Action and Observer dependence in Euclidean quantum gravity

Abstract

Given a Lorentzian spacetime $(M, g)$ and a non-vanishing timelike vector field $u(λ)$ with level surfaces $Σ$, one can construct on $M$ a Euclidean metric $g_E^{ab} = g^{ab} + 2 u^a u^b$. Motivated by this, we consider a class of metrics $\hat{g}^{ab} = g^{ab} - Θ(λ)\, u^a u^b$ with an arbitrary function $Θ$ that interpolates between the Euclidean ($Θ=-2$) and Lorentzian ($Θ=0$) regimes. The Euclidean regime is in general different from that obtained from Wick rotation $t \rightarrow - i t$. For example, if $g_{ab}$ is the $k=0$ Lorentzian de Sitter metric corresponding to $Λ>0$, the Euclidean regime of $\hat{g}_{ab}$ is the $k=0$ Euclidean anti-de Sitter space with $Λ<0$. We analyze the curvature tensors associated with $\hat{g}$ for arbitrary Lorentzian metrics $g$ and timelike geodesic fields $u^a$, and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime $Θ\to -2$ of $\hat{g}_{ab}$. (ii) For the simplest choice of a step profile for $Θ$, the Ricci scalar Ric$[\widehat{g}]$ of $\hat{g}_{ab}$ reduces, in the Lorentzian regime $Θ\to 0$, to the complete Einstein-Hilbert lagrangian with the correct Gibbons-Hawking-York boundary term; the latter arises as a delta-function of strength $2K$ supported on $Σ_0$. (iii) In the Euclidean regime $Θ\to -2$, Ric$[\hat{g}]$ also has an extra term $2\, {}^3 R$ of the $u$-foliation. We highlight similar foliation dependent terms in the full Riemann tensor. We present some explicit examples and briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.