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Cosmic Solenoids: Minimal Cross-Section and Generalized Flux Quantization Aharon Davidson and David Karasik Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel ([email protected], [email protected]) A self-consistent general relativistic configuration describing a finite cross-section magnetic flux tube is constructed. The cosmic solenoid is modeled by an elastic superconductive surface which separates theMelvin core from thesurroundingflat conic structure. Weshow that a given amount Φ of magnetic flux cannot be confined within a cosmic solenoid of circumferential radius smaller √3G than Φ without creating a conic singularity (the expression for the angular deficit is different 2πc2 from that naively expected). Gauss-Codazzi matching conditions are derived by means of a self- consistentaction. Thesourceterm,representingthesurfacecurrents,issandwichedbetweeninternal and externalgravitational surface terms. Surfacesuperconductivityisrealized bymeansof aHiggs 9 scalarminimallycoupledtoprojectiveelectromagnetism. Tradingthe’magnetic’Londonphasefora 9 e 1 9 dual’electric’surfacevectorpotential,thegeneralizedquantizationconditionreads: Φ+ Q=n hc e 1 withQdenotingsomedual’electric’charge,therebyallowingforanon-trivialAharonov-Bohmeffect. n Ourconclusions persist for dilaton gravity provided thedilaton coupling is sub-critical. a J 2 I. INTRODUCTION keep the total magnetic flux finite, the surface current 1 gets infinitely large as the tube becomes infinitesimally v TheDiracprocedureofsqueezingamagneticfluxtube, narrow. 2 whilekeepingitstotalmagneticfluxΦfixed,playsanim- The situation is conceptually and drastically changed 0 portant role in theoreticalphysics. It is usually assumed once gravity (or string theory) enters the game. In this 0 thatonecanshrinkthesolenoidintoathinmagneticflux paper, a self-consistent general relativistic configuration 1 0 string characterizedby the potential describing a finite cross-section cosmic solenoid is constructed. The cosmic solenoid is modeled by an elastic 9 9 A dxµ =Ndφ= N(xdy ydx) . (1) superconductive surface which separates the curved in- / µ r2 − ner core from the surrounding flat conic structure. Any c attempttosqueezethecosmicsolenoid,whileholdingits q Suchameasurezeroinfinitely-longfluxstringisofcourse - totalmagneticfluxΦfixed,comeswithacosmicpenalty. r classically invisible, but can still allow for a non-trivial Intheextreme,iftryingtoimprisonanygivenamountof g quantummechanicalAharonov-Bohmeffect[1]. Further- : magneticfluxwithinatubeofasub-criticalcross-section, v more, if the total magnetic flux is properly quantized, one pays the ultimate price of closing the surrounding i namely X space and creating a conic singularity. r eΦ Adopting the h¯ = c = 1 units, here are some exact a =n , (2) hc spacetime configurations [3] relevant to our discussion: Thegeneralstationary cylindricallysymmetricvacuum thefluxstringcanberegardedanartifactsinceitcannot • solution of Einstein equations is given by be detected by test particles carrying quantized electric charge in units of e. In which case, eq.(1) is nothing but r2 a pure gauge. By the same token, the semi-infinite mag- ds2 = Ω(dt+Vdφ)2+ω(dr2+dz2)+ dφ2 , (4) − Ω netic flux string attached to a Dirac magnetic monopole [2],carryingmagneticchargeg,becomesphysicallyirrel- where we have used the notations evant in case that ω(r) = r21(n2−1) , eg 1 hc = 2n . (3) Ω(r) = r α βrn , rn − (5) (cid:16) (cid:17) On dimensional grounds, however, it is clear that a β rn+1 string is not really capable of fully representing the case V(r) = . α Ω of an arbitrarily narrow tube. In particular, the inno- r cent looking magnetic flux string configuration eq.