Horizon Multipoles and Distortion

Horizon multipoles describe the intrinsic distortion and rotation of a finite black-hole boundary. For a smooth axisymmetric isolated horizon, the scalar curvature of a cross-section and the curl of its rotation one-form can be expanded in invariantly defined harmonics. The resulting moments are diffeomorphism invariant and reconstruct the horizon’s free intrinsic data under stated hypotheses. They are quasi-local source moments; they need not equal the gravitational field multipoles defined at infinity.

Required background. Hypersurface Normals and Tangential Projectors supplies cross-sectional geometry. Helpful background. Exact Spacetimes supplies Kerr checks; Causality and Global Structure separates finite and asymptotic data; Einstein Equations relates Weyl and intrinsic curvature; Black-Hole Geometry and Mechanics supplies isolated horizons; Variational and Symplectic Geometry supplies horizon mass and angular momentum.

Axisymmetric intrinsic data

Let $S$ be a cross-section of a smooth, nonextremal, axisymmetric isolated horizon, initially with no matter fields on the horizon. Write

$$A_\Delta=4\pi R_\Delta^2$$

and normalize the axial Killing field $\varphi^a$ to have $2\pi$-periodic orbits. The area form and axial field determine an invariant polar coordinate $\zeta\in[-1,1]$, fixed by placing its extrema at the two axial fixed points and setting its mean to zero. This construction does not use an exterior radial coordinate or a coordinate embedding of $S$.

Two gauge-invariant scalar fields carry the relevant free data: the intrinsic scalar curvature $\widetilde{\mathcal R}$ of $S$ and the curvature of the normal bundle. Package them without importing a Newman–Penrose sign as

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

where $\widetilde\omega_a$ is the pullback of the rotation one-form. Changing the cross-section or boosting the null normals adds an exact gradient to $\widetilde\omega_a$ but leaves its curl unchanged. In vacuum with zero cosmological constant, the cited construction writes $\Phi_H=-\Psi_2^{\rm source}$ in its Newman–Penrose convention. Keeping $\Phi_H$ explicit prevents that source-local scalar sign from being mistaken for the site’s $\Psi_2$ convention.

Axisymmetry defines a canonical round metric with the same area, hence normalized harmonics $Y_{n0}(\zeta)$ satisfying

$$\int_S Y_{n0}Y_{m0}\,d^2V =R_\Delta^2\delta_{nm}.$$

The physical metric need not be round; the round structure supplies the invariant comparison harmonics.

Geometric curvature and rotation moments

Define the dimensionless geometric multipoles together by

$$I_n+iL_n=\int_S \Phi_HY_{n0}(\zeta)\,d^2V.$$

The $I_n$ measure intrinsic curvature distortion; the $L_n$ measure rotation- bundle structure. They are invariant under diffeomorphisms preserving the horizon geometry because $\zeta$, the measure, and both scalar fields are intrinsically constructed. Ashtekar, Engle, Pawlowski, and Van Den Broeck define these moments and establish their invariance in Ashtekar, Engle, Pawlowski, and Van Den Broeck 2004, §III.A, pp. 8–10.

They are constrained. Gauss–Bonnet fixes the curvature monopole $I_0=\sqrt\pi$ in this normalization. The rotation monopole vanishes, $L_0=0$, because the curl of a globally defined one-form integrates to zero. The invariant coordinate choice places the mass dipole in the horizon center- of-mass frame, giving $I_1=0$. Smoothness controls the large-$n$ falloff, and regularity of the axisymmetric metric restricts which infinite sequences are admissible.

For the stated nonextremal type-II class, area together with all $I_n$ and $L_n$ reconstructs the intrinsic metric and connection up to diffeomorphism. The reconstruction and uniqueness argument are given in Ashtekar, Engle, Pawlowski, and Van Den Broeck 2004, §III.B, pp. 10–12. A finite truncation is a diagnostic summary, not complete geometric data.

