Cosmic strings in a braneworld theory with metastable gravitons

  • Published on
    28-Mar-2017

  • View
    212

  • Download
    0

Transcript

  • PHYSICAL REVIEW D 66, 043509 ~2002!Cosmic strings in a braneworld theory with metastable gravitons

    Arthur Lue*Department of Physics, New York University, New York, New York 10003

    ~Received 11 December 2001; published 15 August 2002!

    If the graviton possesses an arbitrarily small~but nonvanishing! mass, perturbation theory implies thatcosmic strings have a nonzero Newtonian potential. Nevertheless in Einstein gravity, where the graviton isstrictly massless, the Newtonian potential of a cosmic string vanishes. This discrepancy is an example of thevan DamVeltmanZakharov~VDVZ ! discontinuity. We present a solution for the metric around a cosmicstring in a braneworld theory with a graviton metastable on the brane. This theory possesses those features thatyield a VDVZ discontinuity in massive gravity, but nevertheless is generally covariant and classically self-consistent. Although the cosmic string in this theory supports a nontrivial Newtonian potential far from thesource, one can recover the Einstein solution in a region near the cosmic string. That latter region grows as thegravitons effective linewidth vanishes~analogous to a vanishing graviton mass!, suggesting the lack of aVDVZ discontinuity in this theory. Moreover, the presence of scale dependent structure in the metric may haveconsequences for the search for cosmic strings through gravitational lensing techniques.

    DOI: 10.1103/PhysRevD.66.043509 PACS number~s!: 98.80.Cqs ar,lde

    essle

    seevititfo

    Vmg-toata

    bitd

    nthicthsacee

    ractuchn beodeiton

    ist theel

    t in

    ex-on-nt

    ain-

    on-avearioo-

    s ac-tedI. INTRODUCTION

    General relativity is a theory of gravitation that supportmassless graviton with two degrees of freedom. Howeveone were to describe gravity with a massive tensor fiegeneral covariance is lost and the graviton would possfive degrees of freedom. In the limit of vanishing mass, thfive degrees of freedom may be decomposed into a mastensor~the graviton!, a massless vector~a graviphoton whichdecouples from any conserved matter source! and a masslesscalar. This massless scalar persists as an extra degrfreedom in all regimes of the theory. Thus, a massive gratheory is distinct from Einstein gravity, even in the limwhere the graviton mass vanishes. This discrepancy is amulation of the van DamVeltmanZakharov~VDVZ ! dis-continuity @13#.

    The most accessible physical consequence of the VDdiscontinuity is the gravitational field of a star or other copact, spherically symmetric source. The ratio of the strenof the static~Newtonian! potential to that of the gravitomagnetic potential is different for Einstein gravity comparedmassive gravity, even in the massless limit. Indeed the ris altered by a factor of order unity. Thus, such effectslight deflection by a star or perihelion precession of an oring body would be affected significantly if the graviton haeven an infinitesimal mass.

    This discrepancy appears for the gravitational field of acompact object. An even more dramatic example ofVDVZ discontinuity occurs for a cosmic string. A cosmstring has no static potential in Einstein gravity; however,same does not hold for a cosmic string in massive tengravity. One can see why by using the momentum spperturbative amplitudes for one-graviton exchange betwtwo sourcesTmn and Tmn :

    Vmassless~q2!;2

    1

    M P2

    1

    q2S Tmn2 12 hmnTaaD Tmn ~1.1!*Email address: lue@physics.nyu.edu0556-2821/2002/66~4!/043509~8!/$20.00 66 0435if,

    ssess

    ofy

    r-

    Z-th

    ios-

    ye

    eoren

    Vmassive~q2!;2

    1

    M P2

    1

    q21m2S Tmn2 13 hmnTaaD Tmn.~1.2!

    The potential between a cosmic string withTmn5diag(T,2T,0,0) and a test particle withTmn5diag(2M

    2,0,0,0) is

    Vmassless50, Vmassive;2TM

    M P2

    ln r , ~1.3!

    where the last expression is taken in the limitm0. Thus ina massive gravity theory, we expect a cosmic string to atta static test particle, whereas in general relativity, no sattraction occurs. The attraction in the massive case caattributed to the exchange of the remnant light scalar mthat comes from the decomposition of the massive gravmodes in the massless limit.

