Local gauge invariant operator on isometry breaking background

TL;DR

This work analyzes how to define local gauge invariant operators in gravity on backgrounds that spontaneously break isometries, using a Stückelberg-type mechanism that combines metric fluctuations along the broken direction with matter fluctuations. In quasi-$dS$ space, timelike isometry breaking yields the curvature perturbation ${\cal R}$ as a local gauge invariant scalar, while the graviton remains massless; analogous constructions are extended to spacelike isometries, yielding operators ${\cal V}_i$ and ${\cal S}$. A central tension emerges: despite having local gauge invariants, the locality of operators on a local region like an island is jeopardized by spacetime-point fluctuations that grow as $t^{1/2}$, requiring strong isometry breaking, $\epsilon_H \gg \kappa^2 H^2$, to remain reliable. The analysis connects to black hole information via islands and suggests that late-time depletion of fluctuations might demand new dynamics (e.g., higher dimensions) to preserve locality, thereby linking gauge-invariance, cosmological perturbations, and quantum gravity phenomenology.

Abstract

Whereas local field operators play the crucial role in reconciling quantum mechanics and special relativity, they are not trivially compatible with the diffeomorphism invariance of gravity. In order to address this issue, we consider the background geometry which breaks the isometry spontaneously. Then the local gauge invariant operator can be constructed through the Stückelberg mechanism, where the fluctuation of the metric in the direction of the isometry breaking combines with that of matter whose classical solution breaks the isometry. This is equivalent to introducing the clock and the rod to promote the local field operators to the gauge invariant ones. A typical example is the curvature perturbation in quasi-de Sitter space arising from the spontaneous breaking of the timelike isometry. We also discuss the features of the local gauge invariant operator when the spacelike isometry is spontaneously broken. Meanwhile, even if the local gauge invariant operators exist, it does not guarantee the reliable construction of the gauge invariant operators on the local region like the island, which is regarded as an essential ingredient to resolve the black hole information paradox. This is because the fluctuation of the spacetime point is accumulated in time, which in fact also gives rise to eternal inflation in quasi-de Sitter space. In order to suppress the fluctuation at late time, the isometry must be strongly broken by the background. In the case of the evaporating black hole, it may be achieved by the transition to the higher dimensional black hole.