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On Some New Black String Solutions in Three Dimensions

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Alberta-Thy-06-95/ McGill 95–11

ON SOME NEW BLACK STRING SOLUTIONS IN THREE DIMENSIONS
Warren G. Anderson1 ? and Nemanja Kaloper2 ?

arXiv:hep-th/9503175v1 24 Mar 1995

1 Theoretical

Physics Institute, University of Alberta,

Edmonton, Alberta, Canada T6G 2J1
2 Department

of Physics, McGill University,

Montreal, Quebec, Canada H2A 2T8. (March 1995)

Abstract
We derive several new solutions in three-dimensional stringy gravity. The solutions are obtained with the help of string duality transformations. They represent stationary con?gurations with horizons, and are surrounded by (quasi) topologically massive Abelian gauge hair, in addition to the dilaton and the Kalb-Ramond axion. Our analysis suggests that there exists a more general family, where our solutions are special limits. Finally, we use the generating technique recently proposed by Gar?nkle to construct a traveling wave on the extremal variant of one of our solutions. PACS numbers: 04.20.Jb, 04.50.+h, 12.10.Gq, 97.60.Lf Submitted to Phys. Rev. D

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? anderson@phys.ualberta.ca ? kaloper@hep.physics.mcgill.ca

1

I. INTRODUCTION

Recently we have witnessed a surge of interest in lower-dimensional theories of gravity, after the realization that many of them contain structures with horizons [1]- [7]. Investigation of these models is motivated by the hope that we may be able to gain more information about the physics of realistic four-dimensional black holes, since mathematical di?culties subside dramatically in fewer dimensions. This approach appears to be particularly fruitful in lower-dimensional stringy gravity, where the facilities of string theory provide us with very powerful tools to study black holes in the classical limit and beyond [8]- [20]. Resorting to these techniques, we may be able to tackle in a systematic way some of the long-standing conundrums of black hole physics. For example, it has been demonstrated that string theory may have the potential to cure some of the singularity problems which plague the classical theory1 [21]. In this paper we shall contribute several new black hole-like solutions to the existing bestiary. We shall employ the Abelian duality symmetry as our main tool to obtain them [8]- [15]. Such symmetries represent a stringy generalization of standard toroidal symmetries, stemming from the presence of commuting translational Killing vectors in a gravitational background. They can be combined and employed to derive new background solutions. The general procedure is dubbed twisting, or O (d, d) boosting, after the complete group of twisting transformations [10–12]. At the level of the background ?eld theory on target space, after integrating out Killing coordinates ` a la Kaluza-Klein, this is realized as a symmetry of the action under mixing of the Kaluza-Klein matter ?elds with the metric. Although the action is invariant under this group, solutions are not, because they employ speci?c initial conditions. Therefore, the twisting transformations can generate new classical solutions. It must be kept in mind, however, that dual solutions may not represent di?erent string physics,

1 This

should be taken with some reservations, because the speci?c conclusions obtained so far

may not hold in general, as shown recently by Horowitz and Tseytlin [22].

2

but merely be di?erent pictures of the same string theory, which for example occurs when one dualizes with respect to the translation of a compact coordinate [14]. Moreover, the full O (d, d) group also contains di?eomorphisms and Kalb-Ramond ?eld gauge transformations, which must be modded out [11]. Thus, the space of classical solutions is spanned by the orbits of the O (d, d) group, modulo di?eomorphisms and Kalb-Ramond gauge transformations. This symmetry is further extended to O (d, d + n) in the presence of n Abelian gauge ?elds [9]. We will use this extended boost symmetry to obtain two new three-dimensional (3D) families of asymptotically ?at solutions. Our families are obtained by, respectively, “twisting in” the gauge ?eld on the black string of Horne and Horowitz [3], and the axion on the 2D electrically charged black hole crossed with a ?at line [4]. They are characterized by three parameters, and for certain ranges of the parameters they represent di?erent stationary, gauge charged con?gurations with regular horizons. Some of their properties are quite remarkable. Namely, although all our non-extremal black strings possess a scalar curvature singularity, which must be included in the manifold because it is spacelike geodesically incomplete, this singularity is quite harmless for pointlike observers. The manifold is null and timelike geodesically complete, since for arbitrary initial conditions at in?nity all causal geodesics have a turning point before reaching the singularity except for one null geodesic, which comes arbitrarily close to the singularity but never reaches it for a ?nite value of the a?ne parameter. The ?rst family possesses an interesting non-singular extremal limit, di?erent from the extremal limits of previously known solutions in that it has one hypersurface orthogonal null Killing vector, and nonvanishing gauge hair. Therefore, we can further extend this solution to include a traveling wave, using the generating technique proposed recently by D. Gar?nkle in the context of string gravity [29]. Our ?rst family also contains a subclass of solutions with interesting global properties. These solutions are without curvature singularities, with spatial hypersurfaces looking like a “cigar”, approaching asymptotically R × S1 . However, the angular variable is “bolted” to time, and hence near the origin there appear closed timelike curves. The second family has, curiously, a critical value of the boost parameter and a black 3

string with two di?erent extremal limits. The critical boost gives the stringy version of the 3D black hole [5,6]. Away from this critical boost we ?nd another family of black strings, which displays a peculiar combination of properties of both black strings and four dimensional black holes. Particularly interesting is the presence of the ergosphere, which arises entirely due to the axion and electric charges. Furthermore, this black string has two extremal limits. One of them corresponds to taking the critical value of the boost, but after a coordinate transformation, in a fashion familiar from the static black string case. The other extremal limit is reminiscent of the extremal Kerr-Newman case, and represents a gauge charged black string, with ergosphere but without null Killing vectors. The paper is organized as follows. In the next section, we will lay out mathematical background for the subsequent study, explaining our approach and deriving various forms of the solutions. Detailed investigation of the solutions will be presented in section III. Section IV contains the derivation of the traveling wave solution, using Gar?nkle’s techniques. The ?nal section o?ers our conclusions and presents arguments that suggest the existence of a larger family of 3D black objects which continually interpolate between our two solutions, as well as the Horne-Horowitz and the stringy BTZ solutions.

II. GENERATING SOLUTIONS

The e?ective action of string theory describing dynamics of massless bosonic background ?elds to the lowest order in the inverse string tension α′ is, in the world-sheet frame [10][12], [23], S = α′ 1 √ dd+1 x ge?Φ (R + ?? Φ? ? Φ ? H?νλ H ?νλ ? F N ?ν F N ?ν + 2Λ). 12 4 (2.1)

The action above is written in Planck units κ2 = 1. Here F N ?ν = ?? AN ν ? ?ν AN ? are ?eld strengths of n Abelian gauge ?elds Aj ? , H?νλ = ?λ B?ν + cyclic permutations ?(α′ /2)?M ?νλ is the ?eld strength associated with the Kalb-Ramond ?eld B?ν and Φ is the dilaton ?eld. The Maxwell Chern-Simons form ?M ?νλ =
N

AN ? F N νλ + cyclic permutations appears

4

in the de?nition of the axion ?eld strength due to the Green-Schwarz anomaly cancellation mechanism, and can be understood as a model-independent residue after dimensional reduction from ten-dimensional superstring theory [24]. In fact, this term is a necessary ingredient of the theory if one wants to ensure the O (d, d + n) invariance, as shown by Maharana and Schwarz [12]. The n Abelian gauge ?elds should be thought of as the components of a non-Abelian gauge ?eld A residing in the Cartan subalgebra of the gauge group, while the rest have been set equal to zero. For convenience we will set α′ = 1. In what follows we will be considering only those extrema of (2.1) that possess d commuting isometries, that is, we will be considering ?eld con?gurations of the form ds2 = Γ(r ) dr 2 + Gjk (r ) dxj dxk , B= 1 Bjk (r )dxj ∧ dxk , 2 (2.2)

AN = AN j (r )dxj , Φ = F (r ), where Gjk (r ) is the metric of a d dimensional submanifold of signature d ? 2. In this case the action (2.1) can be rewritten in the manifestly O (d, d + n) invariant form [10–12] Sef f = √ 1 1 dr Γe?φ φ′2 + Tr(M′ L)2 + 2Λ , Γ 8Γ (2.3)

where the prime denotes the derivative with respect to r . Note that the physical dilaton Φ has been replaced by the e?ective dilaton φ = Φ ? (1/2) ln | det G| after dimensional reduction. Matrices M and L which appear in the action (2.3) are de?ned by M =
?

g ?1

? ? ? ?C T g ?1 ? ?

