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An Equivalence Between Momentum

arXiv:hep-th/9110065v1 23 Oct 1991UCSBTH-91-53October, 1991An Equivalence Between Momentumand Charge in String TheoryJames H. Horne, Gary T. Horowitz, and Alan R. SteifInstitute for Theoretical PhysicsUniversity of CaliforniaSanta Barbara, CA 93106-4030jhh@cosmic.physics.ucsb.edugary@cosmic.physics.ucsb.edusteif@cosmic.physics.ucsb.eduABSTRACT: It is shown that for a translationally invariant solution to stringtheory, spacetime duality interchanges the momentum in the symmetry directionand the axion charge per unit length.As one application, we show explicitlythat charged black strings are equivalent to boosted (uncharged) black strings.The extremal black strings (which correspond to the field outside of a fundamentalmacroscopic string) are equivalent to plane fronted waves describing strings movingat the speed of light.

Despite extensive work over the past several years, we still lack a fundamental de-scription of string theory, including a complete understanding of the basic objects thatare involved and the principles which guide its construction. However there are severalhints and suggestive clues which have been uncovered.

Perhaps the most important isspacetime duality [1,2,3]. This is the fact that different spacetime backgrounds correspondto equivalent solutions in string theory.

Spacetime duality is usually thought of in termsof relating small distances to large distances since in the simplest example of flat spacewith one direction identified, the duality relates radius R to radius α′/R. * Although thisis certainly an important aspect of duality, there are many other consequences as well.

Forexample, it has been shown that spacetime manifolds of different topology can be equiv-alent as string solutions (for a recent review, see [5]). In this paper we describe anotherconsequence of duality which concerns asymptotically defined conserved quantities.

For asolution which has a translational symmetry and is asymptotically flat in the transversedirections, one can define the total energy momentum per unit length Pµ and the totalcharge per unit length Q associated with the three form Hµνρ. (We will refer to Q as the“axion charge”.) We will show that duality interchanges Q and the component of Pµ inthe symmetry direction, while leaving the orthogonal components of Pµ unchanged.

Thus,in this sense, linear momentum is equivalent to axion charge in string theory.An immediate consequence of this equivalence is that one can add axion charge toany solution which is static as well as translationally invariant as follows: First boost thesolution to obtain a nonzero linear momentum and then apply a duality transformationto convert this momentum to charge. (This procedure is related to the transformationsdiscussed in ref.

[6].) We will illustrate this by considering the black string solutions.

Blackstrings are one dimensional extended objects surrounded by event horizons. Solutions tothe low energy field equations describing charged black strings in ten dimensions werefound in ref.

[7]. Exact conformal field theories describing charged black strings in threedimensions were found in ref.

[4]. We will consider these three dimensional solutions firstand describe the duality between charge and momentum.

For |Q| < M, duality relatessolutions with an inner horizon and a timelike singularity to solutions with no inner horizonand a spacelike singularity. For |Q| > M, duality relates completely nonsingular solutionsto solutions with naked singularities.

We then construct charged black string solutions inhigher dimensions D ≥5 by boosting the uncharged solution (which is simply the productof a D−1 dimensional black hole and the real line) and applying the duality transformation.It is known [7] that the field outside of a straight fundamental string in ten dimen-sions [8] is just the extremal black string (|Q| = M). We will see that this is true in alldimensions D ≥5.

More importantly, the extremal black strings turn out to be equiva-lent under duality to plane fronted waves, which describe strings moving at the speed oflight. We conclude with some further consequences of the linear momentum – axion charge* In this paper, we will set α′ = 1.

(In ref. [4], α′ = 12 was used.

)2

equivalence.We begin by reviewing the definition of energy-momentum and axion charge. Sincethey are defined as surface integrals at infinity where the fields are weak, it suffices to workwith the low energy equations of motion.

These can be obtained by varying the actionS =Zeφ[R + (∇φ)2 −112H2 + a2]. (1)where H = dB is the three form and a2 denotes a possible cosmological constant.

Letgµν, Hµνρ and φ be an extremum of S (in D dimensions) which is independent of acoordinate x.We assume that at large distances transverse to x, gµν →ηµν + γµν,Bµν →0 and φ →φ0 + χ. The field equations require that φ0 = ay + b for some Cartesiancoordinate y.

