Localization and anomalous reference frames in gravity

TL;DR

This work develops a localized, gauge-invariant phase space for gravitational degrees of freedom along a null ray by introducing a dynamical dressing time built from spin-0 data and edge modes, enabling observables on a null-ray segment that commute with those on its complement. By decoupling the tree-level Raychaudhuri constraint and employing embedding/dressing fields, the authors construct a robust segment-based formalism with an explicit dressing map linking gauge-fixed and gauge-invariant observables. They then extend the classical framework to an effective theory that incorporates quantum diffeomorphism anomalies, deforming the Raychaudhuri equation and symplectic form into Virasoro-type structures with central charges, and identifying three diffeomorphism actions—reparametrizations, reorientations, and dressed reparametrizations—each with distinct central extensions. The work shows how edge modes and the Virasoro coadjoint orbit structure provide a foundation for quantizing gravitational null-ray segments and for treating the dressing time as a genuine quantum reference frame, with implications for locality, horizon thermodynamics, and the generalized second law in a quantum gravitational setting.

Abstract

In this work, we study the classical phase space for the gravitational degrees of freedom along a null ray. We construct gauge-invariant observables localized on a null ray segment that commute with those localized on the complement; thus, the phase space describes a genuine gravitational subsystem compatible with both locality and diffeomorphism invariance. Our construction employs 'dressing time' (a null time coordinate built from spin 0 gravitational degrees of freedom) as a dynamical reference frame. The existence of such a frame depends on the use of edge mode variables, which we argue are generally required to upgrade a local gauge-fixing condition to a global 'frame-fixing'. To analyze the effects of quantum diffeomorphism anomalies on these structures, we then establish an 'effective' classical description in which the Raychaudhuri equation, symplectic form, and edge mode variables all acquire Virasoro-type deformations. Within this framework, we identify three distinct diffeomorphism actions: reparametrizations (gauge transformations), reorientations (physical symmetries of the reference frame), and dressed reparametrizations. Each acquires its own central extension and plays a different crucial role in the effective theory. The resulting structures provide a foundation for quantizing gravitational null ray segments, including promoting dressing time to a genuine quantum reference frame.

Figures (3)

• Figure 2.1: In this paper, we construct an effective classical description of the null ray segments $\mathcal{I}$ of a gravitational caustic-free null surface $\mathcal{N}$. The degrees of freedom are those of matter $\varphi$, perturbative spin 2 gravitational radiation $h_{ab}$, the area element $\Omega$ and its conjugate momentum $\beta$ on cuts $\mathcal{C}$ of the null surface, as well as edge modes: the locations of the endpoints $v_0,v_1$ of the segment, and boost frames $\eta_0,\eta_1$ at each endpoint.
• Figure 2.2: We use embedding fields $X:I\to \mathcal{I}$ and $\bar{X}:\bar{I}\to \bar{\mathcal{I}}$ to construct the phase space of the null ray segment $\mathcal{I}$. Here, $I=[0,1]\in\amsmathbb{R}$ is a reference interval for $\mathcal{I}$ via $X$, while the complement $\bar{I}$ is a reference for the complement $\bar{\mathcal{I}}$ via $\bar{X}$. On $I\cup\bar{I}$ we use the coordinate $\mv$, while on $\mathcal{I}\cup\bar{\mathcal{I}}$ we use the coordinate $v$. We will take $X$ to be the inverse of the 'dressing time' $V:\mathcal{I}\to I$, but leave $\bar{X}$ general, only requiring $X(a)=\bar{X}(a)=v_a$ at $a=0,1\in\partial I$. The mismatches between the derivatives of $X$ and $\bar{X}$ at $\partial I$ manifest as edge modes in the segment phase space.
• Figure 3.1: A naïve canonical quantization of the tree-level theory results in a non-zero central charge $c\ne0$. The classical limit of this quantized theory is the effective theory. It replaces the original tree-level description.