Quasi-Local Mass, Spin, and Multipoles

Area is intrinsic to a horizon cross-section, but numerical mass, spin, and multipoles require additional structures whose uniqueness weakens away from equilibrium. A reliable computation therefore reports the surface, the charge definition, the symmetry construction, and their separate numerical errors. The goal is not to manufacture a single “black-hole parameter,” but to identify which quasi-local statements survive distortion and time dependence.

Helpful background. Formulations of General Relativity supplies hypersurface data and constraints; Differential Equations for Relativity supplies elliptic eigenproblems and numerical quadrature; Black-Hole Geometry and Mechanics defines isolated and dynamical horizons; Gravitational Radiation and Waveforms supplies asymptotic balance quantities; and Computational Relativity supplies the shared evidence standard.

Geometry carried by a horizon section

Let $S$ be a smooth marginal surface with induced metric $q_{ab}$, area element $dA$, and area

$$A=\int_S dA.$$

Area is invariant under a relabeling of points on the same geometric surface, although its numerical value still inherits horizon-location, volume-interpolation, and quadrature error. Define the irreducible mass by

$$M_{\mathrm{irr}}=\sqrt{\frac{A}{16\pi G^2}},$$

with $c=1$ and $G$ retained. This definition alone does not determine a total mass for a distorted, dynamical horizon. To identify the following area–spin expression with the physical rest mass, the assumptions are an uncharged, stationary Kerr black hole in Einstein gravity. In that regime a spin $J$ supplies the Christodoulou relation

$$M_{\mathrm H}^2=M_{\mathrm{irr}}^2+ \frac{J^2}{4G^2M_{\mathrm{irr}}^2}.$$

This expression also supplies a useful, explicitly chosen horizon-mass definition away from Kerr. Its use there does not prove that the horizon is stationary or that this is a unique dynamical mass.

For example, a Kerr black hole of physical mass $M$ and $a_*=J/(GM^2)=0.6$ has $\sqrt{1-a_*^2}=0.8$ and $A=14.4\pi G^2M^2$. Hence $M_{\mathrm{irr}}^2=0.9M^2$ and $J^2/(4G^2M_{\mathrm{irr}}^2)=0.1M^2$: the two contributions recover $M_{\mathrm H}=M$. The spin term is necessary even though area is measured exactly. In the nonrotating limit it vanishes, giving $M_{\mathrm H}=M_{\mathrm{irr}}$.

We use the hypersurface convention $K_{ij}=-h_i{}^a h_j{}^b\nabla_a n_b$. Given a tangential axial vector $\varphi^i$ on $S$ and outward spatial normal $s^i$, a standard vacuum angular momentum is

$$J[\varphi]=\frac{1}{8\pi G}\int_S K_{ij}\varphi^i s^j\,dA.$$

Matter and alternative-theory terms must be added when the corresponding balance law requires them. More importantly, $J[\varphi]$ depends on the choice and normalization of $\varphi^i$ unless an exact axial symmetry fixes both. The usual trace term $-Kh_{ij}$ makes no contribution because $\varphi^i s_i=0$. This is a useful implementation check: adding that term should change the result only by the numerical tangency residual.

Approximate symmetry is part of the answer

On a generic merger horizon there is no exact Killing field. One practical construction searches for a divergence-free tangential vector that minimizes the shear of the surface metric. Writing $\varphi^a=\epsilon^{ab}D_b z$ reduces the problem to an elliptic eigenproblem for a potential $z$. Cook and Whiting derive a specific approximate-Killing-vector construction in Cook and Whiting 2007, Eqs. (1)–(15) and Figs. 1–3. The lowest admissible eigenmode is a candidate rotational direction, but the algorithm must still address nearly degenerate modes, the placement of poles, and normalization.

Normalization is consequential because replacing $\varphi^a$ by $c\varphi^a$ replaces $J$ by $cJ$. For a clean axial geometry, normalize the parameter along closed orbits to period $2\pi$. For a distorted surface whose integral curves do not close cleanly, state the substitute convention and treat the spread among defensible choices as a definition systematic. A small eigenvalue indicates approximate symmetry; it does not by itself prove that the chosen mode represents physical rotation rather than a coordinate or tidal distortion.