?g ?1 C g+a+C g C A + C T g ?1 A
? ? ? ? ? ?
T ?1

?g ?1 A A+C g A 1n + AT g ?1A
?
T ?1

?

?g ?1 A

? ? ?, ? ?

(2.4)

0 1d 0

1d 0 0 5

0 0 1n

L =

? ? ?. ? ?

Here g and b are d × d matrices de?ned by the dynamical degrees of freedom of the metric and the axion: g = (Gjk ) and b = (Bjk ). The matrix A is a d × n matrix built out of the gauge ?elds: AkN = AN k . The matrices a and C are de?ned by a = AAT and C = (1/2)a + b respectively, and 1d and 1n are the d and n dimensional unit matrices. Note that MT = M and M?1 = LML. Thus we see that M is a symmetric element of O (d, d + n). Therefore a cogradient O (d, d + n) rotation M → ?M?T is a symmetry of the action, and the equations of motion, because it represents a group motion which changes M while maintaining its symmetry property. In this paper we will apply this technique to several well-known solutions of stringy gravity to the lowest order in the inverse string tension expansion, describing black strings. Speci?cally, we will use the black string family discovered by Horne and Horowitz [3], as well as the 2D electrically charged black hole crossed with a ?at line, discussed by McGuigan et al. [4]. We start with the black string solution, given by [3]: ds2 = Q2 dr 2 m + (1 ? )dx2 ? (1 ? )dt2 , 2 Q mr r 2Λ(r ? m)(r ? m ) Q A = 0, B = dx ∧ dt, r √ e?Φ = 2Λr.

(2.5)

Following the prescription outlined above, and using
?

?=

0 ? √ ? ? ? ? 0 (1 + c)/2 0 (1 ? c)/2 ?s/ 2 ? ? ?
? ? ?0 ? ? ? ?0 ? ?

1

0

0

0

? ? ? ?, ? ? ? ? ? ?

0

1

0

(1 ? c)/2 0 (1 + c)/2 √ √ s/ 2 0 ?s/ 2 0

0 √ s/ 2 c

(2.6)

where c = cosh(2α) and s = ? sinh(2α), we obtain the new solution ds2 = dr ?2 r ? ? ms ?2 ? Q2 /m 2 + dx 2Λ(? r ? mc ?2 )(? r ? ms ?2 ? Q2 /m) r ? ? ms ?2 r ? ? mc ?2 ((? r ? ms ?2 )dt ? s ?2 Qdx)2 , ? 2 r ? (? r ? ms ?2 ) 6

B=

Qc ?2 dx ∧ dt, r ? e?Φ =



√ s ?c ? A = ? 2 (Qdx + mdt), r ? 2Λ? r,

(2.7)

where c ? = cosh α, s ? = ? sinh α, and r ? = r + ms ?2 . Clearly, (2.7) generalizes (2.5), reducing to it when α = 0. It is obvious that this solution, like (2.5), is asymptotically ?at in the limit r ? → ∞. We also ?nd it useful to represent our new solution in terms of the shifted time coordinate τ = t + (Q/m)x instead of t. In this gauge, the axion and the dilaton are the same as above. The gauge ?eld is oriented completely along dτ , and the expressions for it and the metric are given as follows: ds2 = Q2 dr ?2 + (1 ? )dx2 2Λ(? r ? mc ?2 )(? r ? ms ?2 ? Q2 /m) m2 mc ?2 mc ?2 ms ?2 Q )dxdτ ? (1 ? )(1 ? )dτ 2 , ? 2 (1 ? m r ? r ? r ? √ ms ?c ? A=? 2 dτ. r ?

(2.8)

This form is suitable for comparison with our second solution, to be presented shortly, but not useful for the analysis of the causal structure, as can be immediately seen from the fact that τ is not the asymptotic time coordinate. We now consider the electrically charged stringy 2D black hole crossed with a ?at line R [4]: ds2 = ? q2 dρ2 2 )(1 ? )dτ ?2 , + dξ ? (1 ? 2 q ?ρ ρ 2Λ(ρ ? ? )(ρ ? ?) √ 2q B = 0, A=? dτ ?, ρ √ e?Φ = 2Λρ,

(2.9)

To obtain our other solution we apply another O (2, 3) twist to it, with b a real number:
?

0 1

b

0

0

?=

? ? ?1 ? ? ? ?0 ? ? ? ?0 ? ?

0 0 ?b 0 0 0 1 1 0 0 7

0 0 0

? ? 0? ? ? ? 0? ?. ? ? 0? ? ?

?

(2.10)

1

This yields the following expression: ds2 = dρ2 b2 (ρ ? M )2 ? f dξ 2 + q2 2 )ρ + Mb2 )2 ((1 ? b 2Λ(ρ ? ? )(ρ ? ?)

ρ2 ? b2 f 2bq 2 2 + dτ ? ? dξdτ ?, ((1 ? b2 )ρ + Mb2 )2 ((1 ? b2 )ρ + Mb2 )2 √ 2q b(ρ ? M ) dξ ∧ dτ ?, A=? (dξ + bdτ ? ), B= 2 2 2 (1 ? b )ρ + Mb (1 ? b )ρ + Mb2 √ e?Φ = 2Λ((1 ? b2 )ρ + Mb2 ), where f = (ρ ? ?)(ρ ? q 2 /?) and M = ? +
q2 . ?

(2.11)

Note that b2 = 1 represents a special

point in the moduli space of this family, as the dilaton ?eld decouples there. Indeed, after a closer look (and some coordinate transformations) we recognize this case as precisely the stringy version of the Banados-Teitelboim-Zanelli (BTZ) [5,6] black hole. In what follows, for computational purposes we will assume b > 1 (the sign of b only determines the sign of the axion charge), without any loss of generality, as we will now demonstrate. To start with, the solution (2.11) can be simpli?ed considerably with some judicious gauge choices. We begin √ √ ?/ b2 ? 1. with the coordinate transformation r ? = (1 ? b2 )ρ + Mb2 , x = ξ/ b2 ? 1, and t = τ We can also apply the axion gauge transformation which shifts the asymptotic value of the axion to zero. In this gauge, the solution takes the form ds2 = dr ?2 2 r??? 2Λ(? r ? ?b2 ? q? )(? ? (1 ?
q 2 b2 ) ?

+ (1 ?

M q 2 (b2 ? 1) ? )dx2 r ? r ?2

bq 2 (b2 ? 1) Mb2 q 2 (b2 ? 1)b2 2 + ) dt ? 2 dxdt r ? r ?2 √ r ?2 √ q b2 ? 1 bM dx ∧ dt, A=? 2 (dx + bdt), B= r ? r ? √ e?Φ = 2Λ? r.

(2.12)

It is straightforward to verify that if |b| < 1, the same coordinate transformation, followed by Wick rotation t, x → it, ix, rescaling r ?→r ?/b2 , a constant dilaton shift and the replacement of the parameter b → b′ = 1/b > 1 reduces the form of (2.11) again to (2.12). We will defer further discussion of the interpretation of the |b| < 1 solutions until later. The solution (2.12) is also an asymptotically ?at con?guration with both axion and gauge ?elds. It is again useful, for easier comparison, to perform another coordinate change, to put 8

this solution in a form similar to (2.8). The dilaton and axion remain the same as above, while the metric and the gauge ?eld are given as follows: ds2 = dr ?2 2 r??? 2Λ(? r ? ?b2 ? q? )(? + (1 ? 1 )dx2 b2 (2.13)

q 2 b2 ) ?

