Quantum metrics with very low action in $R+R^2$ gravity

TL;DR

This paper presents numerical simulations of Euclidean lattice quantum gravity with $R+R^2$ action, revealing localized metric oscillations and a non-trivial ground state, contributing to understanding quantum gravitational effects.

Contribution

The study introduces large-scale numerical simulations of $R+R^2$ gravity on a lattice, demonstrating the existence of localized oscillations and a non-trivial ground state in quantum gravity.

Findings

Localized metric oscillations at low temperature

Average metric differs from flat space

Scaling behavior of action and metric with lattice size

Abstract

We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as $r=hL_{Planck}$, with $h=0,1,...,N$ and $N$ up to $10^5$. The Lagrangian is of the form $\sqrt{g}(R+\alpha R^2)$ (in units $c=\hbar=G=1$) and the action is positive-definite, allowing the use of a standard Metropolis algorithm with update probability $\exp(-\beta \Delta S)$. By minimizing the $R+R^2$ action with respect to conformal modes, Bonanno and Reuter have recently found analytical evidence of a non-trivial "rippled" ground state resembling a kinetic condensate of QCD. Our simulations at low but finite temperature ($T=\beta^{-1}$) also display strong localized oscillations of the metric, whose total action $S$ remains $\ll \hbar$ thanks to the indefinite sign of $R$. The average metric $\langle g_{rr} \rangle$ is significantly different from flat space. The scaling properties of $S$ and $\langle g_{rr} \rangle$ are investigated in dependence on $N$ and $\beta$.