    Nevertheless, the presence of the VDVZ discontinuitymore subtle than just described. Vainshtein suggests thadiscontinuity is derived from only the lowest order, tree-levapproximation and that this discontinuity does not persisthe full classical theory@4#. However, doubts remain@5#since no self-consistent theory of massive tensor gravityists. One can shed light on the issue of nonperturbative ctinuity versus perturbative discontinuity by studying a receclass of braneworld theories1 with a metastable graviton onthe brane@1012#. The theory we wish to consider hasfour-dimensional brane embedded in a five-dimensional,

    1There has been a recent revival of interest in the VDVZ disctinuity in the context of braneworld theories. These studies hfocused on variations of the Randall-Sundrum braneworld scenwhere the brane tension is slightly detuned from the bulk cosmlogical constant. The localized four-dimensional graviton acquiresmall mass, allowing one to study the VDVZ problem in an effetive massive four-dimensional gravity theory. For examples relato such work, see@69#.2002 The American Physical Society09-1

  • dtuthp

    hemis

    noralb-rea

    tenerdicmco

    sati-inee

    triicinro

    wn

    sete

    eo

    ck-

    t-

    ania-

    the

    ht,yell

    en-w-

    edthe

    er,t,t

    co-

    gy-tter

    h a

    s-

    e-

    mic

    ARTHUR LUE PHYSICAL REVIEW D 66, 043509 ~2002!finite volume Minkowski bulk, where the graviton is pinneto the brane by intrinsic curvature terms induced by quanfluctuations of the matter. The metastable graviton hassame tensor structure as that for a massive graviton andturbatively has the same VDVZ problem in the limit that tgraviton linewidth vanishes. In this model the momentuspace perturbative amplitude for one-graviton exchange

    V~q2!;21

    M P2

    1

    q21 iqr 021 S Tmn2 13 hmnTaaD Tmn,

    ~1.4!

    where the scaler 0 is the scale over which the gravitoevaporates off the brane. But unlike a massive gravity thethis braneworld model provides a self-consistent, genercovariant environment in which to address the nonperturtive solutions in the limit asr 0`. Indeed, exact cosmological solutions@13# in this theory already suggest that theis no VDVZ discontinuity at the nonperturbative classiclevel @14#.

    We would like to continue this program and investigathe gravitational field of compact objects in the same braworld theory with a metastable brane graviton. In this regaone would ideally like to identify the nonperturbative metrof a spherical, Schwarzschild-like source. That problehowever, possess considerable, though not insuperable,putational difficulties.

    Instead, we investigate the metric of a cosmic string aclose alternative formulation of the VDVZ problem forcompact source. The advantage of this system is its relasimplicity, as well as the clarity with which the VDVZ discontinuity manifests itself. After laying out the frameworkwhich the problem is phrased, we identify various regimwhere one can linearize the cosmic string metric. We thargue that there exists a region where these cosmic ssolutions are simultaneously valid and that they are identup to a coordinate redefinition. The resulting cosmic strmetric indicates there is no discontinuity in the fully nonpeturbative theory. It also provides an understanding as to hdifferent phases appear in different regions near and far afrom the string source. We conclude with some commeregarding the consequences of this solution.

    II. THE SOLUTION

    A. Preliminaries

    We wish to address the issues raised in the previoustion using a braneworld theory of gravity with an infinivolume bulk and a metastable brane graviton@10#. Considera four-dimensional braneworld embedded in a fivdimensional spacetime. The bulk is empty; all energy mmentum is isolated on the brane. The action is

    S(5)521

    2M3E d5xAuguR1E d4xA2g(4)Lm1SGH .

    ~2.1!

    The quantityM is the fundamental five-dimensional Planscale. The first term in Eq.~2.1! corresponds to the Einstein04350me

    er-

    y,lya-

    l

    -,

    ,m-

    a

    ve

    snngalg-wayts

    c-

    --

    Hilbert action in five dimensions for a five-dimensional meric gAB ~bulk metric! with Ricci scalarR. The termSGH isthe Gibbons-Hawking action. In addition, we considerintrinsic curvature term which is generally induced by radtive corrections by the matter density on the brane@10#:

    21

    2M P

    2E d4xA2g(4)R(4). ~2.2!Here,M P is the observed four-dimensional Planck scale~see@1012# for details!. Similarly, Eq. ~2.2! is the Einstein-Hilbert action for the induced metricgmn

    (4) on the brane,R(4)

    being its scalar curvature. The induced metric is

    gmn(4)5]mX

    A]nXBgAB , ~2.3!

    whereXA(xm) represents the coordinates of an event onbrane labeled byxm.

