Remnant Kick Accumulation and Frame Choice

A black-hole kick is the remnant’s velocity relative to a declared asymptotic rest frame, inferred from the linear momentum carried by radiation. The flux is well defined once a Bondi frame and waveform normalization are fixed. Its integral needs an initial momentum; a boost changes the quoted kick, while a supertranslation changes the cuts used to compare the time-dependent history. Separating these ingredients prevents coordinate drift from becoming a physical recoil.

Helpful background. Radiative and horizon charges build on Black-Hole Geometry and Mechanics; waveform modes come from Black-Hole Perturbation Theory; Bondi frames and gauge transformations are developed in Formulations of General Relativity; extraction uncertainties belong to Computational Relativity; and the source asymmetries are organized in Gravitational Radiation and Waveforms.

Momentum flux at null infinity

Choose a Bondi system $(u,r,\theta,\phi)$. If $h_+$ and $h_\times$ are dimensionless strains measured at large radius, define the asymptotic strain coefficient and news by

$$h(u,\Omega)=\lim_{r\to\infty}r(h_+-ih_\times), \qquad N=\partial_u h.$$

Thus $h$ has dimensions of length and $N$ is dimensionless in geometric units $G=c=1$. With this normalization, the spatial three-momentum carried by radiation obeys

$$\frac{dP^i_{\rm rad}}{du} =\frac{1}{16\pi}\int_{S^2} n^i\lvert N(u,\Omega)\rvert^2\,d\Omega ,$$

with $n^i$ the outward unit radial vector. Momentum conservation gives $dP^i_{\rm source}/du=-dP^i_{\rm rad}/du$. The angular integral vanishes for radiation patterns with the relevant reflection symmetry; recoil measures anisotropy of the angular energy flux, not simply total luminosity. In a spin-weighted-spherical-harmonic expansion, this includes interference between neighboring modes and diagonal mode powers. In particular, the $z$-momentum flux contains terms proportional to $2m\lvert N_{\ell m}\rvert^2/[\ell(\ell+1)]$, as well as cross terms with neighboring $\ell$ Ruiz et al. 2008, Eqs. (3.14)–(3.19), p. 5, PDF.

For a useful check, a single normalized $_{-2}Y_{22}$ news mode has angular power proportional to $(1+\cos\theta)^4$. Its momentum-to-energy flux ratio is

$$\frac{dP^z_{\rm rad}/du}{dE_{\rm rad}/du} =\frac{\int_{-1}^{1}z(1+z)^4\,dz} {\int_{-1}^{1}(1+z)^4\,dz} =\frac23.$$

Thus one helicity-weighted mode can carry net momentum. Equal powers in the reflection-related $m=\pm2$ modes cancel this contribution, as expected for equatorially symmetric radiation.

Integrating between retarded times $u_0$ and $u$ gives

$$P^i_{\rm rem}(u)=P^i_0-\int_{u_0}^{u}\frac{dP^i_{\rm rad}}{du'}\,du'.$$

The constant $P^i_0$ is the source’s Bondi momentum on the first recorded cut. If it is obtained from an earlier ADM momentum, the previously emitted momentum must be subtracted: $P^i_0=P^i_{\rm ADM}-P^i_{\rm rad,early}$ under the matching assumptions. Starting a finite waveform after junk radiation does not determine this constant. One either includes sufficiently early data, fixes it with an independent ADM/Bondi charge, or estimates it using a justified early inspiral model. In geometrized units the dimensionless velocity is $v^i/c=P^i_{\rm rem}/E_{\rm rem}$; restoring SI units gives $v^i=c^2p^i_{\rm rem}/E_{\rm rem}$. Thus $p^i_{\rm rem}$ is measured in $\mathrm{kg\,m\,s^{-1}}$, $E_{\rm rem}$ in joules, and the physical kick is commonly reported in $\mathrm{km\,s^{-1}}$ in the selected late-time inertial frame. Before the system has settled to a single remnant, this ratio describes the Bondi momentum of the remaining system, not the velocity of a uniquely isolated horizon.

