# On black strings & branes in Lovelock gravity

###### Abstract

It is well known that black strings and branes may be constructed in pure Einstein gravity simply by adding flat directions to a vacuum black hole solution. A similar construction holds in the presence of a cosmological constant. While these constructions fail in general Lovelock theories, we show that they carry over straightforwardly within a class of Lovelock gravity theories that have (locally) unique constant curvature vacua.

## I Introduction

It is straightforward to construct uncharged black strings and branes in Einstein gravity. Starting from a Schwarzschild black hole in -dimensions, for example, one obtains a black string in -dimensions simply by adding a flat direction to the metric. Despite the seemingly trivial nature of this construction, the phenomenology associated with black strings, begining with the Gregory-Laflamme instability Gregory:1993vy , has turned out to be surprisingly rich (see reference Kol:2004ww for a review). One can similarly construct black -branes by adding additional flat directions, while rotating black branes can be obtained by adding flat directions to the -dimensional Myers-Perry metrics Myers:1986un .

Lovelock gravity theories Lovelock:1971yv are fascinating extensions of general relativity that include higher curvature interactions. The Lagrangian density for Lovelock gravity in spacetime dimensions can be written , where

(1) |

and . The symbol above denotes the totally antisymmetric product of Kronecker deltas, normalized to take values and . The term gives the cosmological term in the action, while gives the Einstein term. The curvature squared term, known as the Gauss-Bonnet term, may be expanded to give . In this term can be written as a divergence and does not contribute to the equations of motion. Similarly the term is the Euler density in and hence contributes to the equations of motion only for dimensions . Lovelock theories are distinguished, among the much larger class of general higher curvature theories, by having field equations involving not more than second derivatives of the metric. Consequently, Lovelock gravity theories are free from many of the pathologies that plague general higher derivative gravity theories.

The black hole solutions of Lovelock gravity have been been studied begining with the work of Boulware:1985wk Wheeler:1985nh Wheeler:1985qd some years ago. One would also like to know the black string and black brane solutions, e.g. in order to study the effects of higher curvature interactions on black string phenomenology. A number of workers have tried to find such solutions. However, one immediately encounters the fact that adding flat directions does not work in the general Lovelock theory. This was noted for Gauss-Bonnet gravity in reference Barcelo:2002wz . Thus far, analytic solutions for black branes in general Lovelock theories have not been found, and workers have turned to other methods. In particular, black strings in -dimensional Gauss-Bonnet gravity were studied numerically in reference Kobayashi:2004hq and more general black branes in this theory were studied via near horizon and far field expansions in reference Sahabandu:2005ma .

In this paper, we show that black brane solutions to Lovelock gravity theories including higher curvature terms may, in fact, be simply constructed, but only within a certain class of Lovelock theories. This class of theories has the following property. Assume that is the highest order term in the Lagrangian, i.e. that the coefficients vanish for . Depending on the values of the nonzero coefficients in the Lagrangian, it then turns out that the theory may have up to distinct constant curvature vacuum solutions Boulware:1985wk Wheeler:1985nh Wheeler:1985qd . The different values that the constant curvature may take are the roots of a th order polynomial. There will, of course, generally be roots, but only real values of the curvature are considered to be physical. The coefficients in the Lovelock Lagrangian may be tuned such that there are real roots and that all these roots coincide. The theory then has a (locally) unique constant curvature vacuum solution. We will refer to these as LUV theories - standing for Lovelock-Unique-Vacuum. We show that LUV theories have simple black brane solutions.

The simplest LUV theory with non-trivial dynamics is Einstein gravity with a cosmological constant , and our results on brane solutions will be, roughly, that what works in this theory works in all LUV theories. We can first consider Einstein gravity with . In this case, as described above, brane solutions are simply obtained by adding flat directions. Our first result will be that adding flat directions also works in LUV theories, in the limit that the curvature of the vacuum has been tuned to zero. Taking this limit of a LUV theory with highest order interaction sends the coefficients of all the lower order terms to zero and we have simply . We will call these pure Lovelock gravity theories, with pure Einstein gravity as the first non-trivial example.

