Classical dynamics of brane-world extended objects

Milovan Vasilic*

Institute of Physics, Post Office Box 57, 11001 Belgrade, Serbia(Received 9 April 2010; published 11 June 2010)

We make use of the universally valid stress-energy conservation law to study the motion of various

branelike extended objects in a generic brane-world. Without specifying any particular action, we are able

to derive the world-sheet equations that govern the dynamics of brane-world test branes. In particular, the

brane-world test particles are shown to follow geodesics with respect to the brane-world induced metric.

At the same time, the presence of extended objects is shown to influence the brane-world geometry. It is

demonstrated that codimension-1 branes necessarily violate the brane-world smooth structure, while

lower-dimensional branes violate the very continuity. In particular, the truly zero-size massive particles

are shown not to exist in a continuous brane-world. As an example, static, axially symmetric membrane-

world in 4d Minkowski background is analyzed.

DOI: 10.1103/PhysRevD.81.124029 PACS numbers: 04.40.�b, 11.27.+d

I. INTRODUCTION

The old idea that we live inside a domain wall propagat-ing in a higher dimensional spacetime has been revived inthe last decade. After the suggestion that hierarchy prob-lem can be solved with large extra dimensions [1,2], a lotof attention has been drawn to brane-world models. Theseare basically models which see our universe as a braneembedded in a higher dimensional spacetime called bulk.All ordinary matter is localized on the brane, and onlygravity lives in the bulk.

A variety of brane-world models have been proposed inliterature, each characterized by a specific choice of thebulk action functional, and the bulk dimension. The fea-tures common to all these models are summarized asfollows. The bulk spacetime is considered an arena forthe assumed higher dimensional physics. The action func-tional has the form I ¼ Ig þ Im, where Ig stands for the

higher dimensional gravitational action, and Im is theaction of matter fields. Both, Ig and Im, are invariant under

bulk diffeomorphisms. The bulk matter field equations areassumed to have a localized solution that is well approxi-mated by an infinitely thin brane.

In this paper, we want to explore model independentfeatures of the brane-world concept. Instead of specifying aparticular action, we choose the universally valid stress-energy conservation equations

r�T�� ¼ 0 (1)

as the starting point of our analysis. Indeed, the covariantconservation of the stress-energy tensor T�� � �Im=�g��

is a direct consequence of diffeomorphism invariance, andtherefore, is model independent. In what follows, we shallbe concerned with the branelike solutions of the Eq. (1) inan arbitrary pseudo Riemannian bulk. We shall not specify

the type of matter the brane is made of, but merely assumethat such localized configurations exist. The effectivebrane-world equations are obtained in the limit of aninfinitely thin brane.The method we use is the manifestly covariant multipole

formalism of Refs. [3,4], developed as a generalization ofthe Mathisson-Papapetrou method for pointlike matter[5,6]. It has already been exploited for the study of simplestrings and higher branes in backgrounds of general ge-ometry [3,4,7]. In this paper, we want to examine complexbranes, characterized by an inhomogeneous distribution ofmatter. In particular, we are interested in branes that ac-commodate other branes.The motivation for this kind of research comes from the

observation that the only way to detect spacetime geometryis the usage of some sort of test matter. Conventionally, oneuses test particles, as they provide the most completeinformation on the probed geometry. In this work, wewant to examine the behavior of extended bodies trappedin a generic brane-world. In the probe matter approxima-tion, the world-sheet equations of the extended bodies arecompared to the similar equations in general relativity. Inparticular, probe particles are expected to follow geodesictrajectories with respect to the brane-world induced metric.If test bodies violate the probe matter approximation, theyare expected to curve the brane-world in a similar way asmassive particles of general relativity curve the spacetime.In any case, the analysis along these lines provides a usefultool for a model independent testing of the brane-worldconcept.The new results obtained in this paper are summarized

as follows. First, the world-sheet equations of a probeq-brane trapped by a generic p-brane have been derived.It is demonstrated that trapped probe branes behave thesame way as free probe branes behave in the (pþ 1)-dimensional spacetime endowed with the metric equivalentof the brane-world induced metric. In particular, the probeparticle is shown to satisfy the brane-world geodesic equa-*mvasilic@ipb.ac.rs

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tion. Second, the world-sheet equations of a p-brane ac-commodating a (p� 1)-brane are derived in a manifestlycovariant way. In such brane-worlds, the embeddedcodimension-1 branes are seen to violate the brane-world’-s smooth structure, but preserve the continuity. The resultis easily generalized to describe any number of intersectingbranes. Finally, the presence of higher codimension branesis shown to violate the very continuity of the brane-world.In particular, the truly zero-size particles are shown not toexist in a continuous brane-world. As an example, static,axially symmetric membrane-world in 4d Minkowskibackground is analyzed.

It should be emphasized that, compared to the similarresults in literature, our results have the advantage of beingmodel independent. Precisely, they are obtained from theuniversally valid conservation equations (1), without spec-ifying any particular action. The sole assumption made isthat matter is localized to resemble a brane.

The paper is organized as follows. In Sec. II, we reviewthe basic notions of the multipole formalism developed in[3,4]. One starts with the covariant conservation of thestress-energy tensor, and analyzes it under the assumptionof highly localized matter. In the lowest, single-pole ap-proximation, the matter is viewed as an infinitely thinbrane. The basic steps in the derivation of the world-sheetequations and boundary conditions are explicitly shown. InSec. III, the general word-sheet equations are applied to thecomplex system of a p-brane accommodating a q-brane.This is done by a suitable choice of the free coefficientsthat represent the effective (pþ 1)-dimensional stress-energy of the brane. Specifically, the p-brane stress-energyis chosen to be maximally symmetric everywhere excepton a (qþ 1)-dimensional subsurface. The resulting equa-tions have two parts. The first determines the q-branedynamics relative to the p-brane. The second shows thatp-brane equations are modified by the presence of a�-function force on the right-hand side. In Sec. IV, thecases q ¼ p� 1 and q ¼ 0 are examined in detail. InSec. V, the general form of the static, axially symmetricmembrane-world in 4d Minkowski background is derived.The influence of matter localization on the membraneworld-sheet is analyzed. Section VI is devoted to conclud-ing remarks.

The conventions used throughout the paper are the fol-lowing. Greek indices�; �; . . . are the bulk indices, and runover 0; 1; . . . ; D� 1. Latin indices a; b; . . . are the brane-world indices, and run over 0; 1; . . . ; p. Latin indicesi; j; . . . refer to the test q-brane world-sheet, and takevalues 0; 1; . . . ; q. The coordinates of the bulk, braneworldand the test-brane are denoted by x�, �a and �i, respec-tively. The bulk metric is denoted by g��ðxÞ, the induced

braneworld metric by �abð�Þ, and the induced test-branemetric by �ijð�Þ The signature convention is defined by

diagð�;þ; . . . ;þÞ, and the indices are raised using theinverse metrics g��, �ab, and �ij.

