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 $\mathcal{A}[��_0]=\exp[{-I_E}]$ for a single space-like hypersurface $��_0 \in I^{+}(p_0)$ to emerge at a constant geodesic distance $��_0$ from $p_0$. Implications for classical and quantum gravity are discussed.

6 pages, 2 figures, added comments and expanded discussion of some implications, matches version accepted in Phys. Rev. D