# Homogeneous black strings in Einstein–Gauss–Bonnet with Horndeski hair and beyond

## Abstract

In this paper we construct new exact solutions in Einstein–Gauss–Bonnet and Lovelock gravity, describing asymptotically flat black strings. The solutions exist also under the inclusion of a cosmological term in the action, and are supported by scalar fields with finite energy density, which are linear along the extended direction and have kinetic terms constructed out from Lovelock tensors. The divergenceless nature of the Lovelock tensors in the kinetic terms ensures that the whole theory is second order. For spherically, hyperbolic and planar symmetric spacetimes on the string, we obtain an effective Wheeler’s polynomial which determines the lapse function up to an algebraic equation. For the sake of concreteness, we explicitly show the existence of a family of asymptotically flat black strings in six dimensions, as well as asymptotically \(\hbox {AdS}_{5}\times \mathbb {R}\) black string solutions and compute the temperature, mass density and entropy density. We compute the latter by Wald’s formula and show that it receives a contribution from the non-minimal kinetic coupling of the matter part, shifting the one-quarter factor coming from the Einstein term, on top of the usual non areal contribution arising from the quadratic Gauss–Bonnet term. Finally, for a special value of the couplings of the theory in six dimensions, we construct strings that contain asymptotically AdS wormholes as well as rotating solutions on the transverse section. By including more scalars the strings can be extended to *p*-branes, in arbitrary dimensions.

## 1 Introduction

Higher dimensional general relativity (GR) possesses a broader spectrum of black hole solutions as compared with its four dimensional formulation [1, 2]. Indeed, four dimensional GR is constrained by uniqueness theorems [3, 4, 5] that ensure that any black hole solution of the theory is contained in the Kerr family [6, 7]. Moreover, topological restrictions allow only for horizons with spherical topology [8]. The spectrum of solutions is limited not only quantitatively but also qualitatively, setting the final state of black hole collapse to be described only by a small set of parameters [9]. On the other hand it is well-known that gravity in higher dimensions admits spacetimes with horizons that can have more general topologies than that of the \(\left( d-2\right) \)-sphere [1, 2], being the existence of black strings, a black hole solution with horizon structure \(S^{(d-3)}\times R\), the simplest counterexample [3, 4, 5] as it coexists with the Schwarzschild–Tangherlini black hole [10]. Black strings also paved the road for the construction of more sophisticated asymptotically flat solutions with non-spherical topology such as black rings [11] and diverse black object solutions [2], demonstrating how topological restrictions [8] are weakened in higher dimensions. It was shown that black strings are affected by Gregory–Laflamme (GL) instability [12, 13], a long-wavelength perturbative instability triggered by a mode that travels along the extended direction of the horizon and, moreover in dimension five the numerical simulations indicate that the instability ends in the formation of naked singularities [14, 15], representing an explicit failure of the cosmic censorship in higher dimensions [16].^{1}

Black strings in GR in vacuum are easy to construct. In fact they are obtained by a cylindrical oxidation of the Schwarzschild black hole in *d* dimensions by the inclusion of *p* extra flat coordinates. This can be realized by seeing that the equations of motion along the extended directions are compatible with the field equations on the *p*-brane, due to the fact that the involved curvature quantities vanish on these coordinates.

Notwithstanding it is not hard to find simple scenarios in which the construction of analytic black strings fails. Indeed, the mere inclusion of a cosmological constant spoils the existence of cylindrically extended black strings, since the compatibility of the equations of motion on the flat coordinates with the trace of the field equations along the brane forces \(\Lambda \) to vanish. This implies that there is no simple oxidation of the Schwarzschild (A)dS black hole.^{2}

^{3}To see this explicitly let us take the following black string ansatz in which a

*d*-dimensional black hole is oxidated by including

*p*flat directions

^{4}

A natural question arises: as higher dimensional objects, do homogeneous and cylindrically extended black strings and black *p*-branes exist in higher curvature extensions of GR?

This paper is organized as follows: To fix ideas, Sect. 2 is destined to construct homogeneous black strings in Einstein–Gauss–Bonnet theory dressed by two scalar fields, a minimally coupled one accounting for the inclusion of the cosmological constant and a non-minimally coupled one accounting for the new \(R^{2}\) curvature terms we are including. We present the equations in arbitrary dimension \(D=d+1\), nevertheless for the sake of concreteness we focus on the minimal dimension allowing for new black strings which in this case is \(D=(1+4)+1\) and analyze the causal structures as well as the thermal properties of the black strings. Section 3 is devoted to extend the previous results to theories containing arbitrarily higher curvature terms in the Lovelock family. Remarkably, we show that there is a pattern that allows to construct homogeneous AdS black strings in Lovelock theories supported by a family of scalars with generalized non-minimal kinetic terms, leading to a Wheeler’s like polynomial with effective couplings. Assuming the existence of an event horizon, we compute the temperature, as well as the entropy of the black hole. In Sect. 4 we revisit the Einstein–Gauss–Bonnet theory and show that for special values of the matter couplings, if the transverse section of the black strings and *p*-branes is five dimensional, the theory allows for wormholes as well as rotating solutions. The latter leads to the first example of a rotating solution in Einstein–Gauss–Bonnet theory in arbitrary dimensions. Finally in Sect. 5 we outline our conclusions and further developments that can follow this work.