II. MULTIPOLE FORMALISM

It has been shown in [3,4] that every exponentiallydecreasing tensor valued function has a manifestly cova-riant expansion as a series of �-function derivatives.According to this, a function FðxÞ, which is well localizedaround the (pþ 1)-dimensional surface M in aD-dimensional spacetime, is decomposed as

FðxÞ ¼ZM

dpþ1�ffiffiffiffiffiffiffiffi��

p �M

�ðDÞðx� zÞffiffiffiffiffiffiffi�gp

�r�

�M� �

ðDÞðx� zÞffiffiffiffiffiffiffi�gp

�þ � � �

�: (2)

The surface M is defined by the equation x� ¼ z�ð�Þ,where �a are the surface coordinates, and the coefficientsMð�Þ, M�ð�Þ; . . . are the multipole coefficients. Here, wemake use of the surface coordinate vectors u

�a � @z�

@�a , and

the surface induced metric �ab � g��u�a u�b. The induced

metric is assumed to be nondegenerate, � � detð�abÞ � 0,and of Minkowski signature. The same holds for the targetspace metric g��ðxÞ and its determinant gðxÞ. The cova-

riant derivative r� is defined by the Levi-Civita

connection.The expansion (2) is particularly useful for the descrip-

tion of matter which is localized to resemble a p-brane. Ithas been shown in [4] that one may truncate the series in acovariant way in order to approximate the description ofmatter. The truncation after the leading term is calledsingle-pole approximation. In the single-pole approxima-tion, matter is approximated by an infinitely thin brane.In this paper, we are interested in infinitely thin branes,

and therefore, restrict our analysis to the single-pole ap-proximation. The multipole expansion of the stress-energytensor then takes the form

T�� ¼ZM

dpþ1�ffiffiffiffiffiffiffiffi��

pB�� �

ðDÞðx� zÞffiffiffiffiffiffiffi�gp ; (3)

with B��ð�Þ standing for the lowest order multipole coef-ficient. The decomposition (3) is used as an ansatz forsolving the conservation equations (1). This has alreadybeen done in [4], where manifestly covariant p-braneworld-sheet equations and boundary conditions havebeen derived. For later convenience, we shall recapitulatethe basic steps of this derivation.We begin by introducing an arbitrary vector valued test

function f�ðxÞ of compact support. The conservation equa-

tions (1) are then rewritten in the convenient formZdDx

ffiffiffiffiffiffiffi�gp

f�r�T�� ¼ 0; 8f�ðxÞ: (4)

Now, we use the single-pole approximation (3) to modelthe stress-energy tensor. Owing to the compact support off�ðxÞ, we are allowed to change the order of integrations,

and to drop surface terms. Thus, we arrive at

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ZM

dpþ1�ffiffiffiffiffiffiffiffi��

pB��f�;� ¼ 0; (5)

where f�;� � ðr�f�Þx¼z. The fact that this equation holds

for every f�ðxÞ puts some constraints on the coefficients

B��. To find these, we decompose the derivatives of thevector field f�ðxÞ into components orthogonal and parallel

to the surface x� ¼ z�ð�Þ:f�;� ¼ f?�� þ ua�raf�: (6)

Here, ra stands for the total covariant derivative that actson both types of indices:

raV�b � @aV

�b þ ����u

�aV�b þ �b

caV�c;

with ���� and �b

ca the Levi-Civita connections of the

spacetime and the world-sheet, respectively. The orthogo-nal and parallel components are obtained by using theprojectors P?�

� ¼ ��� � u

�a ua� and Pk�

� ¼ u�a ua�. This

way, the Eq. (5) is decomposed into terms proportional tothe independent derivatives f?�� and f�:

0 ¼ZM

dpþ1�ffiffiffiffiffiffiffiffi��

p ½B��f?�� � f�raðB��ua�Þþ raðB��ua�f�Þ�: (7)

Owing to the arbitrariness of f�, we deduce that the three

terms must separately vanish. The first two yield the world-sheet equations

P�?�B

�� ¼ 0; raðB��ua�Þ ¼ 0:

The third is transformed to a surface integral, and yields theboundary conditions

naB��ua�j@M ¼ 0;

where na is the unit boundary normal. The equationP�?�B

�� ¼ 0 tells us that B�� coefficients lack the orthogo-

nal components. Their general form is, therefore, given by

B�� ¼ mabu�a u�b: (8)

Here, mabð�Þ are the residual free coefficients, which playthe role of the effective stress-energy tensor of the brane. Interms ofmab, the brane world-sheet equations are rewrittenas

raðmabu�b Þ ¼ 0; (9)

while the boundary conditions take the form

namabu

�b j@M ¼ 0: (10)

The world-sheet equations (9) and boundary conditions(10) describe the dynamics of an infinitely thin p-branein a D-dimensional pseudo Riemannian spacetime. It isseen that the brane dynamics depends on the type of matterthe brane is made of. This dependence is encoded in themab coefficients. The brane equations (9) imply their co-

variant conservation,

ramab ¼ 0; (11)

thereby justifying the interpretation of mab as the branestress-energy.We want to emphasize that the obtained dynamics has

rather universal character. This is because our derivationrests upon the mere existence of the stress-energy conser-vation equations. This is the virtue of our approach, as it isindependent of the particular action used. On the otherhand, the conservation equations have the weakness ofbeing incomplete, in the sense that they carry much lessinformation than the full set of field equations. As a con-sequence, the derived world-sheet equations contain freecoefficients.In the next section, we shall exploit the brane world-

sheet equations for the study of generic brane-worlds. Weshall use the arbitrariness of the mab coefficients to modelbrane-worlds that accommodate other branes. The induceddynamics is examined, as a model independent test of thebrane-world concept.

III. BRANE-WORLD EXTENDED OBJECTS

In this section, the study of p-branes in curved back-grounds is adapted for the study of brane-worlds. Thebrane world-sheet is now called the brane-world, and thebrane target space is addressed as the bulk. Being typicallyboundary-less, the generic brane-world is not subject toany boundary conditions. In our approach, this leaves uswith the Eq. (9) of the preceding section, as the onlyequation that governs the brane-world dynamics.In what follows, we want to examine the behavior of the

brane-world extended bodies. Geometrically, this is abrane-within-a-brane physical system. We shall model itby making a suitable choice of the brane stress-energytensor mab.Let us first define the vacuum brane-worlds. We can not

simply put mab ¼ 0 in the brane-world equations, as theabsence of matter destroys the brane itself. The most wecan do is to choose the stress-energy tensor mab to have ashigher symmetry as possible. The obvious choice is

mab ¼ T�ab;

where T is a constant known as the brane tension. With thischoice of the brane constituent matter, we obtain theNambu-Goto equations

rað�abu�b Þ ¼ 0; (12)

as the brane-world vacuum equations [8,9]. The Eqs. (12)are seen as the brane-world alternative to the vacuumEinstein equations. In their general form, they lack thedesirable feature of being in agreement with general rela-tivity. Typically, the desirable features of a brane-world aresought in special bulks with special action functionals. Inour approach, we still have the freedom to choose the bulk

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geometry. Hopefully, the appropriate choice of the bulkcould lead to the brane-world with the correct Einsteinlimit. In this paper, however, we shall not do this kind ofresearch. Instead, we shall keep our considerations asgeneral as possible.

