Homogenous BTZ black strings in Chern-Simons modified gravity

Four dimensional homogeneous anti de-Sitter black string configurations in dynamical Chern-Simons modified gravity, with and without torsion, are presented. These solutions, which are supported by (pseudo)scalar fields depending only on the extended flat coordinate, represent four dimensional black string extensions of the Ba\~nados-Teitelboim-Zanelli black hole. The case with nontrivial torsion is studied within the first-order formalism of gravity, by considering nonminimal couplings to three topological invariants: Pontryagin, Nieh-Yan, and Gauss-Bonnet terms, which are studied separately. These interactions generate torsion in vacuum, in contrast to Einstein-Cartan theory. The field equations impose severe restrictions on the axial and vectorial irreducible components of the torsion along the 3-dimensional transverse manifold, however, they do allow for nontrivial configurations along the extended direction. In all cases, torsion contributes to an effective cosmological constant that, in particular cases, can be set to zero by a proper choice of the parameters.


I. INTRODUCTION
The detection of gravitational waves (GWs) [1,2] and the confirmation of their luminical propagation [3] have placed Einstein General Relativity Theory (GR) as the most successful description of the gravitational interaction, complementing more than one century of experimental success at the solar system length scales [4].In spite of this huge achievement, GR is continuously challenged to give self-contained explanations for several phenomena coming from opposite ends of the length scale range.An iconic example is the accelerated expansion of the Universe [5,6], for which GR with a cosmological constant term cannot account for without facing certain difficulties [7].Several theories beyond GR, known as modified gravity theories [8], have appeared exposing a wide variety of theoretical predictions.However, after the simultaneous detection of GWs and their electromagnetic counterparts [3], very strict constraints have been imposed for any modified theory aiming to describe gravity 1and several models that predicts anomalous GWs speed have been ruled out [9][10][11][12][13][14][15][16].
Among the large set of modified gravity theories, recently, it has been established that for generic ghost-free theories with parity violation, only Chern-Simons modified gravity (CSMG) accomplished for the observed propagation of GWs [17].Such a theory considers nonminimal couplings between gravitational pseudoscalar degrees of freedom and the topological Pontryagin density in four dimensions [18], and it is well-motivated by the anomaly cancelation in curved spacetimes, string theory compactifications, and particle physics [19].
The nonminimal coupling between the pseudoscalar field and the Pontryagin density might explain flat galaxy rotation curves without introducing dark matter [20], and future gravitational wave detections might be sensitive to such a modification through frame dragging, gyroscopic precession, and amplitude birefringence in propagation of gravitational waves [21][22][23].In the nondynamical case, namely in absence of a kinetic term for the pseudoscalar field, all spherically symmetric solutions of GR are solutions of CSMG, since the Pontryagin density and its associated Cotton tensor2 vanishes for the action of such an isometry group.However, this is not the case for axially-symmetric solutions, such as the Kerr black holes, where neither the Pontryagin nor the Cotton tensor vanish [24][25][26].On the other hand, CSMG should be considered as an effective theory, since there seems to be evidence that its Cauchy initial-value problem is ill-posed when the pseudoscalar field is dynamical [27]. 3evertheless, a possible way out would be to consider its first-order formulation by treating the metric and connection as independent fields by introducing torsion.
The first-order formalism of gravity offers a natural framework in which gravity can be considered as a gauge theory [28][29][30].This formalism, in absence of the Hodge dual, gives first-order field equations for the two independent gravitational potentials: the vielbein and Lorentz connection.The geometrical structure is characterized by the curvature and torsion as independent objects and it allows to include the spin density of matter as a gravitational source [31][32][33].The most general four-dimensional action constructed out of the vierbein and Lorentz connection, without the Hodge dual, and that is invariant under diffeomorphisms and local Lorentz transformations is the Einstein-Cartan theory with cosmological constant [34].This theory has vanishing torsion in vacuum and it reduces to GR with cosmological constant on shell.However, deviations from GR are expected in presence of polarized spin density, since the contribution of torsion emerges as contact spinspin interactions at the effective level [32,35].In the realm of the first-order formalism, Chern-Simons modified gravity has been studied from different motivations, by considering nonminimal couplings of gravitational (pseudo)scalar degrees of freedom to the Pontryagin, Nieh-Yan, and Gauss-Bonnet densities in four dimensions, generating torsion in vacuum.