Physical mass and current moments

Let $M_\Delta$ and $J_\Delta$ be the horizon Hamiltonian mass and angular momentum in the chosen isolated-horizon normalization. Rescale the geometric moments to define

$$J_n= \sqrt{\frac{4\pi}{2n+1}} \frac{R_\Delta^{n+1}}{4\pi G}\,L_n,$$ $$M_n= \sqrt{\frac{4\pi}{2n+1}} \frac{M_\Delta R_\Delta^n}{2\pi}\,I_n.$$

Then $M_0=M_\Delta$, $J_0=0$, $M_1=0$, and $J_1=J_\Delta$. The moments have the expected length or mass dimensions and parity. With equatorial reflection symmetry, odd mass moments and even current moments vanish. A spherically symmetric horizon has no moments beyond $M_0$; a static but distorted horizon can have nonzero higher mass moments while all current moments vanish.

The physical rescaling and symmetry checks appear in Ashtekar, Engle, Pawlowski, and Van Den Broeck 2004, §III.C, pp. 12–16. Matter on the horizon modifies the curvature combination and can introduce electromagnetic multipoles. The action, matter content, and charge normalization must then accompany the reported moments.

Source moments versus field moments

These moments characterize the horizon boundary. Geroch–Hansen-type field multipoles characterize a stationary asymptotic gravitational field. Exterior matter and the gravitational field between the horizon and infinity contribute to the latter, so no general equality is expected even when both sets exist.

Kerr is an instructive comparison, not an identification. Its horizon and asymptotic moments are both determined by $M$ and $J$ because the exact vacuum solution is highly constrained, yet their higher numerical moments arise from different definitions. Moreover, the 2004 axisymmetry-based isolated-horizon moments and the later generic nonexpanding-horizon moments do not coincide even for Kerr: in the small-spin limit their first difference occurs at $\ell=2$ Gourgoulhon, Le Tiec, and Casals 2026, §VI.F, Eqs. (6.49) and (6.53), pp. 38–40. The present page uses the 2004 construction and labels it explicitly rather than implying a unique Kerr horizon-multipole sequence.

In a dynamical simulation, curvature moments of successive marginal sections can track relaxation toward Kerr. That use requires a consistent identification of the axial field and polar coordinate between leaves. A coordinate shape in the computational grid is not an invariant distortion measure; the intrinsic curvature spectrum is.

Limits of the construction

Exact axisymmetry supplies the canonical $m=0$ harmonics. Without it, a preferred basis is absent and generalized or approximate constructions require additional choices. Extremal isolated horizons have different free-data properties and are not obtained by silently applying every nonextremal reconstruction step. Low differentiability limits harmonic falloff and the number of robustly extractable moments.

Numerically, high-$n$ moments amplify discretization noise because they probe short angular scales and derivatives of the metric. A responsible report tests resolution, approximate symmetry, cross-section choice, and convergence before interpreting a decaying high-multipole tail as physical relaxation.

Extract a quadrupolar distortion

Suppose an axisymmetric cross-section has, to first order in small $\varepsilon$,

$$\widetilde{\mathcal R} =\frac{2}{R_\Delta^2} \left[1+\varepsilon P_2(\zeta)\right],$$

with unchanged area element at this order and no rotational curvature. Using $Y_{20}=\sqrt{5/(4\pi)}P_2$, compute $I_2$ and $M_2$. Which moments vanish?

Solution

Orthogonality removes the constant term, and $\int P_2^2d\Omega=4\pi/5$. Therefore

$$I_2=\frac14\int_S \frac{2\varepsilon}{R_\Delta^2}P_2Y_{20}\,d^2V =\varepsilon\sqrt{\frac{\pi}{5}}.$$

The physical quadrupole is

$$M_2=\sqrt{\frac{4\pi}{5}} \frac{M_\Delta R_\Delta^2}{2\pi}I_2 =\frac{\varepsilon}{5}M_\Delta R_\Delta^2.$$

All $L_n$ and hence all $J_n$ vanish because $\operatorname{Im}\Phi_H=0$. At the displayed order, orthogonality also makes all mass moments except the fixed monopole and $M_2$ vanish. A complete metric must still satisfy the regularity constraints; this calculation isolates the moment carried by the stated curvature perturbation.

Slicing, Foliation, and Quasi-Local Ambiguity next asks which features survive when the choice of marginal tube or spacetime slicing changes.

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.
• Gourgoulhon, Eric, Alexandre Le Tiec, and Marc Casals. “Horizon Multipole Moments of a Kerr Black Hole.” Physical Review D 114 (2026): 024014. DOI. Open PDF.