(1) is The static solution calls for V(r) = const, and can be apparently sourceless. To keep track of the source, one •put in the familiar (a,b,c) Kasner form shouldconsideramagneticfluxtubeoffiniteradius,and let a surface current constitute the source. This way, to ds2 = r2adt2+r2bdz2+dr2+γ2r2cdφ2 , (6) − 1 1 with the various parameters subject to M(r) 1+ GB2r2 . (14) ≡ 4 a+b+c=1 , (7) It is accompanied by the electromagnetic configuration a2+b2+c2 =1 . Br2 A = , The factor γ acquires a (global) physical meaning once φ 2M(r) the periodicity notation is specified, say (15) Br F = . ∆φ=2π . (8) rφ M2(r) The only asymptotically-flat Kasner solution is the One observes that the magnetic flux is practically con- 2 locally-flatyetgloballyconic(0,0,1)solution. Theconic finedwithinatubeofradius B,whereB isthevalue defect, reflecting the topological charge of the sectional √G 2-metric,measuresthe variationof the invariantcircum- ofthemagneticfieldattheorigin(inourformnotations, 1 ferential radius ρ(r) with respect to the invariant radial B = F ). Asexpected,theMelvinsolutionrepresents distance R(r). Given the flat Kasner metric eq.(6), for z r rφ a soliton which connects the only two Kasner branches which ρ=γr and R=r, one finds which are allowed by partial Lorentz invariance, namely dρ =γ , (9) 2 2 1 dR (0,0,1) ( , , ) . (16) ←→ 3 3 −3 corresponding to a deficit angle of 2π(1 γ). As long as − Whereas the (0,0,1) branch, associated with r 0, is γ ≥0, the outer space is open, but γ <0 takes us to the regular, the (2,2, 1) branch, associated with r→ , back side of the cone. In which case, the outer space is 3 3 −3 → ∞ is notonly singular but furthermoreexhibits a vanishing closed and exhibits a singular point at a finite distance. circumferential radius. A well known example is the space surrounding a cos- micstring[4–6]. ThecorrespondingVilenkinconicdefect [4] reads γ =1 4µG , (10) − whereµistheenergydensityperunitlengthofthestring. Thephysicaldemandγ 0thenleadstotheconsistency 1 ≥ condition µ . ≤ 4G The general cylindrically-symmetric static solution • of Einstein-Maxwell equations involving a longitudinal magnetic field (caused by an angular current) is referred FIG. 1. Melvin bone: The 2-dim surface (as embed- to as the Witten solution [7] ded in flat 3-space) characterized by the sectional 2-metric r2 ds2=M2(r)dr2+r2M−2(r)dφ2. ds2 =ω4W2( dt2+dr2)+W2dz2+ dφ2 , (11) − W2 Inlightofthis,acosmicsolenoidshouldbeconstructed where the Witten factor W(r) is given by bypastingtogethertwomanifolds. Thecoreofthetube, W(r)= α +βrn+1 . (12) which contains the magnetic flux and consequently ex- rn 1 hibits the Melvin geometry, is wrapped by a flat Kas- − ner Universe which carries no magnetic fields but has a This metric can be viewed as a soliton connecting two generic conic structure. These manifolds are separated Kasner regimes, namely nn2(n−n+1)1,n21−nn+1,n2 nn+1 and byanelasticsuperconductivesurfacehostingthe surface − − − its n n companion. (cid:16) (cid:17) currents. In a self-consistent model like ours, these sur- →− Regularity at the r = 0 axis of symmetry (at the ex- face currents are expected to be dynamically created by •pense of a generic singularity at r ) singles out the some surface fields minimally coupled to projective elec- Lorentz invariant (along the z-axi→s) α∞2 = n = 1 case, tromagnetism. The corresponding self-consistent action known as the Melvin solution [8] principle must not only govern the equations of motion, butalsothe matchingconditions(including inparticular r2 the Gauss-Codazzi formalism) of the various fields in- ds2 =M2(r)( dt2+dz2+dr2)+ dφ2 , (13) − M2(r) volved. Thelatterincludethegravitationalfieldgµν,the electromagneticvectorpotentialA ,theoptionaldilaton µ where the Melvin factor M(r) is given by fieldη,andavarietyof(2+1)-dimensionalsurfacefields. 