On Singularities and Instability for Different Couplings between Scalar Field and Multidimensional Geometry

TL;DR

This paper investigates the stability and singularities of multidimensional universe models with scalar fields, analyzing how different coupling constants affect classical solutions and their equivalence to minimal coupling models.

Contribution

It identifies critical coupling values that separate stable and unstable solution regimes in multidimensional scalar-tensor gravity models.

Findings

Critical coupling constants determine solution stability.

Existence of conformal equivalence between models with different couplings.

Instability occurs only within specific coupling ranges.

Abstract

We consider a multidimensional model of the universe given as a $D$-dimensional geometry, represented by a Riemannian manifold $(M,g)$ with arbitrary signature of $g$, $M= \R\times M_1\times \cdots \times M_n$, where the $M_i$ of dimension $d_i$ are Einstein spaces, compact for $i>1$. For Lagrangian models $L(R,\phi)$ on $M$ which depend only on the Ricci curvature $R$ and a scalar field $\phi$, there exists a conformal equivalence with minimal coupling models. For certain nonminimal models we study classical solutions and their relation to solutions in the equivalent minimal coupling model. The domains of equivalence are separated by certain critical values of the scalar field $\phi$. Furthermore, the coupling constant $\xi$ of the coupling between $\phi$ and $R$ is critical at both, the minimal value $\xi=0$ and the conformal value $\xi_c=\frac{D-2}{4(D-1)}$. In different noncritical regions of $\xi$ the solutions behave qualitatively different. Instability can occure only in certain ranges of $\xi$. {This paper is dedicated to Prof. D. D. Ivanenko.}