Cosmic Solenoids: Minimal Cross-Section and Generalized Flux Quantization

11 Pages·0.18 MB·English

by  Aharon Davidson

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 con-
structed. 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 direc-
tion,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 con-
stant ‘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 elec-
tromagnetic 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 the-
ory, 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 circum-
ferential radius of the solenoid.
Afinalremarkisinorder. Thecasek =1issingledout
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ρ2 =GN2 . (100) Astrophys.J.288,422(1985);W.A.Hiscock,Phys.Rev.
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VI. CONCLUSIONS
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and J. Goodman, Nucl. Phys. B291, 847 (1987); N.K.
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M. A. Melvin, Phys. Lett. 8, 65 (1964); K.S. Thorne,
general relativistic configuration which describes such a
Phys. Rev.139, 244 (1965).
finite cross-section magnetic flux tube. The so-called
[9] G. W. Gibbons and S. W. Hawking, Phys. Rev. D15,
cosmic solenoid is modeled by an elastic superconduc-
2752 (1977); J. W. York Jr.,pp. 246-254 in Between
tive surface which separates the inner Melvin core from
Quantum and Cosmos, eds. W. H. Zurek and W. A.
the surroundingflatconic geometry. The Gauss-Codazzi
Miller (Princeton Univ.Press, Princeton N.J., 1988).
(andelectromagnetic)matchingconditionsarederivedby
[10] M.K. Parish and F. Wilczek (gr-qc/9712077).
means of a self-consistent action where the source term,
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J. Math. Phys. 21, 1423 (1980); J. Isper and P. Sikivie,
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10