## 2 Homogeneous black strings in Einstein–Gauss–Bonnet theory

*z*, the equations for the scalars imply a linear dependence on

*z*. The shift symmetry of the scalar field theories can be used to set to zero the additive integration constants that appears in both scalars. So we have

*d*-dimensional manifold and the equation along the

*z*direction respectively read

*d*-dimensional manifold with line element \(d\tilde{s}_{d}\). To avoid incompatibilities these two equations must be proportional term by term, i.e. \(\mathcal {E}_{1}=\xi \mathcal {E}_{2}\) and therefore, when the higher curvature Gauss–Bonnet term is present, one obtains that the proportionality constant must be fixed as well as the integration constants of the scalars, leading to

^{5}

*d*-dimensional metric \(d\tilde{s}_{d}^{2}\) fulfils the Einstein–Gauss–Bonnet field equations with the following rescaled couplings

*z*section of the black string

^{6}(10). Since \(\tilde{H}_{\mu \nu }\) vanishes identically in dimensions \(d\le 4\), to have a non-vanishing contribution from the Gauss–Bonnet tensor, we need \(d\ge 5\) (in \(d=4\) the equation (16) is identically fulfilled for any metric and the system is degenerate at such point).

*K*, and assume that the scalars depend only on the

*z*direction.

*z*-dependence of the scalars is fixed by the scalar equations to be linear, giving rise to

*m*is an integration constant and \(V^{\left( K\right) }_{3}\) stands for the volume of manifold with line element \(d\Sigma _{K,3}\).

*m*, leading to a finite mass contribution (mass density if one considers the extended direction). Considering \(\alpha \) positive and \(l^{2}\) to be the curvature radius of the brane at infinity, leads to the restriction \(\frac{9}{2}<l^{2}<9\).

*m*with the mass density, the thermodynamical quantities fulfil the first law

We have therefore shown that the inclusion of the non-minimal Einstein–kinetic coupling (5) allows to construct black strings in Einstein–Gauss–Bonnet theory, for arbitrary values of the coupling constants of the theory.

As mentioned above, the obstruction to the existence of cylindrically oxidated solutions comes from the incompatibility between the trace of the field equations on the brane and the equation along the extended direction. The scalar fields \(\psi \) and \(\chi \) provide a natural manner to circumvent this incompatibility even in the asymptotically flat case. Below, we give a detailed explanation of the mechanism behind the existence of these solutions for general Lovelock theories, and show that the results can be extended beyond Einstein–Gauss–Bonnet by the inclusion of scalars with non-minimal kinetic coupling to Lovelock tensors, that naturally extend (5).

## 3 Homogeneous black strings in general Lovelock theory

*k*th scalar directly to the

*k*-th order Lovelock tensor \(E_{CD}^{(k)}\) (33). The Lovelock tensor has a divergence that vanishes identically since it corresponds to the Euler–Lagrange derivative with respect to the metric of a diffeomorphism invariant action. The \(\alpha _{k}\) are the dimensionful Lovelock couplings and \(\beta _{k}\) are the matter couplings.

*k*defined as

*k*th scalar \(\phi _{\left( k\right) }\) is given by

We will split the *D*-dimensional indices \(\left\{ A,B,C, \ldots \right\} \) in \(\left\{ \mu ,\nu , \ldots \right\} \) that run on the brane with line element \(d\tilde{s}_{d}\) and *z* along the extended direction. Quantities intrinsically defined on the *d*-dimensional manifold will have tildes on top.

*z*direction \(\mathcal {E}_{zz}\) reads

*n*integration constants \(c_{k}\) for all the scalars in the following manner

*m*is an integration constant and \(V^{\left( K\right) }_{d-2}\) stands for the volume of manifold with line element \(d\Sigma _{K,d-2}\). This leads to a black string in a general Lovelock theory of gravity that can be asymptotically flat or asymptotically AdS depending of whether we include a cosmological term in the action. The temperature and the entropy of the corresponding black hole are respectively given by

*m*as the mass density.

*d*to dimension \(d+1\). This will be exploited in the next section.

## 4 Extra physically interesting solutions

*m*vanishes, one will obtain

*n*different solutions for the metric function \(f\left( r\right) \), leading to

*n*different constant curvature solutions on the transverse section of the string. When the curvature radii of these solutions coincide, it is natural to expect an enlargement of the space of solutions of the theory, as it occurs in vacuum. To see this in a concrete example, let us revisit the Einstein–Gauss–Bonnet theory in six dimensions

*L*, in double oblate coordinates, i.e.

