Extended objects with edges

Riccardo Capovilla*Departamento de Fı´sica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo Postal 14-740, 07000 Me´xico, D.F., Mexico

Jemal Guven†

Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xico, Apdo. Postal 70-543, 04510 Me´xico, D.F., Mexico~Received 12 September 1996!

We examine, from a geometrical point of view, the dynamics of a relativistic extended object with loadededges. In the case of a Dirac-Nambu-Goto~DNG! object with DNG edges, the world sheetm generated by theparent object is, as in the case without boundary, an extremal timelike surface in spacetime. Using simplevariational arguments, we demonstrate that the world sheet of each edge is a constant mean curvature embed-ded timelike hypersurface onm, which coincides with its boundary]m. The constant is equal in magnitude tothe ratio of the bulk to the edge tension. The edge, in turn, exerts a dynamical influence on the motion of theparent through the boundary conditions induced onm, specifically that the traces of the projections of theextrinsic curvatures ofm onto ]m vanish.@S0556-2821~97!02604-0#

PACS number~s!: 11.27.1d

I. INTRODUCTION

The lowest order phenomenological action describing thedynamics of a relativistic extended object, or membrane, isproportional to the area of its world sheetm and is known asthe Dirac-Nambu-Goto~DNG! action~for relevant examples,see@1–3# and, in the related statistical mechanical context,@4#!. The corresponding equations of motion of a closed ob-ject ~without boundary! are completely described by theworld sheet diffeomorphism covariant system of nonlinearsecond order hyperbolic partial differential equations:

Ki50 . ~1!

Here,Ki is the trace of thei th extrinsic curvature ofm em-bedded in spacetime, one for each codimension of the em-bedding. In particular, the classical dynamics is entirely in-dependent of the tension of the membranem0.

In this paper we focus on the modification required to thisgeometrical description when massive edges are admitted.Such edges may consist of several disconnected components.Concrete examples consist of a segment of string with mono-poles attached to its ends~a disconnected boundary! or adomain wall bounded by a string. The former is relevant inhadron physics as an effective description of color flux tubesin QCD @5#. The dynamics of such systems is also relevant incosmology because objects of this type could have been gen-erated if the early Universe underwent an appropriate se-quence of phase transitions@1,6#.

The key observation is that each edge world sheet is itselfan embedded hypersurface in the world sheet of the parentmembrane, which coincides with the boundary of the parentworld sheet. The edges are thus treated as membranes them-selves, one dimension lower than the parent membrane. Theparent world sheet is the spacetime where the edges live.

Since the parent membrane has a dynamics of its own, how-ever, this is no longer a fixed, prescribed background space-time. In the lowest order approximation, the edges will alsobe described by a DNG action of the appropriate dimensionwith its own characteristic tensionmb ~or mass, if pointlike!.The edge world sheet]m will then satisfy@7#

mbk52m0 , ~2!

wherek is the trace of the extrinsic curvature of]m embed-ded inm.

There are two tractable approximations. In the limit thatthe mass of the edge tends to zero, the null boundary dynam-ics associated with the theory of open membranes is recov-ered @2#. This limit is the one adopted for the open stringaction of string theory in its most ambitious form as a theoryof everything. On the other hand, in the limit that the edgetension goes to infinity,mb→`, the edges themselves be-come extremal surfaces of the background spacetime, andthe membrane interpolates accordingly. In particular, if thebackground spacetime is flat, the edges can be assumedfixed. This approximation is frequently exploited in thestring approximation of the inter quark potential@5#.

