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All the solutions of the form M2(warped)x\Sigma(d-2) for Lovelock gravity in vacuum in the Chern-Simons case

In this note we classify a certain family of solutions of Lovelock gravity in the Chern-Simons (CS) case, in arbitrary (odd) dimension greater than four. The spacetime is characterized by admitting a metric that is a warped product of a two-dimensional spacetime M2 and an (a priori) arbitrary Euclidean base manifold Sigma(d-2) of dimension d-2. We show that the solutions are naturally classified in terms of the equations that restrict the base manifold. According to the strength of such constraints we found the following branches in which Sigma(d-2) has to fulfill: a Lovelock equation with a single vacuum (Euclidean Lovelock Chern-Simons in dimension d-2), a single scalar equation that is the trace of an Euclidean Lovelock CS equation in dimension d-2, or finally a degenerate case in which the base manifold is not restricted at all. We show that all the cases have some degeneracy in the sense that the metric functions are not completely fixed by the field equations. This result extends the static five-dimensional case previously discussed in Phys.Rev. D76 (2007) 064038, and it shows that in the CS case, the inclusion of higher powers in the curvature does not introduce new branches of solutions in Lovelock gravity. Finally we comment on how the inclusion of a non-vanishing torsion and matter fields may modify this analysis.


I. INTRODUCTION
Gravity in higher dimensions has proved to be an interesting arena to test how generic are the notions gained in four dimensional gravitational physics. Even in higher dimensional General Relativity (GR), properties as uniqueness and stability of solutions in vacuum may depart completely from their four-dimensional counterpart (for a recent summary of the state of the art see [1]). Maintaining the second order character of the field equations in higher dimensions, it is possible to consider a more general setup than the one defined by Einstein's gravity, since as proved by Lovelock in [2] the most general parity-even Lagrangian in arbitrary dimension d, that gives second order field equations for the metric is given by an arbitrary linear combination of the dimensional continuations of all the lower dimensional Euler densities. This gives rise to the so-called Lovelock gravity, the simplest case after GR being the Einstein-Gauss-Bonnet (EGB) gravity. In this theory, in addition to the Einstein-Hilbert and cosmological terms, one includes a term which is quadratic in the curvature and gives non-trivial field equations in dimensions grater than four. This quadratic combination is very precise, in such a way that the possible higher derivative terms cancel each other and one gets second order field equations. Since the field equations come from a diffeomorphism invariant action, their divergence vanishes identically.
To find exact and analytic solutions of these theories is a non-trivial problem when one departs from spherical symmetry 1 . For example, a problem that is solved in a very simple manner in GR, corresponds to finding the most general solution of the form where Σ d−2 is an arbitrary Euclidean manifold of dimension d − 2. Einstein equations plus a cosmological constant in vacuum 1 Departing from the family of metrics (1), looking for exact rotating solutions is also a more difficult than in GR, since for example considering the Kerr-Schild ansatz that naturally gives rise to the Myers-Perry solution with cosmological constant in GR [4], one finds that in order to have a non-trivial solution in EGB, the coupling constants must be fixed as in the Chern-Simons case [5] and even more the solution turns out to be non-circular [6], making the analysis of the causal structure more cumbersome (for some perturbative and numerical solutions see also [7]).
imply that the metric functions do not depend on t, and are given by where µ is an arbitrary integration constant and Σ d−2 must be an Einstein manifold fulfilling the equationR HereR ij is the Ricci tensor of Σ d−2 andg ij its metric [3].
Solving exactly the same problem in Lovelock gravity is more complicated. For example, in the EGB theory for the static case, the work [8] solves this problem in arbitrary dimension finding a rich set of causal structures. For arbitrary values of the coupling constants of the theory, the analysis done in [8] reduces to the done previously reported in [9], where it was proved that if one assumes Σ d−2 to be Einstein, then one can show that it must also obey a quadratic restriction on the Weyl tensor which includes a new parameter θ. That parameter appears in the lapse function and even more, it modifies the asymptotic behavior of the metric (see also [10]).
For arbitrary Σ, beyond the EGB case not much is known. The static solution in the spherically symmetric case was found in [11]. When Σ d−2 is a constant curvature manifold, a Birkhoff's theorem was proved in [12] (see also [13]). Reference [12] also shows that Birkhoff's theorem is not valid when the coupling constants are fixed in a precise way and some degeneracies may appear since in such cases, some of the metric functions are not determined by the field equations (for some particular cases, this was previously observed in reference [14]). Lovelock theory, being a gravity theory with higher powers in the curvature, could have more than one maximally symmetric solution, and the mentioned degeneracies appear precisely at the regions in the space of couplings in which some of these vacua coincide 2 (for some static black hole solutions, with constant curvature horizons in this case see [16]).
