Horizon Multipole Moments of a Kerr Black Hole

Abstract

The horizon multipole moments of a Kerr black hole are computed from two distinct definitions that have been proposed in the literature. The first one [Ashtekar et al., Class. Quantum Grav. 21, 2549 (2004)] regards axisymmetric isolated horizons, while the second one [Ashtekar et al., J. High Energ. Phys. 2022, 28 (2022)] applies to generic (i.e., not necessarily axisymmetric) non-expanding horizons. We review these definitions in a common frame and perform a detailed study of the resulting multipole moments for the Kerr event horizon. The horizon multipoles are found to share several properties with the (Hansen) field multipoles, including parity constraints and the leading scaling behavior with respect to the Kerr spin parameter a in the regime of small a. For the axisymmetry-based definition, we have obtained a closed-form expression of the multipole moments in terms of a and the spherical harmonic degree l. For the generic definition, we have established closed-form expressions for the conformal unit round metric, the `electric' and `magnetic' potentials related to the multipoles, and the values of the multipoles in the small a limit. We show that the two definitions lead to different values of the Kerr horizon multipoles as soon as l >= 1 (generic nonzero value of a) or l >= 2 (small a limit).

Figures (16)

• Figure 1: Null hypersurface $\mathcal{H}$ of topology $\mathbb{R}\times\mathbb{S}^2$. Some of the null geodesic generators $L_{(\theta_1,\theta_2)}$ ruling $\mathcal{H}$ are depicted as green curves with tangent vectors $\bm{\ell}$, the latter being null normals to $\mathcal{H}$. $\mathcal{S}$ is a cross-section of $\mathcal{H}$; it is drawn as a curve (of topology $\mathbb{S}^1$), instead of a surface (of topology $\mathbb{S}^2$), due to the dimensional reduction of the graphic. The metric null cone is depicted at some point $p\in\mathcal{S}$; this cone is tangent to $\mathcal{H}$ along the null generator $L_{(\theta_1,\theta_2)}$ through $p$, with its past (resp. future) nappe lying outside (resp. inside) $\mathcal{H}$. $\bm{n}$ is a future-directed null vector transverse to $\mathcal{H}$ and normal to $\mathcal{S}$ at $p$, while $\Delta$ is an ingoing null geodesic admitting $\bm{n}$ as a tangent vector.
• Figure 2: Two cross-section slicings $(\mathcal{S}_v)_{v\in\mathbb{R}}$ and $(\mathcal{S}'_{v'})_{v'\in\mathbb{R}}$ of a null hypersurface $\mathcal{H}$, both defining the same null normal $\bm{\ell}$ to $\mathcal{H}$: $\ell^a = \mathrm{d} x^a / \mathrm{d} v = \mathrm{d} x^a / \mathrm{d} v'$ along the generators $L_{(\theta_1,\theta_2)}$ of $\mathcal{H}$.
• Figure 3: Isometry $\Phi$ between two arbitrary cross-sections $\mathcal{S}$ and $\mathcal{S}'$ of a NEH $\mathcal{H}$, each being endowed with the metric induced by the spacetime metric $g_{ab}$. The green vertical lines depict some null geodesic generators $L_{(\theta_1,\theta_2)}$ of $\mathcal{H}$.
• Figure 4: Function $G_\ell(\hat{a}^2)$ defined by Eq. \ref{['bidule']}, for selected values of $\ell$.
• Figure 5: Shape and current horizon multipole moments $I_\ell^{\rm axi}$ and $L_\ell^{\rm axi}$ of the Kerr black hole, resulting from the axisymmetry-based definition, as functions of the Kerr spin parameter $a$.