2 It is only after such a model is analytically constructed, + S detγd3y (19) that one is able to analyze the case from the shrinking L − − IS circumferential radius point of view. 1 p Kout detγd3y+ This paper is organized as follows. First, we present −8πG − thetenableactionprincipleandderivethecorresponding 1 IS p equations of motion and the attached matching condi- + out detgd4x 16πG R − − tions. In this Lagrangian formalism, the surface source ZVout termissandwichedbetweentheinternalandtheexternal 1 (gµλgνσpF F )out detgd4x . gravitational surface terms. This is the prescription be- −16π µν λσ − ZVout hind the field theoretical recovery of the Gauss-Codazzi p and the electromagnetic matching equations. We are Thecanonicalfieldsaregµν(x)andAµ(x),whicharecon- after the cylindrically symmetric static solution, focus- tinuousoverthesurface,andsomesurfacedegreesoffree- ing our attention on two parameters, namely the cir- dom (soon to be specified). In our notations, cumferential radius of the tube and the conic defect of γ =g (x(y))xµ xν (20) the surrounding space-time. Surface superconductivity ab µν ,a ,b is realized by means of a Higgs scalar minimally coupled is the induced (2+1)-dimensional metric, Kin/out is the to projective electromagnetism, thereby establishing the µν extrinsiccurvatureofthesurfaceS embeddedinV , linkage between the circumferential radius and the conic in/out and K = gµνK . Each gravitational action in V defect. The main result of this linkage is that the cir- µν in/out is accompanied by its own surface term [9]. This way, cumferential radius ρ cannot be arbitrarily small unless the surface matter actionis sandwichedbetweenthe two the surrounding space-time suffers a singularity. To be gravitational surface terms. Obviously, the two gravita- more specific (and use momentarily the full units), tional surface terms are expected to cancel eachother in theemptycase,wherethesurfaceisregardedanartifact. √3G ρ Φ (17) Asimilarideahasbeenrecentlysuggestedinthecontext min ≥ 2πc2 of black hole membranes [10]. Thevariationoftheactionwithrespecttog (x)gives Notice that this bound is h¯-independent, indicating that • µν rise to the Einstein equations in the entire spacetime this result is purely gravitational and has nothing to do with quantum mechanics. Trading the ’magnetic’ Lon- 1 don phase for an ’electric’ dual surface vector field, we µν gµν 8πGTµν =0 , (21) R − 2 R− then replace eq.(2) by the generalized quantum mechan- (cid:12)in/out (cid:12) ical quantization condition (cid:12) and produces the Gauss-Codazzi(cid:12)matching condition on the surface eΦ 1 + Q=n (18) hc e + =8πGS , (22) Kab|in Kab|out ab where Q is the dual electric charge involved. Notice the where fact that Φ itself is not quantized, thereby opening the (Kh K )xµxν , (23) door for a non-trivial Aharonov-Bohm effect. Finally, Kab ≡ µν − µν ,a ,b we offer a similar treatment to a cosmic solenoid in dila- andh isthefirstfundamentalformofthesurface. The µν ton gravity background, and verify the validity of our energy-momentum tensor in V is mainresultsaslongasthedilatoncouplingremainssub- in/out critical. ∂ in/out Tin/out = 2 L + in/outg . (24) µν − ∂gµν L µν II. SELF-CONSISTENT ACTION PRINCIPLE Ina similar way,the surface energy-momentumtensoris defined via Up to as yet unspecified model-dependent source, the ∂ S action principle is quite conventional S = 2 L + Sγ . (25) ab − ∂γab L ab Action= Eq. (22) is the matching equation suggested by Israel 1 (gµλgνσF F )in detgd4x+ [11](but has notbeen derivedby meansof a Lagrangian −16π µν λσ − ZVin formalism). In its present form it solely involves surface + 1 in detgd4xp projections, a property which in turn allows