The equations of motion~1! and ~2! are not complete asthey stand. What is missing is a statement about the dynami-cal feedback that the edges have on the parent object span-ning them.~This simple fact was overlooked in Ref.@7#.!This is implemented in the form of constraints on the extrin-sic geometry ofm at the boundary. Specifically, one obtainsthat

HabKabi 50 , ~3!

whereKabi is the i th extrinsic curvature ofm embedded in

spacetime,Hab is the projection operator inm onto ]m, andthis equation is to be evaluated on]m. These constraintsmust be implemented as boundary conditions on Eq.~1!.Suppose we had failed to implement these conditions. Then

*Electronic address: capo@fis.cinvestav.mx†Electronic address: jemal@nuclecu.unam.mx

PHYSICAL REVIEW D 15 FEBRUARY 1997VOLUME 55, NUMBER 4

550556-2821/97/55~4!/2388~6!/$10.00 2388 © 1997 The American Physical Society

any given initial conditions on the boundary which are tan-gent to the world sheet of a closed solution to Eq.~1! wouldsimply generate a timelike hypersurface on this world sheet.The only feedback it would have on the parent membranewould be to determine the limits of the truncation of theclosed solution. On the contrary, the boundary conditions~3!generally place stringent conditions on the motion of themembrane. They are, however, vacuous when the membraneis totally geodesic,Kab

i 50. Such is the case of a planarworld sheet describing, for example, a nonrotating string or aplanar disc of membrane in Minkowski space.

For the case of a string with massive ends, the boundaryequation of motion~2! and boundary conditions~3! can becast in a particularly attractive form. If the trajectory of anend is parametrized with proper timet, then Eqs.~2! and~3!can be combined to give

mbDtS dXm

dt D52m0hm, ~4!

wherehm is the inward normal to the boundary of the stringworld sheet, andDt :5(dXm/dt)Dm is the projection ontothe end world line of the spacetime-covariant derivative. Theacceleration of an end is, therefore, of constant magnitudeand directed into the string world sheet.

This paper is organized as follows: In Sec. II we providea summary of the relevant mathematical formalism. In par-ticular, we discuss the connection between the hierarchy ofembeddings:]m in m, m in spacetime, and the direct em-bedding of the edges in spacetime. In order to simplify ourpresentation, we confine our attention to the case of an ex-tremal membrane, described by the DNG action, with edgesdescribed by a DNG action of one lower dimension. In Sec.III we derive the complete equations of motion for this sys-tem, Eqs.~1!–~3!. What is remarkable is just how efficientvariational principles are in isolating the appropriate bound-ary conditions@8#. We conclude in Sec. IV with a brief dis-cussion that focuses on a rotating string with massive ends.

II. KINEMATICS

To begin with, consider an orientedtimelikeworld sheetm of dimensionD, which corresponds to the trajectory of themembrane in anN-dimensional spacetime$M ,gmn%. Theworld sheetm is described by the embedding@9#

xm5Xm~ja!, ~5!

wherexm are coordinates onM , and ja coordinates onm(m,n,•••50, . . . ,N21, and a,b, . . .50,•••,D21). TheD vectors,

ea :5X,am ]m , ~6!

form a basis of tangent vectors tom, at each point ofm. TheLorentzian metric induced on the world sheet is then givenby

gab5emae

nbgmn . ~7!

Note that in statistical mechanics applications, we are inter-ested in a Euclidean ‘‘spatial’’ metricgmn , and, of course,the induced metricgab is Euclidean as well.

Let the spacetime vectorsnm i denote thei th unit normalto the world sheet (i , j , . . .51, . . . ,N2D), defined, up to alocalO(N2D) rotation, with

gmneman

n i50 , gmnnm inn j5d i j . ~8!

Normal indices are raised and lowered withd i j and d i j ,respectively, whereas tangential indices are raised and low-ered withgab andgab , respectively.

The vectors$ea ,ni% form a basis for spacetime vectorsadapted to the situation of interest here.

The world sheet projection of the spacetime covariant de-rivatives is defined byDa :5ea

mDm , whereDm is the ~tor-sionless! covariant derivative compatible withgmn . Theclassical Gauss-Weingarten equations~see@9,10#! are givenby

Daemb5gab

cemc2Kab

inmi , ~9!

Danmi5Kabie

mb1vaijnm

j . ~10!

