# On Lovelock analogs of the Riemann tensor

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## Abstract

It is possible to define an analog of the Riemann tensor for *N*th order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd \(d=2N+1\) dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock–Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock–Weyl tensor that vanishes.

## 1 Introduction

In order to write an equation of motion for Einstein gravity, one has to obtain a divergence free second rank symmetric tensor constructed solely from the metric and the Riemann curvature – the Einstein tensor. This is usually done by varying the Einstein–Hilbert action, the scalar curvature *R*, relative to the metric tensor. Alternatively one can obtain the same result by invoking the differential geometric property that the Bianchi derivative of the Riemann tensor identically vanishes – the Bianchi identity. From the trace of this identity we can then extract the required Einstein tensor. This is a very neat and elegant purely geometric way to get to the equation of motion.

The most natural generalization of Einstein gravity in higher dimensions is Lovelock gravity, whose equation of motion inherits the basic property of being second order, though polynomial in curvature. The natural question then arises, could the same geometric method used for Einstein gravity also work for Lovelock theories? The answer is yes. In [1], one of the authors defined an *N*th order Lovelock analog of the Riemann curvature, which is a homogeneous polynomial in the Riemann tensor. Even though this tensor does not satisfy the Bianchi identity, the trace of its Bianchi derivative vanishes yielding the *N*th order Lovelock analog of the Einstein tensor. This tensor agrees with the one obtained by varying *N*th order Lovelock Lagrangian and is divergence free.

*kinematic*, i.e. whenever the Love-lock-Ricci vanishes so does the corresponding Riemann. That is, pure Lovelock vacuum in critical odd dimension is Lovelock flat, as is the case for \(N=1\) Einstein gravity in 3 dimension. Based on this result, Dadhich [5] conjectured that this should be true not only for static vacuum solutions but in general for all vacuum spacetimes. It would be a universal gravitational property. However, it later turned out that this is not true in general for Dadhich’s Lovelock–Riemann tensor while it actually holds for Kastor’s analog [2]. This is a purely algebraic property due to the fact that we can write Kastor’s 4Nth rank tensor

^{1}\({}_{(N)}\mathbb {R}^{(4N)}\) (therefore also all its contractions) in terms of the Lovelock–Ricci (or equivalently the corresponding Lovelock–Einstein, \({}_{(N)}\mathcal {E}^{a}_{\ b}\)). As we will describe below, in \(d=2N+1\) we can write

Lovelock gravities are a very interesting set of theories and have been used in many contexts with very diverse applications (see for instance [3] for a recent review). They can be viewed as a model of ghost free higher curvature/derivative gravity as Lovelock gravity captures many of the defining features of those theories while avoiding some of their problems, in particular the existence of higher derivative ghosts. Lovelock gravities have also been instrumental in exploring the role of higher curvature corrections in the holographic context.

The paper is organized as follows: we will first review Kastor’s formulation that can be suitably described in the language of differential forms. This will be particularly convenient for the derivation of the Bianchi identities for the new higher order tensors, and also to introduce some simple tensor identities. Using these we will find a much more direct route to show the kinematicity of pure Lovelock gravity in odd critical dimensions. Next we reconcile the two formulations showing their equivalence and we end with a discussion.

## 2 Kastor’s formulation

*N*, 2

*N*)-rank tensor product of

*N*Riemann tensors, completely antisymmetric, both in its upper and lower indices,With all indices lowered, this tensor is also symmetric under the exchange of both groups of indices, \(a_i\leftrightarrow b_i\). In a similar way we will denote the contractions of \({}_{(N)}\mathbb {R}^{(4N)}\) simply as

*N*) indicating the Lovelock or curvature order except when not clear from the context.

*N*form which is the antisymmetrized wedge product of

*N*curvature 2-forms,

*N*-form it is trivial to construct both the Lovelock action and its corresponding equation of motion. We just need to complete a

*d*-form with vielbeins and contract with the antisymmetric symbol,

*D*, with the corresponding connection 1-form \(\omega ^a_{\ b}\), in addition to the usual exterior operator

*d*. In the Lovelock case, we may take the usual metric variation of the action to obtain the equation of motion or, instead, take independent variations with respect to vielbein and spin connection. The two approaches yield the exact same result, the equation coming from the \(\omega \)-variation being proportional to the torsion and thus zero by assumption.

