Non-Expanding horizons: Multipoles and the Symmetry Group

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 $\mathcal{I}^+$.