Abstract
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.
Highlights
High-precision observational data have confirmed with startling evidence that the universe is undergoing a phase of accelerated expansion [1]
In the standard cosmological model, where the acceleration of the universe is taken into account by a positive cosmological constant term, dark energy is responsible for the acceleration of the universe
An interesting possibility is the existence of extra dimensions
Summary
High-precision observational data have confirmed with startling evidence that the universe is undergoing a phase of accelerated expansion [1]. A very promising way to explain these major problems is to assume that at large scales, or higher dimensional spacetime, Einstein theory of General Relativity breaks down and a more general action should describe the gravitational field In this context, an interesting possibility is the existence of extra dimensions. In the context of black hole physics, we investigate the problem of having a 4-dimensional asymptotically anti-de Sitter (AdS) or de Sitter (dS) charged black hole in the absence of electromagnetic field and the cosmological constant term in the field equations of gravity This idea has been used for the Gauss-Bonnet gravity, but the authors were forced to keep the cosmological constant in higher dimensions which weakens the idea of empty higherdimensional spacetime [10, 11].
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