It would be interesting therefore to classify all the solutions of the form (1) in higher curvature Lovelock theories. In this work we focus on the odd-dimensional case, when the highest possible power of the curvature is present in the Lagrangian and all the vacua coincide. This theory is known as Lovelock-Chern-Simons (LCS) theory (for a recent review see [17]).
The action for a general Lovelock theory can be written as where κ and α p are arbitrary (dimensionfull) coupling constants, ε a 1 ...a d is the Lorentz invariant Levi-Civita tensor, R ab := dω ab + ω ac ω b c is the curvature two-form written in terms Lorentz connection one-form ω ab , and e a is the vielbein. [x] stands for the integer part of x. Wedge exterior product between differential forms is understood. Finally, latin indices The term with p = 0 in (5), corresponds to a volume term that gives the contribution of the cosmological constant, for p = 1 one gets the Einstein-Hilbert term, while for p = 2 the Lagrangian reduces to the Gauss-Bonnet term. As mentioned before, here we will focus on the case d = 2n + 1 and the coefficients α p are given by where l 2 is the squared curvature radius of the unique (AdS) maximally symmetric solution.
For simplicity we will focus on the case l 2 > 0, nevertheless the de Sitter case is trivially obtained by analytically continuing l → il, while the flat limit (up to some subtleties that will be mentioned when necessary) can be obtained by taking l → ∞ .
When torsion vanishes, the field equations coming from (5) with the couplings given by (6) can be written as where we have defined the concircular curvature two-form asR ab := R ab + 1 l 2 e a e b . In terms of tensors, if we use the generalized Kronecker delta of strength one denoted by δ α 1 ...αp β 1 ...βp , by defining the concircular curvature tensorR αβ γδ = R αβ γδ + 1 l 2 δ αβ γδ , the field equations (7) read In the next section we will prove that all the solutions of the form (1), for the field equation (7) (or equivalently (8)) fall into one of the following three different classes: Case I: The manifold Σ d−2 is arbitrary and the metric reads where µ is an integration constant.
Case II: For ξ = 0, if the manifold Σ d−2 satisfies the following (scalar) restriction whereR ij is the curvature two-form intrinsically defined on Σ d−2 and the indices {i, j} run on Σ d−2 , then the metric reads with c 1 (t) and c 2 (t) arbitrary integration functions. In the flat limit (l → ∞) the metric reduces to In the case ξ = 0 (which does not exist in the limit l → ∞) the restriction on Σ d−2 is obtained by setting ξ = 0 in (10) and the metric reads where again c 1 (t) and c 2 (t) are arbitrary integration functions. Note that in all of these cases, by redefining the time coordinate, one can gauge away one of the two integration functions, but not both simultaneously.
Case III: The manifold Σ d−2 satisfies the following tensor restriction and the metric reads with f (t, r) an arbitrary function.
This result extends the static five dimensional case previously analyzed in [18]. In Case I, we see that the manifold Σ d−2 is arbitrary, i.e. it is not fixed by the field equations.
In Case II, the manifold Σ d−2 is fixed by a single scalar equation which, even after using diffeomorphism invariance, in general it is not enough to determine a metric on it. Finally in Case III, we see that the lapse function f 2 (t, r) is left arbitrary by the field equations.
Therefore we conclude that, in the previously mentioned sense, all the cases have some degeneracy.

II. PROOF OF THE CLASSIFICATION
To develop the proof of the classification it is useful to have the components of the curvature two-form with respect to some basis for the metric (1). If we define the components of the vielbein as whereẽ i is the vielbein intrinsically defined on Σ d−2 , then the nontrivial components of the concircular curvature two-formR ab read whereR ij is the curvature two-form intrinsically defined on Σ d−2 and A, B, C, F , H and J are functions of t and r defined by Primes denote derivation with respect to r while dots derivation with respect to t.