There is a conserved component of the energy momentum associated witheach translational symmetry of the background. (Thus, if a ̸= 0, Py is not well defined.

)For D ≥5, we require a = 0 and γµν, Bµν, χ to all fall offlike r4−D where r is a radialcoordinate orthogonal to x. For D = 3, we allow a ̸= 0 and require γµν, Bµν, χ to fall offlike e−ar.

The total energy momentum is obtained by first extremizing eq. (1) with respectto the metric and linearizing the resulting expression about the background solution.

Onethen contracts with a translational symmetry and integrates over a spacelike plane. Theresult is a total derivative which can be reexpressed as a surface integral at infinity.

Dueto the translational symmetry in x, it is convenient to work with the energy momentumper unit length. Its time component (the ADM energy per unit length) is given by*M = CIeφ0[∂iγij −∂j(γ + 2χ) + γij∂iφ0]dSj(2)where C is a dimension dependent constant, i, j run over spatial indices, γ = γii, and theintegral is over the D −3 sphere at large r and constant x.

The spatial components arePi = CIeφ0[∂tγij −∂jγti + ∂iγtj]dSj. (3)Since ∂xγµν = 0, we havePx = CIeφ0[∂tγxr −∂rγtx].

(4)Finally the total charge per unit length associated with H is given byQ = CIeφ0 ∗H = CIeφ0Hxrt = CIeφ0[∂tBxr + ∂rBtx](5)where we have chosen a gauge with ∂xBrt = 0 in the last step. * This formula is valid in an asymptotically Cartesian coordinate system.To obtain aformula valid in any coordinate system, one simply replaces the partial derivatives withcovariant derivatives.3

Given a solution gµν, Hµνρ, φ with a translational symmetry in x, there is a dualsolution [2] given by:˜gxx = 1/gxx,˜gxα = Bxα/gxx˜gαβ = gαβ −(gxαgxβ −BxαBxβ)/gxx˜Bxα = gxα/gxx,˜Bαβ = Bαβ −2gx[αBβ]x/gxx˜φ = φ + log gxx(6)where α, β run over all directions except x. This is sometimes referred to as “sigma modelduality”.

This map between solutions of the low energy field equations exists whether ornot x is compact. It has recently been shown [9] that if x is compact, the original solutionand its dual are both low energy approximations to the same conformal field theory*.With the above asymptotic conditions we see that under this duality transformation, theasymptotic fields transform as˜γxx = −γxx˜χ = χ + γxx˜γxα = Bxα˜Bxα = γxα .

(7)The remaining fields are unchanged to leading order. It is now easy to verify that underduality, Px and Q are interchanged while the orthogonal components of Pµ are unchanged.We now use this duality to generate the three dimensional charged black strings whichwere constructed in ref.

[4].Since the underlying conformal field theory is known, wewill indicate how the duality between the charged solutions and the boosted unchargedsolutions is represented in the exact solution. First, let us construct the |Q| < M blackstring.

We begin with the product of the two dimensional Lorentzian black hole [11] withR, corresponding to the uncharged black string in three dimensions. The low energy metricisds2 = −1 −M0rdˆt2+k4r2(1 −M0/r)dr2 + dˆx2φ = log r + 12 log k,B = 0,(8)where k is related to the cosmological constant a.

(A simple coordinate transformationr = M0 cosh2 ˆr puts eq. (8) in the more familiar form k dˆr2 −tanh2 ˆr dˆt2 + dˆx2.) We nowapply a Lorentz boost ˆt = t cosh α +x sinh α and ˆx = x cosh α +t sinh α to the solution (8)* If x is not compact, there is an equivalent dual description based on eq.