The gabc5gba

c are the connection coefficients compatiblewith the world sheet metricgab . The quantityKab

i is thei th extrinsic curvature of the world sheet defined by

Kabi52gmnn

m iDaenb5Kba

i . ~11!

The extrinsic geometry ofm is determined byKabi , and by

the extrinsic twist potentialvai j , associated with the cova-riance under normal frame rotations.~See, e.g.,@11,10#!.

Not every specification of the intrinsic and of the extrinsicgeometry is necessarily consistent with some embedding.There are integrability conditions, the Gauss-Codazzi,Codazzi-Mainardi, and Ricci equations, which must be sat-isfied by the intrinsic and extrinsic geometry, for an embed-ding to exist. We will return to these equations below in thecontext of the boundary.

We turn now to the definition of the intrinsic and extrinsicgeometry of the world sheet boundary]m. We treat]m as atimelike surface of dimensionD21, described by the em-bedding in the world sheetm,

ja5xa~uA!, ~12!

whereA,B, . . .50,1, . . . ,D22, anduA are coordinates on]m.

The definition of the extrinsic and intrinsic geometry ofthe world sheet boundary provides a special case of the dis-cussion given above for an arbitrary world sheet. In order toestablish our notation, we repeat it, specializing to the caseof codimension one. TheD22 vectors,eA :5xa

,A]a , aretangent to the boundary world sheet]m. The metric inducedon ]m is then

hAB5gabxa,Axb

,B . ~13!

The normal to]m is defined by

gabhaebA50, gabh

ahb51 . ~14!

55 2389EXTENDED OBJECTS WITH EDGES

The Gauss-Weingarten equations take the form

¹AeaB5gABCeaC2kABh

a, ~15!

¹Aha5kABeaB, ~16!

where¹A5eaA¹a is the gradient along the tangential basis$eA%, gAB

C are the connection coefficients compatible withthe boundary world sheet metrichAB , andkAB5kBA is theedge world sheet extrinsic curvature associated with the em-bedding ofdm in m. For a codimension one embedding, theextrinsic geometry is determined completely by the extrinsiccurvature, and the Ricci integrability conditions are vacuous.

For the role it will play in the sequel, it is useful also tocontrast this description with the description of the boundarydm or, which is the same thing, of the edge world sheet,embedded directly in spacetime:

xm5Xm~uA!, ~17!

with tangentseAm :5ea

meAa . This corresponds to the map com-

positionXm„ja(uA)…5Xm(uA). The induced metric is exactly

as before, Eq.~13!. The spacetime normals tom are alsonormal to]m in spacetime. Withhm:5ea

mha, these vectorscomplete the normal basis which we labelnI :5$h,ni%. Wewill use the index 0 to denote the direction alonghm. Itshould not be confused with a timelike index. We can nowwrite down the corresponding Gauss-Weingarten equations(DA :5eA

mDm),

DAemB5gAB

CemC2KAB

InmI , ~18!

DAnmI5KABIe

mB1vAIJnm

J . ~19!

With respect to this adapted basis, it is simple to check thatKABi 5eA

aeBbKab

i , andKAB0 5kAB . In addition,vAi j5eA

avai j .The boundary inherits the extrinsic curvature and twist of theworld sheet. However, note thatvAi05haeA

bKabi . Thus,there is the possibility that the boundary world sheet mighthave a nontrivial twist~associated with its embedding inspacetime!, though the parent world sheet does not. In par-ticular, this might be the case when the parent world sheet isembedded as a hypersurface in spacetime. In the case of aone-dimensional boundary, however, the extrinsic twist willbe pure gauge.

It is also instructive to examine the hierarchy of integra-bility conditions which emerges in these alternative embed-dings of the boundary. We have the Gauss-Codazzi,Codazzi-Mainardi, and Ricci integrability conditions associ-ated with the embedding ofm in spacetime:

Rabcd5Rabcd2KaciKbdi1Kad

iKbci , ~20!

