REVIEW
Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole/slow-motion approximation
General Relativity and Quantum Cosmology · Submitted 13 Jul 2004 · 64 references
Pith's verdict: A careful, genuinely useful methods paper — the time-domain flux formulas are the deliverable, and the abstract oversells the quadrupole interpretation that the Note added already walks back.
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Referee report
The paper
The paper claims
The paper derives time-domain prescriptions for how much mass and angular momentum a black hole absorbs from gravitational radiation, and then solves them analytically for small or slowly moving holes, where the absorption is carried by tid
The reading
What carries the argument
The load-bearing device is the Hawking-Hartle description of a perturbed event horizon: the horizon's intrinsic two-metric $gamma_{AB}$ evolves according to its expansion $Theta$ and shear $sigma_{AB}$, the shear is driven by the horizon Weyl curvature through $(partial_v - kappa)sigma_{AB} = -C_{AB}$, and the expansion is driven by the shear squared. Averaging the expansion over times long compared with $kappa^{-1}$ turns the area growth into a flux, and the averaged first law $(kappa/8pi)langle dot A rangle = langle dot M rangle - Omega_H langle dot J rangle$ converts that single number into mass and angular-momentum rates. The second device is the rewriting of $C_{AB}$ in terms of the Teukolsky scalar $Psi$, which is what makes the whole thing computable from a standard perturbation code. The third is the observation that in the small-hole/slow-motion regime the perturbation equations can be integrated by parts into local expressions, so that the asymptotic tidal field $E_{ab}(v)$, $B_{ab}(v)$ can be propagated to the horizon analytically.
Extended reading notes
Core claim
The central claim is a pair of flux formulae. For a general Kerr hole, one solves the Teukolsky equation for the horizon Weyl scalar $Psi$, forms the two integrated curvatures $Phi^m_+$ and $Phi^m_-$ (one an exponentially weighted integral into the future, one a plain integral into the past), and inserts them into Eqs. (5.27)-(5.29) to get $langle dot M rangle$, $langle dot J rangle$, and $langle dot A rangle$. For a Schwarzschild hole, one solves the Regge-Wheeler and Zerilli equations and inserts $Psi^{lm}_{RW}(v,r_+)$ and $dot Psi^{lm}_{ZM}(v,r_+)$ into Eqs. (7.8)-(7.9); apart from $u to v$ these are the same expressions that give the energy and angular-momentum carried to future null infinity. The author then evaluates these in a small-hole/slow-motion expansion, where $M/R ll 1$ with $R$ the external radius of curvature, and obtains closed forms: for Schwarzschild, $langle dot M rangle = (16M^6/45) langle dot E_{alpha beta} dot E^{alpha beta} + dot B_{alpha beta} dot B^{alpha beta} rangle$ and a corresponding $langle dot J rangle$ scaling as $M^6/R^5$; for Kerr, $langle dot J rangle = O(M^5/R^4)$ with an explicit dependence on the rotation parameter $chi$ and on three invariants of the tidal field, while $langle dot M rangle$ is left at $O(M^5/R^5)$ because the leading-order calculation cannot pin it down. The physical reading offered is that the hole acquires induced mass and current quadrupole moments — proportional to $dot E_{alpha beta}$ and $dot B_{alpha beta}$ for Schwarzschild, to $E_{alpha beta}$ and $B_{alpha beta}$ themselves for Kerr — and that the absorption is the work done by the tide against those moments.
Load-bearing premise
The derivation converts a single number, the averaged growth of horizon area, into separate mass and angular-momentum rates by assuming that each mode carries energy and angular momentum in the ratio $omega/m$ — a relation the paper states holds for matter fields and then explicitly assumes carries over to gravitational perturbations, following Teukolsky and Press.
Editorial extensions
If this is right
- A time-domain numerical integration of the Teukolsky or Regge-Wheeler/Zerilli equations can be post-processed into absorbed mass and angular momentum by evaluating the perturbation variables at $r = r_+$ and performing two one-dimensional integrals in $v$, with no frequency decomposition at all.
- For a small or slowly moving hole, the absorbed energy scales as $M^6/R^6$ and the absorbed angular momentum as $M^6/R^5$ in the non-rotating case, so the effect is sharply suppressed for small holes but grows quickly with mass.