(6), but one mustview x in the dual solution as having zero radius so there are no momentum modes andonly winding modes [9,10].4

to obtain a new solutionds2 = − 1 −M0 cosh2 αr!dt2 +k4r2(1 −M0/r)dr2+ 2M0 cosh α sinh αrdtdx + 1 + M0 sinh2 αr!dx2φ = log r + 12 log k,B = 0(9)which has energy per unit length M = M0 cosh2 α and momentum per unit length Px =M0 cosh α sinh α. *We now apply the duality transformation (6) on x to obtain (aftershifting r →r −M0 sinh2 α)ds2 = −1 −Mrdt2+1 −Q2Mrdx2 +1 −Mr−1 1 −Q2Mr−1 k dr24r2φ = log r + 12 log k ,Bxt = Q/r(10)where M = M0 cosh2 α and Q = M0 cosh α sinh α.

This is the |Q| < M charged blackstring in three dimensions [4], and the momentum and charge have been switched underduality.If x is not periodically identified, then eqs. (9) and (10) presumably represent differentconformal field theories.However, if x is compact, these solutions are equivalent.Atfirst sight this is quite surprising since the global structure of the two spacetimes is quitedifferent.

As discussed in ref. [4], the charged black string (10) has an inner horizon and atimelike singularity, while eq.

(9) has the usual event horizon and a spacelike singularity.However, with hindsight, this result could have been anticipated. Inner horizons in generalrelativity are known to be unstable [12,13].

Since the arguments are quite general, oneexpects that they should apply in string theory as well. (One can show e.g.

that thetachyon field becomes singular near the inner horizon in the metric (10).) Now a solutionto the low energy field equations is a good approximation to an exact solution to stringtheory only if the fields are weak so the higher order corrections are initially small, andif it is stable so that the corrections remain small.

If the solution is unstable the higherorder corrections will become large and change the character of the solution. Thus eventhough the low energy metric (10) has an inner horizon, the fact it is equivalent to eq.

(9)shows the exact solution does not.The derivation above resulted in |Q| < M. As discussed in ref. [4], the metric (10) isnonsingular for |Q| > M as long as r is redefined using r = ˜r2 + Q2/M, and x is identified* Note that even though the two solutions are related by a coordinate transformation, theyare physically different since the coordinate transformation does not reduce to the identityat infinity.5

with the appropriate period. We can obtain it by starting with the product of a negativemass Euclidean black hole with timeds2 = −dˆt2 +1 + M0rdˆθ2 +k4r2(1 + M0/r)dr2,φ = log r + 12 log k,B = 0,(11)where ˆθ is not periodic.

This metric has a naked singularity at r = 0. Now boost thesolution (11) using ˆt = t cosh α + θ sinh α, and ˆθ = θ cosh α + t sinh α to obtainds2 = − 1 −M0 sinh2 αr!dt2 + 1 + M0 cosh2 αr!dθ2+ 2M0 cosh α sinh αrdt dθ +k4r2(1 + M0/r)dr2 ,φ = log r + 12 log k,B = 0.

(12)A duality transformation on θ in eq. (12) along with a shift r →r−M0 cosh2 α again yieldsthe charged black string (10) but now with M = M0 sinh2 α and Q = M0 cosh α sinh α, sothat |Q| > M.Again we see that the global structure of the solutions (12) and (10) is different.

Theboosted solution (12) is singular at r = 0, but the charged solution (10) (with |Q| >M) has bounded curvature everywhere. (Euclidean examples of this phenomenon havebeen noticed previously [14,15,16,17,9].) If θ is periodic, these solutions are equivalent.This shows that certain curvature singularities in the low energy metric do not adverselyaffect string propagation.

However, it is known that there are other types of curvaturesingularities which do affect strings very strongly [18]. It remains an outstanding questionto understand the basic difference between these types of singularities.The exact conformal field theories corresponding to these three dimensional blackstrings are known in terms of gauged WZW models.

In this case, the duality (6) corre-sponds to interchanging vector and axial vector gauging. As shown in ref.

[4], the chargedblack strings are obtained by axial gauging an appropriate one dimensional subgroup ofSL(2, R) × R. One can verify that the boosted uncharged black strings are obtained byvector gauging the same subgroup*.The extremal black string (|Q| →M) does not correspond directly to a WZW model,but can be realized as a limit of either the |Q| > M or |Q| < M constructions. This limitcan be taken in a variety of inequivalent ways [4].

