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Given a Lorentzian spacetime $(M, g)$ and a non-vanishing timelike vectorfield $u(\lambda)$ with level surfaces $\Sigma$, one can construct on $M$ aEuclidean metric $g_E^{ab} = g^{ab} + 2 u^a u^b$. Motivated by this, weconsider a class of metrics $\hat{g}^{ab} = g^{ab} - \Theta(\lambda)\, u^a u^b$with an arbitrary function $\Theta$ that interpolates between the Euclidean($\Theta=-2$) and Lorentzian ($\Theta=0$) regimes. The Euclidean regime is ingeneral different from that obtained from Wick rotation $t \rightarrow - i t$.For example, if $g_{ab}$ is the $k=0$ Lorentzian de Sitter metric correspondingto $\Lambda>0$, the Euclidean regime of $\hat{g}_{ab}$ is the $k=0$ Euclideananti-de Sitter space with $\Lambda<0$. We analyze the curvature tensorsassociated with $\hat{g}$ for arbitrary Lorentzian metrics $g$ and timelikegeodesic fields $u^a$, and show that they have interesting and remarkablemathematical structures: (i) Additional terms arise in the Euclidean regime$\Theta \to -2$ of $\hat{g}_{ab}$. (ii) For the simplest choice of a stepprofile for $\Theta$, the Ricci scalar Ric$[\widehat{g}]$ of $\hat{g}_{ab}$reduces, in the Lorentzian regime $\Theta \to 0$, to the completeEinstein-Hilbert lagrangian with the correct Gibbons-Hawking-York boundaryterm; the latter arises as a delta-function of strength $2K$ supported on$\Sigma_0$. (iii) In the Euclidean regime $\Theta \to -2$, Ric$[\hat{g}]$ alsohas an extra term $2\, {}^3 R$ of the $u$-foliation. We highlight similarfoliation dependent terms in the full Riemann tensor. We present some explicitexamples and briefly discuss implications of the results for Euclidean quantumgravity and quantum cosmology.

Action and Observer dependence in Euclidean quantum gravity