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 four-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. © 2013 American Physical Society.