Euclidean action and the Einstein tensor

We give a local description of the Euclidean regime (M,g,u) of Lorentzian spacetimes (M,g) based on timelike geodesics u passing through an arbitrary event p0M. We show that, to leading order, the Euclidean Einstein-Hilbert action IE is proportional to the Einstein tensor G[g](u,u). The positivity of IE follows if G[g](u,u)>0 holds. We suggest an interpretation of this result in terms of the amplitude A[Σ0]=exp[-IE] for a single spacelike hypersurface Σ0I+(p0) to emerge at a constant geodesic distance λ0 from p0. Implications for classical and quantum gravity are discussed.