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

TL;DR

This paper classifies all six-derivative gravity Lagrangians with second or third order traced field equations across dimensions, identifying special combinations and solutions, including conformal anomalies and spherically symmetric solutions.

Contribution

It provides a comprehensive classification of six-derivative gravity Lagrangians with specific order field equations and derives explicit solutions and anomalies.

Findings

Lagrangians reduce to Euler density and Weyl invariants in higher dimensions.

Identifies special cubic combinations with second-order equations in five dimensions.

Derives static spherically symmetric 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.