In the presence of matter, the brane-world dynamics isgoverned by the Eqs. (9), with mab � T�ab. In what fol-lows, the presence of matter in brane-words will be definedwith respect to the Nambu-Goto background mab ¼ T�ab.Precisely, we shall use the decomposition

mab ¼ T�ab þ�ab (13)

to define the new stress-energy tensor�ab. This new stress-energy tensor is easily checked to be covariantly con-served, and the vacuum equations are defined by its ab-sence, as in general relativity.

Now, we are ready to depart from the vacuum brane-worlds. Specifically, we are interested in brane-worlds thataccommodate other branes. The p-brane-world M, con-sisting of a single q-brane living in the vacuum p-brane, isdefined by the following choice of the p-brane stress-energy tensor:

�ab ¼ZN

dqþ1�ffiffiffiffiffiffiffiffi��

pbab

�ðpþ1Þð�� Þffiffiffiffiffiffiffiffi��p : (14)

Here, the q-brane world-sheet N � M is defined by theequations �a ¼ að�Þ, where �i are coordinates onN , andthe q-brane induced metric is given by �ij ¼ �abv

ai v

bj ,

where vai � @a

@�iare the q-brane coordinate vectors. The

world-sheet tensor babð�Þ is the lowest order coefficient inthe multipole expansion of the stress-energy �abð�Þ. Thedecomposition (14) models the stress-energy tensor �ab tobe zero everywhere except on N � M. It describes amassive q-brane in a vacuum p-brane, and will be used asan ansatz for solving the brane-world equations (9).

Let us start with the conservation equation (11), which isthe brane-world projection of the Eq. (9). In terms of �ab,it retains the simple form

ra�ab ¼ 0: (15)

To solve this equation, we use the ansatz (14), and followthe procedure of Sec. II. The result has the same expectedform. First, the coefficients bab are determined in terms ofthe q-brane coordinate vectors:

bab ¼ �ijvai v

bj :

The residual free coefficients �ijð�Þ play the role of theeffective stress-energy tensor of the q-brane. Second, theq-brane world-sheet equations read

rið�ijvaj Þ ¼ 0; (16)

where ri stands for the total covariant derivative that actson all three types of indices:

riV�bj ¼ @iV

�bj þ ����v

�i V

�bj þ �bcav

ai V

�cj

þ �jkiV

�bk:

Here, the induced connection �ijk is the Levi-Civita con-

nection, and v�i � u

�a va

i are the bulk components ofq-brane coordinate vectors. The world-sheet projection ofthe Eq. (16) yields the conservation equation ri�

ij ¼ 0,thereby justifying the interpretation of �ij as the q-branestress-energy. Finally, the boundary conditions take theform

ni�ijva

j j@N ¼ 0:

These boundary conditions apply to finite open q-branes.In what follows, we shall restrict our considerations toclosed or infinite q-branes. This way, we get rid of theboundary @N , and simplify further exposition withoutcompromising the announced generality.The world-sheet equations (16) describe the dynamics of

a q-brane trapped by a p-brane. In the probe matter ap-proximation (�ij ! 0), the p-brane is not influenced by thepresence of the q-brane, so that the brane-world inducedmetric �ab plays the role of an external field. It is then seenthat trapped probe branes behave the same way as freeprobe branes behave in the (pþ 1)-dimensional spacetimeendowed with the metric equivalent of the brane-worldinduced metric. In particular, probe particles follow geo-desic trajectories relative to the brane-world induced met-ric. Indeed, in the case q ¼ 0, the Eq. (16) reduces to thegeodesic equation rva ¼ 0. If the particle world line isparametrized by the proper distance s, the geodesic equa-tion takes the standard form

d2a

ds2þ �a

bc

db

ds

dc

ds¼ 0: (17)

The Eq. (17) is an important achievement in the modelindependent testing of the brane-world concept. It showsthat the brane-world test particles behave the same way asin general relativity. This justifies the commonly acceptedinterpretation of the brane-world induced metric as themetric of the observed spacetime.Let us now consider more general q-branes, which are

able to influence the brane-world geometry. In this case,the Eq. (16), although still valid, loses its predictive power.Indeed, the brane-world metric �ab is influenced by theq-brane itself, and can not be considered an external filedany more. The additional information that we need isobtained by solving the full set of brane-world equations(9). Again, we use the decomposition (13), and the ansatz(14), to rewrite the Eq. (9) in the form

Trað�abu�b Þ ¼ �Z

dqþ1�ffiffiffiffiffiffiffiffi��

pF� �ðpþ1Þð�� Þffiffiffiffiffiffiffiffi��

p :

(18a)

The force F� on the right-hand side is defined by the

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equation

rið�ijv�j Þ ¼ F�: (18b)

The Eqs. (18) describe the nonvacuum dynamics of the(pþ 1)-dimensional brane-world M relative to theD-dimensional bulk. This is a complex brane-world con-sisting of the p-brane accommodating a q-brane. The firstEq. (18a) differs from the vacuum equations (12) by thepresence of the nonvanishing force on the right-hand side Itdescribes the motion of the p-brane as dragged by theq-brane. The second Eq. (18b) governs the dynamics ofthe q-brane relative to the bulk. The force on the right-handside stems from the p-brane in which the q-brane isembedded.

The force F� is a bulk vector which is orthogonal to thebrane-world,

ua�F� ¼ 0; (19)

as seen by applying (16) to the brane-world projection ofthe Eq. (18b). Owing to this orthogonality condition, theq-brane moves freely through the brane-world (Eq. (16)),but experiences the orthogonal force in the bulk[Eq. (18b)]. The p-brane itself is influenced by the�-function force located at the position of the q-brane[Eq. (18a)].

In what follows, we shall be concerned with the geo-metric interpretation of the right-hand side of the Eqs. (18).This is because the force F� is an undetermined quantity inthe above equations, and the only way to determine it is tocompare the �-function term on the right-hand side of theEq. (18a) with the geometric term on the left-hand side.The type of equation with the �-function source is afamiliar type of equations. It is conventionally solved thefollowing way. First, one uses the fact that the Eq. (18a)reduces to the free Eq. (12) everywhere on M nN . ThesurfaceN is of zero measure onM. Therefore, by solvingthe free equation (12), one finds the solution of (18a) up tothe boundary conditions to be imposed on N � M. Inwhat follows, we shall demonstrate how such boundaryconditions emerge in the q ¼ p� 1 case.