    We wish to find the spacetime around a perfectly straiginfinitely thin cosmic string. With Lorentz boost symmetrand translational invariance along the cosmic string, as was rotational symmetry around the string axis, the most geral time-independent metric can be written with the folloing line element:

    ds25N2~r ,z!~dt22dx2!2A2~r ,z!dr2

    2B2~r ,z!@dz21sin2 z df2#, ~2.4!

    where the string is located atr 50 for all (t,x). These coor-dinates are depicted in Fig. 1. If spacetime were flat~i.e.,N5A51, B5r ), we would choose the brane to be locatat z5p/2. In general, one can choose coordinates withincontext of the line element Eq.~2.4! such that the brane islocated atz5p/2, even when spacetime is not flat. Howevwe will find it useful to apply a less stringent constrainconsidering coordinates where the brane is located az5pa/2, where the parametera is to be specified by thebrane boundary conditions. Again, one can find a set ofordinates within the ansatz Eq.~2.4! in which this is possiblein general.

    Assuming the cosmic string dominates the enermomentum content of the spacetime, we ignore the ma

    FIG. 1. A schematic representation of a spatial slice througcosmic string located atA. The coordinatex along the cosmic stringis suppressed. The coordinater represents the 3-dimensional ditance from the cosmic stringA, while the coordinatez denotes thepolar angle from the vertical axis. In the no-gravity limit, the branworld is the horizontal plane,z5p/2. The coordinatef is theazimuthal coordinate. Note that everywhere except at the cosstring, the unit vector in the direction of thez coordinate extendsperpendicularly from the brane into the bulk.9-2

  • vaby

    lle

    n

    uu

    -th

    si

    tre

    t-e

    nthe

    b-

    al

    by

    and

    di-

    icateter-tedlu-re-in-r-theror

    a-te

    sghat

    COSMIC STRINGS IN A BRANEWORLD THEORY WITH . . . PHYSICAL REVIEW D66, 043509 ~2002!effects of the brane itself, except through the intrinsic curture term Eq.~2.2!. Using the coordinate system specifiedEq. ~2.4!, the energy momentum of the system is2

    Ttt52Txx5N2~r ,z!

    T d~r !

    2pA~r ,z!B2~r ,z!sinz, ~2.5!

    where the parameterT denotes the string tension and aother components of the energy-momentum tensor are zThe Einstein equations dictated by the action Eqs.~2.1! and~2.2! are

    1

    2r 0GAB1

    1

    B~r ,z!dS z2 pa2 DGAB(4)5 1M P2 TAB , ~2.6!

    where GAB is the five-dimensional Einstein tensor,GAB(4) is

    the induced four-dimensional Einstein tensor on the braTAB is the energy-momentum on the brane Eq.~2.5!, and wehave defined a crossover scale

    r 05M P

    2

    2M3. ~2.7!

    This scale characterizes that distance over which metric fltuations propagating on the brane dissipate into the b@10#.

    We assume aZ2-symmetric brane acrossz5pa/2. Underthis circumstance, one may solve Eqs.~2.5!, ~2.6! by solvingGAB50 in the bulk, i.e., whenz,pa/2 andr0, such thatthe following brane boundary conditions apply atz5pa/2:

    S NzN 1 AzA 1 BzB D5 r 0rA2 FNrrN 2 NrN ArA 1 1r S NrN 2 ArA D G2

    A12b2

    b1

    r 0T/M P2

    2pb

    1

    Ad~r !

    S 2NzN 1 BzB D5r 0rA2 FNr2

    N21

    2

    r

    NrN G2 A12b2b

    ~2.8!

    S 2NzN 1 AzA D5r 0rA2 F2NrrN 1 Nr2

    N222

    NrN

    ArA G

    where we have definedb5sin(pa/2) and where the subscript represents partial differentiation with respect tocorresponding coordinate. Equations~2.8! follow from Gtt ,Grr andGff , respectively, and are generated by the intrincurvature term induced by the action Eq.~2.2!. We also im-pose boundary conditions to ensure continuity of the meand its derivatives atz50, and to fix a residual gauge degreof freedom by choosingB(z5pa/2)5r .

