Cosmological constant and AdS spacetimes from Minkowski spheres

Abstract

A theory of gravity without masses can be constructed starting from Minkowski [Formula: see text] spaces. The corresponding adapted [Formula: see text] 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 [Formula: see text] space, the non-Euclidean geometry and the difference between Minkowski [Formula: see text] sphere and the pseudosphere seen as a surface of the [Formula: see text]-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 [Formula: see text] gravity.