Before we proceed, let us make a remark about thesmoothness of the brane-world. It is seen from theEq. (18a) that the generic brane-world M can not be aneverywhere smooth surface. Indeed, if M is everywheresmooth, the left-hand side of (18a) can not have infinitediscontinuities, and therefore, the force F� must vanish.This implies that the two branes are free, which contradictsthe assumption that the q-brane is trapped by the p-brane.Thus, the general solution of the brane-world equations(18) is expected to lack the smooth structure.

IV. MASSIVE BRANE IN THE VACUUM BRANE-WORLD

In this section, we shall reconsider the brane-worlddynamics of the preceding section in a slightly different

manner. Instead of using the ready made p-brane equation(9), we shall start from scratch, as in Sec. II. This way, weshall be able to weaken the assumptions under which theEq. (9) has been derived, and adapt the procedure for thetreatment of complex branes. The new approach does notcompromise the results of the preceding sections. It issimply better suited for the description of motion relativeto the bulk, as opposed to the approach of Sec. III, which isbetter suited for the treatment of motion relative to thebrane-world. In particular, we shall obtain the Eqs. (18) inthe form which is free of the �-function source, but weshall not be able to easily derive the Eq. (16). This is whywe need both approaches.We start with the conservation equations (1), but this

time the stress-energy T�� is modeled so that the brane-within-a-brane structure is explicitly seen. Thus, instead of(3), we use the single-pole expansion of the form

T�� ¼ZMnN

dpþ1�ffiffiffiffiffiffiffiffi��

pB�� �

ðDÞðx� zð�ÞÞffiffiffiffiffiffiffi�gp

þZN

dqþ1�ffiffiffiffiffiffiffiffi��

pb�� �

ðDÞðx� zðÞÞffiffiffiffiffiffiffi�gp : (20)

As before, the p-brane world-sheet M is defined by x� ¼z�ð�Þ, and the q-brane world-sheet N is defined by �a ¼að�Þ. The corresponding multipole coefficients are de-noted by B��ð�Þ and b��ð�Þ, respectively. The expansion(20) describes a pair of branes, one of which belongs to theother. It can be obtained from the expansion (3) by allow-ing the coefficients B�� to be singular, and the surface Mto lack smoothness on N � M.Now, we proceed by using the ansatz (20) to solve the

conservation equations (1). The procedure is the same as inSec. II. First, we multiply the Eq. (1) with the vector valuedtest function f�ðxÞ of compact support. Then, we decom-

pose the derivativesr�f� into components orthogonal and

parallel to the surface M [the Eq. (6)], and perform thenecessary integrations. As a result, we obtain the equation

0 ¼ZMnN

dpþ1�ffiffiffiffiffiffiffiffi��

p ½B��f?�� � f�raðB��ua�Þ

þ raðB��ua�f�Þ� þZN

dqþ1�ffiffiffiffiffiffiffiffi��

pb��r�f� (21)

which holds true for every choice of the test function f�.

This means that terms proportional to the independentderivatives of f� must separately vanish. To make use of

this fact, we first divide the full integration domainM intothe boundary N , and the interior M nN . We see that,apart from the manifest boundary integral in the last termof the Eq. (21), there is a total divergence in the firstintegral, which also gives rise to a boundary term. Owingto the arbitrariness of the function f�, we deduce that the

two interior terms of (21) must separately vanish. Theresulting equations P?�

�B�� ¼ 0 and raðB��ua�Þ ¼ 0 co-

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incide with the corresponding equations of Sec. II. Theyyield the p-brane equations of the form

B�� ¼ mabu�a u�b; raðmabu

�b Þ ¼ 0: (22)

The residual stress-energy tensor mab is smooth every-where on M nN , but may be ill defined on N . Thevacuum equations are defined by the choice mab ¼ T�ab.Now, we are left with the boundary equation

0 ¼ZN

dqþ1�ffiffiffiffiffiffiffiffi��

pb��r�f�

þ lim@M!N

Z@M

dp�ffiffiffiffiffiffiffi�h

pB��n�f�: (23)

The Eq. (23) is obtained by applying the Stokes theorem tothe function possessing a possible singularity onN � M.The boundary @M stems from the hole cut around thesurface N . As we mentioned earlier, the brane-world Mis supposed to have no other boundaries. The boundary

@M is defined by the equation �a ¼ að�Þ, where �ðiÞ,ðiÞ ¼ 0; 1; . . . ; p� 1, parametrize the surface @M. Theinduced metric on @M is defined by

hðiÞðjÞ ¼ �abwaðiÞw

bðjÞ;

where waðiÞ � @a

@�ðiÞ are the surface coordinate vectors. The

bulk components of the unit boundary normal na aredefined by n� ¼ nau

�a .

In what follows, we shall continue the analysis of theboundary equation (23) by separating the case q ¼ p� 1from the case q < p� 1. We shall demonstrate that em-bedded codimension-1 branes violate the brane-worldsmooth structure, but preserve the continuity. The lower-dimensional branes (q < p� 1), on the other hand, violatethe very continuity.

A. Codimension-1 branes

Let us consider the case q ¼ p� 1 in more detail. First,note that the p-dimensional world-sheet N now dividesthe (pþ 1)-dimensional brane-world M into two pieces(see the example in Fig. 1). The boundary @M consists oftwo surfaces, located on the opposite sides of N . In thelimit @M ! N , we shall refer to them as N þ and N �.The surfacesN þ andN � are infinitesimally close toN ,

so that the coordinates �i are naturally chosen to play the

role of �ðiÞ on both sides of N . At the same time, theinduced metrics ½hðiÞðjÞ�þ and ½hðiÞðjÞ�� both approach �ij.

As a consequence, the second integral of the Eq. (23) isrewritten asZ

Ndp�

ffiffiffiffiffiffiffiffi��p ½ðB��n�Þþ þ ðB��n�Þ��f�:

The subscriptsþ and� tell us which side ofN is used forthe evaluation of the discontinuous integrand. As we cansee, the coefficient B��, and the normal n� are the onlyquantities that may have a jump on N � M. The testfunction f�, and the induced metric �ij are continuous.

In the first integral of the Eq. (23), we shall decomposethe derivative r�f� into components orthogonal and par-

allel toN . To this end, we use the projectors p�?� ¼ �

�� �

v�i v

i� and p

�k� ¼ v

�i v

i� to obtain

r�f� ¼ f>�� þ vi�rif�:

In terms of the independent derivatives f>�� and f�, the first

integral in (23) is rewritten asZN

dp�ffiffiffiffiffiffiffiffi��

p ½b��f>�� � f�riðb��vi�Þ�:

The divergence termriðb��vi�f�Þ is dropped, as it reduces

to a boundary integral that vanishes on the account ofcompact support of f�. This is a consequence of the

assumed infiniteness of the q-brane, which takes theboundary @N to spatial infinity.Now, we can return to the full boundary equation (23).

