  Since the advent of new pairwise non-diffeomorphic structures on smooth
manifolds, it has been questioned whether two topologically identical manifolds
could admit different geometries. Not surprisingly, physicists have wondered
whether a smooth structure assumption different from some classical known
models could produce different physical meanings. In this paper, we inaugurate
a very computational manner to produce physical models on classical and exotic
spheres that can be built equivariantly, such as the classical Gromoll--Meyer
exotic spheres. As first applications, we produce Lorentzian metrics on
homeomorphic but not diffeomorphic manifolds that enjoy the same physical
properties, such as geodesic completeness, positive Ricci curvature, and
compatible time orientation. These constructions can be pulled back to higher
models, such as exotic ten spheres bounding spin manifolds, to be approached in
forthcoming papers.
