Higher dimensional spheres, black holes and cosmology

Abstract

We explain in sorne detail the geometric structure of spheres in any dimension. Our approach may be helpful for other homogeneous spaces (with other signatures) such as the de Sitter and anti-de Sitter spaces. As a particular case we consider the 1-Sphere. Assuming the light path as a l-Sphere (circle) in vacuum we develop the corresponding special theory of relativity. We show that the derived metric reproduces time dilation and the length contraction of the special theory of relativity. We argue that this is an interesting result from which one can derive both the Schwarzschild and the de Sitter metrics. Moreover, using a Lagrangian approach we set the bases for a future work towards a theory for black holes and cosmology models as unified concepts From this formalism we show that one can derive the field equations for both the FRW-cosmological model and the Schwarzschild black hole solution from a first order Lagrangian of a constrained system, which is derived from the Einstein -Hilbert action.