If we now consider Einstein gravity with , additional directions may still be added, but they are no longer flat, i.e. the new metric must have nontrivial coordinate dependence on the new direction. A good example of this is the AdS black string of reference Chamblin:1999by , which is given by

(2) |

where is the radius of curvature of -dimensional AdS. This differs from the black string of pure Einstein gravity in the overall conformal rescaling by the function of the new coordinate . Note also that the -dimensional ‘seed’ metric for the construction, the Schwarzschild metric in this case, is a solution to Einstein gravity with vanishing cosmological constant. More generally the -dimensional Schwarzschild metric in (2) may be replaced by any Ricci flat metric. We will show below that in LUV theories with nonzero vacuum curvature, new directions may be added in a similar way. The metric will generally be conformally scaled by a function of the new coordinate. The lower dimensional seed metric can be a solution to a LUV theory with a different value of the vacuum curvature, possibly zero as in the metric (2) above.

LUV theories and their black hole solutions have been extensively
discussed in reference Crisostomo:2000bb . It is argued there that
LUV theories have a number of nice properties that distinguish them as possiby the
most physically relevant Lovelock theories. We note Crisostomo:2000bb that
in odd dimensions the LUV theory with highest term , which is the
highest non-trivial term,
corresponds to Chern-Simons gravity^{3}^{3}3Further properties of these theories are discussed in references Mora:2004kb Banados:2005rz ., while in even dimensions the LUV theory that includes the highest non-trivial interaction follows from a Born-Infeld type action.

## Ii LUV Theories

In this section we recount some useful results in Lovelock gravity, focusing in particular on the LUV theories. The equations of motion following from the Lovelock Lagrangian , with have the form where

(3) |

For the purposes of studying LUV theories, it is also useful to write in an alternative form which we will make use of below,

(4) |

The original form of the equations of motion can then be recovered through repeated applications of the identity

(5) |

The coefficients are given by sums of products of the parameters in (4). The precise relation is given in reference Crisostomo:2000bb . Inverting this relation to get the ’s in terms of the ’s requires solving a polynomial equation of order . Hence the ’s are generally complex parameters.

For our purposes, it is convenient to take the parameters with to be real valued and regard the coefficients in the Lagrangian to be determined by them. It is then clear from the expression (4) for that if the parameters with are all distinct, then the theory will have distinct constant curvature solutions, with Riemann tensors given respectively by

(6) |

The LUV theories discussed above result from setting all the parameters with equal to a common value . There is then, at least locally, a unique constant curvature vacuum. If we further set , we get a pure Lovelock theory with Lagrangian , which has flat spacetime as its unique constant curvature vacuum.

The static, spherically symmetric solutions to Lovelock gravity in -dimensions can be written in the form

(7) |

where . The equations of motion in the form (5) resemble an th order polynomial equation of order. It is then not surprising that the metric function involves solving an th order algebraic equation Boulware:1985wk Wheeler:1985nh Wheeler:1985qd (see also Myers:1988ze for a detailed analysis of the general case). We will examine the solutions in Gauss-Bonnet gravity, in which , in some detail. In this case the solutions for the metric function has two branches, given by

(8) |

As tends to infinity the branches and approach constant curvature metrics with curvatures and respectively. If we take , then the solutions in the brance are asymptotically flat. Keeping the leading order correction to the flat metric in this case, we have as tends to infinity

(9) |

This is the correct fall-off for a black hole of finite ADM mass in -dimensions. However, the metric differs from the Schwarzschild metric at smaller scales.

We can take the LUV limit of the Gauss-Bonnet theory by setting , with . The expressions for the metric functions simplify considerably in this limit Crisostomo:2000bb to be

(10) |

For , the branch then gives black holes in (anti-)deSitter, while the branch gives naked singularities. In each case, the falloff to the vacuum is slower than in Schwarzschild-(A)dS, and the ADM mass would therefore be infinite. However, it was shown in Aros:1999kt Crisostomo:2000bb that an appropriate definition of the mass in LUV theories gives a finite result.

One can further take the limit to get pure Gauss-Bonnet theory. We then have simply

(11) |

For the two branches again yield either black holes, or naked curvature singularities. The case , however, is particularly interesting. In this case the metric function is simply a constant. Rescaling the time and radial coordinates, the metric can be written as

(12) |

where for the corresponding to -spheres with missing solid angle, while for corresponding to an excess of solid angle. In both cases there is a conical singularity at the origin. Other solutions to Lovelock theory with conical singularities at the origin (but with a different geometry in the constant section were obtained recently Dehghani .

