Action and Observer dependence in Euclidean quantum gravity

Abstract

Given a Lorentzian spacetime $(M,g)$ and a non-vanishing timelike vector field $u(\lambda )$ with level surfaces $\Sigma$, one can construct on $M$ a Euclidean metric $g_E^{ab}=g^{ab}+2u^a{u}^b$. Motivated by this, we consider 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 in general different from that obtained from Wick rotation $t\to -it$. For example, if $g_{ab}$ is the $k=0$ Lorentzian de Sitter metric corresponding to $\Lambda >0$, the Euclidean regime of ${\hat{g}}_{ab}$ is the $k=0$ Euclidean anti-de Sitter space with $\Lambda <0$. 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 $\Theta \to -2$ of ${\hat{g}}_{ab}$. (ii) For the simplest choice of a step profile for $\Theta$, the Ricci scalar Ric$[\hat{g}]$ of ${\hat{g}}_{ab}$ reduces, in the Lorentzian regime $\Theta \to 0$, 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 ${\Sigma }_0$. (iii) In the Euclidean regime $\Theta \to -2$, 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.

v2: discussions and references added; to appear in CQG Letters

References in corpus (3)

• Grin of the Cheshire cat: Entropy density of spacetime as a relic from quantum gravity
• How to Wick rotate generic curved spacetime
• Intrinsic and Extrinsic curvatures in Finsler-esque spaces

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