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Exact Primordial Black Strings In Four Dimensions

arXiv:hep-th/9303059v1 10 Mar 1993Alberta-THY-14-1993MARCH 1993Exact Primordial Black Strings In Four DimensionsNemanja KaloperTheoretical Physics InstituteDepartment of Physics, University of AlbertaEdmonton, Alberta T6G 2J1, Canadaemail: kaloper@fermi.phys.ualberta.caAbstractA solution of effective string theory in four dimensions is presented which admitsinterpretation of a rotating black cosmic string. It is constructed by tensoring the threedimensional black hole, extended with the Kalb-Ramond axion, with a flat direction.

Thephysical interpretation of the solution is discussed, with special attention on the axion,which is found to play a role very similar to a Higgs field. Finally, it is pointed out thatthe solution represents an exact WZWN σ model on the string world sheet, to all ordersin the inverse string tension α′.Submitted to Phys.

Rev. Lett.

In recent years we have witnessed a very rapid growth of the family of “black”configurations, representing gravitational fields with event horizons of various topology.They have ranged from various black holes [1-9], over string-like configurations [1,10-13],to p-branes [7,13-14].In this letter, I will attempt to expand this family by constructing a solution of thefour dimensional effective action of string theory which admits the interpretation as a blackcosmic string inside a domain of axion field gradient. The configuration is primordial, inthe sense that the domain is essential for its existence, because the “axion charge” (i.e., thegradient of the pseudoscalar axion) is what stabilizes the string.

Its geometric structure isthat of the three dimensional black hole recently found by Banados, Teitelboim and Zanelli(BTZ)[3], tensored with a flat line which is interpreted as the string axis. The solutioncould also be viewed in light of toroidal black holes investigated by Geroch and Hartle[15], and could be understood as such a black hole, which is bigger than the cosmologicalhorizon of the Universe in which it is imbedded.The dynamics of the background field formulation of string theory is defined withthe effective action which, in the Einstein frame and to order O(α′0), isS =Zd4x√g 12κ2 R −16e−2√2κΦHµνλHµνλ −12∂µΦ∂µΦ + Λe√2κΦ(1)where R is the Ricci scalar, Hµνλ = ∂[λBµν] is the antisymmetric field strength associ-ated with the Kalb-Ramond field Bµν ,Φ is the dilaton field and Λ the cosmologicalconstant.The metric is of signature +2, the Riemann tensor is defined according toRµνλσ = ∂λΓµνσ −.

. ., and the cosmological constant is defined with the opposite signfrom the more usual GR conventions: Λ > 0 denotes a negative cosmological constant.

Ithas been included to represent the central charge deficit. In the remainder of this paper,I will work in the Planck mass units: κ2 = 1.Instead of writing out explicitly Einstein’s equations, I will work in the action, as thisapproach simplifies finding the solutions.

The background ans¨atz is that of a stationary2

axially symmetric metric:ds2 = µ2(r) dr2 + Gjk(r) dxjdxk + η2(r)dz2(2)where the 2×2 matrix Gjk(r) is of signature 0 as the metric (2) is Lorentzian and one of thecoordinates {xk} is timelike. The z coordinate in (2) is noncompact, whereas the spacelikexk is compact.

The “lapse” function µ2 is kept arbitrary as its variation in (1) yieldsthe constraint equation. The dilaton Φ is a function of r only, and the axion equations∂[µHνλσ] = 0 and ∂µ√g exp(−2√2Φ)Hµλσ= 0 are solved in terms of the dual vectorfield V µ byHνλσ = exp (2√2Φ)√gǫµνλσV µ(3)and V = Q dz (the topological charge term).

This solution has been discussed at morelength in [16] (see also [11]). There, it has been argued that the axion equations of motioncan be solved by topological charge terms in cylindrical backgrounds, since such topologiesinclude a nontrivial first cohomology from a non-contractible loop S1 in the manifold.

Thecharge Q above can therefore be thought of as associated with such a loop of string, exceptthat the string size is bigger than the cosmological horizon.The simple form of the axion allows that it be integrated out from the action, with thehelp of a Lagrange multiplier, and treating the solution (3) as a constraint. The resultingeffective action for the dilaton-gravity system isS =Zd4x√g12R −12∂µΦ∂µΦ −Q2η2 e2√2Φ + Λe√2Φ(4)Note that the action (1) has been rewritten in the form similar to Einstein gravity with aminimally coupled self-interacting scalar field.

Also note that the sign of Q2 is negative.This is a consequence of the proper replacement of the axion with its charge using theLagrange multiplier method, and can be verified from the inspection of the equations of3

motion derived from (1). The action (4) will be discussed at more length later, with specialemphasis on the role of the axion in it.

The dilaton equation of motion is just∇2Φ = ∂V (Φ)∂Φ(5)with the potential V (Φ) = (Q2/η2) e2√2Φ −Λe√2Φ. It is obviously desirable to look forthose solutions with the dilaton minimizing the potential V (Φ), since if they exist, beingthe dilaton “vacua” they are the minimum energy solutions of (1).

