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

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 $\beta$.