1 Mb2 Mb2 q 2 (b2 ? 1)b2 ? 2 (1 ? )dxdτ ? (1 ? + )dτ 2 , b r ? √ r ? r ?2 √ qb b2 ? 1 dτ. A=? 2 r ?

Despite the conspicuous similarity between (2.8) and (2.13), we will demonstrate later in our analysis that they are indeed di?erent. This can already be glimpsed, however, by realizing that the matter content of the two con?gurations is exactly the same once the proper coordinate rescalings are performed, and that since they are stationary and contain a scalar, a vector and the volume form in the (x, τ ) subspace, we are left without any freedom to perform further coordinate transformations which do not alter the form of the matter. Speci?cally, the fact that the dilaton of both con?gurations is essentially the radial coordinate, and that x and τ are Killing coordinates restricts the available coordinate transformations to only linear transformations in the (x, τ ) plane. These in general induce the changes of the two gauge ?elds A and B which are proportional to the ?eld components themselves. Since the ?elds are nontrivial, i.e. have non-vanishing ?eld strengths, the changes induced by di?eomorphisms are not pure gauges and hence cannot be removed by gauge transformations. The last step in the argument is the comparison of the two metrics, which shows that they do not match; indeed, if we denote the (x, τ ) parts of the two metrics as g1 and g2 , respectively, we can see that g1 = g2 + Cdx2 , for some given constant C . Since the horizons are determined by the determinant of these matrices, the above shift induces the corresponding shift in the locations of these surfaces. In the next section we will investigate causal properties of these solutions. Our analysis will con?rm and elaborate upon the argument presented above, that they represent di?erent black strings. Before we close this section, however, we should explain an apparent peculiarity which appears in the gauge sector of the two solutions. Namely, we see that in both 9

(2.8) and (2.13) the gauge ?eld looks precisely like the ?eld of a point charge in three spatial dimensions, despite the fact that it lives in two dimensions, where one would expect it to be proportional to the logarithm of the distance from the source. Indeed, such behavior has been noted in Ref. [5], where charged black holes in 3D Einstein-Maxwell theory were studied. The resolution to this lies in the fact that in our background the gauge ?eld acquires the (quasi)topological mass term due to its coupling to the Kalb-Ramond ?eld via the ChernSimons form [25,26]. The Kalb-Ramond ?eld is trivially integrable in three dimensions [26], and if non-zero, yields the gauge ?eld mass term. The standard Maxwell equation for the gauge ?eld should be replaced in this case by, in form notation, d exp(?Φ)? F = 2QA F, where QA is the axion charge, de?ned by QA = ? exp(?Φ)? H; it is straightforward to verify that our backgrounds solve it. It is furthermore interesting to note, that whereas in this case the gauge ?eld is (quasi)topologically massive, the gauge sector of either solution does not represent a gauge anyon, as the Chern-Simons form itself vanishes in both cases.

III. CAUSAL STRUCTURE

Here we will investigate the structure of the two new solutions presented above. Whereas some aspects of the geometry of these two solutions are remarkably similar, there are interesting di?erences. As a warm-up, let us review the static black string (2.5) of Horne and Horowitz, which will be the basis for comparison. This solution has three metric singularities at r = m, r = Q2 /m and r = 0, and obviously there are three di?erent cases, 0 < |Q| < m, |Q| = m, and |Q| > m. When 0 < |Q| < m, r = 0 is a scalar curvature singularity and r = m and r = Q2 /m are the event and Cauchy horizons respectively. The singularity is “real” in the sense that the manifold is null geodesically incomplete. The causal structure of the solution is qualitatively similar to that of the Reissner-Nordstr?m solution, with the exception that the time-like coordinate inside the Cauchy horizon is x rather than t. Thus the 2D Penrose diagrams aren’t completely adequate for the description of the geometry, but they can be 10

used with the proviso that one remembers that the time-like coordinate makes a “right angle” turn on the inner horizon. As a consequence, in this solution there are no static observers inside the Cauchy horizon. This case is summarized in Fig.1. When |Q| = m, the form of the solution (2.5) breaks down at the (degenerate) horizon r = m. It turns out that the coordinate r is not suitable for the extension beyond the event horizon, which appears to be a turning point for all geodesics. To see that the manifold does not end there, the authors use the modi?ed radial coordinate r ′ 2 = r ? m, and show that the geometry contains an event horizon at r ′ = 0 but has no singularity (Fig. 2). It is interesting to note that this is identical to the causal structure seen in the extremal Kerr solution along the axis of symmetry [27]. Finally, for the case |Q| > m the authors ?nd that the manifold is completely regular, when using the appropriate radial coordinate r ?2 = r ? Q2 /m. It terminates at r ? = 0, and the potential conical singularity there is removed by a periodic identi?cation of the spacelike coordinate x. Therefore, the spacelike sections have the structure of a “cigar”, looking ?at near the origin but asymptotically approaching R × S1 . Let us now turn our attention to our new solutions. Both (2.7) and (2.12) share some of the features of the static black string (2.5). They are both asymptotically ?at con?gurations with two Killing ?elds, ?x and ?t , with in?nity described by the limit r ? = ln r ? → ∞, where they approach exponentially fast the linear dilaton vacuum, with ?at Minkowski metric and vanishing gauge ?elds. They also have three metric singularities each, r ? = mc ?2 , ms ?2 + Q2 /m, and 0 for the ?rst, and r ? = ?b2 + q 2 /?, b2 q 2 /? + ?, and again 0 for the second. The surface r ? = ms ?2 in the ?rst solution actually isn’t singular, as can be seen from expanding the squared bracket in (2.7) and collecting the like terms. The nature of the singular points can be examined by investigating the behavior of curvature invariants as these points are approached. This arduous task is in fact easier in three dimensions, because Weyl curvature is identically zero, and the only scalar curvature invariants are R, R?ν R?ν and det(R?ν )/ det(g?ν ) [30], which all blow up as r ? → 0 and are ?nite elsewhere. Thus r ?=0 is the only polynomial curvature singularity for our solutions. We should note here that our 11

choice to rely on the conventional de?nition of curvature singularities of General Relativity is equivalent to assuming that the space-time geometry can be probed only by pointlike observers. Whereas this assumption is obviously of limited validity in string theory, it is a useful working tool in the absence of a more general de?nition, and we will restrict our attention to it (for more general criticism see Ref. [22]). In order to analyze our solutions further, we have to investigate them one by one. The results are summarized in the following ?ve subsections.

A. The ?rst family with 0 < |Q| < m

Here we present our ?rst black string solution. The surfaces r ?+ = mc ?2 and r ?? = ms ?2 + Q2 /m are removable singularities, where coordinates change signature, and thus represent the event and Cauchy horizons, respectively. This can be seen from the fact that they are both null surfaces, and that almost all timelike and null geodesics cross them, as we will demonstrate shortly. The behaviour of the coordinates while crossing these surfaces is somewhat di?erent from the situation enjoyed by the Horne-Horowitz black string. While the radial coordinate behaves the same, being spacelike outside the event horizon and inside the Cauchy horizon, and timelike in between, the time at in?nity t, which turns spacelike after crossing the event horizon, regains the timelike character again after crossing the interior static limit r ?t = ms ?2 , inside the Cauchy horizon. Likewise, x also changes signature, becoming timelike after the surface r ?x = Q2 /2m + (Q2 s ?2 c ?2 + Q4 /4m2 )1/2 . Thus, as in the static black string case, representation of the causal structure by planar Penrose diagrams is not completely accurate, since there is more freedom in choosing the time coordinate, but the situation here is a bit more complicated. As we see, there are regions where the time coordinate is an r -dependent linear combination of t and x. However, if we keep this in mind, we can still employ the diagrammatic technique as a descriptive tool. Before completing the description of the causal structure, we will investigate geodesics of this solution. Again, due to the presence of two Killing vector ?elds, the geodesic equations 12

take a particularly simple form. Introducing two integrals of motion associated with the cyclic coordinates P? = (?E, P ), and the squared rest mass of the particle moving on the geodesic p = 0, 1 (distinguishing null and timelike geodesics), we obtain the following formula for the radial coordinate (the overdot denotes the derivative with respect to the a?ne parameter): ˙2 r ? Q2 Q2 s ?2 c ?2 mc ?2 ms ?2 2 2 = E (1 ? ? ) ? P (1 ? )(1 ? ) 2Λ? r2 mr ? r ?2 r ? r ? mc ?2 mc ?2 ms ?2 + Q2 /m Qs ?2 (1 ? ) ? p(1 ? )(1 ? ). + 2EP r ? r ? r ? r ?

