Abstract
In this paper we perform systematic investigation of all possible solutions with static compact extra dimensions and expanding three-dimensional subspace (“our Universe”). Unlike previous papers, we consider extra-dimensional subspace to be constant-curvature manifold with both signs of spatial curvature. We provide a scheme how to build solutions in all possible number of extra dimensions and perform stability analysis for the solutions found. Our study suggests that the solutions with negative spatial curvature of extra dimensions are always stable while those with positive curvature are stable for a narrow range of the parameters and the width of this range shrinks with growth of the number of extra dimensions. This explains why in the previous papers we detected compactification in the case of negative curvature but the case of positive curvature remained undiscovered. Another interesting feature which distinguish cases with positive and negative curvatures is that the latter do not coexist with maximally-symmetric solutions (leading to “geometric frustration” of a sort) while the former could – this difference is noted and discussed.
Highlights
Action has additional term quadratic in the curvature with respect to general relativity exists for space-time dimensions d ≥ 5
In the case that curvature of the compact dimensions is negative and the couplings of the EGB are chosen from the open region of couplings space where geometric frustration occurs it was shown that realistic compactification scenarios exist where the scale factor of the extra dimensions tends to a constant
No realistic compactification scenario was found for the case when the extra dimensions have positive curvature or when there is no geometric frustration
Summary
EGB gravity, whose action holds -term, Einstein– Hilbert term and quadratic Gauss-Bonnet term, has the remarkable feature that it can possess up to two independent maximally symmetric solutions (even with different sign of the curvature scale). The discriminant of the quadratic equation for the curvature scale can be negative for a range of values of the coupling constants In this case there exist no maximally symmetric space-time solution at all. Zumino [14] extended Zwiebach’s result on higher-than-squared curvature terms, supporting the idea that the low-energy limit of the unified theory should have a Lagrangian density as a sum of contributions of different powers of curvature In this regard the Einstein–Gauss–Bonnet (EGB) gravity could be seen as a subcase of more general Lovelock gravity [1], but in the current paper we restrain ourselves with only quadratic corrections and so to the EGB case. We summarize the results, discuss them and draw conclusions
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