The former case exhibits modifications to the standard four-fermion interaction appearing in Einstein-Cartan theory, whose physical consequences has been discussed in [36].On the other hand, the Nieh-Yan density [37] that measures the difference between SO(5) and SO(4) Pontryagin classes, contributes to the fermionic axial anomaly in Riemann-Cartan spacetimes, and it diverges once the regulator is removed [38].A CSMG model based on the pseudoscalar-Nieh-Yan coupling has been proposed in [39,40], whose shift symmetry allows to regularize such a divergence.When torsion is integrated out, this model resembles the scenarios of Refs.[41,42], and it might offer a solution to the strong CP problem [43][44][45].Finally, first-order cosmological scenarios have been studied in dilaton-Gauss-Bonnet gravity [46][47][48][49], motivated by dimensional reductions of Lovelock gravity, which appears as low-energy corrections of string theory [50].
When considering any modified theory of gravity, it is of primary interest to study the phase diagram of the existing solutions and their properties.In this work it is shown that the spectrum of compact object solutions of CSMG can be enlarged by the existence of black string configurations, for both cases, with and without torsion.Black strings are, in principle, higher dimensional asymptotically flat black hole solutions with an extended horizon of topology S 2 × R n , or (S 1 ) n when compactifying the extra dimensions [51].They are easily constructed by considering extra flat directions on the spacetime metric and they represent the most simple counterexample to the uniqueness theorems for higher dimensional GR [52][53][54], as it is revealed by its coexistence with the Schwarzschild-Tangherlini black hole [55].Even more, they pave the way to construct asymptotically flat solutions with nonspherical topology such as black rings [56] and diverse black object solutions [57], demonstrating that topological restrictions [58] lose their strength in higher dimensions.The black string was demonstrated to suffer from Gregory-Laflamme (GL) instability [59,60], a long-wavelength perturbative instability triggered by a mode traveling along the extended direction.Moreover, it was numerically shown that in five dimensions this instability ends up in the formation of naked singularities [61,62], representing an explicit failure of the cosmic censorship conjecture in dimensions greater than four [63].This instability seems to persist even when black strings are constructed in other gravitational theories apart from GR [64][65][66][67].Nevertheless, the evident simplicity involved in the construction of black strings, namely, the simplicity involved in the oxidation process of the spacetime by adding extra flat directions, there are simple setups in which the construction is not evident.The most illustrative case is when the cosmological constant is included.It is direct to see that, if a D = d + p dimensional spacetime is considered with p flat directions, the field equations force the cosmological constant to vanish.This implies that there is no simple oxidation of the Schwarzschild (A)dS black hole. 4Moreover, similar obstructions are encountered when trying to oxidate the Reissner-Nordstrom black hole.
In [71], a simple approach to construct homogeneous AdS black strings in GR has been developed, by including a set of p minimally coupled scalar fields that depend only on the extra flat coordinates.This strategy allows to obtain the black string oxidation of the Schwarzschild AdS black hole in any dimension, showing that the Bañados-Teitelboim-Zanelli (BTZ) black hole [72] can be uplifted to a black string in four dimensions, despite 4 Nonhomogeneous AdS black strings have been constructed in Ref. [68] by considering warped spacetimes.This result was generalized for Lovelock theories possessing a unique constant curvature vacua [69] and for more general Lovelock theories by generalizing the concept of Einstein spaces [70].AdS black strings and black rings have been constructed only numerically for nonhomogeneous geometries [51,57].
these configurations have been thought to be higher dimensional.The aim of this work is to extend the approach of Ref. [71] to dynamical CSMG, with and without torsion, to obtain exact four-dimensional black string solutions in vacuum.In order to do so, the (pseudo)scalar fields are not assumed to be compatible with the hypersurface orthogonal Killing vector field that foliates the black string geometry, in contrast to Refs.[73,74].Relaxing this assumption, (pseudo)scalar fields with linear dependece on the extended coordinate are found, allowing to obtain solutions that belong to the so-called Chern-Simons sector of the space of solutions of the theory [24], since their Cotton tensor contributes nontrivially to Einstein's field equations.Then, a first-order extension of CSMG is studied by considering the nonminimal couplings of the (pseudo)scalar fields to the Pontryagin, Nieh-Yan, and Gauss-Bonnet terms, generate torsion in vacuum, in contrast to the Einstein-Cartan theory.Then, the method of Ref. [71] is generalized within this framework by treating the vierbein and Lorentz connection as independent fields.Restrictions on the irreducible components of the torsion in the 3-dimensional transverse manifold are found, however, nontrivial torsional configurations with components in the extended direction are obtained in each case.The solutions presented in this work, with and without torsion, represent the black string extension of the BTZ black hole [72], and they can be used to test stability and to see whether torsion can cure the Gregory-Laflame instability [59,60].