    We wish to find the full five-dimensional spacetime meric induced by a thin cosmic string situated within the bran

    2Throughout this paper, we define the distributionald(x) of thevariable x, such that given any well-behaved functionf (x),*dx d(x) f (x)5 f (0).04350-

    ro.

    e,

    c-lk

    e

    c

    ic

    -

    world. The problem defined by Eqs.~2.5!, ~2.6! is dependentonly on the scaler 0 and the dimensionless parameterT/M P

    2 .We are interested in the problem whenr 0` with all otherparameters held fixed. Sincer 0 represents the only scale ithe problem, this statement implies we are interested insystem whenr !r 0 with T/M P

    2 fixed.

    B. The Einstein solution

    Before we attempt to solve the full five-dimensional prolem given by Eqs.~2.5!, ~2.6! and Eq.~2.8!, it is useful toreview the cosmic string solution in simply four-dimensionEinstein gravity@15,16#. For a cosmic string with energymomentum Eq.~2.5!, the exact metric may be representedthe line element:

    ds25dt22dx22S 12 T2pM P2 D22

    dr22r 2df2. ~2.9!

    This represents a flat space with a deficit angleT/M P2 . Thus,

    there is no Newtonian potential between a cosmic stringa static test particle. However, a test particle~massive ormassless! suffers an azimuthal deflection ofT/M P

    2 whenscattered around the cosmic string. With a different coornate choice, the line element can be rewritten as

    ds25dt22dx22~y21z2!2T/2pM P2@dy21dz2#.

    ~2.10!

    Again, there is no Newtonian potential between a cosmstring and a static test particle. However, in this coordinchoice, the deflection of a moving test particle can be inpreted as resulting from a gravitomagnetic force generaby the cosmic string. We can ask whether this Einstein sotion is recovered on the brane in the limit of the theory whethe graviton linewidth vanishes. In this limit, gravity fluctuations originating on the brane are pinned on that surfacedefinitely, implying that gravity should resemble a foudimensional theory. However, the question remains whethe four-dimensional theory that results is Einstein gravitysome massless scalar-tensor theory instead.

    C. Linearized five-dimensional Einstein equations

    Let us examine the linearized form of the Einstein equtions ~2.6!. We will see that the trick is to find an appropriabackground~including boundary conditions! around whichlinearize. We take the following:

    N~r ,z!511n~r ,z!1

    A~r ,z!511a~r ,z!1 ~2.11!

    B~r ,z!5r @11b~r ,z!1#,

    where it is assumed that the function$n(r ,z),a(r ,z),b(r ,z)%!1 in the regimes of interest. Takinthe Rtt component of the Einstein equations, one finds tthe partial differential equation~PDE! for n(r ,z)9-3

  • lg

    er

    ze

    ins

    e

    e

    oc

    eth

    is

    ne

    ed

    ab-qs.

    e

    g-be-

    a

    e-

    vere-

    the-

    l

    s

    aith

    ARTHUR LUE PHYSICAL REVIEW D 66, 043509 ~2002!r 2nrr 12rnr1nzz1cosz

    sinznz50, ~2.12!

    may be decoupled from the others. Equation~2.12! is simplyLaplaces equation for the Newtonian potential,n(r ,z), re-ducing the determination ofg00 to a linear static potentiaproblem~albeit, with unusual boundary conditions resultinfrom the presence of the brane!. In order to determinea(r ,z)and b(r ,z), one can directly integrate theGzz and theGzzcomponents of the Einstein equations, leaving

    a~r ,z!522n~r ,z!1r f 8~r !cosz1rg8~r !1g~r ! ~2.13!

    b~r ,z!522n~r ,z!1 f ~r !cosz1g~r !. ~2.14!

    The functionsf (r ) and g(r ) are to be determined by thbrane boundary conditions, as well as the last remainingsidual gauge. This technique for decomposing the lineariEinstein equations is the direct analog of that used in@17#.

    The brane boundary conditions Eqs.~2.8! in the linearizedEinstein equations may be used to complete the determtion of the metric components. The second two equationEqs.~2.8! both yield

    b f ~r !5A12b2

    b22r 0nr~r !usin z5b . ~2.15!