The limit that preserves the asymptotic* We have been told that Ginsparg and Quevedo have made a similar observation.6

behavior has the low energy solutionds2 =1 −Mr(−dt2 + dx2) +1 −Mr−2 k dr24r2φ = log r + 12 log k ,Bxt = Mr . (13)The correct extension past r = M is given in terms of a new radial coordinate ˜r2 = r −M.The resulting metric has a horizon but no singularity.

We can dualize on x and then makethe coordinate transformationx = 12(u −v),t = 12(u + v),r = M + e2ˆr/√k,(14)to obtain the following solutionds2 = −du dv + dˆr2 + Me−2ˆr/√kdu2φ = 2√kˆr + 12 log k ,B = 0 . (15)This metric describes a plane fronted wave.

It corresponds to boosting either eq. (8) oreq.

(11) to the speed of light. It is possible to show that eq.

(15) is a solution to thestring equations of motion to all orders in α′ [19,20,18]. This is a result of the fact thatthe curvature is null, so that all higher order curvature terms vanish.

(The solution wehave obtained is actually a slight generalization of the plane fronted waves that have beendiscussed in the literature, since it includes a linear dilaton. )Most of our results about black strings in D = 3 can be extended to D ≥5*.

Sincethe exact conformal field theories are not known explicitly, we work with the low energyfield equations. These should be a good approximation to the exact solution away fromthe singularity.

The uncharged black string solution for D ≥5 is simply the product ofthe higher dimensional Schwarzschild solution with R:ds2 = −(1 −M0/rn)dˆt2+(1 −M0/rn)−1dr2 + r2dΩ2n+1 + dˆx2φ = 0,B = 0(16)where n = D −4. As before, we boost and dualize.

The result isds2 = −(1 −M0/rn)(1 + M0 sinh2 α/rn)dt2+dx2(1 + M0 sinh2 α/rn)+dr2(1 −M0/rn) + r2dΩ2n+1φ = log(1 + M0 sinh2 α/rn) ,Bxt =M0 cosh α sinh αrn + M0 sinh2 α. (17)* There do not appear to be static, asymptotically flat black strings in four dimensions [7,8].7

This solution describes a charged black string. For D = 10 (n = 6) it is precisely thecharged black string solution found in ref.

[7]. It has mass M = M0(1 +nn+1 sinh2 α) andcharge Q =nn+1M0 cosh α sinh α.The field outside of a fundamental macroscopic string in D ≥5 was found in ref.

[8].It was shown in ref. [7] that for D = 10, this field is simply the extremal limit Q = M ofthe black string.

From the above formulas for M and Q, the extremal limit correspondsto takingM0 →0,α →∞(18)such that M0 sinh2 α stays constant. It is easy to verify that for any D ≥5, the extremallimit of eq.

(17) is the field outside of a fundamental string. It follows from eq.

(18) thatthe solutions dual to the extremal strings are precisely the black strings (16) boosted tothe speed of light, which are described byds2 = −(1 −M/rn)dt2 + 2Mrn dt dx + (1 + M/rn)dx2 + dr2 + r2dΩ2n+1 . (19)With new coordinates x = 12(u −v) and t = 12(u + v), the metric becomesds2 = −du dv + dr2 + r2dΩ2n+1 + Mrn du2.

(20)This is precisely the form of a plane fronted wave [19,20,18]. If we were to add a δ(u)factor to guu, then eq.

(20) would be the Aichelburg–Sexl metric describing a point particleboosted to the speed of light [21]. Without this factor, the metric describes a string boostedto the speed of light.The equivalence between charge and linear momentum undoubtedly has broad implica-tions.

We have discussed applications to black string solutions. We now conclude with twopossible further consequences.

First, from the conventional viewpoint, linear momentumis associated with a spacetime symmetry and axion charge is associated with an internalsymmetry. The fact that they are equivalent is a concrete indication of a unification ofthese symmetries in string theory.

Second, if x is compact, Px should be quantized. Thisimplies that axion charge must be quantized as well.

This appears to be a new argumentfor quantization of charge.AcknowledgementsWe would like to thank N. Ishibashi, M. Li, J. Polchinski, and A. Strominger for usefuldiscussions. This work was supported in part by NSF Grant PHY-9008502.8

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