The article is organized as follows: in Sec.II, the method for constructing black strings in GR in arbitrary dimensions is reviewed, the Riemannian (torsion-free) dynamical CSMG theory is presented, and the BTZ black string solution is obtained.In Sec.III, the first-order formulation of CSMG is considered by taking the nonminimal couplings of (pseudo)scalar fields to Pontryagin, Nieh-Yan, and Gauss-Bonnet terms.The method of Ref. [71] is generalized within this framework, imposing on-shell restrictions on the torsion components.
Then, black strings solutions are found with nontrivial torsion by studying each case separately.Conclusions and comments are presented in Sec.IV.Finally, Appendix A shows how to connect the results of Sec.II with the first-order formulation by imposing the torsion-free condition through a Lagrange multiplier.The notation used here considers greek and latin characters as spacetime and Lorentz indices, respectively.In the Subsec.II A, the method for constructing black strings and branes is presented in D dimensions, however, all the solutions presented throughout this work are four-dimensional.The metric signature under consideration is (−, +, ..., +).

II. BLACK STRINGS IN CHERN-SIMONS MODIFIED GRAVITY
In this section, the method for constructing black strings in D-dimensional GR is reviewed [71], and it is employed to obtain a BTZ black string in CSMG with a dynamical pseudoscalar field.Configurations of this class have been studied in the nondynamical case [74], by assuming that the pseudoscalar field is compatible with the isometry group of the black string.In such a case, the pseudoscalar field remains as an arbitrary function of the radial coordinate, while acting as a Lagrange multiplier that imposes the Pontryagin density to be zero.This latter constraint is trivially fulfilled by virtue of the isometry group, and the Chern-Simons term does not contribute to the Einstein's field equations since its associated Cotton tensor vanishes.The solutions presented throughout this work differ from [74] in that the (pseudo)scalar fields possess a kinetic term and their compatibility with the hypersurface orthogonal Killing vector that foliates the black string geometry is not assumed.The Klein-Gordon equation, alongside the nondiagonal components of the field equation for the metric, is then solved by (pseudo)scalar fields that depend linearly on the extended coordinates.The shift symmetry of CSMG guarantees finite energy density of the pseudoscalars and the energy-momentum tensor is compatible with the symmetries of the metric.These configurations belong to the so-called Chern-Simons sector of the space of solutions [24] since, even though the Pontryagin density vanishes by the action of the isometry group, it contributes nontrivially to the Einstein's field equations through its associated Cotton tensor.This class of solutions has not been considered in the dynamical case and it presents the main goal of this section.

A. Homogeneous AdS black strings in GR
Here, the construction of AdS black strings in GR with cosmological constant proposed in [71] is reviewed and the four-dimensional BTZ black string is presented.Consider a spacetime with D = d + p dimensions foliated with p space-like hypersurface orthogonal Killing vector fields ξ i with i = 1, ..., p, such that g µν ξ µ i ξ ν j = δ ij and [ξ i , ξ j ] = 0.These conditions guarantee a vanishing extrinsic curvature and they define the projection where barred symbols denote projection on the d-dimensional spacetime, ḡµ ν ḡν λ = δ µ λ and ξ µ i ḡµν = 0.A convenient set of coordinates associated to the integral curves of these Killing vectors can be introduced to write the D-dimensional spacetime metric as Thus, p represent the extra flat directions that oxidate the d-dimensional metric.The existence of ξ i imply that this metric is homogeneous, in the sense that it remains invariant under translations along the z i coordinates.