us to use 16πG R − − differentcoordinate systemsinside andoutside the tube. ZVin 1 p The variationofthe actionwithrespectto A (x)gives Kin detγd3y+ • µ −8πG − rise to the Maxwell equations in the entire spacetime IS p 3 Fµν =0 , (26) The surface equation is obtained by the embedding ;ν in/out xµ (y). Each of the metrics (29) and (30) was writ- (cid:12) in/out andproducesthe elect(cid:12)romagneticjunctionconditions on teninaconvenientcoordinatesystem;onemustthususe the surface differentcoordinate systemsinside andoutside the tube. Both line elements exhibit Lorentz invariance in the z Fµνnµxν,a in+ Fµνnµxν,a out =4πγabjb , (27) direction, and ∆φ=2π periodicity. Therefore, the most generalcoordinatetransformationbetweentheinnersys- (cid:12) (cid:12) where nin/out is(cid:12)the unit outsid(cid:12)e/inside-pointing normal tem and the outer system is given by µ vector. The surface current is given by tout = tS = λtin . ∂ S zout = zS = λzin . (32) ja = L , (28) ∂Aa φout =φS =φin . withA =A xµ denotingtheprojectiveelectromagnetic a µ ,a The locationofthe separatingsurface is defined by both vector potential. rin =Rin , (33) rout =Rout , III. COSMIC SOLENOID butthereisnoreasonwhyshouldRin/out betakenequal. Thecosmicsolenoidsolutionissubjecttothefollowing Substitute Eqs. (29,30) in Eqs. (21-28), one derives requirements: the following matching relations: 1. Theconfigurationisstatic,cylindricallysymmetric, Keeping the metric continuous over the surface deter- • and partially Lorentz invariant. It admits three mines the coordinate transformation and the conic de- ∂ ∂ ∂ fect, namely Killing vectors: , , . ∂t ∂z ∂φ 1 λ=1+ GB2R2 , (34) 2. The surface is stable, defined by r =R=const. 4 in 3. The inner space is conic singularity free. R γ = in . (35) 4. The outerspacetimeis asymptoticallyflatandfree R 1+ 1GB2R2 of electromagnetic fields. out 4 in (cid:0) (cid:1) Calculate the extrinsic curvature of the surface, with SolvingEinstein-Maxwellequations,giventheabovecon- • respecttotheinside/outsideembeddings,anduseeq.(22) straints, one can immediately verify that to obtain The inner solution is the Melvin Universe • 1 1 1 r2 St =Sz = , (36) ds2in =M2(r)(−dt2+dz2+dr2)+ M2(r)dφ2 , (29a) t z 8πG Rout − Rin 1+ 41GB2Ri2n ! (cid:0) (cid:1) B2R Br Br2 Sφ = in . (37) Friφn = M2(r) , Aiφn = 2M(r) . (29b) φ −8π 1+ 14GB2Ri2n 2 (cid:0) (cid:1) Notice that, as r 0, Ain vanishes (no residual flux The continuity of the electromagnetic vector potential → φ • strings) and ds2 approaches Minkowski line element. across the surface leads to: in • The outer solution is the (0,0,1) Kasner Universe with N = BRi2n . (38) a conic structure 2 1+ 1GB2R2 4 in ds2out =−dt2+dz2+dr2+γ2r2dφ2 , (30a) • The junction cond(cid:0)ition for the el(cid:1)ectromagnetic fields reads Fout =0 , Aout =N . (30b) B rφ φ 4πjφ = . (39) R 1+ 1GB2R2 The total magnetic flux confined in the tube is in 4 in Thesetofmatchinge(cid:0)quationstells u(cid:1)sthatthe yetun- Φ= A dφ=2πN . (31) specifiedsurfacefieldsmustfulfill twoconsistencycondi- φ tions, namely I 4 Stt =Szz =−σ , (40a) γ =(1−6µinG)2/3−4µsG (47) thereby constituting the flux-tube generalization of the Sφ+Njφ =0 . (40b) φ cosmic string conic defect eq.(10). For large ρ, that is in the Nambu-Goto limit, we do recover Eq.(40a) reflects the Lorentz invariance in the z direction,anddefines thesurfaceenergydensityσ. Eq. (40b) γ 1 4(µ +µ )G . (48) in s expresses the equilibrium between the gravitational at- ≈ − traction and the electromagnetic repulsion. An observer living in the outer spacetime is actually aware of only three parameters. They are: IV. SURFACE SUPERCONDUCTIVITY 1. The total flux confined in the tube Φ=2πN, The missing pieces of the puzzle areof coursethe (2+ 2. Theconicdefectγ,whichcanbemeasuredbygrav- 1)-dimensionalsurface fields. By virtue of their coupling itational lensing, and to the (3+1)-dimensional fields, they serve as sources. And by exhibiting their own equation of motion, whose 3. The circumferential radius ρ of the tube, given by solution must obey eqs.