Using the above results, it is rewritten in the form

0 ¼ZN

dp�ffiffiffiffiffiffiffiffi��

p fb��f>�� � ½riðb��vi�Þ � ðB��n�Þþ

� ðB��n�Þ��f�g; (24)

which holds for every test function f�ðxÞ. This means that

the terms proportional to the independent derivatives f>��and f� must separately vanish, which leads to

b�� ¼ �ijv�i v

�j ;

rið�ijv�j Þ ¼ ðnamabu�b Þþ þ ðnamabu�b Þ�: (25)

The Eqs. (22) and (25) govern the dynamics of thecomplex system consisting of two branes. The first is ap-brane characterized by the stress-energy tensormab. Thesecond is a (p� 1)-brane trapped by the first brane. Thesecond brane is characterized by its own stress-energytensor �ij. By inspecting the form of these equations wesee that they describe a motion relative to the bulk. TheEq. (25) describes the motion of the (p� 1)-brane underthe influence of the p-brane in which it is embedded. At thesame time, it plays the role of the boundary conditions forthe Eq. (22). The Eq. (22), subject to the boundary con-

==

n+n-

FIG. 1. The case p ¼ 1, q ¼ 0, for particle on the string.

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ditions (25), governs the dynamics of the p-brane asdragged by the embedded (p� 1)-brane.

In this paper, we examine the behavior of extendedbodies in vacuum brane-worlds. As explained in Sec. III,this means that the brane-world stress-energy is defined bymab ¼ T�ab. If this is the case, the Eqs. (22) and (25) yield

raua� ¼ 0; rið�ijv

�j Þ ¼ Tðn�þ þ n��Þ: (26)

The Eqs. (26) provide a complete description of the p-dimensional vacuum brane-world under the influence ofthe embedded massive (p� 1)-brane. By comparing themwith the results of Sec. III, we find the following. First, theEq. (26) implies the covariant conservation of the stress-energy �ij, in complete agreement with Sec. III. Second,the comparison with the Eq. (18b) provides a geometricinterpretation of its source term:

F� ¼ Tðn�þ þ n��Þ: (27)

As we can see, the force F� depends on the brane-worldtension, but also on the intersection angle between its twosmooth pieces. If the brane-world M is everywheresmooth, the two normals cancel each other, and the forcebetween the branes disappears. The two branes becomefree despite the assumption that they are attached to eachother. In what follows, we shall address this situation astrivial. In general, the presence of a massive (p� 1)-braneviolates the brane-world’s smooth structure.

Before we continue our general analysis, let us brieflycomment on the intriguing relation between the Eqs. (26)and (16). The basic difference between these two equationsis that (16) describes a motion relative to the brane-world,while (26) refers to the motion relative to the bulk. Thelatter carries the complete information on the brane dy-namics, and therefore, we should be able to derive (16)from (26). This is usually done by projecting (26) to Mwhile using the fact that the force on the right-hand side isorthogonal toM. However, ourM is not smooth, and doesnot have a well defined normal. Although the force (27) is awell defined vector, we can not rigorously prove its or-thogonality toM. What one should do is to think ofM asan approximation of an everywhere smooth surface. Then,the sum n

�þ þ n�� is seen as an average normal, and the

sum u�aþ þ u

�a� as an average tangent vector. The projec-

tion of the boundary equation (26) to the average tangentvectors can then be shown to imply the average version ofthe Eq. (16).

Finally, let us note that the Eqs. (26) can easily begeneralized to describe any number of intersecting branes.In such situations, the first part of (26) remains the same,and applies to every smooth piece in the system of inter-secting branes. The second part is modified by the inclu-sion of all the normals that characterize the intersection.For example, three vacuum branes with identical tensions,and a massive common intersection are described by

raua� ¼ 0; rið�ijv

�j Þ ¼ Tðn�1 þ n

�2 þ n

�3 Þ:

In the absence of matter on the intersection (�ij ¼ 0), theboundary equation takes the form n�1 þ n�2 þ n�3 ¼ 0.

B. Point particles

In this subsection, we shall examine the case q < p� 1,and demonstrate how higher codimension branes violatethe continuity of the brane-world. In particular, we shallconcentrate on point particles (q ¼ 0). This way, we sim-plify the exposition without losing the basic features of thegeneric case.Let us first note that, as opposed to the q ¼ p� 1 case,

the 1-dimensional world line N does not violate theconnectedness of the brane-world M. The p-dimensionalboundary @M, that appears in (23), has the topology of acylinder R� Sp�1 (see the example in Fig. 2). In the limit@M ! N (now seen as the limit Sp�1 ! 0), the parame-

trization of @M is naturally chosen in the form �ðiÞ ¼ð�;!hiiÞ, hii ¼ 1; . . . ; p� 1. Here, � is the parameter of

the world line N , and !hii are identified with the angularcoordinates that parametrize the sphere Sp�1. In whatfollows, we shall restrict to reference frames in which the! and � coordinate lines are mutually orthogonal, as ex-pressed by the gauge condition hð0Þhii ¼ 0. In such refer-

ence frames, the metric determinant h � detðhðiÞðjÞÞ is

decomposed as

h ¼ � detðhhiihjiÞ � �hhi;with �ð�Þ standing for the induced metric onN . Thus, thefirst integral of the Eq. (23) is transformed toZ

Nd�

ffiffiffiffiffiffiffiffi��p

f�ISdp�1!

ffiffiffiffiffiffiffihhi

pB��n�

because the test function f� can be treated as a function of

the sole � coordinate. Indeed, the function f� is every-

where smooth, and the ! dependence of smooth functionson @M becomes trivial in the limit S ! 0.The second integral of the Eq. (23) is transformed by the

decomposition of the derivative r�f� into components

orthogonal and parallel to N . This is done the same wayit was done in the preceding subsection, resulting in the

ξ

ξ ξ

0

1 2

==

n

n

n

FIG. 2. The case p ¼ 2, q ¼ 0, for particle on membrane.

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same expression. The whole Eq. (23) is then rewritten as

0 ¼ZN

d�ffiffiffiffiffiffiffiffi��

p fb��f>�� � f�½riðb��v�Þ

� limS!0

ISdp�1!

ffiffiffiffiffiffiffihhi

pB��n��g; (28)

and holds for every test function f�ðxÞ. Thus, the terms

proportional to the independent derivatives f>�� and f�must separately vanish, which leads to the world-lineequations

b�� ¼ �v�v�;

rð�v�Þ ¼ limS!0

ISdp�1!

ffiffiffiffiffiffiffihhi

pu�a mabnb:

(29)

The residual coefficient �ð�Þ is interpreted as the particlemass, while mab stands for the effective stress-energy ofthe brane-world. In vacuum brane-worlds (mab ¼ T�ab)the second Eq. (29) reduces to

rð�v�Þ ¼ TlimS!0

ISdp�1!

ffiffiffiffiffiffiffihhipn�: (30)

It implies the conservation of the particle mass � along theparticle world line.