Rabci5¹̃aKbci2¹̃bKaci , ~21!

and

Rabi j5Vabi j2KaciKbcj1KbciKa

cj . ~22!

Here, ¹̃a is the covariant derivative associated with the ex-trinsic twist potentialva

i j , andVabi j is its curvature. Theleft-hand sides of these equations denote the contraction ofthe background spacetime Riemann tensorRm

nrs with the

basis $ea ,ni%. Ra

bcd is the Riemann tensor of the worldsheet covariant derivative¹a . We also have the Gauss-Codazzi and Codazzi-Mainardi integrability conditions asso-ciated with the embedding of]m in m:

RABCD5RABCD2kACkBD1kADkBC , ~23!

and

RABCdhd5DAkBC2DBkAC . ~24!

The left-hand sides of these equations denote the contractionof the world sheet Riemann tensorRa

bcd with the basis$eA ,h%. We use the notation RABCD to denote the Riemanntensor of the boundary covariant derivativeDA . Finally,there are the Gauss-Codazzi, Codazzi-Mainardi, and Ricciintegrability conditions associated with the direct embeddingof the boundary in spacetime:

RABCD5RABCD2KACIKBDI1KAD

IKBCI , ~25!

RABCI5D̃AKBCI2D̃BKACI , ~26!

and

RABIJ5VABIJ2KACIKBCJ1KBCIKA

CJ . ~27!

D̃A is the twist covariant derivative associated withvAIJ. We

note that consistency between Eq.~22! and ~27! implies

VABi j5eAaeB

bVabi j ,

VABi05eCc @eA

aKacikBC2eB

bKbcikAC#.

III. EXTREMAL OBJECTS WITH LOADED EDGES

The dynamics of the membrane is specified by the choiceof an appropriate phenomenological action, constructed withscalars built with the quantities that characterize the intrinsicand extrinsic geometry of the membrane world sheet. In thepresence of edges, one needs also to specify some dynamicalrule for the edges themselves. We choose the DNG action forthe membrane, and the same action for its edges.

The action we consider is

S5S01Sb , ~28!

where

S0@X,x#52m0EmdDjA2g, ~29!

Sb@x,X#52mbE]mdD21uA2h, ~30!

m0 is the membrane tension,mb is the tension of the edgemembrane,g the determinant of the membrane world sheetmetricgab , andh is the determinant of the boundary worldsheet metrichAB . This action is a functional both of theembeddingXm of m in M , and of the embeddingxa of ]m inm. There may well be many disconnected edges. To avoidclutter we ascribe the one tensionmb to all.

2390 55RICCARDO CAPOVILLA AND JEMAL GUVEN

To derive the equations of motion arising from the action~28!, consider first a variation of the embedding ofm,Xm→Xm1dXm. The displacement is assumed to vanish ontwo spacelike hypersurfaces ofm, which play the role ofinitial and final times.

We decompose the displacement with respect to thespacetime basis$ea,ni%, as

dX5Faea1F ini . ~31!

We now have that under this displacement, the intrinsic met-ric change is@10,12#

dXgab52Kabi F i1¹aFb1¹bFa . ~32!

The variation of the membrane actionS0 gives

dXS052m0EmdDjAggab@Kab

i F i1¹aFb#

52m0EmdDjAg@KiF i1¹aF

a# ~33!

52m0EmdDjA2gKiF i2m0E

]mdD21uA2hhbFb .

~34!

The last line obtains from the preceding one by applyingStokes’ theorem to the second term. Here,ha is the outward-pointing normal to]m introduced in Eq.~14!. We find thatonly the normal projection of the variationF i contributes tothe equations of motion of the membrane, this is generallytrue regardless of the form of the actionS0 so long as it isconstructed in a world sheet (m) diffeomophism-invariantway. There is no boundary term associated withF i . This isnot, however, generally true, it is an artifact of extremal dy-namics.