*D*is not nilpotent. The Bianchi identities can be written simply as

*N*th order Lovelock gravity. In the same way as for the curvature 2-form, we can take the exterior covariant derivative of \(\mathbb {R}\) and write

## 3 Tensor identities and kinematicity

*N*indices. Thus this tensor vanishes for dimensions \(d<2N\) and that is the reason for the corresponding Lovelock term to become trivial. We can now antisymmetrize over bigger sets of indices respecting the symmetry properties of the above tensor, i.e. we may define

*N*indices. This new tensor, which can be written explicitly in terms of \(\mathbb {R}^{(4N)}\) and its contractions, vanishes for dimensions \(d<4N\). Interestingly, we get a way of writing \(\mathbb {R}^{(4N)}\) completely in terms of its contractions below that dimensionality. Further, we can define similar tensors reducing the number of free indices on each set,

*J*th contraction of \(\mathbb {R}^{(4N)}\) in terms of lower contractions below that threshold dimension. These identities were used by Kastor [2] in order to prove that in all odd \(d=2N+1\), the 4Nth rank tensor \(\mathbb {R}^{(4N)}\) can be written completely in terms of its corresponding Lovelock–Ricci. Therefore whenever the latter vanishes so does the former. This is, however, not the case in the next even \(d=2N+2\) dimension or higher. Depending on the number of free indices these tensors \({}_{(N)}\mathbb {A}^{(2n)}\) correspond to various structures appearing in Lovelock gravity, for instance the equations of motion themselves, \({}_{(N)}\mathbb {A}^{(2)}\), or the tensor multiplying the linearized Riemann in the linearized equations of motion, \({}_{(N-1)}\mathbb {A}^{(6)}\).

The above construction based on identities looks simple, however, it may become very cumbersome when it comes to derive explicit expressions. In \(d=2N+1\) dimensions, for example, in writing explicitly the 4Nth rank tensor in terms of the corresponding Lovelock–Ricci, one has to write down the explicit expressions of the \(2N-1\) identities available for that dimensionality and combine them to get the desired expression. In the next few paragraphs we will describe a much more elegant and efficient way of deriving these expressions based on the use of the Hodge duality. In fact, we will introduce a single tensor identity, basically \((\star \!\star \!\mathbb {R}^{(4N)}\!\star \!\star )=\mathbb {R}^{(4N)}\) (see details below), that will imply all of those used by Kastor and has lots of potential applications in the context of Lovelock gravity. In particular, as stated in the introduction, this will lead directly to our Eq. (1).

*n*and \((d-n)\) have the same number of independent components. In fact there is a reversible map between the two equivalent representations, namely the Hodge duality, which basically amounts to a contraction with the antisymmetric symbol,

To sum up, in any dimension we can always write \(\mathbb {R}^{(4N)}\!=\star (\!\star \mathbb {R}^{(4N)}\star \!)\star \) and thus express \(\mathbb {R}^{(4N)}\) in terms of \((\!\star \mathbb {R}\star \!)^{(2d-4N)}\). For low enough dimension, \(d<4N\), the dual tensor itself will be a combination of contractions of \(\mathbb {R}^{(4N)}\) with the same number of free indices. In \(d=2N\), \((\star \mathbb {R}\star )^{(0)}\sim \mathcal {L}\) is just a scalar, which means that \(\mathbb {R}^{(4N)}\) can be written solely in terms of the Lovelock scalar, in \(d=2N+1\) it can be written in terms of the Lovelock–Ricci and its trace, in \(d=2N+2\) we need to include also the rank four contraction and so on.

