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

TL;DR

This paper numerically constructs stationary, spherically symmetric metrics with near-zero Einstein action in quantum gravity, revealing configurations that differ from traditional spacetime foam models and may represent vacuum fluctuations.

Contribution

It introduces a novel numerical method to generate and analyze metrics with extremely small Einstein action, expanding understanding of quantum gravitational vacuum states.

Findings

Generated metrics with $S_E o 0$ using a new Monte Carlo approach.

Discovered spontaneous polarization in scalar curvature regions.

Compared these metrics with known zero-action solutions, highlighting differences.

Abstract

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 $\exp(-\beta^2 S^2_E)$ and $\beta \gg \hbar^{-1}$. 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".