Charges and Fluxes on (Perturbed) Non-expanding Horizons

TL;DR

This work develops a covariant phase space framework for general relativity with an internal null boundary that is an NEH or a perturbed NEH, introducing a 1-parameter NEH symmetry group $\mathfrak{G}$ and defining charges and fluxes for its generators. By fixing a preferred symplectic potential on the boundary and extending boundary symmetries into the bulk, the authors obtain finite, physically meaningful horizon charges, which vanish for the NEH background and acquire positive-definite fluxes at second order for perturbed NEHs. The analysis yields explicit expressions for dilation, supertranslation, rotation, and boost charges, with Schwarzschild and Kerr providing consistency checks against Komar and ADM quantities; importantly, fluxes are nonnegative for perturbations, aligning with horizon energy balance. The results pave the way for gravitational-wave tomography from null infinity data and motivate extensions to dynamical horizons and potential quantum considerations, while clarifying the role of the NEH boundary in avoiding spurious Minkowski-like charges present in more general null-boundary formalisms.

Abstract

In a companion paper we showed that the symmetry group $\mathfrak{G}$ of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group $\mathfrak{G}$ at $\mathcal{I}^{+}$. For each infinitesimal generator of $\mathfrak{G}$, we now define a charge and a flux on NEHs as well as perturbed NEHs. The procedure uses the covariant phase space framework in presence of internal null boundaries $\mathcal{N}$. However, $\mathcal{N}$ is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of $\mathfrak{G}$ are free of physically unsatisfactory features that can arise if $\mathcal{N}$ is allowed to be a general null boundary. In particular, all fluxes vanish if $\mathcal{N}$ is an NEH, just as one would hope; and fluxes associated with symmetries representing `time-translations' are positive definite on perturbed NEHs. These results hold for zero as well as non-zero cosmological constant. In the asymptotically flat case, as noted in \cite{akkl1}, $\mathcal{I}^\pm$ are NEHs in the conformally completed space-time but with an extra structure that reduces $\mathfrak{G}$ to $\mathfrak{B}$. The flux expressions at $\mathcal{N}$ reflect this synergy between NEHs and $\mathcal{I}^{+}$. In a forthcoming paper, this close relation between NEHs and $\mathcal{I}^{+}$ will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at $\mathcal{I}^{+}$.

Figures (1)

• Figure 1: Part of the space-time that is relevant for the sub-manifold ${\Gamma}_{\rm cov}^{\rm NEH}$ of the full phase space ${\Gamma}_{\rm cov}$ that is of primary interest to our discussion. The inner boundary $\Delta$ is an NEH for each solution $g_{ab}$ in ${\Gamma}_{\rm cov}^{\rm NEH}$. $\Sigma_1$ and $\Sigma_2$ are partial Cauchy surfaces that intersect $\Delta$ in 2-spheres $\partial\Sigma_1$ and $\partial\Sigma_2$, respectively. Although this figure includes the asymptotic region including $\mathcal{I}^{+}$ and $i^o$, only a neighborhood of $\Delta$ is relevant for our detailed calculations.