Horndeski Gravity as the (D\rightarrow4) Limit of Gauss-Bonnet Gravity
In the study of higher-dimensional gravities, the identification of lower-dimensional limits that retain the essential features of the original theories is of significant interest. The paper by H. Lü and Yi Pang investigates a specific procedure to obtain a (D\rightarrow4) limit of Einstein-Gauss-Bonnet (EGB) gravity, which results in a well-defined action principle in four-dimensional spacetime. This approach involves compactifying (D)-dimensional EGB gravity on a ((D-4))-dimensional maximally symmetric space and redefining the Gauss-Bonnet coupling as (\alpha \rightarrow \alpha/(D-4)). The result is a model within the Horndeski gravity framework, a class of scalar-tensor theories with second-order field equations.
Main Findings and Methodology
The investigation demonstrates how the proposed limiting procedure results in a scalar-tensor theory where the static black hole solutions are particularly noteworthy. The metric profiles obtained through this method are independent of the curvature of the compactified internal space. Interestingly, this independence suggests that the curvature information is encoded in the additional scalar degree of freedom introduced in the theory.
The paper extends the procedure to define additional limits by varying the dimensionality of the compactified space, characterized by (p\leq 3). This leads to novel (D\rightarrow4), (D\rightarrow3), and (D\rightarrow2) dimensional limits of EGB gravity, translating into different Horndeski-like theories in lower dimensions.
Key Results
- Static Black Hole Solutions: The black hole solutions found are consistent with the (D\rightarrow4) limit of EGB black hole metrics. Despite the usual dependence on internal curvature in higher-dimensional solutions, these metrics emerge intrinsically from the scalar field’s behavior.
- Black Hole Entropy: When applying the Iyer-Wald formula, a finite contribution to the entropy is obtained, which matches the entropy structure derived from (D)-dimensional limits. This addresses previous issues of divergent entropy expressions in a direct (D\rightarrow4) limit of EGB gravity.
- Generalization: The technique may apply to arbitrary higher-dimensional Lovelock gravity theories, suggesting a method to derive various lower-dimensional Horndeski models potentially applicable to cosmological problems or quantum gravity considerations.
Implications and Future Directions
The work effectively showcases that a well-constructed dimensional reduction process can lead to physically consistent and mathematically rigorous lower-dimensional gravitational theories. The implications are twofold—providing a theoretical underpinning for scalar-tensor theories like Horndeski gravity while offering potentially manageable models for scenarios where direct higher-dimensional descriptions are cumbersome.
Future research should consider exploring the cosmological implications of the derived Horndeski models, which could provide insights into non-standard cosmological models or quantum gravity. Additionally, further investigations may crystallize how these lower-dimensional theories conform to existing observational data in astrophysics and cosmology.
Conclusion
Through a systematic reduction and redefinition of parameters, this paper advances our understanding of how higher-dimensional gravitational theories can yield viable lower-dimensional counterparts. The exploration of these limits not only broadens the theoretical landscape for modified gravity theories but also bridges traditional gravitational frameworks with modern scalar-tensor formulations, promising rich avenues for both theoretical investigations and phenomenological applications.