Action and Observer dependence in Euclidean quantum gravity

Understand

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 $g_E^{ab} = g^{ab} + 2 u^a u^b$.

• 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 $t \rightarrow - i t$.
• 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 $\Lambda<0$.