By including p minimally coupled scalar fields ϕ i , with i = 1, 2, ..., p, the equations of motion of GR with cosmological constant and for the scalar fields reads where = g µν ∇ µ ∇ ν , and Einstein equations along the d-dimensional manifold and Einstein equations projected along the p flat directions read Observe that no restriction on the bare cosmological constant arises when the trace of Eq. ( 6) is taken, whenever λ and Λ are related properly.In absence of the scalar fields, the equations of motion would have required a vanishing cosmological constant.
In addition, if the d-dimensional metric is static and endowed with either ISO(d − 2), SO(d − 1), or SO(1, d − 2) isometries, the D = d + p dimensional spacetime metric can be written as where dΩ 2 d−2,γ represents a (d − 2)-dimensional base manifold of constant curvature with γ = 0, ±1, representing flat, spherical, and hyperbolic section, respectively.Assuming the scalar fields to be invariant only under the isometries of ḡµν , the Klein-Gordon equations ( 4) alongside the nondiagonal components of the Einstein's equations admit a branch of solutions for scalar fields depending linearly on the z i coordinates, that is with λ an integration constant.Even though the flat directions are extended, the energy density associated to the scalar fields remains finite.Moreover, although the scalar field is not compatible with the symmetries generated by the hypersurface orthogonal Killing vector, the energy-momentum tensor is compatible with such a symmetry.This kind of scalar fields have been used to construct planar hairy black holes that exhibit momentum relaxation in their dual representations.By the explicit breaking of translational symmetry, it is possible to obtain well behaved holographic conductivities in the dual field theory [75].
The solution to the field equations ( 3) is given by where and M is an integration constant related to the mass. 5Observe that Λ must be negative and that the AdS radius gets a modification given by the number of extra flat directions p.
These results imply that black strings, which are originally thought to be higher dimensional objects, also exist in four dimensions.In fact, the BTZ black string metric reads which is supported by a single scalar field ϕ = λz, with λ 2 = −Λ/κ.In what follows, these ingredients will be used to construct homogenous AdS black strings in CSMG by considering d = 3 and p = 1.

B. Chern-Simons modified gravity
Chern-Simons modified gravity considers two independent gravitational fields: the metric g µν and the pseudoscalar field ϕ.The action principle for dynamical CSMG is given by [19] where κ = 8πG N is the gravitational constant and α is a dimensionful coupling constant, and the Pontryagin term is with ǫ µνλρ being the Levi-Civita tensor.The field equations for this theory are obtained by performing stationary variations of the action (13) with respect to the metric and the pseudoscalar field, respectively giving where and * R µνλρ = 1 2 ǫ λρστ R µν στ .Notice that the contribution of the Cotton tensor C µν in the Einstein's field equations involves covariant derivatives of the Riemann tensor, giving, in general, third order field equations for the metric.Importantly, since the field equations of CSMG involve only derivatives of the pseudoscalar field, they are invariant under the shift in field space δφ = c, where c is a constant, while the metric remains invariant.This is a key feature of CSGM that allows for the pseudoscalar fields to have a linear dependence on the extended coordinate without breaking the isometries of the 3-dimensional metric.

C. BTZ black string
In this subsection, the construction of the BTZ black string in CSMG is presented.The metric ansatz under consideration is This form of the metric ansatz imply that the Pontryagin term vanishes identically, that is, * RR = 0. Therefore, one ends up with a free Klein-Gordon equation ( 16) that, assuming the scalar field to be invariant only under the isometries of ḡμν and using the nondiagonal part of Eq. ( 15), can be integrated giving a pseudoscalar field with a linear dependence on the extended coordinate, i.e., ϕ = λz.Projecting Eq. ( 15) along the 3-dimensional coordinates and taking its trace, while, on the other hand, projecting the same equation on the zz components, respectively give where the fact that ḡμν C μν = 0 = C zz has been used.Equations ( 20) and ( 21) impose λ 2 = −Λ/κ, forcing the bare cosmological constant Λ to be negative.The contribution of the Cotton tensor in Eq. ( 15) is nontrivial due to the linear dependence of the pseudoscalar field on the z-coordinate, however, if the pseudoscalar field would have been compatible with the symmetry of the hypersurface orthogonal Killing vector, the Cotton tensor would have vanish as noticed in Refs.[73,74].
