Cosmological constant and AdS spacetimes from Minkowski spheres

A theory of gravity without masses can be constructed starting from Minkowski M(p,q) spaces. The corresponding adapted (p,q) Minkowski potentials, gradients and Laplacians built on each signature lead to a field equation similar to the Newton-Laplace one. In this framework, the anti-de Sitter spacetime is a hypersurface described by a constant potential of Minkowski gravitational force in absence of matter. We show that the cosmological constant of the anti-de Sitter spacetimes is related to a geometric property of Minkowski spheres: the centro-affine conservation of volumes defined from the so-called Minkowski-Tzizteica affine spheres. It is possible to show the connection between the anti-de Sitter AdS(2, 3) space, the non-Euclidean geometry and the difference between Minkowski M(2,3) sphere and the pseudosphere seen as a surface of the 3-Euclidean space. According to a suitable parameterization, the relation between de Sitter and anti-de Sitter spacetimes in any dimension is fixed. As a consequence, the light can travel in any anti-de Sitter spacetime defining the geodesic structure. These considerations can be recast in terms of Einstein field equations with cosmological constant and extended to f(R) gravity.