The programs of the Extended Relativity in C-spaces: towards the physical foundations of String Theory

TL;DR

This paper explores the framework of Extended Scale Relativity in C-spaces, unifying various dimensions and linking to string theory, black hole physics, and cosmology through a geometric approach involving poly-vector coordinates.

Contribution

It introduces a geometric formulation of C-spaces with poly-vector coordinates, extending relativity and connecting to string theory, black hole entropy, and cosmological phenomena.

Findings

Unified treatment of dimensions via poly-vector coordinates

Derivation of string uncertainty relations and black hole entropy corrections

Natural emergence of higher derivative gravity and cosmological implications

Abstract

An outline is presented of the Extended Scale Relativity (ESR) in C-spaces (Clifford manifolds), where the speed of light and the minimum Planck scale are the two universal invariants. This represents in a sense an extension of the theory developed by L. Nottale long ago. It is shown how all the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using poly-vector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,... In addition, we review the derivation of the minimal-length string uncertainty relations; the logarithmic corrections (valid in any dimension) to the black hole area-entropy relation. We also show how the higher derivative gravity with torsion and the recent results of kappa-deformed Poincare theories of gravity follow naturally from the geometry of C-spaces. In conclusion some comments are made on the cosmological implications of this theory with respect to the cosmological constant problem, the two modes of time, the expansion of the universe, number four as the average dimension of our world and a variable fine structure constant.