*l*is a constant defined by the theory, while

*L*is an integration constant as well as

*a*and

*b*. The latter are the so-called oblateness parameters. Since the vector field

*k*defines a null and geodesic congruence of the background metric \(d\bar{s}_{L}\), the metric (56) is a cylindrical oxidation of a Kerr-Schild metric in \(\hbox {AdS}_{5}\). The five dimensional metric in the section of the string defines the first known exact, analytic, rotating solution of Einstein–Gauss–Bonnet in five dimensions and it was originally found in [53]. It was also latter proven to be of the non-circular type in [54].

^{7}

## 5 Final remarks

Until now in higher curvature gravity, exact and homogeneous black string solutions have been constructed only for special values of the coupling constants [56, 57] also including *p*-form fields [58], and the general problem of the construction of Lovelock branes was studied in [59], and for arbitrary values of the couplings only numerical or perturbative solutions were available (see e.g. [60, 61]). In this paper we have constructed new, exact, homogeneous black strings in arbitrary Lovelock theories. The solutions are supported by scalar fields with non-minimal kinetic couplings constructed with Lovelock tensors, ensuring that the field equations are of second order. These scalars have been previously considered as part of the higher dimensional extension of Horndeski theories in [62, 63]. Here, the scalars being dependent only on the coordinate along the extended direction, turn out to be linear, and the proportionality constant gets fixed by the requiring the compatibility of the whole system. The transverse section of the string can be asymptotically flat or *AdS*. For concreteness we computed the thermodynamic quantities and showed that the entropy of the black string receives a contribution from the matter part. This is interesting because the pattern of transitions between black string and black hole can change due to the presence of the new Horndeski fields. The gravitational stability of the black string and *p*-branes in the presence of a single quadratic or cubic Lovelock term has been studied in [64, 65, 66], while the effect of quartic corrections coming from M-theory have been explored in [67]. It is interesting to mention that the Large D approach [68] allows to keep all the terms in the Einstein–Gauss–Bonnet Lagrangian [69].

For simplicity we focus on the case with a single extended direction, but this construction also works with *p*-branes. For example, in the Einstein–Gauss–Bonnet theory, with a cosmological constant, one would have to consider *p* minimally coupled scalars mimicking such of reference [23], as well as *p* scalars with Einstein–kinetic couplings which would also turn out to be linear and depending on a single extended direction. From this, the extension to arbitrary Lovelock theories, with flat *p*-branes is clear.

Interpreting our solutions as compactifications with non-trivial scalar fluxes along the extended direction, one obtains an effective Lovelock theory induced on the brane. We exploited this idea to construct cylindrically extended solution with wormholes on the transverse section, which are asymptotically \(\hbox {AdS}_{5}\times R\), in both directions. Within the same realm we also constructed a cylindrical oxidation of the rotating spacetime of Einstein–Gauss–Bonnet gravity, constructed from a Kerr–Schild ansatz in [53]. The wormhole solution can of course be extended to \(\hbox {AdS}_{2n-1}\times R\) provided one considers Lovelock theory with all the possible terms in dimension 2*n*. In such case, the wormhole on the string will be the one reported in [51, 70].

## Footnotes

- 1.
- 2.
- 3.
Previously, AdS black strings in GR were constructed considering warped spacetimes [24] providing non-homogenous configurations.

- 4.
- 5.
Note that when \(\alpha =0\), one obtains only two equations for the three constant \(\xi \), \(c_{0}\) and \(c_{1}\), leading to \(c_{0} ^{2}=-\frac{4\Lambda _{0}\left( \xi d-1\right) }{\xi d+1}\) and \(c_{1} ^{2}=\frac{4}{\gamma }\frac{\left( \left( d-2\right) \xi -1\right) }{\left( \left( d-2\right) \xi +1\right) }\) for an arbitrary \(\xi \). Since we are interested in the inclusion of higher derivative terms, we do not elaborated further in the case \(\alpha =0\).

- 6.
Note that the equations that determine the geometry (16) do not depend on \(\gamma \), since the energy-momentum tensor is proportional to the coupling \(\gamma \) and at the same time is quadratic in the field \(\chi \sim \gamma ^{-1/2}\).

- 7.
For a recent extension of the rotating metric in Chern–Simons theories in odd dimensions, within the Lovelock family in vacuum, see [55].

## Notes

### Acknowledgements

The authors would like to acknowledge E. Babichev, C. Charmousis, N. Grandi and J. Rocha for valuable comments and remarks. J.O. is partially supported by FONDECYT Grant 1181047. A. C. is supported by Fondo Nacional de Desarrollo Científico y Tecnológico Grant no. 11170274 and Proyecto Interno Ucen \(I+\hbox {D}\)-2016, CIP2016.

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