  In recent years, a growing momentum has been gained by the emergent gravity
framework. Within the latter, the very concepts of geometry and gravitational
interaction are not seen as elementary aspects of Nature but rather as
collective phenomena associated to the dynamics of more fundamental objects. In
this paper we want to further explore this possibility by proposing a model of
emergent Lorentzian signature and scalar gravity. Assuming that the dynamics of
the fundamental objects can give rise in first place to a Riemannian manifold
and a set of scalar fields we show how time (in the sense of hyperbolic
equations) can emerge as a property of perturbations dynamics around some
specific class of solutions of the field equations. Moreover, we show that
these perturbations can give rise to a spin-0 gravity via a suitable
redefinition of the fields that identifies the relevant degrees of freedom. In
particular, we find that our model gives rise to Nordstrom gravity. Since this
theory is invariant under general coordinate transformations, this also shows
how diffeomorphism invariance (albeit of a weaker form than the one of general
relativity) can emerge from much simpler systems.