- In a slow-motion binary the results reduce to $langle dot M rangle = (32/5) eta^2 [M/(M+M_{ext})]^4 V^{18}$ and a matching $langle dot J rangle$, which the author notes agree with earlier work by Alvi and by Poisson and Sasaki in their respective limits.
- For a rotating hole the leading behaviour is $langle dot J rangle = O(M^5/R^4)$, and in rigid rotation $langle dot M rangle = -(8/5) eta^2 [M/(M+M_{ext})]^3 chi(1+3chi^2) V^{15}$, so a spinning hole in an identical environment absorbs more than a non-spinning one.
- The classic spin-down of a rotating hole in a static tidal field, $dot J = -(2/5) E^2 M^5 chi sin^2alpha [1 - (3/4)(1-5sin^2alpha)chi^2]$, drops out of the general Kerr formula as a special case.
Reading between the lines
- If the time-domain formulae hold, the same machinery should apply to a horizon-absorbed flux computed inside a full numerical-relativity evolution, giving a horizon-side energy budget that can be checked against the wave-zone flux without ever taking a Fourier transform.
- The sharp contrast between the $M^6/R^6$ (Schwarzschild) and $M^5/R^5$ (Kerr) scalings suggests the small-hole expansion is not uniformly valid as spin is lowered: the condition $M/R ll chi$ degrades as $chi to 0$, so any numerical scheme spanning both regimes needs to interpolate between two different orderings rather than take a limit.
- The paper's remark that a non-rotating hole's induced quadrupole tracks the time derivative of the tide, while a rotating hole's tracks the tide itself, looks like a statement about which timescale sets the response — the tide's own variation period versus the hole's rotational period — and should generalise to any tidal response problem with two competing clocks.
- The undetermined $O(M^5/R^5)$ piece of $langle dot M rangle$ for Kerr is the natural next target; getting it requires the subleading terms in the near-horizon expansion of the radial Teukolsky solution, which is a finite but unglamorous calculation.
Original abstract
The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. Two such prescriptions are presented. The first is based on the Teukolsky equation and it applies to general (rotating) black holes. The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. For a rotating black hole they are proportional to the tidal fields themselves.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. Two such prescriptions are presented. The first is based on the Teukolsky equation and it applies to general (rotating) black holes. The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. For a rotating black hole they are proportional to the tidal fields themselves.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. | contribution 0.98 | 0.64 | 0.08 | asserted 0.92 |
| s1 | These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. | method 0.86 | 0.81 | 0.04 | asserted 1.00 |
| s2 | Two such prescriptions are presented. | contribution 0.97 | 0.88 | 0.04 | asserted 0.99 |
| s3 | The first is based on the Teukolsky equation and it applies to general (rotating) black holes. | method 0.64 | 0.70 | 0.05 | asserted 1.00 |
| s4 | The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. | method 0.63 | 0.67 | 0.04 | asserted 1.00 |
| s5 | The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. | contribution 0.97 | 0.77 | 0.05 | asserted 0.96 |
| s6 | In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. | result 0.50 | 0.82 | 0.23 | asserted 0.88 |
| s7 | The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. | interpretation 1.00 | 0.66 | 0.26 | asserted 0.88 |
| s8 | For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. | result 0.79 | 0.74 | 0.09 | asserted 1.00 |
| s9 | For a rotating black hole they are proportional to the tidal fields themselves. | result 0.77 | 0.72 | 0.10 | asserted 1.00 |
What was examined, and what was not
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References
How later papers used it
75 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 1 of 5 papers with this label · 1 by the same authors
By the same authors
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Applied this paper's formalism to a small black hole on a circular equatorial orbit of a large Kerr black hole to calculate the rates at which the tidal interaction changes its mass and angular momentum.
Tidal interaction of a small black hole in the field of a large Kerr black hole (2009) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Compares with it 1 of 15 papers with this label
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Found that its result, setting the imaginary part of the frequency to zero, matches this paper's pure mode result up to a coordinate choice.
Nonlinear effect of absorption on the ringdown of a spinning black hole (2024) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
69 arXiv papers cite this in 186 sentences
Papers citing it · 119 sentences without a confident label
Contests it 1 paper
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Finally, this prescription implies vanishing tidally-induced multipole moments for a Kerr black hole, which appears to be in contradiction with the tidal torquing calculation of [ref] ; see Sec. [ref] above.