The above discussion implies that, for pure Lovelock gravity in the critical dimensionality, the vacuum equation of motion (with or without cosmological constant) \(\mathcal {E}^a_{\ b}=\lambda \delta ^a_b\) completely fixes \(\mathbb {R}^{(4N)}\) and all its contractions. Thus we have proved in a very direct way that pure Lovelock gravity is kinematic in all odd \(d=2N+1\) dimensions, its Lovelock–Riemann tensor is completely fixed, generalizing the well-known three dimensional property. However, unlike in \(d=3\) this does not fix completely the Riemann curvature, thus our solutions are not locally maximally symmetric spaces and we have in general propagating degrees of freedom. For zero \(\lambda \) we can rephrase this by saying that any solution of pure Lovelock is Lovelock flat even though the Riemann curvature is not necessarily zero. For nonzero \(\lambda \) the Lovelock–Weyl tensor is zero but still the Weyl tensor does not necessarily vanish. The implications of this for the dynamics of pure Lovelock theories are still unclear.

## 4 Kastor–Dadhich reconciliation

As we have described in the previous section, Kastor’s tensors contain all the relevant information from which we can reconstruct action and equation of motion in any pure Lovelock theory. In this way, any other formulation that cannot be obtained from this one would contain more information on the spacetime that does not enter either the action or the corresponding Lovelock–Einstein tensor. In particular, Kastor’s 4Nth rank tensor is totally antisymmetric on each set of 2*N* indices and symmetric under exchange of both sets. In addition, the dualization procedure allowed us to write \(\mathbb {R}^{(4N)}\) in terms of its contractions in low dimensions, \(d<4N\). This reduces the information contained in \(\mathbb {R}^{(4N)}\) to the minimal possible amount still capturing the whole dynamics. Any extra information is thus irrelevant from this point of view. We can rephrase the dynamical information contained in \(\mathbb {R}^{(4N)}\) in terms of its contraction of rank \((d-2N,d-2N)\), i.e. the Lovelock–Ricci tensor in the critical \(d=2N+1\), the fourth rank Lovelock–Riemann tensor in \(d=2N+2\), and so on.

*N*th order homogeneous polynomial in the Riemann curvature. It is given by

*F*looks very similar to the Riemann contraction of \(\mathbb {R}\), except that upper indices are not completely antisymmetrized. The contracted indices can be considered as antisymmetrized as lower indices are, but not the whole set. We shall now extract the difference between the two classes of tensors. Comparing the above

*F*with the fourth rank Lovelock–Riemann in Kastor’s formulation,

*F*is not symmetric under the exchange of both pairs of indices (when all lowered), whereas this contraction of \(\mathbb {R}^{(4N)}\) is. By repeatedly using Eq. (17), \(\mathbb {R}^{cd}_{ab}\) can be rewritten as

*F*is the second term in the bracket, which is the only other tensor that can be written respecting all the relevant symmetries. Both structures are equal in the trivial case of \(N=1\). Note that Lovelock–Ricci tensors arising from

*F*and \(\mathbb {R}\) are the same,

### 4.1 Pure Gauss–Bonnet check

*F*and the second term in Eq. (42) exactly cancel out each other. For spherically symmetric solutions it turns out that

*F*and the extra term vanish separately in \(d=2N+1\), and that is the basis for Dadhich’s conjecture [4, 5] for the kinematicity of pure Lovelock gravity in odd critical dimensions. As an example for which both contributions to the Lovelock–Riemann are separately nonzero we can consider a pure GB Kasner vacuum in five dimensions [8]. One such metric is, for instance,

*F*is canceled by the extra term to give \(\mathbb {R}^{ab}_{cd}=\mathbb {R}^{(4N)}=0\). In fact, in the context of Kasner type metrics, the different properties displayed by Kastor’s and Dadhich’s analog Riemann tensors allow for a characterization of this family of solutions in the context of pure Lovelock, splitting them into different classes or isotropy types [8].

