Non-Expanding horizons: Multipoles and the Symmetry Group

TL;DR

This paper extends the understanding of non-expanding horizons by introducing multipole moments without axisymmetry, characterizes their symmetry group as an extension of BMS, and discusses implications for asymptotic structures and gravitational wave analysis.

Contribution

It introduces multipole moments for NEHs beyond axisymmetry and characterizes their symmetry group as a 1D extension of the BMS group, with implications for gravitational wave studies.

Findings

NEHs can be characterized by multipole moments without axisymmetry.

The symmetry group of NEHs is a 1D extension of the BMS group.

Asymptotic infinity can be viewed as a NEH with additional structure.

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 $\mathcal{I}^+$.