Dark Energy and Matter in 4 Dimensions From an Empty Kaluza-Klein Spacetime

TL;DR

This paper demonstrates that higher-dimensional empty Kaluza-Klein spacetimes governed by Lovelock gravity can induce effective dark energy and matter in four dimensions, leading to black hole solutions without matter or cosmological constant.

Contribution

It shows that vacuum Lovelock equations in higher dimensions impose constraints that produce dark energy and matter effects in 4D, revealing a geometric origin for these phenomena.

Findings

Derived a 4D black hole solution without matter or cosmological constant.

Showed higher-dimensional curvature induces dark energy and matter in 4D.

Established the uniqueness of the 4D metric satisfying Lovelock vacuum equations.

Abstract

We consider the third order Lovelock equations without the cosmological constant term in an empty $n(\geq 8)$-dimensional Kaluza-Klein spacetime $\mathcal{M}^{4}\times \mathcal{K}^{n-4}$, where $\mathcal{K}^{n-4}$ is a constant curvature space. We show that the emptiness of the higher-dimensional spacetime imposes a constraint on the metric function(s) of 4-dimensional spacetime $\mathcal{M}^{4}$. We consider the effects of this constraint equation in the context of black hole physics, and find a black hole solution in 4 dimensions in the absence of matter field and the cosmological constant (dark energy). This solution has the same form as the 4-dimensional solution introduced in [H. Maeda and N. Dadhich, Phys. Rev. D 74 (2006) 021501(R)] for Gauss-Bonnet gravity in the presence of cosmological constant, and therefore the metric of $\mathcal{M}^{4}$ which satisfies the vacuum Lovelock equations in higher-dimensional Kaluza-Klein spacetime is unique. This black hole solution shows that the curvature of an empty higher-dimensional Kaluza-Klein spacetime creates dark energy and matter with non-traceless energy-momentum tensor in 4 dimensions.