1957, Phys. Rev. 108 1063

View on Pith 1957
[6]

Zerilli F 1970, Phys. Rev. Lett. 24 737

View on Pith 1970
[7]

This is because we only have expressions for K (2),t and H (2)0 ,t. Since we are solving the vacuum Einstein equations, this on ly implies that we should also give appropriate initial data fo r these functions. In practical calculations this is automa tically assured when we impose that the data satisﬁes the initial val ue constraints of general relativity

View on Pith
[8]

Cunningham C, Price R, Moncrief V 1979, Ap. J. 230 870

View on Pith 1979
[9]

Cutler C, Apostolatos T, Poisson E 1993, Phys. Rev. D47 1511; Poisson E, Sasaki M 1995, Phys. Rev. 51 5753 and references therein

View on Pith 1993
[10]

P. Anninos, R. Price, J. Pullin, E. Seidel, W.-M. Suen Ph ys. Rev. D52, 4462 (1995). 6

View on Pith 1995

Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring
gr-qc 2025-09 conditional novelty 7.0 of 10

Second-order gravitational perturbations on plane waves are solved with a GHP master equation and tensor harmonics, yielding quadratic quasinormal mode ratios and selection rules.
Black Hole Ringdown Nonlinearities in the Large-D Limit
gr-qc 2026-06 unverdicted novelty 6.0 of 10

In the large-D limit, analytic third-order nonlinear corrections to quasinormal modes improve ringdown modeling accuracy by several orders of magnitude for head-on black hole collisions.
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.
Non-linear Black Hole Ringdowns: an Analytical Approach
gr-qc 2023-08 conditional novelty 6.0 of 10

Second-order black hole ringdown modes can be computed analytically from a light-ring-localized quadratic source, and the ratio of second-order to squared first-order amplitude is nearly independent of initial conditions.
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...

Show 7 more citing papers

Second Order Quasi-Normal Mode of the Schwarzschild Black Hole
gr-qc 2007-08 conditional novelty 6.0 of 10

Second-order quasi-normal modes of a Schwarzschild black hole oscillate at twice the first-order frequency with an amplitude about 10% of the first-order ringdown.
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 impact of initial conditions on quasi-normal modes
gr-qc 2024-12 conditional novelty 5.0 of 10

Using an exactly solvable delta-function potential, the authors show linear quasi-normal mode amplitudes are generically time-dependent and derive stabilization times that can greatly exceed the usual 10 solar mass ru...
Nonlinear Quasi-Normal Modes: Uniform Approximation
hep-th 2023-09 conditional novelty 5.0 of 10

A uniform (comparison-equation) approximation yields linear quasinormal frequencies that remain accurate at high overtone number and a closed-form ratio between nonlinear and linear ringdown amplitudes.
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.
Can We Detect Deviations from Einstein's Gravity in Black Hole Ringdowns?
gr-qc 2024-11 conditional novelty 4.0 of 10

Higher-derivative corrections to black hole ringdown spectra and quadratic modes are either absent or suppressed below detectability once causality is enforced, leaving Einstein's gravity as the sole observable description.
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...

How the literature uses it

38 arXiv papers cite this in 58 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 · 33 sentences without a confident label

Uses its method or tool 1 paper

Following [ref] , it is useful to take advantage of this freedom, to modify the source terms so they have the desired fall-off at large distances and close to the horizon, namely: [equation] Since the linear QNM solutions $^e,oΨ^(1)$ behave as $exp{-i ω(t±r_*)}$ with constant amplitude in the $r_* →±∞$ limit, the source terms chosen thus have an analogous scaling with radius as the potentials in Eqs. ([ref])-([ref])[Keeping $t±r_*$ fixed.].

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

Compares with it 1 paper

Our solution is very similar to the one adopted by Nakano and Ioka [ref] for the specific case of the dominant QQNM, while it differs from the one chosen by [ref] .

Amplitudes and Polarizations of Quadratic Quasi-Normal Modes for a Schwarzschild Black Hole · Introduction · 0.98

Cites it as support 1 paper

Indeed, one can check that, were that term not present, the RWZ variables would exhibit more divergent behaviors at second order compared to the first-order QNM [ref] .

Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole · Introduction · 0.97

Cites it as background 22 papers

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
At second order in the GW amplitude one finds similar equations for the second-order perturbations $h^(2)$, now sourced by first-order quantities [ref] .

Nonlinear effects in black hole ringdown · Introduction · 0.83
Focusing on eigenfunctions at definite frequency $ω̅$ and angular momentum $(ℓ̅,m̅)$, and denoting by $Ψ̅(r_*)$ the dimensionless amplitude of the propagating degree of freedom at linear order (either in the even or the odd sector), we have [equation] where $r_*$ is the tortoise coordinate $r_* = r + 2 G M log( r/(2GM)-1 )$ (while $r$ is the standard radial coordinate), and $V(r)$ is the Regge-Wheeler or Zerilli potential, given by [equation] At second order in perturbation theory, it turns out that the second-order perturbations of the metric can also be encoded into one parity-odd and one parity-even function which we will denote by $Ψ$ (also chosen to be dimensionless in the following) [ref] .

Nonlinear Quasi-Normal Modes: Uniform Approximation · Nonlinear perturbations of a black hole · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Davis M, Ruﬃni R , Press W H and Price R H 1971, Phys. Rev. Lett. 27 1466; Petrich L, Shapiro S, and Wasserman I 1985, Astrophys. J. Suppl. Ser. 58 297

View on Pith 1971

[2] [2]

Price R H and Pullin J 1994, Phys. Rev. Lett. 72 3297

View on Pith 1994

[3] [3]

Suen W M, private communication

View on Pith

[4] [4]

Abrahams A and Price R 1996, Phys. Rev. D53 1963

View on Pith 1996

[5] [5]

Regge T and Wheeler J A 1957, Phys. Rev. 108 1063

View on Pith 1957

[6] [6]

Zerilli F 1970, Phys. Rev. Lett. 24 737

View on Pith 1970

[7] [7]

This is because we only have expressions for K (2),t and H (2)0 ,t. Since we are solving the vacuum Einstein equations, this on ly implies that we should also give appropriate initial data fo r these functions. In practical calculations this is automa tically assured when we impose that the data satisﬁes the initial val ue constraints of general relativity

View on Pith

[8] [8]

Cunningham C, Price R, Moncrief V 1979, Ap. J. 230 870

View on Pith 1979

[9] [9]

Cutler C, Apostolatos T, Poisson E 1993, Phys. Rev. D47 1511; Poisson E, Sasaki M 1995, Phys. Rev. 51 5753 and references therein

View on Pith 1993

[10] [10]

P. Anninos, R. Price, J. Pullin, E. Seidel, W.-M. Suen Ph ys. Rev. D52, 4462 (1995). 6

View on Pith 1995