  In the Ashtekar and geometrodynamic formulations of vacuum general
relativity, the Euclidean and Lorentzian sectors can be related by means of the
generalized Wick transform discovered by Thiemann. For some vacuum
gravitational systems in which there exists an intrinsic time variable which is
not invariant under constant rescalings of the metric, we show that, after such
a choice of time gauge and with a certain identification of parameters, the
generalized Wick transform can be understood as an analytic continuation in the
explicit time dependence. This result is rigorously proved for the Gowdy model
with the topology of a three-torus and for a whole class of cosmological models
that describe expanding universes. In these gravitational systems, the analytic
continuation that reproduces the generalized Wick transform after gauge fixing
turns out to map the Euclidean line element to the Lorentzian one multiplied by
an imaginary factor; this transformation rule differs from that expected for an
inverse Wick rotation in a complex rescaling of the four-metric. We then prove
that this transformation rule for the line element continues to be valid in the
most general case of vacuum gravity with no model reduction nor gauge fixing.
In this general case, it is further shown that the action of the generalized
Wick transform on any function of the gravitational phase space variables, the
shift vector, and the lapse function can in fact be interpreted as the result
of an inverse Wick rotation and a constant, imaginary conformal transformation.
