Final Mass and Spin

The final mass and spin are best determined redundantly. Radiative balance subtracts energy and angular momentum reaching null infinity from the initial ADM charges; quasi-local geometry measures area and angular momentum on the remnant horizon; late-time waveform frequencies provide a third Kerr-based estimate. Agreement among these methods is a stringent test because their dominant systematic errors are different.

Helpful background. Horizon charges and the Kerr relation are developed in Black-Hole Geometry and Mechanics; spectral inference uses Black-Hole Perturbation Theory; ADM and Bondi charges build on Formulations of General Relativity; numerical error control belongs to Computational Relativity; and flux normalization follows Gravitational Radiation and Waveforms.

Three determinations of the remnant

For an asymptotically flat vacuum binary in a fixed inertial frame, use $G=c=1$, so masses and energies are represented by lengths and angular momenta by lengths squared. Balance gives

$$E_f^{\rm bal}=E_{\rm ADM}-E_{\rm rad}, \qquad \mathbf P_f^{\rm bal}=\mathbf P_{\rm ADM}-\mathbf P_{\rm rad},$$

and hence the Lorentz-invariant remnant rest mass is

$$M_f^{\rm bal} =\sqrt{(E_f^{\rm bal})^2-|\mathbf P_f^{\rm bal}|^2}.$$

The distinction matters conceptually even when the kick is small: $E_f=M_f/\sqrt{1-v_f^2}$, so replacing $M_f$ by the energy introduces a relative error $v_f^2/2+O(v_f^4)$. The angular-momentum difference $\mathbf J_{\rm ADM}-\mathbf J_{\rm rad}$ is the final total angular momentum in the selected frame. Comparison with intrinsic horizon spin requires a consistent final center, Bondi-cut prescription, and rest-frame transformation; a displaced moving remnant also has an orbital contribution about the original origin.

The radiative integrals require all resolved modes and a treatment of early junk radiation. On a sufficiently settled horizon, let $A$ be the area, $M_{\rm irr}=\sqrt{A/(16\pi)}$, and $J_H$ a specified quasi-local angular momentum. Assuming the exterior has approached Kerr, the Christodoulou relation gives

$$M_f^{H}=\sqrt{M_{\rm irr}^2+\frac{J_H^2}{4M_{\rm irr}^2}}, \qquad \chi_f=\frac{J_H}{(M_f^{H})^2}.$$

In representative SI dimensions, $m_{\rm irr}=(c^2/G)\sqrt{A/(16\pi)}$, the same relation reads

$$m_f^2=m_{\rm irr}^2+\frac{c^2J_H^2}{4G^2m_{\rm irr}^2}, \qquad \chi_f=\frac{cJ_H}{Gm_f^2},$$

where $m_f$ and $m_{\rm irr}$ are in kilograms, $A$ in square metres, and $J_H$ in $\mathrm{kg\,m^2\,s^{-1}}$.

This is not a generic mass formula for a violently distorted horizon: approximate axial symmetry, settling, and a controlled spin definition are part of the inference. A third estimate fits late-time complex frequencies to the Kerr quasinormal spectrum. Centrella and collaborators describe these independent balance, horizon, and ringdown checks Centrella et al. 2010, §§ IV.H and VIII.B.

Numerical inputs, invariants, and benchmark

The continuum inputs are initial ADM charges, Bondi news or strain at null infinity, and intrinsic horizon geometry. The invariant outputs are the charge balance residuals, horizon area, quasi-local spin magnitude and direction in a declared frame, and dimensionless Kerrness residuals. Coordinate radius and puncture angular velocity are diagnostics, not remnant parameters.

An equal-mass nonspinning binary with the usual exchange and orbital-plane reflection symmetries is a clean benchmark: these symmetries require zero final spin component orthogonal to the orbital angular momentum and zero recoil, while published results constrain the radiated-energy scale. A deliberately asymmetric benchmark is the nonspinning $q=M_1/M_2=0.33$ sequence of González and collaborators: resolutions $h=1/45$, $1/51$, and $1/58$ exhibit second-order convergence of the recoil components, within a survey whose fitted maximum kick is $175.2\pm11\ {\rm km\,s^{-1}}$ near $q=0.36$ González et al. 2007, Eq. (1), Figs. 1–3, and pp. 1–3. A second benchmark uses a single stationary Kerr hole to check horizon charges and vanishing radiative flux. To test frequency fitting, add a controlled small perturbation and compare its ringdown with the known background spectrum; exact stationary Kerr has no excited ringdown signal. Empirical final-state fits can be a cross-check, but they are calibrated interpolants rather than independent conservation laws; their construction and calibration domain are given in Lousto et al. 2010, §§ 2–5.

Convergence, failure modes, and reproducibility boundary

Perform a resolution study on the waveform, horizon integrals, and balance residuals separately. Vary extraction radius or compare characteristic extraction with extrapolation; increase angular-mode truncation; shift the ringdown fit window; and verify a plateau in horizon quantities. The reported uncertainty should encompass the spread between convergent independent methods, not just the finest-grid difference.

One important failure mode is missing low-frequency memory in the radiative integral; others include a spin axis that wanders because approximate symmetry is poor, cancellation between ADM and flux errors, and an overly early ringdown fit that absorbs nonlinear signal into extra modes. Near extremality, small area or spin errors are amplified in $1-\chi_f$ and require special care.

This page states the continuum calculation and validation protocol; it does not rerun a simulation or reproduce a catalog fit locally. Reproduction requires versioned initial data, evolution and extraction software, gauge choices, resolutions, waveform modes, horizon finder settings, and analysis scripts. Without those inputs, the equations define checks but not a numerical result.

Exercise

In the geometrized units used in this section, a settled horizon has $M_{\rm irr}=0.9M$ and $J_H=0.6M^2$, where $M$ is a reference length scale. Compute $M_f^H$ and $\chi_f$.

Solution

The squared mass is $0.9^2+0.6^2/(4\times0.9^2)=0.81+0.1111=0.9211$, so $M_f^H\simeq0.9597M$. Hence $\chi_f=0.6/0.9211\simeq0.651$. The calculation is meaningful only after the horizon has settled enough for the chosen $J_H$ to represent Kerr angular momentum.

References

• Centrella, Joan, John G. Baker, Bernard J. Kelly, and James R. van Meter. “Black-Hole Binaries, Gravitational Waves, and Numerical Relativity.” Reviews of Modern Physics 82 (2010): 3069–3119. doi:10.1103/RevModPhys.82.3069.
• González, José A., Ulrich Sperhake, Bernd Brügmann, Mark Hannam, and Sascha Husa. “Total Recoil: The Maximum Kick from Nonspinning Black-Hole Binary Inspiral.” Physical Review Letters 98 (2007): 091101. doi:10.1103/PhysRevLett.98.091101.
• 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.