Abstract
In this work we show that Einstein gravity in four dimensions can be consistently obtained from the compactification of a generic higher curvature Lovelock theory in dimension $D=4+p$, ($\mathrm{p}\ensuremath{\ge}1$). The compactification is performed on a direct product space ${\mathcal{M}}_{D}={\mathcal{M}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathcal{K}}^{p}$, where ${\mathcal{K}}^{p}$ is a Euclidean internal manifold of constant curvature. The process is carried out in such a way that no fine tuning between the coupling constants is needed. The compactification requires us to dress the internal manifold with the flux of suitable $p$-forms whose field strengths are proportional to the volume form of the internal space. We explicitly compactify Einstein-Gauss-Bonnet theory from dimension six to Einstein theory in dimension four and sketch out a similar procedure for this compactification to take place starting from dimension five. Several black $\mathrm{string}/p$-branes solutions are constructed, among which, a five dimensional asymptotically flat black string composed of a Schwarzschild black hole on the brane is particularly interesting. Finally, the thermodynamic of the solutions is described and we find that the consistent compactification modifies the entropy by including a constant term, which may induce a departure from the usual behavior of the Hawking-Page phase transition. New scenarios are possible in which large black holes dominate the canonical ensemble for all temperatures above the minimal value.
Highlights
The original Kaluza-Klein scheme [1,2] provides, via dimensional reduction to four dimensions, a consistent method to unify gravity and electromagnetism by starting from a purely geometrical higher dimensional theory [3]
In this work we show that Einstein gravity in four dimensions can be consistently obtained from the compactification of a generic higher curvature Lovelock theory in dimension D 1⁄4 4 þ p, (p ≥ 1)
Two fundamental assumptions are made: First, gravity is described by Einstein general relativity (GR) and second, the higher dimensional spacetime is empty
Summary
The original Kaluza-Klein scheme [1,2] provides, via dimensional reduction to four dimensions, a consistent method to unify gravity and electromagnetism by starting from a purely geometrical higher dimensional theory [3]. On might be tempted to dress the internal manifold with pforms, we have shown in [14] that this only eliminates the tuning of the cosmological constant with the Gauss-Bonnet coupling, but the internal manifold remains hyperbolic To solve these problems it is mandatory to include higher dimensional matter in the shape of p-forms nonminimally coupled to the curvature tensor, keeping the second order character of the theory as a guiding principle. First described by Horndeski in [15] and later generalized in [16] provides the only nonminimally coupled gauge-invariant electrodynamics with second order field equations which when going to flat spacetime reduces to Maxwell equations This model renders possible a generic compactification of EinsteinGauss-Bonnet gravity on an internal manifold of positive constant curvature [14], a fact that is extendable to any Lovelock theory.
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