Euclidean Action and the Einstein tensor

Understand

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 $p_0$.