Then, the equations of motion (15) are solved by the following metric functions where M and j are integration constants denoting the mass and angular momentum, respectively, and ℓ −2 = −Λ/2.This solution is locally AdS 3 × R, and it represents the cylindrical extension of the BTZ black hole [72].The metric (19) possesses three Killing vectors: ξ 2 = ∂ φ , and ξ 3 = ∂ z .Moreover, the metric function f (r) vanishes for and, for a horizon to exist, the condition M 2 −4j 2 /ℓ 2 ≥ 0 must be met.The extremal case is obtained when this last condition is saturated, where the two roots coincide.The curvature invariants constructed out of ( 19) with ( 22) remain constants, however, a singularity at r = 0 arises from the identification of points of anti-de Sitter space by a discrete subgroup of SO(2, 2) [76].
One could pursue charged BTZ black string solutions by considering the cylindrical extension of the charged 3-dimensional black hole studied in [77].This solution, however, exhibits some pathological aspects.For instance, it allows for arbitrary negative values of the mass and there is no upper bound on the electric charge.Besides this, it has been shown that Reissner-Norstrom black hole cannot be cylindrically extended in Einstein-Maxwell theory (see [78] and references therein).This stems from the fact that the field equations impose severe restriction on the form of the electric field, forcing the electric charge to be zero.The same impossibility is found here for CSMG with and without torsion.It is worth mentioning that this can be circumvented in higher dimensional gravity by considering Einstein-Gauss-Bonnet theory, p-forms instead of Maxwell fields [78], or in four-dimensional Einstein-SU(2) Skyrme model where the charge like term comes from the inclusion of the Skyrme fields rather than from the Maxwell fields [79].
Chern-Simons modified gravity can be extended by including extra scalar gravitational degrees of freedom coupled to additional topological densities, such as the Gauss-Bonnet term in four dimensions.This modification resembles the dilaton-Gauss-Bonnet gravity, which can be motivated from dimensional reduction of Lovelock gravity, considered as the low energy limit of string theory [50].However, its addition is trivial in the sense that the Gauss-Bonnet term and their associated Cotton tensor vanish for the black string ansatz (19), provided the linear dependence on the z-coordinate of the scalar field solution.Thus, Eqs. ( 22) trivially satisfy the field equations of the theory ( 13) with the addition of dilaton-Gauss-Bonnet term.In spite of this, such a modification becomes nontrivial in the first-order formalism with nonvanishing torsion, as it will be shown in the next section.

III. BLACK STRINGS IN FIRST-ORDER FORMULATION OF CHERN-SIMONS MODIFIED GRAVITY
The first-order formalism of gravity considers two independent gravitational potentials: the vierbein 1-form e a = e a µ dx µ related to the spacetime metric g µν through g µν = η ab e a µ e b ν , where η ab = diag(−, +, +, +); and the Lorentz connection 1-form ω ab = ω ab µ dx µ , encoding its affine structure.These objects transform as 1-forms under diffeomorphisms and as vector and gauge connection under local Lorentz transformations, respectively.The Lorentz curvature and torsion 2-forms are defined by the Cartan's structure equations where d is the exterior derivative, ∧ is the wedge product of differential forms, and the last equality defines the exterior covariant derivative D with respect to ω.The curvature and torsion satisfy the Bianchi identities DR ab = 0 and DT a = R a b ∧ e b .The Lorentz connection can be decomposed in terms of their Riemannian and non-Riemannian pieces, namely, where ω denotes the Levi-Civita connection satisfying de a + ωa b ∧ e b = 0, and the contorsion 1-form is defined through T a = K a b ∧ e b .Using the decomposition (26), the Riemannian (torsion-free) curvature 2-form Rab = dω ab + ωa c ∧ ωcb is related to the Lorentz curvature 2-form through where D denotes the exterior covariant derivative with respect to the Levi-Civita connection.
The contorsion can be written in terms of their irreducible components as where V a and A a are Lorentz-valued 0-forms denoting the vectorial and axial pieces, respectively, while the mixed piece Q ab = Q ab c e c is defined such that i a Q ab = 0 = Q ab e a ∧ e b , where i a is the inner contraction along the vector basis and e a µ E ν a = δ µ ν .Using the contorsion's definition T a = K a b ∧ e b , the torsional irreducible components can be read off directly from Eq. (28).From the definition of torsion in Eq. ( 25), three quadratic invariants can be constructed out of T a bc = i b i c T a as which have been reported in Refs.[80][81][82].
