Foliated asymptotically safe gravity in the fluctuation approach

TL;DR

This paper investigates the asymptotic safety of quantum gravity using the ADM decomposition and the Wetterich equation, demonstrating a non-Gaussian fixed point and phase structure consistent with covariant approaches, thus supporting the viability of foliated spacetime models.

Contribution

First fluctuation computation of asymptotically safe gravity within the ADM framework, establishing the existence of a suitable non-Gaussian fixed point and phase diagram with foliation structure.

Findings

Identification of a non-Gaussian fixed point for gravity in the ADM setting

Phase diagram governed by interplay of fixed points ensuring positive graviton mass

Qualitative agreement with covariant fluctuation computations

Abstract

The gravitational asymptotic safety program envisions a high-energy completion of gravity based on a non-Gaussian renormalization group fixed point. A key step in this program is the transition from Euclidean to Lorentzian signature spacetimes. One way to address this challenge is to formulate the quantum theory based on the Arnowitt-Deser-Misner decomposition of the metric field. This equips the Euclidean spacetime with a preferred direction which may serve as the time-direction in the Lorentzian setting. In this work we use the Wetterich equation in order to compute the renormalization group flow of the graviton two-point function. The resulting beta functions possess a non-Gaussian renormalization group fixed point suitable for rendering the theory asymptotically safe. The phase diagram underlying the flow of the two-point function is governed by the interplay between this non-Gaussian fixed point, the Gaussian fixed point, and an infrared fixed point. The latter ensures that the renormalized squared graviton mass cannot take negative values. These results are in qualitative agreement with fluctuation computations carried out in the covariant setting. We take this as non-trivial evidence that the asymptotic safety mechanism remains intact when considering quantum gravity on spacetimes carrying a foliation structure. Technically, our work constitutes the first fluctuation computation carried out within the ADM-framework. Therefore, we also provide a detailed discussion of the conceptual framework, highlighting the elements which differ from fluctuation computations in the covariant setting.