The tangential variation gives only a boundary term. Thisis a consequence of the fact that tangential deformations cor-respond modulo a displacement of the boundary to infinitesi-mal world sheet diffeomorphisms. This is why we could ig-nore such variations in our study of objects withoutboundary.

We note that the boundary contribution to Eq.~34! is~minusm0 times! the change in the world sheet volume un-der a normal deformation of the boundary,dxa5hbFbh

a.As we will see, it will contribute to the equations of motionof the edge. The projections ofFb onto ]m do not contrib-ute.

Before considering the variational analysis ofSb , wecomment briefly on the case of an open membrane with amassless boundary. The normal term will vanish wheneverg(h,h)50 or the boundary is null. Physically, no momen-tum may cross the surface. This can only occur if the bound-ary is a null surface, moving at each point at the speed oflight @2#. In the textbook treatment, this is arranged by de-manding that the normal projection of the world sheet de-

rivative of the embedding function vanishes on the boundary.~For a geometric treatment of such boundary conditions, see@13#.!

Let us consider now the variation of the edge action underthe infinitesimal variation in the world sheet~31!. This varia-tion is transmitted to the geometry of the boundary throughits effect ongab , given by Eq.~32!. We have

dXSb521

2mbE

]mdD21uA2hhABdXhAB

52mbE]mdD21uA2hHab~Kab

i F i1¹aFb!,

~35!

where we have introduced the projector onto]m,Hab:5hABx ,A

a x ,Bb and the fact thatdXhAB5x ,A

a x ,Bb dXgab .

The vanishing of the variation of the total actionS ~28!under arbitrary normal deformationsF i gives the equationsof motion for the membrane, Eq.~1!, Ki50. F i is not fixedon the boundary, so there is a boundary term appearing inEq. ~35! to contend with. It vanishes whenever Eq.~3!,HabKab

i50, evaluated on]m, is satisfied. The variationalprinciple has, therefore, also provided the natural boundaryconditions on the embeddingX. We will discuss the inter-pretation of these conditions below.

The vanishing of the variation ofS, under arbitrary tan-gential deformationsFa , with support on the boundary,gives the equation of motion for the boundary~2!, if mbÞ0, k52m0 /mb . To see this, we note that

Hab¹aFb5DAFA1khaFa , ~36!

where we have exploited the fact thatk5Hab¹ahb and wedefineFA5gabF

aeAb , so that

dXSb52mbE]mdD21uA2h~HabKab

i F i1DAFA1khaFa!.

~37!

The first term appearing on the right-hand side of Eq.~36! isa divergence, corresponding to an infinitesimal boundary dif-feomorphism,dxA5FA. In the case of asmoothphysicalboundary, this term does not contribute~the boundary of aboundary is zero!. The latter term appearing on the right-hand side of Eq.~36!, however, adds to the surface termappearing in Eq.~34! to give Eq.~2!.

We have not had to vary the action with respect to theboundary embedding to obtain Eq.~2!. For completeness,and consistency, let us now consider the variation inS in-duced by a displacement of the boundary world sheet]m:

dx5Ch1CAeA . ~38!

We obtain

55 2391EXTENDED OBJECTS WITH EDGES

dxS52m0E]mdD21uA2hC

2mbE]mdD21uA2hhABkABC, ~39!

modulo the same divergence appearing in Eq.~36!. We againreproduce Eq.~2!, nothing new is obtained. It is worth notingthat this variation does not pick up Eq.~3!.

It is instructive to compare the equations of motion de-scribing the dynamics of an isolated boundary~imagine thespanning membrane removed! with Eq. ~2!. The dynamics isnow simply extremal and we have~in the notation of Sec. II!hABKAB

I 50, or alternatively,k50 and HabKabi 50. The

former differs from Eq.~2! in the manner we would expect.The latter set of equations, however, reproduce the boundaryconditions given by Eq.~3!. The departure from extremalitywhen the boundary is spanned by a membrane occurs alongthe normal which is tangent to the membrane world sheet.