Multipoles sharpen this issue. Intrinsic curvature and the normal-bundle connection can be projected onto harmonics adapted to the selected symmetry axis. Owen develops coordinate-independent horizon multipoles and their numerical behavior in Owen 2009, §II, Eqs. (2)–(9), and §III, Figs. 1–13. The harmonic basis, centroid or poles, phase, and areal-radius normalization must travel with the reported coefficients. On a dynamical surface these numbers diagnose shape and rotation; they need not obey the stationary Kerr multipole relation.

Kerr and merger benchmarks

The first benchmark uses versioned Kerr–Schild slices with lengths measured in $GM$ and several spins, for example $a_*=0$, $0.5$, and $0.9$. At each of at least three volume and surface resolutions, compute the horizon area, approximate-symmetry residual, $J$, $M_{\mathrm{irr}}$, and $M_{\mathrm H}$. Translate the analytic Kerr area and angular momentum into the same $G$, mass, and orientation conventions. Rotate and translate the coordinates while leaving the physical slice unchanged. Invariant charges should agree within the combined interpolation and quadrature error, whereas coordinate locations need not.

The second benchmark follows a binary through common-horizon formation. Compare at least two defensible symmetry constructions and, where possible, surface and asymptotic angular-momentum balances. Resolution refinement should decrease the surface residual and stabilize the charge, but definition spread may remain larger than truncation error during the most dynamical interval. Report both; adding them invisibly into a single error bar hides useful physics.

A discriminating independent check reconstructs the stationary Kerr mass from area and spin and compares it with the known input mass. For a merger remnant, first subtract radiated energy and momentum from the initial ADM four-momentum in a common asymptotic frame. If the remaining system is one settled black hole with negligible exterior radiation or matter, its rest mass is

$$M_{\mathrm{bal}}=\sqrt{E_{\mathrm{bal}}^2-\lvert\mathbf P_{\mathrm{bal}}\rvert^2}.$$

Compare this with $M_{\mathrm H}$: the remaining energy $E_{\mathrm{bal}}$ includes recoil kinetic energy and is not itself the horizon rest mass. Similarly, compare intrinsic horizon spin with asymptotic angular momentum only after fixing the origin, frame and any remnant orbital contribution. The energy, momentum and angular-momentum balance discussion explains these distinctions and the treatment of initial spurious radiation. Agreement between two quadratures using the same faulty $\varphi^a$ is not independent evidence.

Reproducible charge data

The environment record should pin the horizon surface coefficients or mesh, the parent slice and checksum, interpolation and differentiation operators, approximate-symmetry eigensolver, quadrature, floating-point precision, and software revision. Inputs include the sign of $K_{ij}$, outward normal, charge formula, matter terms, mode-selection and normalization rules, and surface resolution. Outputs should retain the selected vector field or potential, eigenvalue or symmetry residual, area, spin, mass, multipoles, and refinement table.

A clean rerun would read a declared Kerr–Schild dataset, evaluate $N=24,48,96$ angular representations, apply coordinate shifts and rotations, and emit analytic discrepancies plus observed convergence. No executable calculation is claimed here. The primary failure mode is to mistake a well-resolved convention for a unique geometric charge. Reject a result if charge errors plateau under joint volume/surface refinement, if the selected eigenmode swaps without being tracked, if spin changes under coordinate rotation beyond uncertainty, if a chosen area–spin mass away from Kerr is presented as a unique invariant dynamical mass, or if definition spread is omitted near merger.

Check your understanding

Two approximate axial fields have equally small symmetry residuals on a newly formed common horizon. They give dimensionless spins $0.69$ and $0.74$, while resolution error within either construction is $0.003$. Which spin should be reported?

Solution

Neither should be promoted alone as a definition-independent value. The $0.05$ spread is much larger than truncation error and reflects the unresolved approximate-symmetry choice. Report both constructions, their normalization rules and residuals, and the spread as a definition systematic. Continue the modes in time rather than reselecting them independently on every slice. A single value becomes defensible only when the horizon settles, the modes cease competing, and both definitions converge toward the same Kerr charge within the stated uncertainty.

Continue with Curvature-Based Wave Extraction.

References

• Cook, Gregory B., and Bernard F. Whiting. 2007. “Approximate Killing Vectors on $S^2$.” Physical Review D 76, 041501. arXiv.
• Owen, Robert. 2009. “The Final Remnant of Binary Black Hole Mergers: Multipolar Analysis.” Physical Review D 80, 084012. arXiv.