Mass, Momentum, Spin, and Constraint Checks

An initial-data solver can make local constraint residuals small while still returning the wrong mass, momentum, spin, or asymptotic frame. Global checks integrate information over surfaces and volumes, so they expose normalization, boundary, parity, and domain-decomposition errors that a pointwise residual may miss. No single charge identity is decisive in every spacetime; the reliable strategy combines constraint norms, multiple extraction surfaces, horizon quantities, and relations whose assumptions are explicitly satisfied.

Required background. Hypersurfaces and Foliations supplies the slice normal, induced metric, extrinsic curvature, and surface geometry used in the charges.

Helpful background. Formulations of General Relativity supplies the constraint systems and asymptotic conditions, Ordinary Differential Equations supplies convergence and extrapolation concepts, and Relativistic Matter and Compact Objects supplies compact-object masses, spins, and material charges.

Local constraints and integrated charges

Evaluate Hamiltonian and momentum constraints with an operator independent of the elliptic solver, reporting both $L_2$ and $L_\infty$ norms. The volume measure, excluded regions, and normalization must be stated. A norm divided by a sum of term magnitudes can aid comparison across scales, but it should accompany the dimensional residual rather than replace it.

For an asymptotically flat slice, the ADM energy is obtained from a large closed surface. Gourgoulhon 2007, §§7.3–7.6, pp. 105–123 gives the ADM mass, momentum, angular-momentum qualifications, and Komar relations in 3+1 form. In asymptotically Cartesian coordinates,

$$E_{\rm ADM}=\frac{1}{16\pi G} \lim_{r\to\infty}\oint_{S_r} (\partial_j\gamma_{ij}-\partial_i\gamma_{jj})s^i\,dS.$$

With $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$, momentum and angular momentum involve $K_{ij}-K\gamma_{ij}$ and the appropriate asymptotic translation or rotation vector. The discrete computation must specify coordinate center, surface geometry, normal, quadrature, and parity assumptions. Cook 2000, §§3.1–3.4 connects these quantities with standard black-hole initial-data constructions.

For example, insert $\gamma_{ij}=\psi^4\delta_{ij}$ with $\psi=1+m/(2r)$ and $m=GM$ into the displayed ADM expression. On the asymptotically Cartesian coordinate sphere, use the Euclidean normal and coordinate area of that ADM formula. The finite-radius integrand gives

$$E(r)=-\frac{r^2}{2G}\frac{d\psi^4}{dr} =\frac{m}{G}\left(1+\frac{m}{2r}\right)^3.$$

Thus $E(r)\to M$ but is not equal to $M$ at finite $r$. The conformally reduced expression $-(2\pi G)^{-1}\oint\partial_r\psi\,dS$ happens to give $M$ at every radius for this exact solution. They define the same asymptotic charge with different finite-radius extensions. A radius-dependent discrepancy between them is therefore not automatically a constraint violation; it is an excellent check of the extrapolation and normalization.

Surface sequences, not one radius

Compute ADM quantities on several nested surfaces and repeat as the outer boundary moves. Fit a motivated asymptotic form, for example

$$E(r)=E_\infty+\frac{a_1}{r}+\frac{a_2}{r^2}+\cdots,$$

varying the fit order and included radii. The spread is part of the uncertainty. A plateau at one grid can result from cancellation between finite-radius and truncation errors; resolution and radius sequences must be crossed.

Compare surface and volume forms when available. Their difference can be related to integrated constraints, but only if inner-boundary contributions and matter sources are included consistently. For binary data, also report component apparent-horizon areas and spins. A quasi-local spin depends on the chosen approximate rotational vector; vary its construction or quantify its residual when the horizon is not axisymmetric.

Relations with restricted assumptions

For stationary asymptotically flat configurations, Komar and ADM masses agree under the appropriate field equations and normalization. In quasi-equilibrium binary data their difference is a useful virial indicator, not proof of exact stationarity. For an isolated stationary black hole, the Smarr relation connects mass, surface gravity, area, angular velocity, and angular momentum. A mismatch may reveal normalization or horizon-integration errors, but a dynamical horizon need not satisfy the stationary relation exactly.

The Christodoulou mass

$$M_H^2=M_{\rm irr}^2+ \frac{J_H^2}{4G^2M_{\rm irr}^2}, \qquad M_{\rm irr}=\sqrt{\frac{A_H}{16\pi G^2}},$$

provides a consistent horizon summary once $A_H$ and $J_H$ are defined Christodoulou 1970, pp. 1596–1597. Compare $M_H$ with the asymptotic mass only in a setting where exterior field and binary binding contributions are understood.

Benchmark and rerun record

Use isolated Schwarzschild and Kerr data. Across at least three resolutions and outer radii, report independent constraint norms, ADM surfaces and extrapolation, horizon area and spin, Christodoulou mass, and the stationary relation appropriate to the data. Then evolve briefly to check that invariant charges remain stationary within the initial uncertainty. A local constraint converging at order $p$ while ADM mass drifts with radius identifies an outer-domain or charge-integral problem.

Pin the data decomposition, coordinate map, surface families, quadrature, derivative operator, horizon finder, spin vector, precision, solver tolerances, and code revision. A reproducible interface could be check-initial-data --config isolated-kerr.toml --output run/, emitting raw surface integrals, constraint fields, fits and covariance, refinement data, and environment metadata. This specifies a rerun rather than claiming an executable is supplied.

Reject results with surface charges lacking an asymptotic plateau, omitted inner terms, constraint norms normalized by vanishing quantities, or inconsistent $G$ and mass conventions. A common failure mode is reporting only the most favorable radius. Another is treating ADM–Komar agreement achieved by tuning as an independent physical validation when both integrals share the same erroneous asymptotic metric.

Archive every surface value before extrapolation. Publishing only the fitted infinity limit prevents others from detecting a poor radial model, a contaminated inner surface, or covariance between finite-radius and resolution errors.

Diagnostic

The Hamiltonian and momentum norms converge at fourth order, but $E_{\rm ADM}$ measured at fixed coordinate radius drifts by one percent as resolution increases. Horizon mass is stable. What should be checked first?

Solution

Check the surface quadrature, derivative operator, coordinate radius versus physical surface, and the joint radius–resolution limit. The stable local constraints and horizon mass do not validate the asymptotic integral. Evaluate ADM energy on multiple surfaces, move the outer boundary, use an independent surface or volume formula, and accept only a stable extrapolated value with an uncertainty covering fit and discretization effects.

References

• Christodoulou, D. (1970). “Reversible and irreversible transformations in black-hole physics.” Physical Review Letters 25, 1596–1597. DOI.
• Cook, G. B. (2000). “Initial data for numerical relativity.” Living Reviews in Relativity 3, 5. DOI.
• Gourgoulhon, E. (2007). “3+1 formalism and bases of numerical relativity.” Lecture notes, arXiv:gr-qc/0703035. arXiv.