There are three kinds of equations depending on whether the free index in (7) goes along the time direction, radial direction or along the manifold Σ d−2 , which respectively reduce to After introducing explicitly in these equations the components of the concircular curvature two-form (17)- (20), we get the following three equations Considering the combinations e 0 G 0 + e 1 G 1 = 0 and e 1 G 0 + e 0 G 1 = 0 one respectively gets This immediately splits the analysis in two cases defined by the (would be) constraint on If (23) doesn't hold, then we need to impose F = B and H = C, the former implies that g (t, r) = g (r), while the later implies f (t, r) = S(t)g (r). The function S (t) can be set to 1 without lost of generality by means of a redefinition of the time coordinate.
Therefore in this branch (i.e. provided (23) doesn't hold), we have that (21) and (22) imply f (t, r) = g (t, r) = f (r) = g (r). If (23) holds, then G 0 = 0 = G 1 without imposing any restriction on the function f and g at the moment. Note that the quantities with tilde on top depend only on the coordinates in Σ d−2 , while the combination Jr 2 , could depend on both t and r. At the moment this is not relevant since equations (21) and (22) are factorized in any case, but later we will see that the consistency of equation (23) strongly constraints the metric functions.
If we consider now equation G 0 = 0, in the case in which (23) doesn't hold and therefore f 2 (r) = g 2 (r) then we can see that H identically vanishes, while the vanishing of the function F implies that g 2 = r 2 l 2 − µ, where µ is an integration constant. As mentioned, in this branch we also have f 2 = r 2 l 2 − µ (since f 2 = g 2 ), and therefore one can see by direct evaluation that A identically vanishes also. Therefore H = F = A = 0 and then equation On the other hand if (23) holds (as mentioned before) G 0 and G 1 vanish identically and then at the moment, the functions f (t, r) and g (t, r) are not restricted. Before continuing to equation G i = 0, let us go back to the problem of the consistency of equation (23).
Considering the derivative of this equation with respect to the t and r, we respectively obtain (n − 1) therefore (Jr 2 ) ′ = (Jr 2 ) · = 0 and consequently Jr 2 = −ξ with ξ a constant, or the second term in both (24) and (25) vanishes, implying a new scalar restriction on Σ d−2 that contains terms of order n−2 in the curvature and might also depend on t, r, therefore its compatibility must be analyzed as well. The first case (Jr 2 = −ξ), implies g 2 (t, r) = g 2 (r) = r 2 l 2 + ξ where ξ is an integration constant. Note also that when n = 2, we are forced to set (Jr 2 ) ′ = (Jr 2 ) · = 0 otherwise the volume element on Σ d−2 would vanish. On the other hand (for n > 2), if we assume (Jr 2 ) ′ and (Jr 2 ) · to be nonvanishing we can divide these factors obtaining the new mentioned scalar restriction on Σ d−2 , which reads Again, we must consider the consistency of this equation by taking its derivative with respect to the parameters r and t. This respectively gives If n = 3 we are forced again to set (Jr 2 ) ′ = (Jr 2 ) · = 0 (otherwise the volume element of Σ d−2 should vanish) which fixes g 2 = r 2 l 2 + ξ, while for n > 3 we can consider (Jr 2 ) ′ and (Jr 2 ) · to be nonvanishing and divide by these expressions, therefore obtaining another scalar restriction on Σ d−2 , which this time, includes powers of the curvature of order n − 3.
Repeating this procedure n − 1 times one eventually gets and if the expressions (Jr 2 ) ′ and (Jr 2 ) · are nonvanishing then we would have that the volume form of Σ d−2 must vanish, arriving to a contradiction. We have proved then that in the case F = B and H = C, the consistency of equation (23) implies that (Jr 2 ) ′ = (Jr 2 ) · = 0 which in turn implies that g 2 = r 2 l 2 + ξ with ξ an integration constant, and consequently (23) reads which now depends only on the coordinates of Σ d−2 .
The remaining structure comes from the analysis of equation G i = 0. Note that g 2 = r 2 l 2 +ξ further implies H = 0 = F , therefore G j reduces to If A = 0, equation (32) implies a tensor restriction on Σ d−2 , which naturally, is stronger than its trace given by (31). When this tensor restriction holds, the metric reads This concludes the proof of the classification.

On the causal structures
For the Case 2 and Case 3, the (t, r)-part of the metrics obtained depend on arbitrary functions of the time coordinate, therefore the causal structure of this spacetimes is not fixed. Note that this dependence cannot be gauged away completely by a diffeomorphism.
Nevertheless, a few comments on the causal structures are in order in all of the three cases when the integration functions are chosen to be constants, i.e. c 1 (t) = c 1 and c 2 (t) = c 2 .
