Non-Expanding horizons: Multipoles and the Symmetry Group

TL;DR

This work develops a gauge-invariant, axisymmetry-free characterization of non-expanding horizons (NEHs) by introducing horizon geometric and current multipoles derived from a universal NEH structure. It shows that the symmetry group preserving this universal NEH structure, $\mathfrak{G}$, is a one-dimensional extension of the Bondi–van der Burg–Metzner–Sachs (BMS) group, featuring a dilation alongside the BMS symmetries. The paper provides explicit definitions and reconstruction procedures for the multipoles $I_{\ell m}$ and $L_{\ell m}$, linking horizon geometry to intrinsic curvature and angular momentum via $\Re\Psi_2$ and $\Im\Psi_2$, and it sets the stage for future work on horizon charges, fluxes, and gravitational-wave tomography that connects strong-field horizon dynamics to asymptotic data at $\mathcal{I}^+$. By identifying null infinity $\mathcal{I}^\pm$ as NEHs in conformally completed spacetimes and outlining how their symmetry is reduced to BMS with extra structure, the work bridges local horizon geometry with global asymptotic symmetries, enabling new avenues for extracting physics from horizon dynamics.

Abstract

It is well-known that blackhole and cosmological horizons in equilibrium situations are well-modeled by non-expanding horizons (NEHs). In the first part of the paper we introduce multipole moments to characterize their geometry, removing the restriction to axisymmetric situations made in the existing literature. We then show that the symmetry group $\mathfrak{G}$ of NEHs is a 1-dimensional extension of the BMS group $\mathfrak{B}$. These symmetries are used in a companion paper to define charges and fluxes on NHEs, as well as perturbed NEHs. They have physically attractive properties. Finally, it is generally not appreciated that $\mathcal{I}^\pm$ of asymptotically flat space-times are NEHs in the conformally completed space-time. Forthcoming papers will (i) show that $\mathcal{I}^\pm$ have a small additional structure that reduces $\mathfrak{G}$ to the BMS group $\mathfrak{B}$, and the BMS charges and fluxes can be recovered from the NEH framework; and, (ii) develop gravitational wave tomography for the late stage of compact binary coalescences: reading-off the dynamics of perturbed NEHs in the strong field regime (via evolution of their multipoles), from the waveform at $\mathcal{I}^+$.

Figures (2)

• Figure 1: Examples of NEHs in the asymptotically flat (i.e. $\Lambda=0$) case. Left panel: Non-spherical gravitational collapse. Since the radiation falling into the black hole rapidly decreases at late time, the portion $\Delta$ of the event horizon is well modeled by a perturbed NEH. Right panel: Spherical collapse of a star followed by the collapse of a spherical shell some time later. $\Delta_1$ and $\Delta_2$ are both NEHs. The event horizon lies outside $\Delta_1$and grows to join on to $\Delta_2$, although there is no matter or gravitational radiation falling cross it.
• Figure 2: Examples of NEHs in solutions with positive cosmological constant $\Lambda$. Left panel: Depiction of a compact binary emitting gravitational waves. The future event horizon $E^+(i^-)$ of $i^-$ is an NEH because there is no incoming radiation. If the emitted radiation is weak, the past horizon $E^-(i^+)$ of $i^+$ can be regarded as a perturbed NEH. Right panel: Conformal diagram depicting a spherical collapse. Shaded (yellow) region corresponds to the collapsing spherical star. The dashed (red) lines with arrows represent integral curves of the 'static' Killing field. Lines at $45^\circ$ denote Killing horizons that are, in particular, NEHs. The inner NEH is the black hole horizon and the outer ones are cosmological horizons.