Tidal Love Numbers of Kerr Black Holes · An alternative tidal/response split · contest 0.71
Builds on it 4 papers
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The calculations we present in this paper mainly build upon the line of work of Poisson and collaborators, who in a series of works studied horizon absorption by tying it to the tidal interaction of a bh with its gravitational environment [ref] .
Spin the black circle: horizon absorption on non-circular, planar binary black hole dynamics · Introduction · 0.83
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The results displayed in Eqs. ([ref]) and ([ref]) generalize those presented in Secs. VIII I and IX G of Ref. [ref] , which apply when the large black hole is nonrotating ($Ξ= 0$).
Tidal interaction of a small black hole in the field of a large Kerr black hole · Introduction and summary · 0.96
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In the relativistic regime this expression can be generalized as follows [ref] :
On the Vanishing of Love Numbers for Kerr Black Holes · Definition · 0.86
Uses its method or tool 4 papers
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We take the small black hole to move on a circular orbit in the equatorial plane of a large Kerr black hole, and we apply the formalism developed in Ref. [ref] to calculate the rates at which the tidal interaction changes the mass and angular momentum of the small black hole.
Tidal interaction of a small black hole in the field of a large Kerr black hole · Introduction and summary · 0.99
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In this section, we obtain a perturbative description of the event horizon's radial profile in terms of the metric perturbation, following the method of Refs. [ref] .
Deformed Schwarzschild horizons in second-order perturbation theory: mass, geometry, and teleology · Event horizon · 0.98
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With the solution of the axial and polar perturbation equation, we can calculate the energy flux at infinity using the following relation [ref] [equation] where [equation] and $n=l(l+1)/2-1$.
Detecting dark matter with extreme mass-ratio inspirals · Dark matter profile and perturbation equations · 0.97
Uses its data or values 1 paper
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The induced quadrupole of a tidally distorted Schwarzschild black hole contains the contribution $3245G^5M^6Ė_ab$ [ref] , which, in the normalization conventions of Eqs. [ref] and [ref], identifies the tidal-heating coefficient as [equation] strictly positive, as Eq. [ref] demands.
A Bound on the Dynamical Love Number · Low-frequency consistency checks for black hole responses · 0.81
Compares with it 15 papers
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This expression correctly reproduces known results in the literature for the few low-order dissipation numbers (note our different choice of normalization in ([ref])) [ref] .
Tidal Deformation and Dissipation of Rotating Black Holes · Tidal effects of the Schwarzschild black hole · 0.98
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Equations ([ref]), ([ref]), and ([ref]) differ from the original expressions displayed in Ref. [ref] by terms involving the gravito-magnetic tidal moments $_ab(v)$.
Tidal interaction of black holes and Newtonian viscous bodies · Flux formulae for black holes · 0.93
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This result matches others in literature [Eqs. (5.14) and (5.22) in Ref. [ref] , as well as [ref] and [ref] ] for a normal mode (i.e. $ℑω= 0$) up to coordinate and tetrad changes.[ Note that our result disagrees with Ref. [ref] slightly for the complex mode case.
Nonlinear effect of absorption on the ringdown of a spinning black hole · Flux for a frequency-space perturbation · 0.97
Cites it as support 1 paper
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This observed change is smaller than the numerical errors in the simulations, and analytical estimates [ref] suggest that the increase in irreducible mass due to tidal heating should be even smaller in this case.
Horizon Surface Gravity in Corotating Black Hole Binaries · Innermost circular orbit · 0.96
Cites it as background 22 papers
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This calculation has been discussed in detail in Refs. [ref] , and some important technical results were previously obtained in Refs. [ref] .
On the Vanishing of Love Numbers for Kerr Black Holes · Spin-2 Response Coefficients · 0.97
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Until recently, the calculation of the horizon fluxes for generic, slowly varying tidal environments had only been carried out to leading order in an expansion in inverse powers of the radius of curvature of the external universe [ref] .
Improved next-to-leading order tidal heating and torquing of a Kerr black hole · Introduction · 0.88
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(See Refs. [ref] for details.)
Tidal interaction of black holes and Newtonian viscous bodies · Flux formulae for black holes · 0.82
Discussion
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