In Case I, the solution describes a black hole. This solution reduces to the one found in [19]. In such case also, its thermodynamics and causal structure coincide with that of the three-dimensional Bañados-Teitelboim-Zanelli (BTZ) black hole [20] where, for generic values of µ, the causal structure singularity at r = 0 of the three-dimensional case is now replaced by a curvature singularity as can be seen by evaluating, for example, the Ricci scalar.
In Case II, the metric might describe the traversable wormhole found in [21], which is asymptotically AdS at both asymptotic regions. This is the case when ξ = −1 and | c 2 lc 1 | < 1, which can be seen directly by performing the change of coordinates r = l cosh ρ and allowing the coordinate ρ to go from −∞ to +∞. In this case, the metric reduces to where ρ 0 = − tanh −1 c 2 lc 1 and we have properly rescaled the time coordinate. The conditions under which the propagation of a scalar field on this background is stable, was studied in [22], and some holographic properties of strings attached to the boundaries have been explored in [23]. For ξ = 0 the metric reduces to When c 1 = 0 this spacetime is asymptotically locally AdS, while if c 1 = 0, the (t, r)-part of the metric reduces to a Lifshitz geometry (geometry with an anisotropic scaling symmetry), with a dynamic exponent equals to z = −1.
Since in Case III the lapse function is arbitrary, the causal structure is also undefined even in the static case.
Does torsion help removing the degeneracy?
The field equations coming from the variation with respect to the spin connection in Lovelock theory, do not necessarily imply that torsion should vanish (for some explicit solutions see e.g. [24]). For example in five dimensions, in first order formalism for the Lovelock CS case, the field equations coming from the variation with respect to the vielbein and the spin connection are respectively given by where we have introduced the torsion two-form T e := De e := 1 2 e e α T α µν dx µ ∧ dx ν . Therefore choosing the Levi-Civita connection is ad-hoc. Then it is natural to wonder whether the equations coming from the torsion may help removing the degeneracy. Posing the question in a different manner one could ask : is there a non-degenerate branch of solutions of (39)-(40) in which the vielbein and the spin connection are compatible with the local isometries of Σ d−2 ?. It is clear that there are particular cases in which the torsion may not be vanishing and anyway the system is degenerated since, if for example we choose the (non-Riemannian) curvature to be constant R ab = − 1 l 2 e a e b , then the torsion is left completely arbitrary by the field equations. Note also that, since this theory has an extra symmetry that mixes the spin connection and the vielbein (see [17]), the arbitrariness in the torsion can be transformed into an arbitrariness of the line element constructed out from the corresponding vielbein. A thorough analysis with the inclusion of torsion will be presented elsewhere [25].

Further comments
As studied for the static quadratic case in [8], when one considers Lovelock theories that do not belong to the subclass of Lovelock CS, but nevertheless the couplings are related in such a way that there is a unique vacuum, there are also sectors in which some of the metric functions are arbitrary. Therefore this phenomenon seems to be more related to the fact of having degenerate maximally symmetric solution than with the appearance of an extra symmetry. In such non-Lovelock CS theories, as well as in the Lovelock CS ones, this degeneracy allows to have interesting causal structures as solutions (see e.g. [26]). Nevertheless in the former cases, there are more restrictions on Σ d−2 , which on one hand can be thought of as helping to remove the degeneracy, while in the other hand could be not compatible beyond the constant curvature case. A simple set of geometries beyond constant curvature manifolds (or their products) are product of the homogenous three dimensional Thurston geometries, which have been recently found to provide simple examples of transverse sections of hairy black holes for some Lovelock theories in even dimensions [27]. In the context of compactifications of Lovelock CS theories, involving metrics that are products of constant curvature spaces, the degenerate behavior is also present as it was proved in reference [28] back in the early 90's. The inclusion of matter fields seems to help removing the mentioned degeneracies (see for example references [29]).
If one departs from the underlying (A)dS symmetry group, static spherically symmetric solutions of gravitational CS theories with matter fields, have also been recently considered in [30]. In this reference, the authors considered a Chern-Simons theory evaluated on a Lie algebra that is obtained by performing what the authors called an S-expansion procedure [31] from the AdS algebra and a particular semigroup S, which provides an approach to obtain GR in odd-dimensions from a CS theory. It would be interesting to study further the properties of these theories and to integrate them in the general ansatz (1) classifying the possible non-degenerate sectors.