Einsteinian cubic gravity (1607.06463v1)

Abstract: We drastically simplify the problem of linearizing a general higher-order theory of gravity. We reduce it to the evaluation of its Lagrangian on a particular Riemann tensor depending on two parameters, and the computation of two derivatives with respect to one of those parameters. We use our method to construct a D-dimensional cubic theory of gravity which satisfies the following properties: 1) it shares the spectrum of Einstein gravity, i.e., it only propagates a transverse and massless graviton on a maximally symmetric background; 2) the relative coefficients of the different curvature invariants involved are the same in all dimensions; 3) it is neither trivial nor topological in four dimensions. Up to cubic order in curvature, the only previously known theories satisfying the first two requirements are the Lovelock ones: Einstein gravity, Gauss-Bonnet and cubic-Lovelock. Of course, the last two theories fail to satisfy requirement 3 as they are, respectively, topological and trivial in four dimensions. We show that, up to cubic order, there exists only one additional theory satisfying requirements 1 and 2. Interestingly, this theory is, along with Einstein gravity, the only theory which also satisfies 3.

An Overview of Einsteinian Cubic Gravity

The paper "Einsteinian cubic gravity" by Pablo Bueno and Pablo A. Cano offers a significant advancement in the paper of higher-order gravity theories. The authors present a novel approach to linearizing general higher-order gravity theories on maximally symmetric spacetimes, which streamlines the complexity typically inherent in such analyses.

Methodology and Linearization Approach

The central contribution of this paper is the simplification of the linearization process for higher-order gravity theories. The authors propose evaluating the gravitational Lagrangian on a specially defined Riemann tensor, $\tilde{R}_{\mu\nu\rho\sigma}(\Lambda,\alpha)$, which depends on two parameters. This approach reduces the problem to a straightforward calculation of derivatives with respect to one of those parameters, significantly expediting the process compared to traditional methods that rely on cumbersome derivative computations related to the Riemann tensor.

The linearization procedure is applied to a D-dimensional cubic gravity theory, resulting in the identification of coefficients that characterize the spectrum of physical modes propagated by metric perturbations. The methodology is demonstrated to be consistent with various established theories, including Lovelock and quasi-topological gravities, confirming its broad applicability and accuracy.

Einsteinian Cubic Gravity

A notable finding in the paper is the identification of a novel class of cubic theories, labeled as Einsteinian cubic gravity. These theories are distinguished by dimension-independent couplings and the exclusive propagation of a massless graviton, sharing the spectrum with Einstein gravity. Einsteinian cubic gravity is defined using two cubic curvature invariants: the dimensionally extended Euler density $\mathcal{X}_6$ and the invariant $\mathcal{P}$, the latter of which is non-trivial in four dimensions.

The authors provide the explicit form of the Einsteinian cubic gravity Lagrangian:

$$\mathcal{L} = \frac{1}{2\kappa}\left[-2\Lambda_0 + R\right] + \alpha \mathcal{X}_4 + \kappa \left[\beta \mathcal{X}_6 + \lambda \mathcal{P} \right]$$

Implications and Future Research Directions

The findings have profound implications for theories of quantum gravity and cosmology, enabling models that extend beyond traditional Einstein gravity while maintaining its desired spectral characteristics. By confirming the non-trivial nature of Einsteinian cubic gravity in four dimensions, the paper paves the way for further exploration of its applications, including potential solutions in holographic contexts and cosmological modeling.

Future directions include investigating the unitarity of Einsteinian cubic gravity on more generalized backgrounds and exploring its holographic implications. The potential for constructing spherically symmetric black hole solutions within this framework is also noted, suggesting a rich field for further paper and application.

In summary, Bueno and Cano's paper provides a rigorous framework and novel insights into cubic gravity theories, considerably advancing the understanding and practical computation of higher-order models. Their work challenges pre-existing boundaries and invites the scientific community to explore new dimensions of gravitational research with a simplified methodological approach.