(3.1)

Upon the inspection of this equation, we note that while all causal geodesics cross the event horizon, the subset for which ms ?2 E = QP terminates at the Cauchy horizon. All other causal geodesics pass through the Cauchy horizon too, and speci?cally null geodesics with P = 0 terminate at the surface r ?x < r ?? , where ?x becomes null, (but does not vanish, as can be seen from computing the component of the tangent along x there). We note that the behavior of the ms ?2 E = QP geodesics is related to the P = 0 case studied in the static solution by Horne and Horowitz. They found that P = 0 geodesics terminate at the Cauchy horizon, and ascribed this to the fact that there is a world line along which the ?eld ?x must be identically zero, and not just null. Note that for our solution, the P = 0 null geodesics are protected from this by the terms proportional to s ?2 > 0, but that we recover the pathology in the limit s ?2 → 0, when the above two geodesics coincide. Therefore we see that the cross-term in our metric has caused the pathological class of geodesics to shift from P = 0 to ms ?2 E = QP . They end at the equivalent region of the black string, with the only di?erence that now it is the vector Q?x + ms ?2 ?t , that vanishes there, because it is orthogonal everywhere to the class ms ?2 E = QP but becomes null on the Cauchy horizon and timelike inside of it. In order to see what happens in the region near the singularity, it is helpful to rewrite the radial geodesic equation (3.1) by collecting the terms of the same order of divergence: ˙2 (QE + mP )(QE ? m(? c2 + s ?2 )P ) ? pm(? c2 + s ?2 ) r ? 2 2 = E ? P ? p ? 2Λ? r2 mr ? 13

?

(QE + mP )2 s ?2 c ?2 + pmc ?2 (ms ?2 + Q2 /m) . r ?2

(3.2)

The coe?cient of the O (1/r ?2) term in this equation is nonpositive for all causal geodesics. As a consequence, no causal geodesic with this term being nonzero, beginning outside of the black string, can reach the singularity at r ? = 0, because the O (1/r ?2) term forces it to stop and turn. Thus, the only geodesics which don’t turn away from the singularity are null geodesics with QE + mP = 0. This is somewhat reminiscent of the behavior of geodesics in the Kerr solution. Noting that Q is similar to angular momentum in this geometry, we can de?ne a = Q/m in analogy with the angular momentum parameter in the Kerr solution. For the geodesics in the equatorial plane of the Kerr solution we can de?ne the impact parameter b = l/E , where l is the conserved angular parameter analogous to our P . For the value of the impact parameter l/E = a these geodesics hit the ring singularity, and in essence behave in the same way as all radial geodesics do in static black hole spacetimes. Thus, we see that our condition QE + mP = 0 is analogous to the Kerr case, again singling out only those radial geodesics which reach the singularity. There is a startling di?erence between our case and Kerr, however. In Kerr, geodesics with b = a are linear in the a?ne parameter, r = Eλ + const. In contrast, in our case the equation (3.1) reduces to the standard linear homogenous equation, with exponential solutions r ? ∝ exp(λ), (analogous to the case |P/E | = |Q/M | for the static solution (2.5)). Thus, although the singularity is the attractor for these geodesics, as they can come arbitrarily close to it, they cannot reach it for any ?nite value of the a?ne parameter. As a consequence, our spacetime is timelike and null geodesically complete. The singularity still must be included in the manifold, which is spacelike geodesically incomplete. In addition, it can also be reached by nongeodesic causal curves. Yet, it is quite harmless for pointlike observers, living serenely along causal geodesics. Remarkably, it would appear to an observer inside the Cauchy horizon as some eerie but ultimate warning against dangerous living! Calculation of the Hawking temperature for this solution is complicated by the presence of cross-terms in the metric. Employing the approach of [31], designed for such situations, 14

we can obtain it by rewriting the metric (2.7) in the ADM form, and then Wick-rotating the time coordinate t = i?. Requiring that the horizon is a regular point, we must identify √ ? with the period 2 2πmc ?2 / Λ(m2 ? Q2 ). This gives the following expression for the Hawking temperature: T = Λ 2 √ m2 ? Q2 . 2πmc ?2 (3.3)

As in the static black string and the Reissner-Nordstr?m case, as Q2 → m2 , the temperature vanishes. Thus the string would settle down to |Q| = m in the absence of charge-dissipating processes. Finally, as we have indicated above, the solution has a static limit at r ?t , where the coordinate t again becomes timelike. Thus, inside this surface it is again possible to ?nd observers at rest with respect to the asymptotic in?nity, much like the Reissner-Nordstr?m solution, and unlike the static black string of Horne and Horowitz. In conclusion, the causal structure of this solution up to the Cauchy horizon, is qualitatively similar to that of the Horne-Horowitz black string, with the di?erences arising near the singularity. The corresponding diagram is presented in Fig. 1.

B. The extremal limit |Q| = m of the ?rst family

As usual, we de?ne the extremal limit of our black string by a choice of parameters which ensures the coincidence of the two horizons r ?? = r ?+ . Naively, we would then expect to obtain a solution with a singularity enclosed by a single horizon. However, Horne and Horowitz found that in the corresponding static case (2.5), the coordinate r ? was not the proper extension across the horizon r ? = m. A hint that a di?erent extension was needed was provided by the radial geodesic equation, which indicated that the horizon is a radial turning point for all causal geodesics. In analogy with this situation, we ?nd that (2.7) does not give the correct extension across the horizon r ? = mc ?2 in the extremal case. Namely, the radial geodesic equation (3.1) for the extremal case |Q| = m can be rewritten as 15

˙2 r ? = (? r ? mc ?2 ) (E + P )((E ? P )? r + ms ?2 (E + P )) ? p(? r ? mc ?2 ) . 2Λ

(3.4)

The right hand side of this equation vanishes at the horizon, and thus it appears that no timelike or null geodesics can cross this horizon. To rectify this problem, we follow the approach of [3] and de?ne the new radial coordinate r ?2 = r ? ? mc ?2 . In terms of it, the radial equation becomes ˙ 2r ? = (E + P )((E ? P )(? r 2 + mc ?2 ) + ms ?2 (E + P )) ? pr ?2 , Λ
2

(3.5)

and thus we see that all causal geodesics (p ≥ 0) in fact cross the horizon, located at r ? = 0. In terms of this coordinate the metric takes the form ds2 = ?2 r ?2 r ?2 2 dr 2 + dx ? ((? r 2 + m)dt ? ms ?2 dx)2 . Λ r ?2 r ?2 + m (? r 2 + m)(? r 2 + mc ?2 )2 (3.6)

As in the static solution (2.5), the only metric singularity is at the horizon r ? = 0. This, of course, is a removable singularity, and the metric can be extended beyond it, to the region r ? < 0. Furthermore, (3.6) is invariant under the re?ection r ? → ?r ?. Thus, after passing through r ? = 0 a particle ?nds itself in a universe identical to the one which it just left. As a consequence, the maximal extension of this solution is based on a zig-zag event horizon along which an in?nite number of asymptotically ?at regions are connected, with the causal structure identical to the extremal black string of Horne and Horowitz (Fig. 2).

