Mass, Spin, Area, and Charge Budgets

A merger calculation becomes physically persuasive when independent measurements close the same conservation laws. Initial ADM charges, fluxes through future null infinity, matter or electromagnetic fluxes where present, and quasi-local remnant quantities must agree within a declared uncertainty. Area growth and subextremality then provide additional inequalities rather than substitutes for conservation. A visually plausible waveform with an unclosed budget is not a validated spacetime.

Helpful background. Quasi-local horizon quantities are developed in Black-Hole Geometry and Mechanics; radiative response and normalization are treated in Black-Hole Perturbation Theory; ADM and characteristic charges come from Formulations of General Relativity; error control belongs to Computational Relativity; and the input binary parameters are organized in Relativistic Two-Body Dynamics.

Continuum balances

For an asymptotically flat vacuum evolution ending in one isolated remnant, choose compatible initial ADM and final Bondi frames, including a center and a prescription for the cuts of null infinity. Let $(E_{\rm ADM},\mathbf P_{\rm ADM})$ be the initial four-momentum and $(E_{\rm rad},\mathbf P_{\rm rad})$ the total outgoing radiative four-momentum. The final balance prediction is

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

The settled horizon determines the rest mass $M_H$, whereas $E_f$ also includes recoil kinetic energy. Consequently the independent energy residual is

$$\Delta E=E_{\rm ADM}-E_{\rm rad} -\sqrt{M_H^2+|\mathbf P_f|^2}.$$

Here $c=1$ as in the site conventions. Equivalently, compare $M_H$ with $\sqrt{(E_f^{\rm bal})^2-|\mathbf P_f^{\rm bal}|^2}$. Using $E_f=M_H$ is valid only in the final rest frame or as a declared small-kick approximation. An independently measured remnant momentum tests more than reusing the same flux integral in both sides of the residual.

Angular-momentum balance similarly gives $\mathbf J_f^{\rm total}=\mathbf J_{\rm ADM}-\mathbf J_{\rm rad}$ in the chosen frame. To compare it with the intrinsic horizon spin, remove the remnant’s center-of-mass orbital contribution and transform to its rest frame. A shift of spatial origin changes total angular momentum when $\mathbf P_f\ne0$; Bondi-cut choices also matter. The aligned residual then compares like quantities. These frame issues are developed in Remnant Kick Accumulation and Frame Choice.

With electromagnetic charge, check $Q_{\rm initial}-Q_{\rm escaped}-Q_f=0$, where escaped charge is carried by charged matter, not by electromagnetic waves themselves. Matter evolutions also account for escaping material and any surviving exterior energy. Flux through a horizon changes the energy already included in the remnant; adding it as a further loss in this global balance would count absorption twice. It is a separate term only in an exterior-region balance whose inner boundary is that horizon.

The horizon area $A$ defines an irreducible mass $M_{\rm irr}=\sqrt{A/(16\pi G^2)}$ in four dimensions. For an approximately axisymmetric uncharged horizon, the Christodoulou relation gives

$$M_H^2=M_{\rm irr}^2+\frac{J_H^2}{4G^2M_{\rm irr}^2}.$$

The dimensionless spin is $\chi=J_H/(GM_H^2)$. Area nondecrease along an appropriate future outer or dynamical horizon and $|\chi|\le1$ are important checks, but both require their geometric and energy-condition hypotheses.

Numerical data and invariant outputs

The continuum input is a constraint-satisfying Einstein or Einstein–matter initial-data family with declared masses, spins, momentum, eccentricity, asymptotics, and boundary conditions. A numerical run must identify formulation, gauge, mesh or spectral resolution, outer boundary, excision or puncture treatment, radiation extraction, and horizon finder. The invariant outputs are ADM and Bondi charges in compatible frames, news or strain multipoles with explicit normalization, horizon area, quasi-local spin, multipoles, and constraint norms.

A clean benchmark is an equal-mass nonspinning binary: symmetry requires zero net recoil and constrains spin components, while independent energy measurements should agree. A head-on nonspinning case adds a simple common-horizon area history. Centrella and collaborators describe benchmark binary evolutions and the associated radiation and remnant checks Centrella et al. 2010, §§ IV–V and VIII. Dynamical-horizon flux laws provide a quasi-local comparison between changing horizon geometry and ingoing gravitational flux Ashtekar and Krishnan 2004, § 3.2, Eqs. (24)–(26). A distinct failure mode is an extraction-frame mismatch; others are an inconsistent spin definition on a distorted horizon and cancellation of individually large errors in a deceptively small closure residual.

