Abstract
We propose a procedure for the D→4 limit of Einstein-Gauss-Bonnet (EGB) gravity that leads to a well defined action principle in four dimensions. Our construction is based on compactifying D-dimensional EGB gravity on a (D−4)-dimensional maximally symmetric space followed by redefining the Gauss-Bonnet coupling α→αD−4. The resulting model is a special scalar-tensor theory that belongs to the family of Horndeski gravity. Static black hole solutions in the scalar-tensor theory are investigated. Interestingly, the metric profile is independent of the curvature of the internal space and coincides with the D→4 limit of the usual EGB black hole with the unusual Gauss-Bonnet coupling αD−4. The curvature information of the internal space is instead encoded in the profile of the extra scalar field. Our procedure can also be generalized to define further limits of the Gauss-Bonnet combination by compactifying the D-dimensional theory on a (D−p)-dimensional maximally symmetric space with p≤3. These lead to different D→4 limits of EGB gravity as well as its D→2,3 limits.
Highlights
The resulting model is a special scalar-tensor theory that belongs to the family of Horndeski gravity
Half a century ago, Lovelock showed that Einstein gravity could be extended by an infinite series of higher curvature terms such that the equations of motion remain second order [1]
Our procedure begins with the compactification of D-dimensional EGB gravity on a maximally symmetric space of (D − 4) dimensions, keeping only the breathing mode characterizing the size of the internal space
Summary
Half a century ago, Lovelock showed that Einstein gravity could be extended by an infinite series of higher curvature terms such that the equations of motion remain second order [1]. Our procedure begins with the compactification of D-dimensional EGB gravity on a maximally symmetric space of (D − 4) dimensions, keeping only the breathing mode characterizing the size of the internal space This Kaluza-Klein reduction ansatz is guaranteed be consistent in that the field equations of the four dimensional theory are compatible with those in higher dimensions. After removing a total derivative term, the limit (5) can be smoothly applied, resulting a finite action taking the form of some special Horndeski gravity [14] or generalized Galileons [15,16], i.e. a non-minimally coupled scalar-tensor theory with at most second order field equations. The D → 4 limit of the EGB black holes (3) obtained through (5) emerge as solutions independent of the curvature of the “internal” space on which original EGB gravity is compactified. Both classes include new D → 4 limits that yield lower p-dimensional theories
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