Euclidean Action and the Einstein tensor

TL;DR

This paper relates the Euclidean Einstein-Hilbert action to the Einstein tensor in Lorentzian spacetimes, providing insights into the emergence of spacelike hypersurfaces and implications for gravity theories.

Contribution

It offers a local description of the Euclidean regime in Lorentzian spacetimes and links the Euclidean action to the Einstein tensor, with implications for gravity.

Findings

Euclidean action proportional to Einstein tensor component

Positivity of action linked to Einstein tensor positivity

Interpretation of action as amplitude for hypersurface emergence

Abstract

I give a local description of the Euclidean regime $(M, g_{ab}, u^a)$ of Lorentzian spacetimes $(M, g_{ab})$ based on timelike geodesics $u^a$ passing through an arbitrary event $p_0 \in M$. I show that, to leading order, the Euclidean Einstein-Hilbert action $I_E$ is proportional to the Einstein tensor $G_{ab}u^a u^b$ of $g_{ab}$. The positivity of $I_E$ follows if $G_{ab}u^a u^b>0$ holds. I suggest an interpretation of this result in terms of the amplitude $\mathcal{A}[\Sigma_0]=\exp[{-I_E}]$ for a single space-like hypersurface $\Sigma_0 \in I^{+}(p_0)$ to emerge at a constant geodesic distance $\lambda_0$ from $p_0$. Implications for classical and quantum gravity are discussed.