A. First-order Chern-Simons modified gravity In this subsection, the first-order formulation of CSMG in four dimensions is considered by including extra (pseudo)scalar fields nonminimally coupled to the Nieh-Yan, Pontryagin, and Gauss-Bonnet terms.The action for such a theory is where κ = 8πG N , Λ is the cosmological constant, ⋆ is the Hodge dual, and denote the Nieh-Yan, Pontryagin, and Gauss-Bonnet densities, respectively.Here, the index i denote different nonminimal couplings to topological invariants rather than extended coordinates as in Subsec.II A. The nonminimal coupling of the (pseudo)scalar fields ϕ i are measured by the coupling constants α i , where α 1 has length units, while α 2 and α 3 have length units to the third power.
The field equations are obtained by performing stationary variations of (30) with respect to the vierbein, Lorentz connection, and (pseudo)scalar fields giving respectively, where no sum over i is assumed unless stated otherwise.From Eq. (32b) it can be seen that torsion is sourced by the exterior derivative of the (pseudo)scalar fields.The energy-momentum 3-form of each (pseudo)scalar fields is defined as The symmetries of the first-order formulation of CSMG has been recently analyzed in Ref. [83] showing that, besides diffeomorphism and local Lorentz symmetry (and therefore local translations), the action ( 30) is quasi-invariant under the shift symmetry δφ i = c i , while the vierbein and Lorentz connection remain invariant.This is analogous to the shift invariance discussed in Sec.II B and it also play a key role in finding the black string solutions within the first-order formalism.Their Nöther current, J i = ⋆dϕ i + α i 4κ C i , is conserved on shell by virtue of the field equation (32c), where the C i have been defined as I i = dC i according to Eq. ( 31), giving rise to a first integral of motion of the form J i = dK i , where Additionally, diffeomorphism invariance imply the on-shell conservation laws for each (pseudo)scalar field where i = 1, 2, 3.The invariance under local Lorentz transformations, on the other hand, imply a trivial condition for the energy-momentum 3-form of the (pseudo)scalar fields.

B. BTZ black strings with nontrivial torsion
In this subsection, BTZ black strings configurations with nontrivial torsion as solutions to the field equations ( 32) are presented.The vierbein basis compatible with the metric structure ( 2) is chosen such that where ā = 0, 1, 2 denote 3-dimensional Lorentz indices.The Levi-Civita connection can be solved in terms of the vierbein from the torsion-free condition de a + ωa b ∧ e b = 0, and it turns out that ωā3 = 0 = ωab z .Nevertheless, in the first order formalism the Lorentz connection contains torsional degrees of freedom beyond the metric ones.The Lorentz connection compatible with the vierbein (35), in the sense that incorporates its perpendicular decomposition and yet is independent of it, is given by This Lorentz connection is compatible with the isometries of ( 2) and it depends only on the 3-dimensional spacetime coordinates {x}.When circular symmetry is assumed on the 3-dimensional manifold, it incorporates 24 independent components through ωā b, α āb , β ā, and γ ā, where the last three are purely torsional.The piece ωā b is recognized as the Lorentz connection of the 3-dimensional spacetime manifold, α āb = −α bā and γ ā are Lorentz-valued 0-forms, while β ā = β ābe b is a Lorentz-valued 1-form.It is worth noticing that, even though the topological invariants constructed out of the Levi-Civita connection vanish by virtue of the isometries of (19), this is not the case when a torsionful connection compatible with such isometries is considered.
Given the decompositions ( 35) and (36), and using the linear dependence on the extended flat coordinate of the (pseudo)scalar's solution, i.e., ϕ i = λ i z, the transverse part of the field equation for the vierbein, E ā = 0, gives while Similarly, decomposition of the field equation for the Lorentz connection (32b), by first taking E āb = 0, imply while the remaining part where Rā b = dω āb + ωā c ∧ ωc b and T ā = de ā + ωā b ∧ e b have been defined, as well as the covariant derivative D = d + ω.The Levi-Civita symbol is also decomposed according to ǫ āb c ≡ ǫ āb c3 .Interestingly, the perpendicular decomposition of the field equations impose where ⋆ denotes the Hodge dual associated to ēā .These conditions imply that vectorial and axial components of the torsion vanish on the 3-dimensional manifold, although no restriction is imposed on their components along the extended flat coordinate.