Convergence and uncertainty

Budget closure is tested across at least three resolutions or a comparable spectral-tolerance sequence. Extrapolate each charge, not just the waveform phase, and propagate the uncertainty in recoil momentum as well as horizon mass. Suppose $\delta M_H$ bounds the absolute mass error and $\delta P_f$ bounds the Euclidean norm of the momentum error. Since $E_f=\sqrt{M_H^2+|\mathbf P_f|^2}$ is the Euclidean norm of $(M_H,\mathbf P_f)$, the triangle inequality gives the deterministic bound

$$|E_f^{\rm measured}-E_f^{\rm true}| \leq\sqrt{(\delta M_H)^2+(\delta P_f)^2} \leq\delta M_H+\delta P_f.$$

With absolute error bounds $\delta E_{\rm ADM}$ and $\delta E_{\rm rad}$, a conservative closure tolerance is

$$\delta_{\rm budget} =\delta E_{\rm ADM}+\delta E_{\rm rad} +\delta M_H+\delta P_f.$$

If these bounds cover all errors and the physical balance is complete, then $|\Delta E|\leq\delta_{\rm budget}$. This bound does not assume independent errors. Standard deviations, however, are statistical uncertainties rather than guaranteed bounds. For small statistical errors, use the full covariance matrix $\Sigma$ of $x=(E_{\rm ADM},E_{\rm rad},M_H,\mathbf P_f)$:

$$\sigma_{\Delta E}^2\simeq g^T\Sigma g, \qquad g=\left(1,-1,-\frac{M_H}{E_f},-\frac{\mathbf P_f}{E_f}\right).$$

The last entry represents three momentum components. Shared extraction data can correlate the inputs, especially when the final momentum is inferred from radiated momentum. Adding their standard deviations does not create an absolute error guarantee.

The error budget separates constraint violation, finite outer boundary, extraction or extrapolation, unresolved early junk radiation, horizon-finder tolerance, incomplete mode sums, and frame alignment. Finite-radius waveforms can carry significant phase and amplitude offsets, while high-order extrapolation can amplify noise, so extraction radius and extrapolation order require their own convergence tests Boyle and Mroué 2009, §§ II–V, pp. 3–13. Angular-momentum errors can be substantially more delicate than energy errors because small phase or center-of-mass drifts feed the flux.

Failure modes include evaluating ADM and Bondi quantities in incompatible frames, truncating radiative modes before convergence, mistaking coordinate spin for quasi-local spin, missing matter outside the horizon, comparing horizon data before relaxation, and hiding a closure residual inside an overly conservative combined error. A converged bulk solution can still yield a biased flux if extraction is not converged independently.

Reproducibility boundary and interpretation

This page specifies a verification protocol and summarizes published calculations; it does not claim that a simulation was rerun. A reproducible record would include initial-data files, code revision, compiler and libraries, gauge and formulation, resolution hierarchy, boundary locations, extraction method, mode list, horizon-finder tolerances, integration constants, and machine-readable charge and residual tables. An independent implementation or characteristic extraction provides a stronger cross-check than repeating the same postprocessing.

When all balances close, the result supports a consistent allocation of energy, angular momentum, linear momentum, area, and charge in the stated data family. It does not prove a universal final-state theorem. Conversely, failure to close a budget does not by itself identify new physics; it first localizes a missing flux, inconsistent definition, or numerical systematic.

Once these balances close, Recoil, Remnants, and Final-State Formation turns the same asymptotic and quasi-local quantities into recoil, remnant-parameter, Kerrness, and final-state tests.

Check the energy accounting

In units with initial ADM energy one, a vacuum merger radiates energy $0.04$ and momentum $0.0096$ along one axis. The initial momentum vanishes. Find the final speed and rest mass. Should an independently measured absorbed energy be subtracted again?

Solution

The final energy is $E_f=0.96$ and the momentum is opposite the radiation, with magnitude $0.0096$. Thus $v_f=|P_f|/E_f=0.01$ and $M_f=0.96\sqrt{1-0.01^2}\simeq0.959952$. The energy exceeds the rest mass by approximately $4.8\times10^{-5}$, the recoil kinetic energy. Absorption is already in the final remnant mass and is not an additional loss from the entire system.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. doi:10.12942/lrr-2004-10.
• Boyle, Michael, and Abdul H. Mroué. “Extrapolating Gravitational-Wave Data from Numerical Simulations.” Physical Review D 80 (2009): 124045. doi:10.1103/PhysRevD.80.124045.
• 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.