Note that is the lowest dimension for which the Gauss-Bonnet term is dynamically relevant. The situation is similar Crisostomo:2000bb for all pure Lovelock theories, , in dimension . The static, spherically symmetric solutions have spheres with missing (or excess) solid angle. The first example with , i.e. pure Einstein gravity, in is a familiar one. Mass in this theory corresponds to point conical defects in an otherwise flat spacetime. In the general case spacetime is not flat, but satisfies the pure Lovelock field equations.

## Iii Black strings and branes in LUV theories

In pure Einstein gravity, we know that we can construct black brane solutions by adding flat directions to any black hole solution. More precisely, if we have a spacetime metric that solves the vacuum Einstein equations in -dimensions, then the metric

(13) |

solves the vacuum field equations in dimensions. If is the metric for a black hole with horizon topology , then the new metric is a black string with horizon topology . The procedure can be iterated to generate black branes with any number of flat spatial dimensions tanget to the horizon.

It is straightforward to check that this simple construction does not work in general Lovelock theories. In fact, it fails already in Einstein gravity with a nonzero cosmological constant. Consider Einstein’s equations for the -dimensional metric (13). Assume that satisfies these same equations. Note that all components of the Riemann tensor of the metric (13) with one, or more, indices necessarily vanish. In particular, this implies that and where the hatted quantities are those computed from . The components of the field equations are then identical to those for which are satisfied by assumption. The component of the field equations, however, requires , which conflicts with the trace of the lower dimensional field equations which implies . We then conclude that the -dimensional metric (13) satisfies the field equations only for , pure Einstein gravity.

Our first new result is to observe that, while adding flat directions does not generally work in Lovelock gravity, it does work in all pure Lovelock theories, with Lagrangians . The result above for pure Einstein gravity can then be seen as the particular case of this more general statement. If solves the field equations of the pure Lovelock theory in -dimensions, then the metric (13) solves the pure Lovelock field equations in dimensions. This is easily seen from the field equations (3), with only nonzero, in the following way. Note that as before all the components of the Riemann curvature tensor with one, or more, indices necessarilly vanish. The field equations are then again identical to the -dimensional field equation for and hence are satisfied by assumption. The equation is given by the trace and hence also vanishes. Note that, as in the pure Einstein case, the trace of the pure Lovelock field equations for is equal to times a constant.

This result leads to some intriguing possibilities. In pure Einstein theory, flat directions can be added to the point-like conical defects of gravity to get cosmic strings in and branes in higher dimensions, which are surrounded by circles with deficit (surfeit) angle. These defects play important roles in, for example, the Weyl solutions and, in particular, in the C-metric. There is great interest in higher dimensional generalizations of these spacetimes (see e.g. Emparan:2001wk Charmousis:2003wm ). If we take the case of the pure Gauss-Bonnet gravity, , then we have in a static, spherically symmetric solutions with missing (excess) angle on an . If we add one flat direction, we get a string in surrounded by an with missing (excess) solid angle. It is tempting to think that this string might serve as a starting point for finding analogues of the Weyl solutions or the C-metric in .

The next step is to attempt to find a construction of black strings and branes that allows us to add back in the lower order terms in the Lovelock Lagrangian. Here, we are able to make some partial progress by again take results from Einstein gravity as a guide. If one considers Einstein gravity with a negative cosmological constant , then we know that if a metric in dimensions is Ricci-flat, then the metric

(14) |

solves the field equations in dimensions (see e.g. Chamblin:1999by ). If we try this in Lovelock theory, we find the following result. If solves the field equations of pure -dimensional Lovelock gravity with Lagrangian , then the metric (14) solves the field equations of the -dimensional LUV theory, also with the highest interaction term , but with .

