Einstein-Gauss-Bonnet gravity in 4-dimensional space-time (1905.03601v3)

Abstract: In this Letter we present a general covariant modified theory of gravity in $D!=!4$ space-time dimensions which propagates only the massless graviton and bypasses the Lovelock's theorem. The theory we present is formulated in $D!>!4$ dimensions and its action consists of the Einstein-Hilbert term with a cosmological constant, and the Gauss-Bonnet term multiplied by a factor $1/(D!-!4)$. The four-dimensional theory is defined as the limit $D!\to!4$. In this singular limit the Gauss-Bonnet invariant gives rise to non-trivial contributions to gravitational dynamics, while preserving the number of graviton degrees of freedom and being free from Ostrogradsky instability. We report several appealing new predictions of this theory, including the corrections to the dispersion relation of cosmological tensor and scalar modes, singularity resolution for spherically symmetric solutions, and others.

• The paper modifies gravitational dynamics by employing a singular limit from higher dimensions to induce a nontrivial Gauss-Bonnet contribution in four dimensions.
• It uncovers dual solution branches in maximally symmetric spacetimes, reproducing features similar to Schwarzschild and Schwarzschild–de Sitter geometries.
• The framework offers insights on avoiding singularities and refining cosmological perturbations, with potential implications for high-energy gravitational phenomena.

Einstein-Gauss-Bonnet Gravity in Four Dimensions

The paper by Dražen Glavan and Chunshan Lin presents a novel approach to Einstein-Gauss-Bonnet gravity formulated in a four-dimensional spacetime context. This approach bypasses Lovelock's theorem, which traditionally restricts the admissible form of general relativity in four dimensions, by introducing a singular limiting process from higher dimensions.

Lovelock's theorem asserts that in four-dimensional spacetimes, under the assumptions of diffeomorphism invariance, metricity, and second-order equations of motion, the theory of gravity is uniquely described by the Einstein-Hilbert action supplemented by a cosmological constant. The background review in this paper notes that including higher-dimensional Lovelock invariants typically does not modify the dynamics in four dimensions due to their vanishing contributions. This paper, however, suggests a modification—involving the Gauss-Bonnet term scaled by the inverse of the dimension selector $(D-4)$ and considering the limit as $D \rightarrow 4$—which results in a nontrivial contribution to the gravitational dynamics.

Novel Contributions and Theoretical Implications

Key theoretical contributions and predictions are discussed:

1. Modified Dynamics: The authors propose that in the aforementioned limit, the Gauss-Bonnet invariant effectively contributes to the dynamics in four dimensions without introducing extra degrees of freedom. Notably, it avoids the Ostrogradsky instability, which is known to plague systems with higher derivative terms.
2. Existence of Multiple Solutions: The theory naturally leads to two branches of solutions in a maximally symmetric space-time. They recover properties resembling those of Schwarzschild and Schwarzschild-de Sitter solutions.
3. Impacts on Cosmology: The Gauss-Bonnet modification introduces corrections to the standard cosmological tensor and scalar modes, potentially observable during high-energy epochs such as inflation. These corrections are encapsulated in modified Friedmann equations and perturbation dynamics. The alterations include changes in the gravitational wave speed and Hubble parameter dependent modifications to damping terms in perturbations.
4. Singularity Resolution in Spherical Symmetry: The paper also highlights that this methodological revision permits avoiding singularities in certain spherically symmetric solutions, unlike classical general relativity where singularities are unavoidable. The gravitational force becomes repulsive at short distances, preventing the formation of singularities.

Practical Implications

The theoretical framework has several practical implications within and potentially beyond the scope of classical and quantum gravity:

• Small-Scale Gravity: The ability to avoid singularities could inform the paper of small-scale gravitational systems and their transition into quantum regimes.
• Verification of Cosmological Models: The behavior of cosmological perturbations predicted by this theory could provide new theoretical underpinnings to interpret data from gravitational wave observatories and high-energy cosmological probes.

Future Directions

The authors propose that extending this approach to include higher-order Lovelock invariants could further challenge the foundational status of general relativity as the sole framework for describing non-linear gravitational interactions in four dimensions. This raises provocative questions on the universality of Einstein’s theory under conditions different from those originally considered.

In conclusion, the introduction of a dimensionally reduced Einstein-Gauss-Bonnet theory in four dimensions represents an intriguing theoretical development. It invites further scrutiny and validation, from both mathematical and physical perspectives, as researchers explore its broader implications and potential applications in gravitational physics.