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(⩾8)-dimensional Kaluza-Klein spacetime M4×Kn−4, where Kn−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 M4. 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, N. Dadhich, Phys. Rev. D 74 (2006) 021501(R)] for Gauss-Bonnet gravity in the presence of cosmological constant, and therefore the metric of M4 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.