Gromoll--Meyer's actions and the geometry of (exotic) spacetimes

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.