Algebraic Geometry Approach in Theories with Extra Dimensions II. Tensor Length Scale, Compactification and Rescaling

TL;DR

This paper introduces the concept of tensor length scale and explores rescaling and compactification in theories with extra dimensions, deriving and solving differential equations for scale functions in string theory contexts.

Contribution

It presents a novel physical notion of tensor length scale and analyzes rescaling combined with compactification, deriving solutions for scale functions in extra-dimensional models.

Findings

Derived quasilinear differential equations for scale functions.

Obtained explicit solutions for flat 4D Minkowski space embedded in 5D.

Investigated scalar curvature equality under rescaling.

Abstract

In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor components. Then the notion of "compactification" in low energy type I string theory is supplemented by the operation of "rescaling" of the contravariant metric components. For both the cases of "rescaling + compactification" and "compactification + rescaling", quasilinear differential equations in partial derivatives have been obtained and the corresponding solutions have been found for the scale (length) function and for the case of a flat 4D Minkowski space, embedded into a 5D space with an exponential warp factor. A differential equation has been obtained and investigated also from the equality of the "rescaled" scalar curvature with the usual one.