To keep the formulas manageable, we will focus on the case of Gauss-Bonnet gravity. The proof straightforwardly generalizes to higher order LUV theories. Our -dimensional spacetime is then assumed to satisfy the equations of motion of pure Gauss-Bonnet gravity

(15) |

where is the Riemann tensor of . If we let denote the metric in (14) then its nonzero curvature components are given by

(16) |

The field equations of the LUV theory are given by . After plugging in the curvature components (16) the component of the field equations becomes

(17) |

Having made the choice , we see that this is simply proportional to the trace of the equations of motion (15) for . Plugging into the components of the field equations gives

For the first term again reduces to the -dimensional equations of motion for , while the second term simply vanishes. The equations are trivially satisfied, and we have thus shown that adding the construction (14) of new solutions holds for LUV theories of maximum order starting with solutions of pure pure Gauss-Bonnet theory. The structure of our derivation makes it clear that the result holds for higher order LUV theories as well.

We now prove a related, but more general result as well. Assume now that the -dimensional metric solves the equations of motion of LUV theory with maximum interaction and curvature parameter ,

(19) |

Now consider a -dimensional metric of the form

(20) |

and ask if it can solve the field equations of the same LUV theory, but with vacuum curvature parameter instead. Our previous result corresponds to the special case that the curvature parameter of the -dimensional metric vanishes. The curvature of the -dimensional metric (20) is given by

(21) |

The field equations for the -dimensional metric (20) are again given by . Upon plugging in the components of the Riemann tensor, the equation becomes

(22) |

This will be proportional to the trace of the field equations (19) for the -dimensional metric if the function satisfies the relation

(23) |

Plugging in the components of the Riemann tensor into the equations gives the equation

The first of terms will vanish by virtue of the equations of motion (19) for the -dimensional metric if the relation (23) is satisfied. The second term will vanish, if the function also satisfies the additional relation

(25) |

If we write the and -dimensional curvature parameters as and respectively, then both relations are satisfied if the conformal factor in the -dimensional metric (14) is taken to be

(26) |

Note that we can take the limit of this result in which the curvature parameter of the -dimensional metric vanishes. Our previous result for this case, that , is then seen as the small angle approximation to the more general result above. We also note that, for the sake of definiteness, we have focused on LUV theories with AdS vacua. However, the above result is only simply modified for LUV theories with deSitter vacua. If we take and then we find that the conformal scaling function must be given by

(27) |

## Iv Concluding Remarks

We have shown that the construction of black string and brane solutions in pure Lovelock gravity theories and in LUV theories parallels the simple constructions in pure Einstein theory and in Einstein theory with a nonzero cosmological constant respectively. While this does not fully solve the problem of constructing black brane solutions in Lovelock gravity theories, it represents useful progress in a number of ways.

On one hand, it has been argued in reference Crisostomo:2000bb that LUV theories represent the most physically interesting class of Lovelock gravity theories. From this perspective, our results cover precisely this most interesting case. Further, the simplicity of our results, and their parallels with Einstein gravity, suggest that there may well be other methods of constructing solutions in Einstein gravity that have simple generalizations to pure Lovelock and LUV theories. In particular, as we commented above, the appearance of string solutions with deficit solid angle on odd dimensional spheres, suggests that we might be able to find analogues of the Weyl solutions and the C-metric in certain of these theories.

On the other hand, one’s primary interest may be in corrections to solutions to pure Einstein gravity due to higher curvature Lovelock terms. In this case, our results are limited, but provide some clues towards finding black string and brane solutions in the general case. Consider the black hole solutions of Gauss-Bonnet gravity which have the form (7) with the metric function given in equation (8). Take , so that we have an asymptotically flat solution. For large this approaches Schwarzschild as in equation (9). As , however, where the curvature becomes large, the solution matches onto a solution to pure Gauss-Bonnet gravity. If we want to construct a black string solution, in this case, it should asymptote for large to Schwarzschild with an extra flat direction, the black string of pure Einstein gravity. Our results suggest that it should also look simple at small radii. There it should approach the pure Gauss-Bonnet limit of (8) again with a flat direction added. For intermediate radii, the solution will presumably depend on the coordinate along the string in a non-trivial way.

###### Acknowledgements.

The authors would like to thank the organizers of the Scanning New Horizons: GR Beyond 4 Dimensions workshop at the KITP where this work was carried out. DK would like to thank Fay Dowker, Satyanarayan Mohapatra and Jennie Traschen for helpful discussions on this subject. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949 and Grant No. PHY 0244801 and by the Natural Sciences & Engineering Research Council of Canada.## References

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