A slight complicationis the presence of η in V (Φ). However, a solution can be sought for which η = 1.

Thisshould be done carefully because it must be verified that η = 1 represents a solution of theequations of motion. An easy way to check it is as follows.The background has three toroidal coordinates {xk} and z which are dynamicallyirrelevant.

Hence the problem is effectively one-dimensional and the Kaluza-Klein reduc-tion [17] can be employed to simplify the action (1). It is instructive here to performthe Kaluza-Klein reduction in two steps, in order to isolate the dynamics of the mode η.The first step is to integrate out the coordinate z.

The resulting effective action in threedimensions is, after the rescaling S3Deff= 2S/Rdz,S3Deff=Zd3x√G12η ¯R −12η∂µΦ∂µΦ −Q2ηe2√2Φ + ηΛe√2Φ(6)The mode η (the “compacton”) in this form of the action is obviously just a Lagrangemultiplier. Its Euler-Lagrange equation however involves the 3D part of the metric.

Thus,to investigate it it is neccessary to write out the complete set of Einstein’s equations inaddition to it. This can be avoided with a conformal redefinition of the 3D metric suchthat the 3D Ricci curvature disappears from the η equation.The conformal rescalingwhich ensures this is Gµν = (1/η) ˜Gµν.

The resulting action is justS3Deff=Zd3xp˜G12˜R −1η2 ∂µη∂µη −12∂µΦ∂µΦ −Q2η4 e2√2Φ + Λη2 e√2Φ(7)4

This action represents ordinary 3D General Relativity with two minimally coupled self-interacting fields Φ and ln η. The effective potential is nowVeff(Φ, η) = Q2η4 e2√2Φ −Λη2 e√2Φ(8)and the equations of motion for the scalars are (5) (where V (Φ) is replaced by Veff(Φ, η))and2∇2 ln η = ∂Veff(Φ, η)∂ln η(9)Interestingly, the system of equations (5), (9) is simultaneously solved by a “vacuum”Φ = Φ0, η = η0 provided thatQ2 = Λ2 η20 e−√2Φ0(10)This equation can always be satisfied, and actually can be viewed as the definition ofthe dilaton vacuum expectation value given the other parameters.

The resulting effectiveaction for gravity in three dimensions under the assumption that the “matter” modes arein “vacuum” can be obtained upon substitution of (10) in (7). It isS3Deffvac =Zd3xp˜G12˜R + λ3D(11)and represents just the normal 3D Einstein-Hilbert action with an effective (negative)cosmological constant λ3D = (Λ/2η20) e√2Φ0.

Its unique black hole solution is the BTZsolution, as shown in [3,9]. Therefore, the next step of dimensionally reducing (11) to aone-dimensional problem can be skipped, and as a consequence, the metric part of theblack string solution can be written as ds2 = ds2BT Z + dz2 after setting η0 = 1.

Thecomplete rotating black string solution of (1) is thusds2 =dρ2λ3D(ρ2 −ρ2+) + R2(dθ + N θdt)2 −ρ2R2ρ2 −ρ2+λ3Ddt2 + dz2Htρθ = ρQΦ = Φ0(12)5

with ρ2+ = M(1−(J/M)2)1/2, R2 = (√λ3D/2)ρ2+M −ρ2+and N θ = −J/2R2, and wherethe identity (10) has been used. The physical black strings should also satisfy the constraint| J |≤M.

If this were not fulfilled, one would end up with a singular structure, manifest bythe appearance of closed timelike curves in the manifold accessible to an external observer,crossing the point R = 0. Such a voyage has been investigated in [11] for the spinless case,and also in [10] for the vacuum.

Moreover, it has been argued that, although the solution(12) does not have curvature singularities (Rµνλ3 = 0, Rµνλσ = −λ3Dgµλgνσ −gµσgνλ),they can develop if the metric is slightly perturbed by a matter distribution [3]. Thus, thesingularities are hidden by a horizon if the spin is bounded above by the mass.

By analogywith the BTZ black hole, the solution with J = M is understood as the extremal blackstring, and J = M = 0 as the vacuum. There is a local correspondence between these twocases, as discussed in [9].The solution (12) describes a rotating black string as is rather obvious from themetric.

However, it is the axion which gives further clues regarding the nature of the string.As was mentioned above, one way to think of the solution is to imagine it as a loop of stringwith its length parametrized by z, which is bigger than the cosmological horizon of theuniverse where it is imbedded. In this sense, the solution represents an explicit example ofa toroidal black hole [15].

Such an interpretation obviously puts limits on the validity of theapproximations underlying the assumption that the string is straight. A more interestingpicture is obtained if one retains the image of the string as infinitely long and straight.The dual axion field strength V = Qdz = da(z) can be integrated between any two space-like (t = const) hypersurfaces z1,2 = const to give a(z2) −a(z1) = Q∆z.

Therefore, theaxion solution can be understood as a constant gradient of the pseudoscalar axion field.As z1,2 →∞, the axion diverges. But this is easy to explain: it is merely a consequenceof the assumption that the string is infinitely long.