As a normalization check, the energy flux is the same integral without $n^i$. Since $|\mathbf n|=1$, the triangle inequality gives $|d\mathbf P_{\rm rad}/du|\leq dE_{\rm rad}/du$. Radiation cannot carry more momentum than energy in units with $c=1$; violating this inequality signals inconsistent units, extraction, or mode normalization Ruiz et al. 2008, Eqs. (2.8) and (2.11).

Frame choice and center-of-mass drift

A rotation merely rotates the kick components, while a boost changes the remnant rest frame itself. A supertranslation changes the cut on which waveform modes are compared and can mix time and angular dependence. Consequently, mode-by-mode fluxes from two simulations should be compared only after mapping both data sets to a common Bondi frame. A pure supertranslation is not an arbitrary extra boost of a stationary final remnant: when the entire radiative history and corresponding nonradiative endpoint cuts are included, shifting the retarded-time origin separately at each angle does not change the integrated outgoing four-momentum. Finite windows and intermediate cuts need explicit matching. Boyle develops the transformation of asymptotic waveform data under the full Bondi–Metzner–Sachs group Boyle 2016, §§ II–IV and Eqs. (20)–(30).

Coordinate center-of-mass drift in a Cauchy evolution is a separate issue. It can contaminate finite-radius extraction and the inferred mode decomposition, but puncture velocity is not the remnant kick. A useful closure test compares three quantities in one frame: initial ADM momentum, integrated radiative momentum, and an independent late-time remnant momentum or boost estimate. The vector residual, not agreement of kick magnitudes alone, tests conservation.

Antikicks and uncertainty

The accumulated velocity can peak and then decrease because late radiation carries momentum opposite to the earlier flux. This “antikick” is an interference effect during relaxation, not a force pulling the hole backward. Its size depends on plunge geometry and ringdown phasing. Numerical surveys of generic spins exhibit both large kicks and substantial late-time cancellation Lousto et al. 2010, §§ 3–5.

Report truncation and extraction errors, omitted-mode error, the uncertainty in $P^i_0$, and frame-fixing uncertainty separately before combining them. Agreement after changing extrapolation order or extraction method is especially important for small kicks, where residual drift may be comparable to the signal.

Check your understanding

A simulation begins recording at $u_0$ after an unmeasured burst. It finds $\int_{u_0}^{\infty}dP^x_{\rm rad}=3\times10^{-4}M$, while an independent early-time charge gives $P^x_{\rm rem}(u_0)=10^{-4}M$. Find the final $x$-momentum and explain why setting the integration constant to zero would be wrong.

Solution

Conservation gives $P^x_{\rm rem}(\infty)=10^{-4}M-3\times10^{-4}M=-2\times10^{-4}M$. Setting the constant to zero would instead give $-3\times10^{-4}M$ and would silently discard the physical momentum already present at the first recorded cut. The sign reversal alone does not establish a conventional antikick: that requires the time-resolved velocity magnitude to decrease from an earlier peak during relaxation. Final momentum also does not determine a speed until the final energy is supplied.

References

• Boyle, Michael. “Transformations of Asymptotic Gravitational-Wave Data.” Physical Review D 93 (2016): 084031. doi:10.1103/PhysRevD.93.084031.
• Lousto, Carlos O., Manuela Campanelli, Yosef Zlochower, and Hiroyuki Nakano. “Remnant Masses, Spins and Recoils from the Merger of Generic Black-Hole Binaries.” Classical and Quantum Gravity 27 (2010): 114006. doi:10.1088/0264-9381/27/11/114006.
• Ruiz, Milton, Miguel Alcubierre, Darío Núñez, and Ryoji Takahashi. “Multipole Expansions for Energy and Momenta Carried by Gravitational Waves.” General Relativity and Gravitation 40 (2008): 2467–2480. doi:10.1007/s10714-007-0570-8.