Now, we want to evaluate the force on the right-handside of the Eq. (30). It stems from the interaction thatprevents the particle from leaving the p-brane, and isnaturally expected to be orthogonal to the brane-world.As in the general case, it is immediately seen that this forcevanishes if the brane-world is everywhere smooth. Indeed,on smoothM, the unit normals n� cancel each other in thelimit S ! 0. This leaves us with singular brane-worldswhose smooth structure is violated at the position of theparticle world line. In what follows, we shall examine whatkind of singularity we are dealing with.

Let us consider the simplest case of violated smooth-ness, but preserved continuity. For simplicity, we can thinkof surfaces with conical singularities. If this is the case, theforce on the right-hand side of the Eq. (30) is proven to bezero. Indeed, the bulk components of the spacelike unitvectors n� are finite quantities, and remain finite in thelimit S ! 0. In the inertial rest frame, they satisfy jn�j �1, and therefore,

jI

dp�1!ffiffiffiffiffiffiffihhip

n�j �I

dp�1!ffiffiffiffiffiffiffihhip

:

The right-hand side of this inequality is nothing else but thevolume of the boundary sphere Sp�1. In the limit S ! 0,this volume goes to zero, and we obtain

limS!0

ISdp�1!

ffiffiffiffiffiffiffihhipn� ¼ 0: (31)

Thus, the force on the right-hand side of the Eq. (30)vanishes even in nonsmooth brane-worlds. In generic situ-ations, this contradicts the fact that the particle is attached

to the brane. We can say that generic point particles cannot live in continuous brane-worlds.The obtained result is an important achievement in test-

ing the brane-world concept. It gives us instructions of howto search for point particles in a brane-world. According tothese, point particles are found in places where singularityof the brane-world is worse than that of violated smooth-ness. In fact, the example of the next section suggests thatpoint particles violate the manifold structure itself. Conicalsingularities, as candidates for point particles, areexcluded.By comparing our results with literature, we have dis-

covered that some codimension-2 brane-world models arereported to possess only conical singularities in the bulk.This seems different from what one might expect on thebasis of our work. Namely, if the bulk in these models isviewed as a surface in some higher-dimensional back-ground, the structure under consideration seems to bereduced to a brane-within-a-brane system of the type con-sidered in this paper. If it is true, we are facing a seriouscontradiction. In what follows, we shall demonstrate that itis not so.First, we want to emphasize that our bulk metric g�� is

everywhere well defined. This follows from the assumedprobe character of the brane-world matter. Our startingEq. (1) is valid irrespective of the probe matter approxi-mation, but it loses its predictive power without this as-sumption. Indeed, without the probe matter approximation,the bulk metric g�� is influenced by the brane-world itself,

and can not be considered an external filed. To deal withthis situation, one needs the full set of field equations, inwhich case the analysis loses its model independent char-acter. As opposed to this, the brane-world models found inliterature almost exclusively deal with singular bulk ge-ometries. The metric singularity is a consequence of theimposed field equations, and therefore, is fully modeldependent. One could try to view the singular bulk as asingular surface embedded in some regular higher-dimensional background. The obtained brane-within-a-brane structure, however, is not of the type considered inthis paper. Indeed, the new background is just an auxiliarybulk with no physical meaning, and there is no a-priorireason that its metric obeys the conservation equation (1).As a final note in this section, let us elaborate on the

probe matter approximation we use in this paper. First, wemust emphasize that our probe matter approximation ex-clusively refers to matter relative to the bulk. This meansthat the parameters that characterize the brane-world mat-ter, such as tension or particle mass, are assumed weakenough not to influence the bulk geometry, but nothing isassumed about the relation between the parameters them-selves. For example, we can assume � � T to define probeparticle relative to the brane-world, or � T to defineparticle which significantly influences the brane-worldgeometry. In both cases, however, we additionally assume

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T ! 0 and � ! 0 to ensure the probe character of thebrane-world relative to the bulk. This way, we have beenable to derive how embedded objects modify the brane-world geometry, irrespectively of the fact that the brane-world’s influence on the bulk has been completely ne-glected. In our approach, the brane-world’s influence onthe bulk can be studied by viewing the bulk as a physicalbrane in some higher-dimensional background. The brane-world extended objects are then seen as complex structuresthat involve three or more physical branes.

V. STATIC AXISYMMETRIC MEMBRANE

As an example, we shall examine the behavior of asimple membrane embedded in flat 4-dimensional bulk.The bulk coordinates are denoted by x� (� ¼ 0, 1, 2, 3),and the bulk metric is chosen to be Minkowskian (g�� ¼���). To keep the analysis simple, we shall restrict to a

static, axially symmetric membrane, and parametrize itsworld-sheet with polar coordinates �a (a ¼ 0, 1, 2). Inwhat follows, we shall use the conventional notation

�0 ¼ �; �1 ¼ r; �2 ¼ ’:

The membrane world-sheet is defined by the equationx� ¼ z�ð�Þ, which, in the case under consideration, issearched for in the form

z0 ¼ �; z1 ¼ rcos’; z2 ¼ r sin’; z3 ¼ hðrÞ:(32)

The Eqs. (32) are used as an ansatz to solve the world-sheetequations (9), and boundary conditions (10). The mem-brane altitude hðrÞ is the only variable in the theory, and thecoefficients mab are considered arbitrary functions of r.Their independence of � and ’ is a consequence of theassumed symmetries.

We start with the calculation of the coordinate vectorsu�a . Their nonvanishing components are found to be

u00 ¼ 1; u11 ¼ cos’; u12 ¼ �r sin’;

u31 ¼ h0; u21 ¼ sin’; u22 ¼ r cos’;(33)

where ‘‘prime’’ denotes derivative in the r direction. Then,the induced metric �ab � ���u

�a u�b is shown to have the

diagonal form

�ab ¼�1 0 00 1þ h02 00 0 r2

0@

1A: (34)

We search for the solution of the world-sheet equations (9)in two steps. First, we analyze the conservation equationsram

ab ¼ 0, and succeed in rewriting them in a moreconvenient form. Then, the world-sheet equations (9) arereduced to mabrau

�b ¼ 0. The � ¼ 0 equation is identi-

cally satisfied, while the remaining three equations areshown to be equivalent to each other. The four differential

equations thus obtained read

ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

prm01Þ0 ¼ ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

pr3m12Þ0 ¼ 0;

½ð1þ h02Þrm11�0 ¼ r2m22; h00m11 þ h0rm22 ¼ 0:

These equations can not be solved with the generic form ofthemab coefficients. It is, however, possible to perform oneintegration, and simplify the expressions. This way, theabove equations are reduced to the lower order differentialequations of the form

rm01 ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

p ; r3m12 ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

p ; (35a)

rm11 ¼ cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih02ð1þ h02Þp ; (35b)

r2m22 ¼ c

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

h02

s �0: (35c)

The parameters a, b, and c are the integration constants tobe specified later.The Eqs. (35) describe a stationary, axisymmetric mem-

brane in 4-dimensional Minkowski spacetime. As we cansee, only one unknown variable (the membrane altitudehðrÞ) is subject to four equations. As a consequence, themembrane stress-energymab is constrained, and can not befreely chosen. In what follows, we shall be concerned withasymptotically vacuum membranes. This means that themembrane constituent matter �ab, as defined by theEq. (13), approaches zero in the limit r ! 1. The regionof nonvanishing �ab is assumed to be highly localizedaround r ¼ 0. What we are particularly interested in isthe limit of pointlike matter.