The boundary conditions~3! are still not exactly the stan-dard~Robin! kind of boundary condition we are accustomedto handle. It is worthwhile, therefore, to demonstrate explic-itly that they are sensible boundary conditions on Eq.~1!.We note that

Kabi 52nm

i ~¹a¹bXm1Gab

m X,aa X,b

b !, ~40!

so that Eq.~3! reads

nmi @~D2hahb¹a¹b!X

m1Gabm HabX,a

a X,bb #50 . ~41!

We now exploit the fact that the LaplacianDC of any worldsheet scalar~such asXm) can be decomposed as

DC5DADAC1~ha¹a!2C1kha¹aC, ~42!

and the fact that

hahb¹a¹bC5~ha¹a!2C, ~43!

to express Eq.~41! in the alternative form

nmi @DADAX

m1Gabm Hab#50 , ~44!

where we have definedHab:5HabX,aa X,b

b In this form, theEq. ~3! involves only derivatives ofXm along]m, and thus itprovides sensible boundary conditions for Eq.~1!.

Moreover, the form~44! of the boundary conditions sug-gests to reexpress the edge equations of motion, Eq.~2!, as

hm@DADAXm1Gab

m Hab#52m0

mb, ~45!

and we can now combine Eqs.~44! and ~45! as

DADAXm1Gab

m Hab52m0

mbhm. ~46!

This equation exhibits clearly the effect of the spanningmembrane on the dynamics of the edges, via the driving termon the right-hand side. In the case of a string, with propertime t along the trajectory of a boundary point, Eq.~46!reduces to@Dt5(dXm/dt)Dm#

mbDtS dXm

dt D52m0hm, ~47!

so that the acceleration of a boundary pointDt(dXm/dt) is

constant in magnitude and directed intom.

IV. DISCUSSION

Consider the example of a rigidly rotating string boundedby point particles. It is clear that there is no solution of Eq.~1! corresponding to a straightnonrotatingsegment of stringwith massless ends. Energy conservation would imply thatsuch a configuration has a fixed proper length which is in-consistent with the nullity of the ends. With masses loadingthe ends, however, a solution exists because energy can betransferred from the string to its boundary. The monopolesare accelerated towards one another by the constant forceprovided by the tension in the string, the string collapses to asingularity.

When the string rotates, the massive ends experience acentrifugal acceleration. Our nonrelativistic intuition sug-gests that stable bound states exist. In particular, circularorbits with a fixed radiusR ~corresponding to a fixed stringlength!, and fixed angular velocityv exist. These orbits areconstrained by the requirement thatvR<1. The correspond-ing world sheet of the string is simply a truncation at thisradius of the circular timelike helicoid of a rigidly rotatingstring with massless ends. Geometrically, this is possible be-cause the boundary conditions, Eq.~3!, are automatically sat-isfied whenv andR are constants.

In the higher-dimensional case of a membrane boundedby a string, a nontrivial interplay between the tension in themembrane and that in the boundary is possible. These forcesmight operate in opposite directions. This is the case for acircular hole in a planar sheet of membrane. The tension onthe circle tends to restore the membrane, that in the mem-brane to self-destruction. There is clearly a critical radiusdetermining which one will prevail. This competition is ex-pected to play a role in topology-changing processes.

In a subsequent publication, we examine perturbationtheory pointing out, in particular, how we must modify thetreatment in@11# or @10# when dynamical boundaries aretaken into account@14#. In @15# the analysis undertaken herefor DNG-extended objects is generalized to arbitrary phe-nomenological actions, both for the membrane and for theboundary.

Note added in proof.For an alternative derivation of Eq.~4!, see Ref.@16#.

ACKNOWLEDGMENT

We gratefully acknowledge support from CONACyTGrant No. 211085-5-0118PE.

2392 55RICCARDO CAPOVILLA AND JEMAL GUVEN

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55 2393EXTENDED OBJECTS WITH EDGES