Action and observer dependence in Euclidean quantum gravity

Abstract

Given a Lorentzian spacetime and a non-vanishing timelike vector field with level surfaces Σ, one can construct on a Euclidean metric (Hawking and Ellis 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge University Press)). Motivated by this, we consider a class of metrics with an arbitrary function Θ that interpolates between the Euclidean () and Lorentzian () regimes, separated by the codimension one hypersurface defined by . Since can not, in general, be obtained from by a diffeomorphism, its Euclidean regime is in general different from that obtained from Wick rotation . For example, if is the k = 0 Lorentzian de Sitter metric corresponding to , the Euclidean regime of is the k = 0 Euclidean anti-de Sitter space with .We analyze the curvature tensors associated with for arbitrary Lorentzian metrics and timelike geodesic fields , and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime of . (ii) For the simplest choice of a step-profile for Θ, the Ricci scalar Ric of reduces, in the Lorentzian regime , 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 . (iii) In the Euclidean regime , Ric also has an extra term of the -foliation. We highlight similar foliation dependent terms in the full Riemann tensor.We present some explicit examples for FLRW spacetimes in standard foliation and spherically symmetric spacetimes in the Painleve–Gullstrand foliation. We briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.