Remnant Multipoles and Kerrness

Mass and spin identify a stationary Kerr black hole, but during relaxation they do not say how far the geometry remains from Kerr. Multipole moments and curvature invariants provide complementary answers: horizon moments track the intrinsic distorted surface, while exterior invariants test the algebraic structure of the four-dimensional curvature. Both require normalization, slicing, and comparison choices to be stated before a small residual is interpreted.

Helpful background. Isolated horizons and Kerr geometry are developed in Black-Hole Geometry and Mechanics; quasinormal decay belongs to Black-Hole Perturbation Theory; curvature decompositions use Formulations of General Relativity; numerical evaluation belongs to Computational Relativity; and radiative multipoles are distinguished in Gravitational Radiation and Waveforms.

Kerr multipole relations

Use $G=c=1$ in this section: $M$ and $a$ have dimensions of length, $J$ has dimensions of length squared, and $M_\ell$ and $S_\ell$ have dimensions of length to the power $\ell+1$. For a stationary, asymptotically flat, axisymmetric vacuum black hole, the Geroch–Hansen mass and current moments satisfy

$$M_\ell+iS_\ell=M(ia)^\ell, \qquad a=J/M.$$

Thus $M_0=M$, $S_1=J$, $M_2=-J^2/M$, and every higher moment is fixed. Representative SI quantities map to these variables as $M=Gm_{\rm SI}/c^2$, $J=GJ_{\rm SI}/c^3$, $a=J_{\rm SI}/(m_{\rm SI}c)$, and $\chi=cJ_{\rm SI}/(Gm_{\rm SI}^2)$. These are asymptotic field moments, defined using stationarity near infinity. A dynamical horizon does not possess the stationarity needed for that construction.

For moments obtained by that same asymptotic definition, dimensionless Kerr residuals are

$$\delta_2=\frac{M_2+J^2/M}{M^3}, \qquad \delta_3=\frac{S_3+J^3/M^2}{M^4}.$$

These formulae must not be applied to a raw horizon quadrupole. Horizon multipoles are moments of intrinsic surface curvature and rotation data; even in stationary Kerr, their higher moments differ from the Geroch–Hansen moments. They approach the Kerr horizon values in the same definition during relaxation. The distinction and its size for Kerr are explicit in Ashtekar et al. 2004, § III.C and Fig. 1, pp. 13–16.

A directly computable horizon comparison

A simple example shows how to compare like with like. On an axisymmetric spherical section, set $R_A=\sqrt{A/(4\pi)}$ and choose an area coordinate $z\in[-1,1]$ and an axial angle $\varphi$ of period $2\pi$, so $dA=R_A^2\,dz\,d\varphi$. The poles fix $z=\pm1$; reversing the axis reverses odd moments. Define the dimensionless curvature moments

$$H_\ell=\frac{1}{8\pi}\int_S {}^{(2)}R\,P_\ell(z)\,dA,$$

where $P_\ell$ is a Legendre polynomial. This explicitly chosen normalization is a geometric diagnostic, not the dimensionful mass-multipole normalization above. Gauss–Bonnet gives $H_0=1$ for every smooth spherical section. A round sphere has $H_{\ell\ge1}=0$.

For a Kerr horizon, $z=\cos\theta$, $r_+=M+\sqrt{M^2-a^2}$, and $R_A^2=r_+^2+a^2$. Its intrinsic curvature is

$${}^{(2)}R_{\rm Kerr}(z) =\frac{2(r_+^2+a^2)(r_+^2-3a^2z^2)} {(r_+^2+a^2z^2)^3}.$$

This follows from the horizon two-metric $q_{\theta\theta}=r_+^2+a^2\cos^2\theta$ and $q_{\varphi\varphi}=(r_+^2+a^2)^2\sin^2\theta/q_{\theta\theta}$. Insert it into the same integral to calculate $H_\ell^{\rm Kerr}(M,J)$, then compare

$$\Delta H_\ell=H_\ell^{\rm measured}-H_\ell^{\rm Kerr}.$$

For example, $\chi=0.7$ gives $H_2^{\rm Kerr}\simeq-0.130397$, whereas the asymptotic field moment is $M_2/M^3=-0.49$. Their difference is not a distortion away from Kerr. Rotation moments require the normal connection as well as intrinsic curvature; curvature alone cannot recover the spin’s sign. For a nonaxisymmetric dynamical surface, the chosen axis, basis, and transport between sections must also be specified, as in Chen et al. 2022, § II, Eqs. (4)–(17).

