On the Nature of the Cosmological Constant Problem

TL;DR

This paper addresses the cosmological constant problem by proposing a geometric fix using Nash's theorem, which introduces extra dimensions and results in a more general gravitational theory that separates vacuum energy from the cosmological constant.

Contribution

It introduces a novel approach to the cosmological constant problem by applying Nash's theorem to incorporate extra dimensions, leading to a generalized gravity theory beyond general relativity.

Findings

Vacuum energy remains confined to four dimensions.

The cosmological constant propagates in extra dimensions.

The vacuum energy influences gravitational perturbations, not the cosmological constant.

Abstract

General relativity postulates the Minkowski space-time to be the standard flat geometry against which we compare all curved space-times and the gravitational ground state where particles, quantum fields and their vacuum states are primarily conceived. On the other hand, experimental evidences show that there exists a non-zero cosmological constant, which implies in a deSitter space-time, not compatible with the assumed Minkowski structure. Such inconsistency is shown to be a consequence of the lack of a application independent curvature standard in Riemann's geometry, leading eventually to the cosmological constant problem in general relativity. We show how the curvature standard in Riemann's geometry can be fixed by Nash's theorem on locally embedded Riemannian geometries, which imply in the existence of extra dimensions. The resulting gravitational theory is more general than general relativity, similar to brane-world gravity, but where the propagation of the gravitational field along the extra dimensions is a mathematical necessity, rather than being a a postulate. After a brief introduction to Nash's theorem, we show that the vacuum energy density must remain confined to four-dimensional space-times, but the cosmological constant resulting from the contracted Bianchi identity is a gravitational contribution which propagates in the extra dimensions. Therefore, the comparison between the vacuum energy and the cosmological constant in general relativity ceases to be. Instead, the geometrical fix provided by Nash's theorem suggests that the vacuum energy density contributes to the perturbations of the gravitational field.