(40), they make self-consistent R the double relation ρ= in =γR . sources. For simplicity, we consider first the prototype 1+ 1GB2R2 out case of a complex scalar field [12] minimally coupled to 4 in projective electromagnetism, and then discuss the dual Holdingthetotalmagneticflux2πN fixed,inaccordwith case. Dirac procedure, the circumferential radius of the tube gets related to the surface current A. Complex scalar field N2 3 N ρ+G = , (41) ρ 2πjφ (cid:18) (cid:19) The simplest source term and the conic defect S = γab(D Ψ) (D Ψ) V(Ψ Ψ) , (49) a † b † L − − 1 γ = N2 2 −4µsG (42) involves a complex scalar field Ψ(y) = ξ(y)eiχ(y) min- 1+G imally coupled to the projective electromagnetic field. ρ2 This is realized by means of the covariant derivative (cid:18) (cid:19) gets related to µs, the surface energy per unit length ∂ D Ψ=( ieA )Ψ , (50) a ∂ya − a µ =2πρσ . (43) s whereA =A (x)xµ anddimensionlessebeingtheelec- The quantity µ need not be confused with µ , the en- a µ ,a s in tromagnetic coupling constant. Invoking the gauge in- ergy per unit length in the tube. The latter can be variant quantity calculated by integrating the T0 component of the en- 0 ergymomentumtensoroveraplaneperpendiculartothe ∆ χ eA , (51) a ,a a solenoid, namely ≡ − the equations of motion are given by Rin 2π rdφ µin =−Z0 M(r)drZ0 M(r)T00 . (44) (γabξ;a);b−ξ(γab∆a∆a+ ddξV2)=0 , (52a) Substituting T0 = 1 F0µF 1 δ0FµνF = 1 B2 , (45) j;aa ≡ 2eξ2γab∆b ;a =0 . (52b) 0 4π 0µ− 16π 0 µν −8πM4(r) (cid:0) (cid:1) The only static solution of these equations which does with M(r) denoting the Melvin factor, we find not upset the consistency conditions eqs.(40) is of the London type, that is ξ = m=const , 1 1 (53) µ = 1  . (46) χ = nφ , in 6G − N2 3 (cid:26)  1+G   ρ2  with integer n keeping the extremal field configuration  (cid:18) (cid:19)  single-valued. Thefactthatm=0meansthatU(1)EM is 6 In turn, the conic defect takes the final form spontaneouslyviolatedon(andonlyon)thetubesurface. 5 This is the group theoretical origin of the superconduc- The demand γ 0 sets a lower bound on the size of a ≥ tive currents, the generators of the imprisoned magnetic tube which carries a given amount of magnetic flux flux. A crucial role is played here by the effective potential 4n ρ2 =GN2 1 (62) min eN − Veff =V +ξ2γab∆a∆a , (54) (cid:18) (cid:19) having the properties that This establishing one of our main results. V = σ , eff|ξ=m  dVeff = 0 . (55)  dξ2 (cid:12)ξ=m (cid:12) Using the definition of S(cid:12)(cid:12)ab and the consistency relations eq.(40), we infer that m2 σ =2 n(n eN) . (56) ρ2 − In turn, we can write m2 V Λ = n2 e2N2 , |ξ=m ≡ ρ2 −  (57)  dV = (n(cid:0)−eN)2 . (cid:1) dξ2 − ρ2 (cid:12)ξ=m Theinsertionofa(cid:12)(cid:12)(cid:12)surfacefieldestablishesthedesiredlink between the surface current ja and the surface energy FIG. 2. Dirac-Melvin contour plot: Lines of constant density σ minimal circumferential radius ρmin range from Dirac line σ = njφ = 2nΛ . (58) (ρmin →0) up toMelvin line (ρmin→∞). e n+eN We now claim that the physical solution dictates Taking into account eq.