Dynamics of Dimensions in Factor Space Cosmology

TL;DR

This paper explores multidimensional cosmological models where the dimensions of factor spaces are dynamic and fractal-like, analyzing their classical and quantum behaviors with a focus on models with two factor spaces.

Contribution

It introduces a framework where factor space dimensions are treated as dynamical variables, extending traditional models with fractal and time-dependent dimensions.

Findings

Classical equations of motion derived for variable-dimensional factor spaces

Quantum Wheeler-de Witt equation formulated in this context

Qualitative analysis of models with two factor spaces

Abstract

We consider multidimensional cosmological models with a generalized space-time manifold M = R x M_1 ...x M_n, composed from a finite number of factor spaces M_i, i=1,..n. While usually each factor space M_i is considered to be some Riemannian space of integer dimension d_i, here it is, more generally, a fractal space, the dimension of which is a smooth function d_i(t) of time. Hence, besides the scale factor exponents ln a_i and their derivatives, we consider also the dimensions d_i of the factor spaces as classical dynamical variables. The classical equation of motions and the corresponding Wheeler-de Witt equation are set up generally, and the qualitative behaviour of the system is discussed for some specific model with 2 factor spaces.