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

We consider the third order Lovelock equations without the cosmological constant term in an empty $n(\ge 8)$-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®] 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.