Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions

TL;DR

The paper tackles the problem of classifying six-derivative gravity Lagrangians whose traced field equations have reduced order across dimensions. It systematically builds the space of degree-6 invariants, derives trace-order constraints, and identifies a small basis of independent invariants: in $D\ge6$ the second-order-trace sector is spanned by $\mathcal{E}_{6}$ and the two Weyl invariants $W_{1},W_{2}$, with a special $\mathcal{N}_{6}$ arising in $D=5$, while $\Sigma$ and $\Theta$ yield third-order traces; in $D=6$ all conformal anomalies are recovered. The authors also present explicit static, spherically symmetric solutions for several theories, including a distinctive five-dimensional cubic invariant $\mathcal{N}_{6}$ and a three-dimensional Cotton-squared theory, illustrating the physical implications of reduced-trace gravity. They conclude with a conjecture generalizing the construction to higher-order invariants and discuss potential extensions such as Birkhoff-type results. These results provide tractable toy models for higher-derivative gravity and insights into holography via exact solutions and conformal anomalies.

Abstract

We classify all the six derivative Lagrangians of gravity, whose traced field equations are of second or third order, in arbitrary dimensions. In the former case, the Lagrangian in dimensions greater than six, reduces to an arbitrary linear combination of the six dimensional Euler density and the two linearly independent cubic Weyl invariants. In five dimensions, besides the independent cubic Weyl invariant, we obtain an interesting cubic combination, whose field equations for static spherically symmetric spacetimes are of second order. In the later case, in arbitrary dimensions we obtain two combinations, which in dimension three, are equivalent to the complete contraction of two Cotton tensors. Moreover, we also recover all the conformal anomalies in six dimensions. Finally, we present some static, spherically symmetric solutions for these Lagrangians.