Horizon Multipoles in Evolution

Horizon multipoles replace coordinate pictures of a distorted black hole by diffeomorphism-invariant numbers on each marginal surface. Intrinsic curvature measures shape, while the curl of a normal-bundle connection measures rotation. Comparing those projections from leaf to leaf shows how a dynamical horizon sheds distortion and approaches an isolated Kerr horizon, provided the leaves are identified by a consistent geometric map.

Required background. Hypersurface normals and tangential projectors supplies intrinsic curvature, area form, and normal connection.

Helpful background. Black-Hole Geometry and Mechanics develops isolated-horizon source multipoles; Black-Hole Perturbation Theory supplies harmonic modes and Kerr relaxation; Formulations of General Relativity explains the extrinsic-curvature data; and Computational Relativity develops approximate symmetries and surface integration.

Geometry encoded on one leaf

Let $S$ be a spherical leaf with intrinsic metric $\widetilde q_{ab}$, derivative $\widetilde D_a$, scalar curvature $\widetilde{\mathcal R}$, area form $\widetilde\epsilon_{ab}$, and areal radius $R$. On an axisymmetric isolated horizon, let $\varphi^a$ be the rotational Killing field normalized to period $2\pi$. A canonical latitude coordinate $\zeta\in[-1,1]$ is fixed by

$$\widetilde D_a\zeta =\frac{1}{R^2}\widetilde\epsilon_{ba}\varphi^b, \qquad \oint_S\zeta\,dA=0.$$

This supplies canonical axisymmetric spherical harmonics $Y_{\ell0}(\zeta)$. Choose future null normals $\ell\cdot k=-1$ and define the leaf’s normal connection by $\omega_a=-k_b\widetilde q_a{}^c\nabla_c\ell^b$. Under a smooth positive rescaling $\ell\to f\ell$, $k\to f^{-1}k$, it changes by

$$\omega_a\to\omega_a+\widetilde D_a\ln f.$$

Its curl is unchanged because the curl of a scalar gradient vanishes. This is why rotation is encoded in the curl rather than in individual components of $\omega_a$. In the isolated-horizon convention of the cited source, the gauge-invariant complex curvature used for geometric multipoles is

$$\Phi_H=\frac14\widetilde{\mathcal R} -\frac{i}{2}\widetilde\epsilon^{ab} \widetilde D_a\omega_b.$$

The geometric multipoles are projections

$$I_\ell+iL_\ell =\oint_S\Phi_HY_{\ell0}(\zeta)\,dA.$$

The sign of $L_\ell$ follows the declared axial and area-form orientations. In vacuum with zero cosmological constant, $\Phi_H=-\Psi_2$ for the source’s Newman–Penrose tetrad convention; writing $\Phi_H$ avoids silently importing a different Weyl-scalar sign. Fixed powers of $R$ and the horizon mass convert $I_\ell,L_\ell$ into conventional mass and angular-momentum source multipoles $M_\ell,J_\ell$. The complete normalization and reconstruction theorem are given in Ashtekar et al. 2004, §§III.A–III.C, pp. 2555–2563, PDF.

These are horizon source multipoles, not the Geroch–Hansen field multipoles defined at infinity. They differ even for stationary vacuum Kerr at higher orders. Therefore the asymptotic Kerr relation $M_j+iS_j=M(ia)^j$ in conventional geometrized units must not be inserted as a formula for these horizon moments. Compare an evolving horizon with a Kerr horizon computed using the same projections and normalization Ashtekar et al. 2004, §III.C and §IV, PDF.

Carrying the construction through evolution

On a spacelike dynamical horizon, each leaf again supplies $\widetilde{\mathcal R}$. Let $q^{(H)}_{ab}=g_{ab}+\hat\tau_a\hat\tau_b$ be the horizon metric and adopt $K_{ab}=q^{(H)}_a{}^c q^{(H)}_b{}^d\nabla_c\hat\tau_d$. The rotational one-form can then be built from this horizon extrinsic curvature,

$$\widetilde\omega_a =\widetilde q_a{}^bK_{bc}\hat r^c.$$

The main difficulty is comparing different leaves without introducing coordinate drift. In exact axisymmetry, the axial fields and latitude functions provide a natural identification. More generally, one selects a late isolated leaf, transports preferred weighting fields backward along a geometrically defined evolution vector, and projects curvature and rotational data using the transported basis. This keeps a changing coordinate grid from masquerading as physical mode mixing.

Ashtekar, Campiglia, and Shah construct multipoles adapted to the dynamical-to-isolated transition and derive their evolution in Ashtekar, Campiglia, and Shah 2013, §§II–IV, pp. 2–22, PDF. Their flux equations provide checks between the change of each multipole and local horizon fields, rather than treating a fitted damped sinusoid as the definition.

A practical hierarchy of checks

First verify Gauss–Bonnet: $\oint_S\widetilde{\mathcal R}\,dA=8\pi$ for a smooth spherical leaf. The curvature monopole is therefore fixed topologically, and the rotational monopole vanishes without NUT-type singularity. In spherical symmetry all nontrivial multipoles vanish. Reflection symmetry forces odd mass and even spin multipoles to vanish in the usual axial convention.

Second, compute the same multipoles at several surface resolutions and with independent approximate-Killing-vector tolerances. Monitor the residual $\mathcal L_\varphi\widetilde q_{ab}$; when it is not small, calling the result an axial multipole requires a prescription label. Reversing the chosen axis sends $\varphi\to-\varphi$ and $\zeta\to-\zeta$. With the surface orientation fixed, $Y_{j0}(-\zeta)=(-1)^jY_{j0}(\zeta)$, so both $I_j$ and $L_j$ acquire the factor $(-1)^j$. In the reflection-symmetric case, only even mass and odd spin moments survive; then mass moments stay fixed and spin moments change sign.

Third, compare late-time ratios with a Kerr horizon having the measured $M_H,J_H$. Agreement of $M_0$ and $J_1$ alone is not a no-hair test; higher multipoles must approach the Kerr relations. Their decay rates and phases can then be compared with horizon perturbation theory and the waveform at infinity, allowing for propagation delay and different source/field definitions.

Limits of interpretation

Multipole decay describes the chosen marginal-horizon branch. Near a branch jump or leaf with poor stability, transporting the basis may become discontinuous. Multiple approximate axial vectors can produce systematic differences larger than surface quadrature error. Matter on or outside the horizon can also keep source and asymptotic moments distinct.

A small higher multipole does not prove the exterior spacetime is Kerr, and an oscillatory coordinate shape does not prove a physical multipole oscillates. The invariant projections, flux residuals, symmetry residual, and resolution study must agree.

Check your understanding

An axisymmetric, reflection-symmetric horizon is perturbed without changing its symmetry. Which of $M_1,M_2,J_1,J_2$ must vanish by symmetry, and which can be nonzero?

Solution

Reflection symmetry makes odd mass multipoles and even spin multipoles vanish, so $M_1=0$ and $J_2=0$. The mass quadrupole $M_2$ and spin dipole $J_1$ can be nonzero. The mass dipole also vanishes in the canonical horizon rest frame.

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.
• Ashtekar, Abhay, Miguel Campiglia, and Samir Shah. “Dynamical Black Holes: Approach to the Final State.” Physical Review D 88 (2013): 064045. doi:10.1103/PhysRevD.88.064045.