C. The ?rst family with |Q| > m

In this case, the signature of the metric changes at the surface r ? = ms ?2 + Q2 /m, since the change of sign of the metric component gr ?r ? is accompanied by the change of both eigenvalues of g2 to negative values, as can be seen from (2.7). The overall signature change is from (-,+,+) to (-,-,-). This indicates that the metric (2.7) cannot be extended beyond r ? = ms ?2 + Q2 /m. To obtain the correct picture, we must rede?ne the radial coordinate. This time, we employ r ?2 = r ? ? ms ?2 ? Q2 /m. With this, our metric becomes 16

?m r ?2 + Q dr ?2 2 r ?2 Q2 2 2 m )dt ? Qs ?2 dx)2 . ds = + ((? r + 2 2 2 dx ? 2 Q Q Q Q 2 2 2 2 2 2 Λr m r + m + ms ?) ? + m ?m r ? + m (? r + m )(?
2

2

(3.7)

In these coordinates, the surface r ? = 0 is singular, since the metric is degenerate there. In fact, if we expand this metric near the origin, after introducing a new radial coordinate by r ?= (Q2 /m ? m) sinh( Λ/2z ), we obtain Λ Q2 ? m2 2 2 z dx 2 Q2 Q2 (Q2 ? m2 ) m2 c ?4 m2 ms ?2 Λ Q2 ? m2 2 2 ? 1 + Λ z ( ? ) dt ? (1 ? z )dx (Q2 + m2 s ?2 )2 Q2 + m2 s ?2 2Q2 Q 2 Q2 + O (z 4 ), as z → 0. This metric looks like the metric of a spinning point source in three dimensions [32], and hence r ? = 0 represents the standard coordinate singularity at the origin, provided √ that we have smoothed it by identifying x with the same period Πx = 2 2Qπ/ Λ(Q2 ? m2 ) as discussed in [3]. Otherwise, we would have ended up with a conical singularity there. The global structure of this manifold is considerably di?erent from the static case. The manifold can still be thought of as consisting of in?nite “cigar”-shaped spatial hypersurfaces de?ned by adjusting the time coordinate such that (? r 2 + Q2 /m)dt ? Qs ?2 dx = 0, (or generated ˙=x by spacelike geodesics t ˙ = 0), planar near the origin and deforming towards R × S1 as r ? → ∞. However, in this solution the angular coordinate x is “bolted” to time in a nontrivial manner, and hence there now appear closed timelike curves. This can be seen by realizing
2 that the coordinate x becomes null at the surface r ?x =

ds2 = dz 2 +

2

(3.8)

Q4 /4m2 + Q2 s ?2 c ?2 ? Q2 /2m ? ms ?2 >

0, and thus for r ?<r ?x the loops (? r = const., t = const.) are timelike. Furthermore, there are geodesics which can reach this region. The geodesic equations in this case can be rewritten in a particularly convenient form by introducing local coordinates x ? = x ? Pλ ? = t ? ms and t ?2 x/Q. These coordinates span a helical frame along each geodesic, twisting around it as the a?ne parameter λ changes. Then, using L = Q(QP ? ms ?2 E )/m, we get

17

Q2 ? m2 L2 m 2 2 ˙ = (? ?4 E 2 ? L2 ) ? . r ? r 2 + Q2 /m ? m)(E 2 ? P 2 ? p) + 2 (Q2 c Λ Q Q2 r ?2 ? and r If we ignore the ?rst terms in the t ? equations, we obtain precisely the polar parametrization of straight lines in Minkowski space. The parameter L then represents the conserved angular momentum along the lines, preventing them from hitting the origin unless L = 0. Thus we see that the curvature e?ects are described by the ?rst terms in the last two equations of (3.9), and that their e?ects (other than rescaling the constant parameters) are essentially negligible near the origin, as indicated by the expansion (3.8). Moreover, comparing the L-dependent terms we con?rm our choice of compacti?cation of the coordinate x. To shed more light on this geometry we can look at several typical geodesics. The ?rst natural candidate is, of course, the L = 0 case, generalizing rays through the origin from the ?at background. We note that in terms of the integrals of motion E and P this condition translates to QP = ms ?2 E . In the static case, when s ?2 = 0, these are just the lines of constant x and of in?nite span in r ?, which pass through the origin and escape to in?nity on both sides. In our case, when s ?2 = 0, this picture is correct only for s ?2 ≤ |Q|/m; if reparametrized in terms of the original coordinate x, these trajectories are hyperbolic spirals, approaching the spiral of Archimedes as s ?2 → |Q|/m. The main point is that these geodesics enter and exit the region of space-time with timelike loops. In contrast, when s ?2 > |Q|/m, we have E 2 ? P 2 ? p = E 2 (1 ? m2 s ?4 /Q2 ) ? p < 0, and all causal geodesics of this kind are bound orbits oscillating near the origin, looking like
2 r ? ∝ sin(x) in the original variables. Their amplitude is bounded from above by r ?max =

˙ = L, x ? r ?2 mc ?4 ˙=? ? t + E, r ?2 + Q2 /m ? m

(3.9)

(m2 s ?2 + Q2 )/m(m2 s ?4 ? Q2 ), and for large enough s ?2 they remain within the region with closed timelike curves. Nonetheless, communication between the two regions is still possible. For example, null geodesics with P = E , which correspond to straight lines with impact

18

parameter l2 in the ?at space, can reach into the region with timelike loops. Their closest approach to the origin is given by the minimal impact parameter lmin 2 = (Q + m)(Q +
2 ms ?2 )2 /ms ?2 (2Q + (Q + m)? s2 ), which is less than r ?x , as can be seen from gxx (lmin ) < 0.

Thus we conclude that the two regions are always geodesically connected, and the region with closed timelike curves cannot be smoothly detached away from the manifold. Because we can extend this solution to four dimensions, by simply adding an additional ?at coordinate, it might be interesting as an example of a spacetime which allows time travel. However, its actual physical signi?cance would remain somewhat dubious, due to its asymptotic topology.

D. Second family of black strings

Before proceeding with the analysis, it is useful to rewrite the solution (2.12) using a di?erent set of parameters. Speci?cally, we eliminate the parameters ?, q and b in favor of the world-sheet frame ADM mass and linear momentum in the x direction, as well as the electric and axionic charges, de?ned by Gauss laws for the two ?elds. The ADM parameters are the components of the ?ux of linearized energy-momentum tensor integrated over a spacelike hypersurface at in?nity, where the metric can be expanded around the Minkowski form: g?ν = η?ν + γ?ν . The necessary formulae are given in [28], which in our case give the following expressions for these quantities per unit length x of the string (with the Gauss laws, which are the integrals of conserved currents over the same spacelike hypersurfaces, given here in form notation), M=? 1 ?Φ ′ e (γxx + γrr Φ′ ), 2Λ 1 e = √ e?Φ ? F, 2 2Λ Px = ? Q= 1 ?Φ ′ e γtx , 2Λ (3.10)

1 ?Φ ? e H. 2Λ

√ The prime denotes the derivative with respect to the “?at” radial coordinate z = ln(? r )/ 2Λ, √ and the additional 2 in the de?nition of the electric charge e re?ects our normaliza√ tion of the F 2 term in the action (2.1). This gives M = Mb2 , Px = 0, e = qb b2 ? 1 19

√ and Q = bM . Now we can solve for ?, q and b, to obtain ? = Q( M2 ? Q2 ± √ √ √ M2 ? Q2 ? 4e2 )/2M M2 ? Q2 , q = Q2 e/M M2 ? Q2 , b = M/Q, and ?nally M = Q2 /M. Using these parameters, we can rewrite our second family of solutions (2.12) as ds2 = Q2 e2 Q2 dr ?2 + (1 ? ? )dx2 2Λ(? r?r ?+ )(? r?r ?? ) Mr ? M2 r ?2 M e2 2 Qe2 ? (1 ? + 2 )dt ? 2 dxdt, r ? r ? Mr ?2 √ e Q Q A = ? 2 (dt + dx), B = dx ∧ dt, r ? r ? M √ e?Φ = 2Λ? r, √