In what follows, three cases are studied separately: (i) pseudoscalar-Nieh-Yan coupling, obtained when λ 2 = λ 3 = 0, (ii) pseudoscalar-Pontryagin coupling, obtained when λ 1 = λ 3 = 0, and (iii) scalar-Gauss-Bonnet coupling, obtained when λ 1 = λ 2 = 0.These three cases are solved by different torsional configurations presented separately, while the metric structure is given by the BTZ black string with vierbein components where with M and j being integration constants related to the mass and angular momentum, while the three different AdS Riemannian curvature radii are denoted by li , with i = 1, 2, 3.The solutions are summarized in Table I.
(i) Pseudoscalar-Nieh-Yan (ii) Pseudoscalar-Pontryagin (iii) Scalar-Gauss-Bonnet For each nonminimal coupling considered separately, this table exhibits the solutions to the field equations (32) in terms of the BTZ black string (42).The AdS Riemannian curvature radius li and the (pseudo)scalar fields ϕ i = λ i z are displayed in each case, provided the particular contorsional configurations given in terms of their nontrivial irreducible components (28).The non-Riemannian curvature radii are represented by ℓ i and the torsional invariants (29) are computed in each case.
The appearance of an effective Riemannian AdS radius stem from the fact that α āb , β ā, γ ā, and T ā sum up to compose an effective cosmological constant, which is already affected by the presence of the scalar fields through the λ 2 i , as it can be seen from Eq. (37a).The contribution of torsion to the cosmological constant in 3-dimensions has been already observed in [84], and in black hole solutions [85,86] of the Mielke-Baekler model [87].
Interestingly, the AdS Riemannian curvature radius do not coincide with the one associated to the Riemannian-Cartan geometry denoted by ℓ i , with i = 1, 2, 3, as it can be seen from Eq. ( 27).The Riemannian and Lorentz curvature 2-forms for the black string configurations presented here are locally constant and respectively given by and they vanish whenever the indices a, b = 3. Importantly, this latter fact is a consequence of the field equations and not of the isometry group, in contrast to the Riemannian case.
The roots of the metric function f (r) are and, for a horizon to exist, the condition M 2 − 4j 2 / l2 i ≥ 0 must be met.For pseudoscalar-Nieh-Yan and pseudoscalar-Pontryagin couplings, the equations of motion of these particular systems determine that α āb = β ā = 0, and respectively, and therefore γ ā = 0, which can be translated in each case into the axial irreducible components given in the Table I, while all the other components vanish.Additionally, the existence of a nontrivial axial torsion in these two cases implies that fermions are sensitive to these black string backgrounds.The vanishing of the torsional invariant T 3 stem from the fact that the vectorial piece of the torsion is zero for these configurations.Moreover, these two cases possess negative curvature radii from both Riemannian and Riemann-Cartan viewpoint as it can be seen from Table I.On the other hand, when the pseudoscalar-Nieh-Yan coupling is considered, the proportionality constant λ 1 differ from the others since the axial piece of the torsion is proportional to the gradient of the pseudoscalar field that, once integrated out, it contributes to the kinetic term of the pseudoscalar field, shifting the value of λ 1 in comparison to the other cases.
For scalar-Gauss-Bonnet coupling, the field equations impose that α āb , γ ā, and T ā vanish, whereas This can be translated into the vectorial irreducible component given in the Table I, while all the other irreducible components are trivial.Therefore, fermions will not be sensitive to the torsional part of this black string configuration, since no axial component is present.
Interestingly, this case admits a Riemannian flat geometry when 2κ − α 2 3 Λ 2 = 0, even though the negative bare cosmological constant is nonvanishing.This is possible since the contribution of torsion can cancel the one coming from the bare cosmological constant, making the effective AdS Riemannian curvature radius to vanish.Additionally, the case when 2κ − α 2 3 Λ 2 < 0 and Λ < 0 admits a positive curvature radius, however, it represents a naked singularity.Finally, the torsional invariants (29) are computed in each case, and they turn out to be constant in all cases as shown in Table I.However, as it was previously mentioned, a singularity at r = 0 arises from the identification of points of anti-de Sitter space by a discrete subgroup of SO(2, 2) [76].