In reality, one should expect somecut-offsufficiently far away along the string. The situation is precisely analogous to thatof the electrostatic potential between the plates of a parallel plate capacitor in ordinary6

electromagnetism.There, the cut-offoccurs on the plates of the capacitor, where thepotential assumes constant values. The gradient is just ⃗∇V = (∆V/∆L)⃗z.

This analogyshows that the black string solution (12) should be viewed as a gravitational configurationwhich arose inside a transitory region separating two domains within which the axion isconstant, a1 and a2 respectively. The axion gradient inside this region corresponds to theadiabatic change in the axion vacuum, where the adiabatic approximation is better if thetransitory region (and hence the string) is bigger.

The configuration (12) then evidentlyneeds the domain of axionic gradient for its existence (because the axion gradient stopsthe dilaton from rolling), and thence can justifiably be labelled primordial.The discussion of the previous paragraph illustrates only one aspect of the importanceof the axion in obtaining the solution (12). Besides providing the extra contribution tothe dilaton-compacton self-interactions, the axion also plays role of a Higgs field, whichis evident from the steps leading from Eq.

(1) to Eq. (4).

The axion condensate Q2 in(4) breaks the normal general covariance group GL(3, 1) of (1) down to GL(2, 1) whichis the invariance group of (4). It should be noted, though, that the Higgs-like behaviourof the axion is purely topological; indeed, in the O(α′0) approximation, the axion has noself-interactions, and hence no potential to minimize.

Again, this is not really a surprise.The behavior of the Peccei-Quinn (PQ) axion has been found very much the same, andat tree-level the PQ axion condensate was also purely topological. It was only after theradiative corrections were included, that its self-interaction potential arose.Hence, toinvestigate the Higgs aspect of the axion further it would be necessary to inspect higherorder corrections to (1).This programme could be best conducted via the Wess-Zumino-Witten-Novikov(WZWN) σ model approach [2].Namely, it was demonstrated recently that the BTZsolution can be obtained as either a non-gauged WZWN model on the group SL(2, R)/Por an extremaly gauged WZWN model on the coset (SL(2, R) × R)/(R × P), where Pis a discrete group which represents compactification of one of the space-like coordinates7

to a circle [8-9]. In this light, the solution (12) is obviously obtained by taking either ofthese two σ models and simply tensoring them with an additional flat direction, whichwill be the coordinate along the string.

Thus, specifically, (12) is an extremaly gaugedWZWN model on the coset (SL(2, R) × R2)/(R × P). Higher order corrections could nowbe investigated following the resummation procedure established by Tseytlin [18] and byBars and Sfetsos [19].

It turns out, that the black string configuration actually survives thecorrections, and appears to be an exact solution of string theory to all orders in α′. Theonly effect of the higher order α′ corrections is finite renormalization of the parameters in(12), and in particular, renormalization of the semiclassical expression for the cosmologicalconstant .

The details will be presented elsewhere [20].There still remains the problem of stability of solution (12) under small perturbations.Some indications can be obtained by looking at the “matter” sector of the effective actionS3Deff(7), after the conformal rescaling.The dilaton and the compacton in (7) can beviewed as an O(2) doublet, with the effective potential (8) manifestly breaking O(2). Asa consequence, the linear combination√2Φ −ln η of the dilaton and compacton picks upa mass term of order Λ, whereas its orthogonal complement remains massless.

It wouldhave been preferable if both the dilaton and the compacton became massive, because theirbig masses would de facto decouple them and improve the stability of the solution (12).As is, the solution (12) could actually be spoiled by perturbations of the massless mode,which can accumulate exterior to the black hole, much like the Goldstone modes presentin global cosmic string backgrounds [21]. This remains to be investigated further in thefuture.In closing, it has been shown that the serendipitous BTZ 3D black hole has simplegeneralizations to four dimensions, where it can be interpreted as a primordial spinningblack string, which is singularity-free.

The most attractive generalization is where it repre-sents a vacuum solution of tree level string theory, where the dilaton and compacton havebeen decoupled due to the axion charge. In this respect, the axion plays the role of a Higgs8

field, since it breaks down the invariance group GL(3, 1) of the underlying 4D theory downto the invariance group GL(2, 1) of the resulting three dimensional effective action, andmodifies the effective scalar potential of the model leading to the previously mentioneddecoupling of the scalar modes. The dilaton of the configuration is constant and thus (12)also represents a solution of four dimensional Einstein gravity with a minimally coupled3-form field strength (see also [22]).

Moreover, the solution represents an exact WZWNσ model on the worldsheet, and thence can be easily extended to include higher order α′corrections, as I will show elsewhere [20]. In the end, these do not affect the nature of thesolution, and it remains a well behaved singularity free string configuration with a horizon.AcknowledgementsI would like to thank B. Campbell and V. Husain for comments on the manuscriptand helpful discussions and G. Hayward for pointing out Ref.

[14] to me.This workhas been supported in part by the Natural Science and Engineering Research Council ofCanada.References[1] for a recent review of stringy aspects of black holes and strings see J.A. Harvey andA.

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