Classification of six derivative Lagrangians of gravity and static spherically symmetric solutions

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 the general static, spherically symmetric solution for some of these Lagrangians.1 Note that the only non-quadratic term with degree of differentiation 4 is R, which is boundary term. 2 Since a divergenceless vector J a cannot be constructed locally out of the curvature, the equation ∇aJ a = 0, with J a := ∇ b −4aR ab + g ab 4a + D 2 b + 2(D − 1)c R , does not have any non-trivial solution.

Quintic quasi-topological gravity

“…Therefore a combination of these traces, fixed by a single constraint, can be added to the quasi-topological terms to provide new theories which on spherical symmetry coincide (see the discussion above equation (88) in [7]). As mentioned in [8], since there are…”

“…and can be singled out as the unique cubic combination in five dimensions whose traced field equations lead to a second order constraint and that also has second order field equation on general spherically (planar or hyperbolic) symmetric spacetimes [8]. 1 It was also realized in [7] that this theory belongs to a general family of Lagrangians of order k in the curvature, that can be constructed in dimensions D = 2k − 1 for k ≥ 3, and have the simple form…”

On conformal higher spin wave operators

“…If the factorization requirement is dropped more conformal operators can be found, e.g. by linearizing the conformal invariant densities of [68][69][70][71]. However, all but one of these densities vanish when linearized on (A)dS backgrounds as they consist of more than two Weyl tensors.…”

Generalized quasitopological gravity

“…The inclusion of quadratic terms has been shown to lead to violations of the Kovtun-SonStarinets (KSS) viscosity/entropy ratio bound [9,10]. Holography has even motivated the construction of new higher curvature gravities, such as quasi-topological gravity [11][12][13][14][15][16].…”