A. Vacuum membranes

Let us first consider the vacuum case. It is defined by thechoicemab ¼ T�ab, where T is a constant referred to as themembrane tension. By inserting this into (35), we imme-diately obtain a ¼ b ¼ 0. This leaves us with theEqs. (35b) and (35c), which turn out to be equivalent toeach other. Indeed, if the integration constant c is redefinedas c ¼ T‘, the two remaining equations are reduced to

h0 ¼ ‘ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � ‘2

p ; (36)

where the redefined integration constant ‘ has the dimen-sion of length. This differential equation is readily inte-grated to yield

h

‘¼ ln

�r

‘þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

‘2� 1

s �: (37)

Here, the constant ‘ is chosen to be strictly positive (‘ >0), and the new integration constant is, for convenience, setto zero. The shape of the static, axisymmetric vacuummembrane, as defined by (37), is shown in Fig. 3. As we

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can see, the membrane is boundaryless, which means that itis not subject to the boundary conditions (10). If one,however, removes a piece of the membrane, a boundarywill be created. Such a membrane will still be a solution ofthe world-sheet equations, but will fail to satisfy the bound-ary conditions.

The vacuum membrane-world defined by (37) looks liketwo universes joined by a tunnel. In the limit ‘ ! 0, thetunnel width goes to zero, the two universes overlap, andthe solution approaches h ¼ 0. The plane h ¼ 0 is referredto as the trivial solution of the vacuum world-sheet equa-tions. In what follows, we want to examine how localizedmatter changes the shape of the vacuum membrane.

B. Nonvacuum membranes

We begin the analysis of nonvacuum membranes bychoosing the asymptotic behavior of its stress-energy. Forthat purpose, we make use of the decomposition mab ¼T�ab þ�ab, and choose �ab to decrease exponentiallyfast as r ! 1. This can be done in a variety of ways,each serving our purpose perfectly well. The resultingnonvacuum membrane significantly differs from the vac-uum (37) only in regions close to r ¼ 0. The constructionof such �ab is quite simple, and we shall not elaborate onthis any further. Instead, we shall concentrate on the be-havior near r ¼ 0.

Now, we are facing a serious problem. What we want tohave is a topologically trivial membrane-world with anembedded massive object. We have already seen that ourvacuum membrane is topologically nontrivial. Indeed, it ischaracterized by the presence of a tunnel of radius ‘. Theonly way to get rid of the tunnel is to overlap it with matterby choosing the size of the embedded massive object to beof the order ‘. In the point particle limit, the localization ofmatter must be accompanied by the simultaneous ‘ ! 0limit.

Let us now proceed with the analysis of the world-sheetequations (35). We immediately see that the assumedasymptotics implies a ¼ b ¼ 0, thereby reducing theEqs. (35a) to m01 ¼ m12 ¼ 0. In the static case, the com-ponent m02 must also be zero. This leaves us with the fullydiagonal mab. The remaining Eqs. (35b) and (35c) aresolved in two steps. First, the coefficient m11 is freely

chosen, and the Eq. (35b) is solved for hðrÞ. Then, thecoefficient m22 is calculated from the Eq. (35c). In whatfollows, we shall analyze how the choice of m11 influencesthe behavior of hðrÞ.Let us start with the reminder that we want to describe a

massive particle on the vacuummembrane. This leads us tothe Eqs. (13) and (14) as a guide for modeling the stress-energy mab. In the case under consideration, the Eq. (14)reduces to

�ab ¼ bab�ð�1Þ�ð�2Þffiffiffiffiffiffiffiffi��

p ;

with the coefficients bab some finite constants. In regularcoordinates, this implies the finiteness of the integralRd2�

ffiffiffiffiffiffiffiffi��p

�ab. The polar coordinates, however, are ill

defined in r ¼ 0, and the assumption of finiteness shouldbe appropriately redefined. For example, the component�22 should be replaced with r2�22 in order to make theabove integral finite. The condition of finiteness for the�11

component, on the other hand, remains unchanged. Afterperforming the ’ integration, it reduces to

2�Z 1

0dr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

pr�11 ¼ b11 ¼ finite: (38)

The purpose of the condition (38) is to prevent physicallyunrealistic choices of the membrane constituent matter. Inparticular, the particle mass and tension are supposed to befinite.

C. Smooth membranes

If the membrane is smooth, the power expansion of thefunction hðrÞ lacks the linear term,

hðrÞ ¼ þ �r2 þ � � � ;and implies h0 � r in the vicinity of r ¼ 0. The Eq. (35b)then tells us that �11 � 1=r2, and therefore,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

pr�11 � 1=r. This is how the integrand in the

condition (38) behaves near r ¼ 0. With such a behaviorat the lower bound of integration, the integral logarithmi-cally diverges, and the condition (38) fails. We concludethat static, axisymmetric nonvacuum membranes made ofrealistic matter can not be everywhere smooth. This con-clusion agrees with the predictions of Sec. IV. There, wehave exclusively considered infinitely thin objects em-bedded in other branes. Our present conclusion, however,holds for smeared distribution of matter, too. This is aconsequence of the adopted restriction to static and axiallysymmetric membranes. Of course, the point particle limitcan not cure the violation of smoothness.

D. Membranes with conical singularity

Let us now consider continuous, but not necessarilysmooth membranes. In this case, the power expansion ofthe function hðrÞ has a nonzero linear term,

x3

x1

x2

FIG. 3. Static, axisymmetric vacuum membrane.

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hðrÞ ¼ þ �rþ � � � ;so that h0 � 1 in the vicinity of r ¼ 0. The behavior of �11

is calculated from (35b). The resulting �11 � 1=r is then

shown to implyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

pr�11 � 1 in the vicinity of r ¼ 0.

