Metrics with zero and almost-zero Einstein action in quantum gravity

We generate numerically on a lattice an ensemble of stationary metrics, with spherical symmetry, which have Einstein action $S_E\ll \mathrm{\hslash }$. This is obtained through a Metropolis algorithm with weight $\exp (-{\beta }^2{S}_E^2)$ and $\beta \gg {\mathrm{\hslash }}^{-1}$. The squared action in the exponential allows to circumvent the problem of the non-positivity of $S_E$. The discretized metrics obtained exhibit a spontaneous polarization in regions of positive and negative scalar curvature. We compare this ensemble with a class of continuous metrics previously found, which satisfy the condition $S_E=0$ exactly, or in certain cases even the stronger condition $R(\mathbf{x})=0$ for any $\mathbf{x}$. All these gravitational field configurations are of considerable interest in quantum gravity, because they represent possible vacuum fluctuations and are markedly different from Wheeler’s “spacetime foam”.