(59), one may further observe that Λ>0 ,  n >1 . (59) ρmin N√3G (63)  eN ≥ To see the point, recall the two relations and is thus driven to a provocative conclusion: In the presence of gravitation, the Dirac procedure does not ρ2 e2N2+ Λ=n2 , (60a) make sense. Any attempt to arbitrarily squeeze a mag- m2 netic flux tube, while keeping its total magnetic flux fixed, would eventually create a singularity (and close n N2 1+ the surrounding space). Magnetic flux simply cannot be R3 = N = eN , (60b) confined within flux tubes of sub-Planck cross sections. in 2πjφ (cid:16)4πΛ (cid:17) andimposeR 0. Thelattercomestoensurethatthe in ≥ outer spacetime would not develop a conic singularity. Finally,weareinapositiontoanalyzetheconicdefect asafunctionofthecircumferentialradiusofthetubeand the total magnetic flux confined. We do it by studying theinterplayofeqs.(41,47,58),whichtogetherdetermine the conic defect to be 4n N2 FIG.3. The outer space surrounding a cosmic solenoid is 1 ( 1)G − eN − ρ2 conicandopen(ρ ρmin γ 0)orelseclosedandsingular γ = . (61) ≥ ⇒ ≥ N2 3 (ρ<ρmin ⇒γ <0). 1+G ρ2 (cid:18) (cid:19) 6 B. Dual vector field Notice that the configurationin hand is that of a constant ‘electric’ field in a ‘ring’ capacitor. It is nothing In order to decode the Pythagoreanquantization con- but the closed (2+1)-dimensional analog of the (3+1)- dition eq.(60a),we restrictourselvesto the Londonlimit dimensional plate capacitor. E turns out to be indepen- of the surface scalar field theory (fixed ξ =m). Starting dent of the distance between the Q ‘charges’, normal- ± from ized such that ρE S = m2γab∆ ∆ Λ , (64) Q . (74) L − a b− ≡ 2m2e the χ equation of motion is The distance between the ‘charges’can then be taken to infinity, so that cylindric symmetry is restored without 1 ∂( S√ detγ) Ja = L − =0 . (65) any lose of generality. The resulting generalized quanti- ;a √−detγ (cid:18) ∂χ,a (cid:19),a zation condition reads The most general solution of this equation is simply eΦ 1 + Q=n (75) 2π e εabc εabc Ja = ∂ q = f , (66) √ detγ b c 2√ detγ bc It is the combination (magnetic flux) + (dual electric − − charge), rather than the magnetic flux by itself, which where fab ∂[aqb]. gets properly quantized. ≡ One can perform now a Legendre transformation to The fact that the magnetic flux N does not get quan- exchange the scalar phase χ for a surface vector field qa. tizedcomeswithnosurprise. Intheorieswherethe com- The prescription then calls for a new Lagrangian plex field extends to spatial infinity and has non-zero S∗ = S Jaχ , (67) modulus m there, the requirement of finite energy im- L L − ,a poses a strict relation between Aµ and the gradient of the phase χ of the complex field. This relation, in turn, which up to a total derivative is nothing but leadstofluxquantizationbecausethecomplexfieldmust LS∗ =−8m12γabγcdfacfbd−e√εadbectγAa∂bqc−Λ . bqueasnintgizleed-vahleureedi.sTthhaetrethaseocnotmhpatletxhescmalaagrnfieetlidc,fltuoxwishnicoht − theprojectionofAµ couplesminimally,existsonlyonthe (68) solenoid surface and does not extend to spatial infinity. Inthisduallanguage,q hasbeenelevatedtothelevelof a a (2+1)-dimensional gauge field exhibiting off-diagonal V. DILATON GRAVITY Chern-Simons interaction [13] with the projective electromagnetic gauge field. The corresponding equation of Dilaton gravity usually arises as the low energy limit motion is given by of string theory, or as the 4-dimensional effective theory εabc of higher-dimensional Kaluza-Klein gravity. The dila- fab =2m2e A . (69) ton is a real scalar field that couples to other fields in a ;b √ detγ b,c − very special way. Two different metrics are often used in dilaton gravity, related to each other by means of a The main questionnow is the following: Whatsurface gaugeconfigurationisequivalenttoχ=nφ? Asimpleal- conformal transformation. In the so-called string basis, thedilatoncouplestotheRicciscalarassociatedwiththe gebrarevealsthatthesolutionconsistentwithconditions stringmetric. In the so-calledEinstein basis,the dilaton eqs.