Solutions of Einstein Field Equation for an Extra-Dimensional Anisotropic Metric with Two Scale Factors

TL;DR

This paper explores solutions to Einstein's field equations in a higher-dimensional anisotropic universe with two different scale factors, analyzing the effects of extra dimensions and their pressure differences on cosmological dynamics.

Contribution

It derives higher-dimensional Friedmann-like equations for an anisotropic universe with compact extra dimensions and investigates the influence of pressure differences and dimensionality on gravitational behavior.

Findings

Extra dimensions can act as an effective matter source.

Pressure differences between dimensions affect cosmic evolution.

Estimated limits for higher-dimensional coupling constants.

Abstract

The manuscript studies a 3+N+1-dimensional space in which the N extra dimensions are dynamically compact. The 3 large dimensions, behaving as the spacial part of the FRW metric, possess a different scale factor in comparison with the N extra ones, making the whole space anisotropic. The possible effects caused by the existence of a common time-like coordinate between the compact dimensions and our 3-dimensional hypersurface are investigated. The higher dimensional Friedmann-Like equations of the mentioned model are achieved. The continuity equation is reached at the special case of 3+4+1-dimensional metric. It is shown that not only the existence of the extra dimensions itself but also the pressure difference between the 3-dimensional hypersurface and the compact dimensions might get probed on the hypersurface as an additive source of gravity with the same behavior as baryonic matter. Furthermore, the relation between the coupling constant of the higher-dimensional universe and the Newton's constant of gravitation is investigated to reach an estimated limit for it. As another aim, the literature studies the role of dimensionality on the behavior of the higher-dimensional Friedmann equations.