With such a behavior at the lower bound of integration, theintegral in (38) converges to a finite value. This is because

the behavior offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h02

pr�11 at the upper bound of inte-

gration is also good, owing to the assumed exponentiallyfast decrease of �11. Thus, we conclude that static, axi-symmetric membranes made of realistic matter can haveconical singularities. What we still have to check is howthese singularities develop in the point particle limit. Let usstart with specifying the behavior of �11 in the form

�11 � �

rwhen r ! 0:

The constant � is freely chosen, and determines the valueof the function h0ðrÞ in r ¼ 0. Indeed, utilizing theEq. (35b), we find

h0ð0Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4þ

�T‘

�

�2

s� 1

2

vuut: (39)

As we can see, the value of h0 in r ¼ 0 depends on the ratioT‘=�. In the point particle limit, both � and ‘ go to zerosimultaneously. There are three possible outcomes. First, ifT‘ � �, the membrane slope h0ð0Þ goes to zero in thepoint particle limit. At the same time, the membranebecomes matter-free outside r ¼ 0. In this area, h0ðrÞ isgiven by the vacuum solution (36), and approaches zero inthe limit ‘ ! 0. Thus, h0ðrÞ ! 0 everywhere, leading tothe trivial solution h ¼ 0, as shown in Fig. 4. The solid lineshows the nonvacuummembrane in the point particle limit,whereas dashed line depicts ‘ ! 0 limit of the vacuumsolution. Second, if T‘� �, the membrane slope h0ð0Þremains finite in the point particle limit. However, by thesame arguments as above, h0ðrÞ ! 0 for all r � 0. Settingthe integration constant to zero, we end up with hðrÞ ¼ 0everywhere outside r ¼ 0, and, owing to the finite value ofh0ð0Þ, we obtain hð0Þ ¼ 0, too. This is again the trivialsolution h ¼ 0. The limiting process that leads to it isdepicted in Fig. 4. Finally, let us consider the case T‘ �, which leads to h0ð0Þ ¼ 1 in the point particle limit.Again, the membrane slope h0ðrÞ goes to zero everywhereoutside r ¼ 0, which leads to hðrÞ ¼ 0 in this area. Thistime, however, the point particle limit of the membranealtitude hð0Þ can not be proven zero. Indeed, the infinitevalue of the membrane slope in r ¼ 0 suggests that hð0Þdoes not vanish, leading to the structure depicted in Fig. 5.

It is not a manifold, and is described as a string attached tothe membrane.To summarize, we have considered how objects of finite

mass and tension influence the shape of a static, axisym-metric membrane. Our results are in complete agreementwith the general predictions of the preceding sections. Inparticular, we have demonstrated the violation of mem-brane smoothness and continuity in the point particle limit.We have considered two most promising cases:�11 � 1=r2

and �11 � 1=r. It is not difficult to prove that other matterdistributions have even worse outcomes. The Table I showsthe behavior of �11 and the corresponding hðrÞ in thevicinity of r ¼ 0. As we can see, the more divergent �11

component is, the more smooth is the membrane, but thecondition (38) is always violated. On the other hand, thesmoother �11 in r ¼ 0, the more divergent is the mem-brane, but the condition (38) is satisfied. In this respect, thecases �11 � rn with n � 0 are not worse than �11 � 1=r.All these, however, also lead to the nonmanifold structuredepicted in Fig. 5. We may speculate that massive pointparticles on a brane are generally found in places wherestrings are attached to the brane.

VI. CONCLUDING REMARKS

In this paper, we have considered extended objectsembedded in a generic brane-world. We have not specifiedany particular action. Instead, we have utilized the univer-sal stress-energy conservation law, as a model independenttool for testing the brane-world concept. The basic assump-tion used throughout the paper is the regularity of the bulkmetric. According to this, the brane-world matter has beenassumed to have probe character relative to the bulk. At thesame time, no such assumption has been adopted for thematter relative to the brane-world.The results obtained by the analysis along these lines are

summarized as follows. First, the dynamics of the brane-world’s extended objects relative to the brane-world hasbeen derived in the probe matter approximation. It has beendemonstrated that the geometry these objects feel is theFIG. 4. The trivial point particle limit.

FIG. 5. The nontrivial point particle limit.

TABLE I. Membrane behavior near r ¼ 0.

�11 h b11 �11 h b11

1=r3 r3 1 r lnr finite

1=r2 r2 1 r2 1=ffiffiffir

pfinite

1=r r finite r3 1=r finite

1ffiffiffir

pfinite

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brane-world induced geometry. In particular, point parti-cles are shown to follow geodesics with respect to theinduced metric. This result confirms the common beliefthat the induced metric determines the observed geometry.Second, the influence of the embedded extended objects onthe brane-world dynamics relative to the bulk has beenexamined. Throughout the paper, this complex dynamicalsystem has been addressed as a brane-within-a-brane sys-tem. It has been shown that embedded codimension-1branes necessarily violate the brane-world’s smooth struc-ture. The result has been generalized to any number ofintersecting branes. Finally, the influence of higher codi-mension branes has been considered through the behaviorof embedded point particles. It has been demonstrated that,in addition to smoothness, massive particles also violatethe brane-world’s continuity. As an example, the static,axisymmetric membrane in Minkowski bulk has been an-alyzed in detail. In this case, the localization of matter in apoint has been shown to violate the very manifold structureof the membrane. The resulting structure appears to be astring attached to the membrane.

In conclusion, the brane-world concept proves to beburdened by the existence of massive point particles. Oneway to attack the problem is to abandon the commonlyaccepted continuity of the generic brane-world. If so, thenovel complex structures, like strings attached to the brane,should be examined. Alternatively, one may consider sin-gular bulks to see if they can prevent the development ofbrane-world discontinuities. The analysis in this directionwould, most likely, be model dependent. In any case, it is along way that leads to an acceptable brane-world model.

At the end, let us say something about prospects of ourresearch. First, the study of vacuum branes, as defined bythe Eq. (12), deserves more attention. In particular, we plan

to search for the bulk geometry that ensures the correctEinstein limit. A promising option is the analysis of sin-gular bulks. In this respect, it would be interesting to applyour analysis to the example of Refs. [1,2]. The reason thatthe Randall-Sundrum model has not been considered al-ready in this manuscript is its complexity. Namely, the bulkin the Randall-Sundrum model is singular, as opposed toregular bulks considered in this paper. The only way toconsider singular bulks within our approach is to viewthem as branes in a new (regular) bulk. The problem isthat the solution of our equations crucially depends on thechoice of the new bulk. As it turns out, simple choices,such as flat bulks, do not lead to the Randall-Sundrumsolution. What one should do is to make a comprehensiveanalysis of the problem, but this lies outside the scope ofthis work. Another interesting application of our method isthe analysis of the more realistic static, spherically sym-metric 3-brane, which is shown to share many features withthe membrane example of Sec. V. In particular, its vacuumsolution is also characterized by the tunnel connecting twouniverses. The observer that lives in one of the universesmay be unaware of the existence of the other. The tunnelmay seem as a massive object to him, and we plan to checkon this by performing an exhaustive Hamiltonian analysis.Finally, we should mention the structure of a string at-tached to the brane. It proved to be relevant for the de-scription of the brane-world point particles, and should beseparately investigated.

ACKNOWLEDGMENTS

This work is supported by the Serbian Ministry ofScience and Technological Development, under ContractNo. 141036.

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