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 $S_E \ll \hbar$. This is obtained through a Metropolis algorithm with weight $\exp(-β^2 S^2_E)$ and $β\gg \hbar^{-1}$. 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 $R({\bf x})=0$ for any ${\bf x}$. 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".

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