(3.11)

with the horizons given by r ?± = (M2 + Q2 ±

√ M2 ? Q2 M2 ? Q2 ? 4e2 )/2M. We note

the distinct appearance of the factor of 4 together with e2 here. This is, as we have pointed out above, due to our normalization conventions for the gauge ?eld F. We should also point out that similar variables for our ?rst family of solutions give a representation far less transparent than the one provided by (2.7). This comes about because the ?rst family has nonvanishing momentum Px , which is non-trivially related to the axion charge [28]. From the formula for r ?± we can now determine the range of parameters which split this family into di?erent subclasses. Obviously the possibilities are M2 ≥ Q2 + 4e2 , Q2 < M2 < Q2 + 4e2 and M2 ≤ Q2 . In the last two cases, although the square roots which appear in the de?nition of r ?± become imaginary, the denominator of the lapse function (which is the only part of the solution containing explicit reference to these terms) remains real, as can be readily veri?ed. Thus, in general, these two cases cannot be excluded. We will not study their properties in detail for the following reasons. In the case de?ned by the second inequality, Q2 < M2 < Q2 + 4e2 , we note that both r ?± are complex numbers. Thus the metric is regular everywhere except at r ? = 0, where we have found a curvature singularity. As a consequence, this solution describes a geometry containing a naked singularity, much like the Reissner-Nordstr?m solution with M 2 < e2 . Furthermore, because b = M/Q, and recalling that the b < 1 case is related to b > 1 by a simultaneous Wick rotation of both t, x variables, we see that the case represented by the last inequality is the proper extension of 20

the solution to b < 1. Drawing on the similar relationship between the ?rst and the third subclass of our ?rst family of solutions, we conclude that this case must be similar to the |Q| > m subclass of our ?rst family of solutions, containing closed timelike curves, wherefore we will not elaborate it further. In the remainder of this section, we will concentrate on the ?rst inequality as well as the two equalities, M2 = Q2 + 4e2 , and M2 = Q2 . We will ?rst elaborate the properties of the non-extremal subclass M2 > Q2 + 4e2 . Here we ?nd a surprisingly rich geometric structure, which looks like a hybrid of black holes in four dimensions and our ?rst family of solutions. To start with, we observe that this solution again possesses the event horizon and the Cauchy horizon, r ?+ > r ?? respectively. All causal geodesics starting from in?nity cross r ?+ , while there still exists the pathological class of geodesics which terminates at the Cauchy horizon, much like the previously discussed cases. This can be seen as follows. After the integrals of motion P? = (?E, P ) and the squared rest mass of the particle p are introduced, the radial geodesic equation can be written as (again the overdot denotes the derivative with respect to the a?ne parameter): ˙2 (QE + MP )(QE ? MP ) ? p(M2 + Q2 )/M r ? 2 2 = E ? P ? p ? 2Λ? r2 Mr ? e2 (QE + MP )2 + p(M2 Q2 + e2 (M2 + Q2 )) ? . (3.12) M2 r ?2 The terms proportional to the squared rest mass of the probe p do not a?ect the properties of geodesics near the two horizons. Ignoring them, we can employ the radial coordinate shifted by the value of the event horizon ρ = r ?? r ?+ . We can then rewrite this equation after √ √ introducing the parameters p± = ( M2 ? Q2 ± M2 ? Q2 ? 4e2 )/2 as (p+ ME ? p? QP )2 ρ ˙2 = + (E 2 ? P 2 )ρ2 2 2Λ M √ √ M2 E 2 ? Q2 P 2 (E 2 ? P 2 ) M2 ? Q2 M2 ? Q2 ? 4e2 + ρ+ ρ. M M

(3.13)

For all inwards-oriented geodesics which emanate from in?nity (E 2 ≥ P 2 ) the RHS of this equation never vanishes for any ρ ≥ 0. Thus they all cross the event horizon and fall into the black string. 21

Similarly, we can use the radial coordinate shifted at the Cauchy horizon, de?ning ρ ?= r ?? r ?? . The radial equation becomes

˙2 ρ ? (p? ME ? p+ QP )2 = + (E 2 ? P 2 )? ρ2 2Λ M2 √ √ (E 2 ? P 2 ) M2 ? Q2 M2 ? Q2 ? 4e2 M2 E 2 ? Q2 P 2 ρ ?? ρ ?. (3.14) + M M

Again, we look only at the arcs of geodesics outside of the Cauchy horizon; hence ρ ? ≥ 0. Because the coe?cient of the linear term in ρ ? is always positive, the RHS of this equation vanishes only when ρ ? = 0 and p? ME = p+ QP simultaneously. The last condition is compatible with E 2 ≥ P 2 since p? < p+ . Therefore, we see that the class of geodesics for which p? ME = p+ QP stops at the Cauchy horizon. This corresponds exactly to the case ms ?2 E = QP studied in the ?rst family of black strings, and shows that the pathology found by Horne and Horowitz still persists. Another similarity between this solution and our ?rst family is that this black string is also causally geodesically complete. Once again, the only causal geodesics which approach the singularity without turning are null geodesics for which QE + MP = 0, which come arbitrarily close to the singularity but again according to r ? ∝ exp λ. All other causal geodesics turn at a ?nite r ? > 0, where the repulsive term of order O (1/r ?2) in (3.12) prevails. Thus, the singularity can never be reached by any causal geodesics for ?nite value of the a?ne parameter, and it appears very much the same as in the ?rst family of black strings. There are, however, considerable di?erences between the two families. Namely, the √ second family (3.11) possesses three Killing horizons, r ?E ± = (M ± M2 ? 4e2 )/2 and √ r ?x = Q(Q + Q2 + 4e2 )/2M where the metric is regular but one of the coordinates t, x becomes null. The ?rst two, where t is null, satisfy r ?E + > r ?+ > r ?? > r ?E ? , and resemble the situation found in the Kerr black hole in four dimensions. The location of the last Killing horizon, where x is null, is inside the event horizon, but depending on the values of M, Q and e it can be either inside or outside the Cauchy horizon. The outer Killing horizon r ?E + de?nes the ergosphere, and thus one might expect that there exist Penrose-type processes for energy extraction from this kind of black string. This 22

issue is far from clear-cut, though, because the energy extracted from the Kerr black hole is at the expense of the hole’s momentum, resulting in slow-down of its rotation, and disappearance of the ergosphere. In our case, quite unexpectedly, the ergosphere appears due to the charges of the axion and gauge ?elds, which are protected by the Gauss laws at in?nity. Therefore, energy extraction by a Penrose-type process would seem to be inextricably linked to the diminishing of the string’s charge, which is in contradiction with the Gauss laws. We believe that the consistent resolution of this problem should be sought by postulating the existence of a more general family of solutions, which will be characterized by a non-zero linear momentum along the string Px . This quantity would then be dissipated by Penrose processes, thus opening the channel for eliminating the ergosphere while keeping the gauge charges conserved. A more detailed study of this problem would seem to be merited. Ultimately, one would like to determine the set of all alowed conserved quantities for three-dimensional stationary con?gurations, analogous to the approach of the last of Ref. [17]. Finally we present the Hawking temperature for this solution. It is obtained analogously to (3.3), and is given by T = √ √ Λ M2 ? Q2 M2 ? Q2 ? 4e2 √ √ . 2 M2 ? Q2 + M2 ? Q2 ? 4e2 (3.15)

In this case, the Hawking temperature vanishes for both extremal limits M2 = Q2 + 4e2 and Q2 = M2 . Which of these limiting situations will be reached by evaporation could in principle be determined by a detailed study of linear momentum transfer between the black string and the Hawking radiation, which is beyond the scope of the present work. In sum, the causal structure of this solution is reminiscent of the ?rst family. The associated Penrose diagram is essentially the same. The most important di?erence is the appearance of the ergosphere, which can provide for interesting e?ects in this geometry. The causal structure for this case is also shown in Fig. 1.