(40) is couples only to matter, but does not have a direct grav- f = E =const . (70) itational coupling with the Ricci scalar associated with tz − Einstein metric. In this context, it is worth mentioning It is accompanied by the relations thattheextrinsiccurvaturesurfacetermpreventstheap- pearance of a dilaton surface term during the conformal eE jφ = , (71) transformation. This is why we choose to work here in ρ the Einstein basis, where the self consistent action takes eNE E2 the form σ = + . (72) ρ 2m2 Action= Now,wematchtheformereq.(58)withthelattereq.(72) 1 (e 2k√GηgµλgνσF F )in detgd4x to obtain −16π − µν λσ − − ZVin n−eN = 2ρmE2 . (73) −81π ZVin(gµνη,µη,ν)in −detgd4x+p p 7 1 1 + in detgd4x gµνη + k√Ge 2k√GηF2 =0 , (81) 16πG R − − ;µν 2 − ZVin (cid:12)in/out 1 p (cid:12) −8πG Kin −detγd3y+ and the associated dilaton matchin(cid:12)(cid:12)g condition IS p + LS −detγd3y− (76) nµη;µ|in+ nµη;µ|out =4π̺ . (82) IS 1 p The dilaton surface charge ̺ is defined by Kout detγd3y+ −8πG − IS ∂ S ∂ S 1 p ̺= L L . (83) + out detgd4x ∂η − ∂η 16πG R − − (cid:18) ;µ(cid:19);µ ZVout 1 p −8π (gµνη,µη,ν)in −detgd4x+ Giventhesamesymmetryconstraints,theaboveequa- ZVout tions admit the following solution: 1 p −16πZVout(e−2k√GηgµλgνσFµνFλσ)out −detgd4x . s•trTaihgehtinfonrewrarsdoluwtaioyn generalizes Melvin universe in a p The parameter k is recognized as the dilaton coupling r2 constant; its value depends, however, on the underlying ds2 =D2(r)( dt2+dz2+dr2)+ dφ2 , (84a) parent theory. It is normalized such that k = 1 is dic- in − D2(r) tated by string theory, whereas 5-dimensional Kaluza- where Klein theory [14] suggests k = √3. The canonical fields afarceenfioewldgsµaνs(xw),elAl.µ(Wx)e, thhaevedisloamtoeniηd(exa),haonwddsoomsuerfsaucre- D(r)=M(r)1+1k2 = 1+ 1GB2r2 1+1k2 . (84b) 4 fields serve as electromagnetic sources, but the dilaton (cid:18) (cid:19) coupling to the surface fields is still an open question. It is accompanied by Now, in analogy with the previous discussion, Thevariationoftheactionwithrespecttogµν(x)gives Fin = ek√Gη0 Br , (84c) r•ise to the Einstein equations in the entire spacetime rφ √1+k2M2(r) 1 Rµν − 2gµνR−8πGTµν =0 , (77) ek√Gη0 Br2 (cid:12)in/out Ain = , (84d) (cid:12) φ √1+k22M(r) (cid:12) and produces the Gauss-Codazzi(cid:12)matching condition on the surface and the dilaton configuration k Kab|in+ Kab|out =8πGSab , (78) ηin =η0 lnD(r) . (84e) − √G only with dilaton modified T and S . µν ab In accord with Aφ(0) = 0, it seems reasonable to insist The variationofthe actionwithrespecttoA (x) gives µ • on η =0 as well. the generalized Maxwell equations in the entire space- 0 time, The outer solution has again a flat conic structure • e−2k√GηFµν =0 , (79) ds2out =−dt2+dz2+dr2+γ2r2dφ2 , (85a) ;ν(cid:12)in/out (cid:16) (cid:17) (cid:12) and the junction conditions(cid:12)(cid:12)on the surface modified to Froφut =0 , (85b) include the dilaton factor 4πγ jb = Aout =N , (85c) ab φ = e−2k√GηFµνnµxν,a + e−2k√GηFµνnµxν,a . but is furthermore characterized by the constant in out (cid:12) (cid:12) (cid:12) (cid:12) (80) ηout η(r )=η . (85d) (cid:12) (cid:12) ≡ →∞ ∞ The variation of the action with respect to the dilaton Substitutingeqs.(84,85)ineqs.(77-83)andkeepingthe • field η(x) leads to g (x), A ,andη(x) continuousoverthe surface,we ob- µν µ tain a bunch of matching conditions: 8 λ= 1+ 1GB2R2 1+1k2 , (86a) LS =−el√Gη γab(DaΦ)†(DbΦ)+V(Φ†Φ) . (90) 4 in (cid:18) (cid:19) The coupling con(cid:0)stant l depends on the under(cid:1)lying theory, the character of the field, and the dimension of the γ = Rin , (86b) surface. Tracingourstepsfromeq.(52)toeq.