Exterior speciality and local curvature

Let $I$ and $J_W$ be the conventional quadratic and cubic complex invariants of the self-dual Weyl tensor. Where $I\ne0$, the speciality index

$$\mathcal S=\frac{27J_W^2}{I^3}$$

equals unity for algebraically special type-D curvature, including Kerr, and also for type II where the ratio is defined. Deviations of $\mathcal S$ from one can locate a failure of algebraic speciality. For a perturbation of amplitude $\epsilon$ about Kerr, however, $\mathcal S-1$ begins at second order: the absence of a first-order deviation is built into this diagnostic. A small speciality residual therefore does not directly measure a small metric perturbation or establish that all first-order radiation has decayed. Baker and Campanelli derive this behavior in Baker and Campanelli 2000, §§ II–III, Eqs. (1)–(4), pp. 1–3.

The test has sharp limitations. Algebraic speciality at one point does not establish global Kerr geometry; $I$ may vanish or make the ratio ill-conditioned; tetrad-independent scalars still vary from point to point and must be compared using a geometrically defined mapping between slices. A robust analysis therefore combines speciality with horizon multipoles, approximate Killing symmetry, charge closure, and ringdown frequencies.

A hierarchy of checks

First verify that $M$, $J$, and area plateau and satisfy the Kerr relation within uncertainty. Next measure several horizon moments and test whether their deviations decay at compatible quasinormal rates. Then evaluate exterior curvature invariants on geometrically selected world tubes or level sets, rather than at fixed coordinate locations. Finally compare the inferred $M$ and $a$ with the frequencies seen in radiation. Chen and collaborators demonstrate how horizon geometry and waveform information can be aligned in a common remnant analysis Chen et al. 2022, § II, Eqs. (4)–(17), and §§ III–IV.

Agreement of these layers is stronger than any one scalar becoming small. Disagreement is informative: it may reveal a bad axis choice, gauge drift, transient mode mixing, insufficient resolution, or matter remaining outside the horizon.

Exercise

For a horizon with independently measured $\chi=0.7$, suppose the curvature integral above gives $H_2=-0.128$. Compute $\Delta H_2$. Would comparing this number with $-\chi^2$ be a Kerrness test?

Solution

Using the Kerr horizon reference gives $\Delta H_2=-0.128+0.130397\simeq0.002397$, about $1.84\%$ of the reference magnitude. Whether this is a resolved distortion depends on uncertainty in the curvature integral, spin, and axis. Comparison with $-\chi^2=-0.49$ would mix a dimensionless curvature moment with a differently normalized asymptotic field moment and would incorrectly reject exact Kerr. Even the correctly defined residual tests only one moment; it does not prove global stationarity.

References

• Ashtekar, Abhay, Jonathan Engle, Tomasz Pawlowski, and Chris Van Den Broeck. “Multipole Moments of Isolated Horizons.” Classical and Quantum Gravity 21 (2004): 2549–2570. doi:10.1088/0264-9381/21/11/003.
• Baker, John, and Manuela Campanelli. “Making Use of Geometrical Invariants in Black Hole Collisions.” Physical Review D 62 (2000): 127501. doi:10.1103/PhysRevD.62.127501.
• Chen, Yitian, et al. “Multipole Moments on the Common Horizon in a Binary-Black-Hole Simulation.” Physical Review D 106 (2022): 124045. doi:10.1103/PhysRevD.106.124045.