## 5 Discussion

It is interesting that there are two parallel but distinct definitions of a higher order Lovelock–Riemann tensor leading to the same equation of motion. That is, the two constructions describe precisely the same gravitational dynamics, even though there is a nontrivial difference at a kinematic level. This became apparent when a pure GB Kasner vacuum solution was found in five dimensions [8] for which Dadhich’s Lovelock–Riemann tensor did not vanish. Besides this was in contradiction with a previous kinematicity conjecture [4, 5]. Dadhich’s tensor did indeed vanish for spherically symmetric pure GB vacuum solutions [4], and based on that it was proposed that any pure Lovelock vacuum in all odd \(d=2N+1\) dimensions would be Lovelock flat. A precise realization of this kinematicity property was nonetheless provided by an alternative formulation put forward by Kastor [2]. Kastor’s Lovelock–Riemann tensor does indeed vanish for the Kasner vacuum solution in question. In fact this tensor vanishes in all odd critical dimensions for any vacuum solution of pure Lovelock gravity, as we discussed. It became thus pertinent to reconcile both formulations, and this is therefore the main motivation of this investigation. Both Lovelock–Riemann tensors differ in a piece that, remarkably, vanishes in the spherically symmetric case and that is how it was not noticed at first [4].

We have also revisited the kinematicity property and rederived it in a much more direct way by making extensive use of the properties of the Hodge dual map. This is a unique and universal distinguishing feature of pure Lovelock gravity in all odd \(d=2N+1\) dimensions which is shared by no other theory. It stems from the fact that, for that critical dimensionality, we can write the higher rank tensor \(\mathbb {R}^{(4N)}\) as the double Hodge dual of the corresponding Lovelock–Einstein tensor. Thus \(\mathbb {R}^{(4N)}\) is completely fixed by the equations of motion. It is important to note that this kinematicity is relative to the Lovelock–Riemann tensor and not to the Riemann curvature. That is, the Lovelock–Riemann tensor vanishes in \(d=2N+1\) whenever the corresponding Lovelock–Ricci vanishes, but the Riemann curvature may be nonzero. In turn, in \(d=2N+2\), the Lovelock–Weyl tensor is a priori unrelated to the equation of motion. This is in complete analogy with the behavior of Einstein gravity in three and four dimensions and it can also be generalized in the presence of a nonzero cosmological constant. Pure Lovelock gravity thus unravels a new universal feature of gravity in higher dimensions.

The fact that Dadhich’s Riemann analog vanishes, for instance, for spherically symmetric pure Lovelock solutions seems to indicate that this tensor being zero might identify special properties of particular classes of solutions. This intuition has been strengthen by the analysis of pure Lovelock Kasner metrics [8]. These family of vacuum solutions can be divided into several classes and turns out that we can use a set of fourth rank tensors, \(R_{abcd}\), \(\mathcal {R}_{abcd}\) and \(\mathbb {R}_{abcd}\), to characterize them. Given a particular solution in this family, one may identify which class it belongs to by analyzing which tensors among those in the set vanish.

Pure Lovelock theories posses many interesting properties. Besides the ones already mentioned, thermodynamic parameters of pure Lovelock static black holes bear a universal relation to the horizon radius [9] and bound orbits exist in all even \(d=2N+2\) dimensions [10] in these spacetimes. It should be pointed out that for Einstein gravity bound orbits around a static black hole exist in 4 dimensions only. All this strongly suggests that pure Lovelock equation is the right equation to describe the gravitational dynamics in higher \(d=2N+1,\, 2N+2\) dimensions [5] such that we preserve many interesting properties that Einstein gravity has only for three or four dimensions.

## Footnotes

- 1.
The lower index indicates the curvature order of the Lovelock term in the action, whereas the upper index corresponds to the tensor rank. Thus contractions of \({}_{(N)}\mathbb {R}^{(4N)}\) will be denoted \({}_{(N)}\mathbb {R}^{(2n)}\), for \(n<2N\).

## Notes

### Acknowledgments

We wish to thank David Kastor for valuable discussions and for first verifying the vanishing of his Riemann tensor for the Kasner solution [8] in question. XOC is grateful to Jamia Millia Islamia and IUCAA for hospitality at the initial stages of this work. ND thanks the Albert Einstein Institute, Golm for a visit during which the project was formulated.

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