23

E. Extremal limit(s) of the second family

As we have indicated above, we will look here at the two special cases of the second family of solutions. We refer to these as the extremal limits in a somewhat tentative manner, because they represent such choices of parameters where the two horizons become degenerate. Yet, the case Q2 = M2 deserves its label as an extremal black string only in an indirect fashion, as we will indicate below, and show in the next section. Our ?rst extremal limit is given by the condition M2 = Q2 + 4e2 , resembling the extremality condition for dyonic Reissner-Nordstr?m black holes. This case is very di?erent from the previously studied extremal limits. There is now a singularity at r ? = 0, a single degenerate horizon r ?h = (M2 + Q2 )/M and, in general, three Killing horizons, located at r ?E ± = (M ± Q)/2 and r ?x = (MQ + Q2 )/2M. By comparing the values of the parameters, we see that there is an ergosphere r ?E + outside of the event horizon r ?h , and that the remaining two Killing horizons are inside of r ?h . Their relative locations however depend on √ the ratio Q/M < 1, and they coincide for Q/M = 2 ? 1. To see that all of these are indeed contained in the manifold, in contrast to our ?rst extremal limit and the extremal limit in the static case, we need to look at the geodesic equations and demonstrate that there are geodesics which extend to all of the above surfaces. This is again controlled by the radial equation. Qualitatively the behavior of geodesics is the same as in the non-extremal case. Here we will just show that all geodesics cross the event horizon in original coordinate r ?, meaning that the proper extension is given by including in the manifold the sector with r ?<r ?h . Since in the extremal limit, the parameters p± degenerate to e, we can rewrite the radial equation for null geodesics, in terms of the radial coordinate shifted by the horizon ρ=r ?? r ?h , as

e2 M2 E 2 ? Q2 P 2 ρ ˙2 2 2 2 2 = ( M E ? Q P ) + ( E ? P ) ρ + ρ. 2Λ M2 M

(3.16)

The RHS of this equation does not vanish for any ρ ≥ 0 (i.e., outside of the event horizon), and since similar conclusion also holds for timelike geodesics, we see that all inwards-oriented geodesics fall into the black string, proceeding to ρ < 0, as claimed. 24

To see that the other characteristic surfaces are also reachable, we need only observe that there still exists the class of null geodesics with QE + MP = 0, discussed in the non-extremal black string background. These are the only causal geodesics in the manifold that come arbitrarily close to the singularity at r ? = 0. However, since their descent towards the singularity is controlled by the exponential of the a?ne parameter, they do not reach it in any ?nite range of the parameter, and thus this manifold is also causally geodesically complete. Thus we conclude that the causal structure of this geometry is similar to the extremal Reissner-Nordstr?m solution, the di?erences being the ergosphere and the causal geodesic completeness of our solution. This comparison of causal structure is graphically summarized by the Penrose diagram in Fig. 3. We should also point out here that this solution does not have any null Killing vectors, in contrast to other extremal black strings. This can be seen from noting that any Killing vector must be a linear combination α?x + β?t . The null condition for this solution then translates into α2 = β 2 and αQ + β M = 0. These two equations can be simultaneously solved only if Q = ±M, which is not the case here. The other extremal limit, Q2 = M2 , as we have mentioned above, can be interpreted as a black string only indirectly. For example, we can see from the geodesic equation for the radial coordinate (3.16) for this case, that no causal geodesics starting from in?nity can ever cross the horizon, although some may come arbitrarily close to it before bouncing back. Nevertheless, if the horizon were probed by non-geodesics world lines, one would discover a structure akin to the horizon of the extremal static case or the extremal limit of our ?rst family of solutions, where the geometry consists of two mirror images divided by the horizon (Fig. 2). In e?ect, in this case the geodesics encounter an in?nite potential barrier at the boundary, due to the terms in the metric proportional to the electric charge. We will present these arguments in mathematical form in the next section, in a slightly more general context. The analogy of this case with an extremal black string can be further strengthened if √ √ we employ the null coordinates u = (t ± x)/ 2, v = (t ? x)/ 2, where the signs are chosen according to whether Q/M is positive or negative unity, respectively. With these 25

coordinates, we can rewrite the solution as ds2 = dr ?2 M e2 2 ? 2(1 ? ) dudv ? du , 2Λ(? r ? M )2 r ? r ?2 M e B= du ∧ dv, A = ?2 du, r ? r ? √ r, e?Φ = 2Λ?

(3.17)

which we recognize as a plane fronted wave carrying electric charge, traveling on the extremal black string of Horne and Horowitz. This interpretation will be given a thorough justi?cation in the next section, where we will demonstrate that this solution is in fact directly related to the extremal limit of our ?rst family by a wave transformation due to Gar?nkle [29].

IV. TRAVELING WAVE ON GAUGE-CHARGED BLACK STRING

We begin by brie?y reviewing the wave generating technique of [29]. This technique allows us to superimpose traveling wave contributions on solutions of Einstein-like theory of gravity with matter couplings of quite general nature (including stringy gravity) which have null hypersurface orthogonal Killing vectors. It works as follows. Let (? g?ν , B?ν , A? , Φ) be a solution of the equations of motion derived from the action (2.1) and given in the Einstein frame, with a null hypersurface orthogonal vector k . The Einstein frame is de?ned by conformally transforming the world-sheet metric g?ν to g ??ν = exp(?2Φ)g?ν . Then, there exists a scalar ?eld F such that 1 ?? kν = (k? ?ν F ? kν ?? F ). 2 Without changing the matter, we can de?ne the new Einstein frame metric by
′ g ??ν =g ??ν + F Ψk? kν ,

(4.1)

(4.2)

where Ψ satis?es k ? ?? Ψ = 0 , ?? ?? Ψ = 0 . 26 (4.3)

′ We can show that the con?guration (? g?ν , B?ν , A? , Φ) also represents a solution of the same

equations of motion. The key is to demonstrate that the equations of motion are invariant under this transformation. An easy way to see that this is true in the cases we consider is to recall that the determinant of the metric changes under (4.2) by a shift proportional to k ? k? . Since k is null, this is zero and the determinant is invariant: det(? g ′) = det(? g ). Furthermore, the ?eld strength of the axion H is a three-form, and thus its dual is invariant under (4.2), because the metric appears in it only through the determinant. Additional constraints must be imposed on the gauge ?eld, however. They are k ? A? = 0 [k, A] = 0 (4.4)

where the last equation represents the requirement that k Lie-derives the gauge ?eld A. These identities then guarantee the invariance of the gauge ?eld sector under (4.2) in the equations of motion. In the cases we consider these identities hold, as we will show below. Finally, if we compute the Ricci tensors, which govern the graviton dynamics, we can show ? ′? ν ? R ? ? ν is proportional to ?2 Ψ, which vanishes by the second condition of (4.3). that R Thus, the Ricci tensor with one contravariant and one covariant index is also invariant under (4.2), and so is the Ricci scalar. This in turn means that all the separate metric-dependent terms which appear in the equations of motion are invariant, and that (4.2) indeed represents a motion in the space of solutions, as claimed above. We remark that the interpretation of the modi?ed solution as a wave traveling in the original background rests on the property that the vector k remains a null Killing vector of the ?nal solution too. This is because the function Ψ is independent of the Killing coordinate according to the ?rst of the conditions (4.3). Thus all disturbances in the metric generated by it must propagate at the speed of light, without changing its shape. Now we can apply this technique to the extremal limit of our ?rst family of solutions. It is evident from the form of (2.8) that in the extremal limit |Q| = m the Killing vector ?x becomes null. For our purposes it is more convenient to use the shifted Killing coordinate 27

χ = x ? τ /2, because in the limit when s ?2 = 0, this solution reduces to the extremal limit of Horne and Horowitz, and then our coordinates χ, τ are exactly the null coordinates u, v . The Killing vector ?χ remains null, as can be seen from ?χ = (?x )τ . After the conformal transformation to the Einstein frame, we can rewrite the solution as r ?2 dr ?2 mc ?2 mc ?2 2 2 ds ? = ? 4Λ? r (1 ? )dχdτ + 2Λms ?r ?(1 ? )dτ 2 , 2 2 (? r ? mc ?) r ? r ? 2 √ ms ?c ? mc ? dχ ∧ dt, A=? 2 dτ, B= r ? r ? √ e?Φ = 2Λ? r.
2