(59),weare 1 led to the following relations Rout 1+ 14GB2Ri2n 1+k2 2eΛ (cid:0) (cid:1) jφ =el√Gη∞ , n+eN 1 1 1 n St =Sz = , σ = jφ , (91) t z 8πGRout − Rin 1+ 14GB2Ri2n 1+1k2  ̺= el√Gσ ,  (cid:0) (cid:1) (86c) − and the subsequent consistency condition 1 B2R n k 8πSφ = in , (86d) = 1 . (92) φ −1+k2 2+k2 eN l ≥ 1+ 1GB2R2 1+k2 4 in Giving both N and η seems a bit too much, as the (cid:0) (cid:1) circumferential radius ρ∞and consequently the conic de- 1 BR2 N = in , (86e) fectγgetfixed. Ourcurrentinterest,however,istostudy √1+k22 1+ 1GB2R2 γ(N,ρ) while holding only N fixed. This way, for each 4 in value of ρ, in particular ρ , one can calculate the at- (cid:0) (cid:1) min tached η . Our goal now is to derive the ρ formula 4πjφ = 1 B , (86f) based on∞the physicalrequirement γ 0. Wemipnroceed in √1+k2Rin 1+ 14GB2Ri2n 1+1k2 steps: ≥ e (cid:0) (cid:1) Startbysubstitutingtherelationjφ = σintoeq.(86f) • n k 1 to obtain ln 1+ GB2R2 =√Gη , (86g) − 1+k2 4 in ∞ 1 (cid:18) (cid:19) 1 1+k2 nB 4πσR 1+ GB2R2 = . (93) in 4 in e√1+k2 k √GB2R (cid:18) (cid:19) 4π̺=−(1+k2) in 2+k2 . (86h) This enters nicely into the γ =0 criticalconditionwhich 2 1+ 1GB2R2 1+k2 now reads 4 in The surface fields are(cid:0)subject to the(cid:1)consistency con- 2nB 1=G . (94) ditions: e√1+k2 Stt =Szz =−σ , (87a) • Next, solve eqs.(86e,89) to obtain an expression for B Sφ = Njφ = ̺ . (87b) B = 2N√1+k2 Rin k2−1 , (95) φ − k√G ρ2 ρ (cid:18) (cid:19) Altogether, we can calculate the relevant physical which can be used to rewrite the critical condition as quantities, the conic defect 1 1=G4nN Rin k2−1 . (96a) γ = 1−8πGσRin 1+ 14GB2Ri2n 1+k2 , (88) eρ2 (cid:18) ρ (cid:19) 2 1+ 14G(cid:0)B2Ri2n 1+k2 (cid:1) Rin To finish the calculationwe only need as a function and the circumfer(cid:0)ential radius o(cid:1)f the flux tube ρ of ρ (and the fixed N). But such a relation is precisely R what eqs.(86e,89) are capable of producing, namely ρ= in . (89) 1 1+ 41GB2Ri2n 1+k2 R 1+k2 1 R 2k2 in =1+GN2(1+k2) in . (96b) Eq.(86h) tells u(cid:0)s that the dila(cid:1)ton surface charge must ρ ρ2 ρ (cid:18) (cid:19) (cid:18) (cid:19) be differentfrom0,thereforethe surfacefields mustcou- ple to the dilaton in some way. The simplest way to do The interplay of the two last equations leads us finally • so is to add a dilaton factor in the following way to our main result 9 1−k2 gauge field with off-diagonal Chern-Simons interaction ρ2 =G4nN 1 eN(1+k2) 1+k2 (97) with projective electromagnetism. It is surface field the- min e − 4n ory which dictates the vital connection between surface (cid:18) (cid:19) current and surface energy density, which in turn links An immediate test of this formula is the recovery of the conic defect to the circumferential radius. Our ana- eq.(62) at the k 0 limit. lytic analysis produces a generalizedquantization condi- → Notice that if the dilaton coupling is such that tion, namely k2 > 4n 1>3 , (98) (magnetic flux)+(dual electric charge)=integer , eN − therebyallowingforanon-trivialAharonov-Bohmeffect. thereisnolowerboundonρ. Itisworthmentioningthat the class of (4+D)-dimensional Kaluza-Klein theories, wheretheextradimensionsformaD-dimensionalsphere, ACKNOWLEDGMENTS induce a 4-dimensional dilaton gravity characterized by It is our pleasure to thank Prof. Yakir Aharonov for 2 k = 1+ . (99) the valuable discussions and enlightening remarks. D r The maximal value of k happens to be precisely √3 (achieved for D = 1). Thus, in this family of dilaton gravity theories there is always a bound on the circumferential radius of the solenoid. Afinalremarkisinorder. Thecasek =1issingledout [1] Y.AharonovandD.Bohm, Phys.Rev.115,485(1959). by string theory, and as such deserves special attention. [2] P.A.M.Dirac,Proc.Roy.Soc.LondonA133,60(1931). However, in the context of the present work, it does not [3] D. Kramer, H. Stephani, E. Herlt, and M. MacCallum, seem to play any exclusive role in the cosmic solenoid ExactSolutionsofEinstein’sFieldEquations(Cambridge game. 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