Metrics with Zero and Almost-Zero Einstein Action in Quantum Gravity

Abstract

We generate numerically on a lattice an ensemble of stationary metrics, with spherical symmetry, which have Einstein action SE « ћ. This is obtained through a Metropolis algorithm with weight exp(−β2S2E) and β » ћ−1. The squared action in the exponential allows to circumvene the problem of the non-positivity of SE. 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 SE = 0 exactly, or in certain cases even the stronger condition R(x) = 0 for any 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”.