(4.5)

Our null Killing vector is k = ?χ , with the only nonzero covariant component k0 = ?2Λ? r (? r? mc ?2 ). We can see that the conditions (4.4) for the gauge ?eld A hold, since it is directed along dτ and does not depend on χ. Therefore, we can apply the wave generating technique. To proceed, we can see that the scalar F = 1/r ?(? r ? mc ?2 ) solves the condition (4.1). The ?nal step of our calculation consists of ?nding a scalar ?eld Ψ which solves the constraints (4.3). The ?rst constraint requires Ψ = Ψ(? r , τ ), and consequently the d’Alembert operator of the second acquires a particularly simple form due to this and the properties of the metric. This equation can be written as ?Ψ ? (? r ? mc ?2 )2 = 0. ?r ? ?r ? The most general solution to (4.6) is Ψ = g (τ ) + f (τ ) . r ? ? mc ?2 (4.7) (4.6)

The g term here can be dropped because its contribution to (4.2) is only a di?eomorphism. The only nonvanishing component of the matrix F Ψk? kν is F Ψk0 k0 = 4Λ? rf (τ ), and thus the new Einstein frame metric is ds ?′2 = mc ?2 mc ?2 r ?2 dr ?2 2 2 ? 4Λ? r (1 ? ) dχdτ + 2Λ m s ? r ? (1 ? ) + 2Λ? rf (τ ) dτ 2 . (? r ? mc ?2 )2 r ? r ? (4.8)

Consequently, we can rewrite the new solution in the world-sheet frame by conformally transforming back, using the (unchanged) dilaton, to get 28

ds′2 =

mc ?2 mc ?2 ms ?2 2Λ dr ?2 ? 2(1 ? ) dχdτ + (1 ? ) + f (τ ) dτ 2 . 2Λ(? r ? mc ?2 )2 r ? r ? r ? r ? √ ms ?c ? mc ?2 dχ ∧ dt, A=? 2 dτ, B= r ? r ? √ r. e?Φ = 2Λ?

(4.9)

The wave behavior is completely determined by the function f (τ ). In the remainder of this section, we will focus on those solutions (4.9) where f = const. To start with, we note that if f = ?ms ?2 /2Λ, the solution is precisely the second extremal limit of the second family of solutions, discussed at the end of the previous section. This coincidence rea?rms our choice to label that solution as a traveling wave on an extremal black string. By the same token, our starting solution (4.5) can also be thought of as a wave of constant amplitude, carrying electric charge and traveling along an extremal black string. In contrast, no choice of f will lead to the ?rst extremal limit of the second family of black strings. We see this because that extremal limit does not have null Killing vectors, whereas (4.9) has one. Another interesting observation related to (4.9) with f = const is the e?ect of this term on the global structure of solutions. Here the value of f plays the crucial role, dividing the solutions into three categories, distinguished by the accessibility of the horizon, located at r ?h = mc ?2 , to geodesics probes. If we look at the geodesic equation for the radial coordinate, with Pχ , Pτ the components of the conserved probe momentum in χ and τ directions, and p its squared rest mass, ˙ r ? 2 = (ms ?2 (? r ? mc ?2 ) + 2Λ? r f )P χ + 2? r(? r ? mc ?2 )Pχ Pτ ? p(? r ? mc ?2 )2 , 2Λ (4.10)

2 ˙ )2 = 4Λ2 mc we see that at the horizon, (r ? ?2 f Pχ . Thus, depending on the sign of f , the

geodesics either cross the horizon, showing that the proper extension across it is by including r ?<r ?h (when f > 0, and is similar to our ?rst extremal limit of the second family), cross the horizon but with a mirror-like extension as in our ?rst extremal limit and the static extremal limit of Horne and Horowitz (when f = 0), or bounce back to in?nity before reaching the horizon (when f < 0, as in our last extremal limit). Therefore, the admissible 29

global structure is similar either to the extremal Reissner-Nordstr?m black hole, the extremal static black string, or the completely nonsingular geometry of our last extremal limit. In fact, all of these solutions can also be thought of as extremal black strings, as can be seen from the fact that their Hawking temperature is identically zero.

V. CONCLUSION

In this paper, we have presented several new solutions of stringy gravity in three spacetime dimensions. We have found that several of those solutions admit interpretation as electrically charged black strings, thus generalizing the static, electrically neutral solutions previously found by Horne and Horowitz. Our black strings have shown a surprisingly rich geometric structure, and the existence of several di?erent extremal limits, which are possible ?nal states which strings can reach by Hawking radiation. This situation is somewhat akin to that found in the case of gauge-charged stringy black holes in four dimensions, where the inclusion of the additional gauge and axion charges has introduced additional dimensions in the space of allowed parameters describing a black hole, resulting also in a collection of new extremal limits [33]. One of our well-de?ned extremal limits has a null Killing vector, which we utilized for generating a traveling wave solution on the string background, employing Gar?nkle’s techniques. In the special cases when the wave pro?le is constant, we have found that the solutions can be interpreted as yet new extremal black strings, since their associated Hawking temperature is zero. We have also found that two of our three extremal limits found directly are mutually related by such wave transforms. Our results are highly supportive of the existence of an even more general family of black objects in three dimensions, which we believe could be obtained by including an additional independent parameter, describing the conserved linear momentum in the direction of the string. Many properties we have observed indicate this; to name just a few, we could quote multiple extremal limits, di?erent non-extremal solutions, etc. We think that perhaps the strongest evidence for this conjecture is our observation of the apparent inconsistency 30

encountered in our second black string family. There we have indicated that on one hand, the presence of the ergosphere should indicate the possibility of energy extraction via Penrose processes, resulting in the disappearance of the ergosphere, which, on the other hand, would be in contradiction with gauge Gauss laws, since the ergosphere in this case is carried solely by the electric and axion charges. A possible resolution of this would be the existence of a more general family with an arbitrary linear momentum along the string, which would be the quantity to dissipate in the energy extraction by neutral probes. In this scenario, the string would eventually evolve towards the extremal limit of our ?rst family, with the gauge charges conserved, but without the ergosphere, and hence with no possibility of further energy extraction. We believe that this issue deserves further attention.

ACKNOWLEDGMENTS

We would like to thank B. Campbell, R. Myers, R. Madden, E. Martinez and W. Israel for helpful discussions. This work has been supported in part by the National Science and Engineering Research Council of Canada. In addition, the work of W.G.A. has been supported in part by an Province of Alberta Graduate Fellowship, and the work of N.K. has been supported in part by an NSERC postdoctoral fellowship.

31

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35

FIGURES
SINGULAR

III

CH

III

II
EH
EH

SINGULAR

CH

I +
I

I +
I

I II
SINGULARITY SINGULARITY

I -

III

CH

III
CH

II
FIG. 1. Causal structure for the two non-extremal black strings (2.7) and (3.11), as well as for the Reissner-Nordstr?m and Horne-Horowitz solutions. The hyperbolae denote the static limits present in our solutions, which do not appear in the previous two cases. Whereas the static limit inside region III is present in both of our two cases, the limit in the asymptotically ?at region I is the ergosphere present only in the second solution (3.11).

36

H

I +

I I +

H
H

I I +

I I +

H
H

I I +

I FIG. 2. Causal structure of the extremal limit of the ?rst family of black strings (2.7) and the extremal Horne-Horowitz solution.

37

II

H

I +
I

H

I SINGULARITY

II
H

I +
I

FIG. 3. Causal structure of the Q2 + 4e2 = m2 extremal limit of our second family (3.11) of black string solutions, as well as of the extremal Reissner-Nordstr?m solution. The hyperbolae here depict the ergosphere (region I) and the inner static limit (region II) present in our case, in contrast to Reissner-Nordstr?m.

H

I
II
H

I +
I

38




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