  The Lorentzian metric structure used in any field theory allows one to
implement the relativistic notion of causality and to define a notion of time
dimension. This article investigates the possibility that at the microscopic
level the metric is Riemannian, i.e. locally Euclidean, and that the Lorentzian
structure, that we usually consider as fundamental, is in fact an effective
property that emerges in some regions of a 4-dimensional space with a positive
definite metric. In such a model, there is no dynamics nor signature flip
across some hypersurface; instead, all the fields develop a Lorentzian dynamics
in these regions because they propagate in an effective metric. It is shown
that one can construct a decent classical field theory for scalars, vectors and
(Dirac) spinors in flat spacetime. It is then shown that gravity can be
included but that the theory for the effective Lorentzian metric is not general
relativity but of the covariant Galileon type. The constraints arising from
stability, the equivalence principle and the constancy of fundamental constants
are detailed and a phenomenological picture of the emergence of the Lorentzian
metric is also given. The construction, while restricted to classical fields in
this article, offers a new view on the notion of time.