IV. CONCLUSIONS
In this work, different four-dimensional black string solutions have been presented in vacuum within dynamical CSMG, with and without torsion, by extending the method of Ref. [71] and exploting the shift symmetry of these theories.The solutions represent the black string extension of the BTZ black hole [72] with one additional extended flat coordinate.For a horizon to exist, the same conditions as in the BTZ black hole must hold.By considering the pseudoscalar field to be compatible only under the isometries of the 3-dimensional metric and not compatible with the symmetries generated by the hypersurface orthogonal Killing vector, it is found that the case with vanishing torsion differ from the one reported in [40] in that the contribution of the Cotton tensor is nontrivial in the Einstein's field equations due to the linear dependence of the pseudoscalar field solution on the extended coordinate.In this sense, the solutions presented here belong to the Chern-Simons sector of the space of solutions according to [24].Moreover, the linear dependence on the extended coordinate of the pseudoscalar field solution does not spoil the isometries of the 3-dimensional metric by virtue of the shift symmetry of CSMG.This solution is also supported in the torsion-free case by the addition of the dilaton-Gauss-Bonnet coupling to the CSMG action (13), since the Gauss-Bonnet term and its associated Cotton tensor vanish by virtue of the isometry group, provided that the dilaton has a linear dependence on the extended coordinate.The addition of the dilaton-Gauss-Bonnet term, however, is nontrivial in the case with nonvanishing torsion.
Next, the first-order formulation of CSMG is studied by considering nonminimal couplings to different topological densities in four dimensions, i.e., the Nieh-Yan, Pontryagin, and Gauss-Bonnet terms, that generate torsion in vacuum.Similar to the Riemannian case, the theory is endowed with a shift symmetry in the field space whose conservation law gives for a first integral of motion.In order to find four-dimensional black strings in this framework, the method of Ref. [71] is generalized, and restrictions on the torsion components are found by considering each nonminimal coupling separately.Still, these restrictions do allow for nontrivial torsion supporting the BTZ black strings that, to the best of the authors' knowledge, represent the first black strings with nontrivial torsion reported in the literature.
It is found that either axial or vectorial components of the torsion arise as nonvanishing solutions, while the mixed part is zero in all cases.Torsional and curvature invariants remain locally constant everywhere, however, the nature of the BTZ singularity persists according to [76].In some cases, torsion contributes to an effective cosmological constant, and shifts the Riemannian AdS curvature radius from its non-Riemannian counterpart.Interestingly, when the dilaton-Gauss-Bonnet coupling is considered, there exists a particular choice of the coefficients that allows for a flat Riemannian geometry in presence of a nonvanishing cosmological constant.This is possible due to the presence of a vectorial component of the torsion that can cancel the contribution of the bare cosmological constant.Finally, it is shown that Dirac spinors are sensitive to two of the BTZ black string backgrounds reported here: (i) in pseudoscalar-Nieh-Yan coupling, and (ii) in pseudoscalar-Pontryagin coupling.
Interesting questions remain open.For instance, conserved charges within the first-order formalism and their connection with black hole thermodynamics have been studied in [88][89][90][91][92][93].In order to apply these techniques for the black string solutions presented here, the gravitational degrees of freedom should be extended by considering the (pseudo)scalar fields present in CSMG.Such an extension is certainly of great interest and it is left for future a contribution.On the other hand, although four dimensional Schwarzschild black hole is stable under linear perturbations [94][95][96], it has been shown that its cylindrical extensions, alongside a variety of black strings and branes in D ≥ 5, suffer from the so-called Gregory-Laflame instability [59,60].An interesting way out, besides compactification of the extended coordinate, would be that torsion allows to cure such an instability.On the other hand, if these black string configurations are unstable, it is very interesting to figure out which would be the final state of the instability from both Riemannian and non-Riemannian viewpoints.
This could led to the formation of naked singularities in four dimensions.It is therefore interesting to address this issue in the future for the black string configurations presented in this work, and to study the role of torsional perturbations of the solutions.
V. ACKNOWLEDGEMENT