    Combining this equation forf (r ) and fixing the gauge choicb(r )usin z5b50 gives

    g~r !52n~r !usin z5b1A12b2

    br 0nr~r !usin z5b2

    12b2

    b2.

    ~2.16!

    The remaining equation in Eqs.~2.8! is then used to set thbrane condition for the Newtonian potentialn(r ,z):

    nzusin z5b5r 0F S r 1 2r 03 A12b2b D nrr 1nr Gsin z5b

    1r 0

    3b2d~r !

    3Fb1S T2pM P2 21D G , ~2.17!where the coefficient of thed-function contribution to thiscondition comes from the matter source@as reflected in thefirst equation of Eqs.~2.8!# and from the step inA(r ,z) atr 50 necessary to maintain elementary flatness at the ltion of the string. Once one determinesn(r ,z) using Eq.~2.12! and the boundary conditions Eq.~2.17! as well asnz(r )uz5050, then one can automatically read offa(r ,z)andb(r ,z) using Eqs.~2.13!~2.16!.

    D. The weak brane limit

    Let us first identify the solution to Eqs.~2.5!, ~2.6! andEq. ~2.8! in the weak field limit. Here, we presume that thmetric deviates from a flat metric with a flat brane whereperturbations~of the bulk and the brane! are proportional tothe strength of the source,T/M P

    2 , assuming this parameter04350e-d

    a-in

    a-

    e

    small. With the coordinate choice under consideration, omay keep the brane atz5p/2 while still allowing for abrane extrinsic curvature ofO(T/M P2 ). We refer to this limitwhere the extrinsic curvature of the brane is perturbaround af lat brane as the weak brane limit.

    In this limit one may use the linearized equations estlished in the last subsection. The explicit solution to E~2.12! and ~2.17! with b51 is

    n~r ,z!52r 03 S T2pM P2 D E0

    ` dk

    11r 0ke2kr coszJ0~kr sinz!,

    ~2.18!

    whereJ0 is the usual Bessel function of the first kind. Oncan then solve fora(r ,z) and b(r ,z) directly using Eqs.~2.13!~2.16!. One may also arrive at this result by applyinthe graviton propagator@10,14# and approximating the gravitational potential through one-particle graviton exchangetween the cosmic string source and a test particle inMinkowski spacetime with a flat braneworld.

    Two limits are of interest. The regime wherer @r 0 repre-sents the crossover from four-dimensional to fivdimensional behavior expected at the scaler 0. Gravitonmodes localized on the brane evaporate into the bulk odistances comparable tor 0. The presence of the brane bcomes increasingly irrelevant asr /r 0` and a cosmicstring on the brane acts as a codimension-three object infull bulk. Here the metric is asymptotically spherically symmetric ~i.e., z-independent! while the Newtonian potentiaEq. ~2.18! becomes

    n~r ,z!521

    3 S T2pM P2 D r 0r 1O~r 02/r 2!. ~2.19!Using the metric Eq.~2.4!, we find the metric on the brane ispecified by the line element

    ds25N2~r !usin z51~dt22dx2!2A2~r !usin z51dr22r 2df2~2.20!

    with

    N~r !usin z515121

    3S T2pM P2 D r 0r 1O~r 02/r 2! ~2.21!A~r !usin z51511

    2

    3S T2pM P2 D r 0r 1O~r 02/r 2!,~2.22!

    recovering the Schwarzschild-like solution forcodimension-three object in five-dimensional spacetime w

    r G5r 0T

    2pM P2 5

    T

    4pM3~2.23!

    acting as the effective Schwarzschild radius.In the complementary limit whenr !r 0, we find that Eq.

    ~2.18! becomes9-4

  • is

    trin

    ec

    he

    s

    s

    .-eathan

    thofe

    th

    tn

    leroe

    -u

    thesic

    ck-of

    ne..tial

    -lds-it,

    to

    ish,

    e

    -n is

    heace,This

    ne-

    COSMIC STRINGS IN A BRANEWORLD THEORY WITH . . . PHYSICAL REVIEW D66, 043509 ~2002!n~r ,z!51

    3 S T2pM P2 D lnF rr 0 ~11cosz!G1OS T2

    M P4 D . ~2.24!

    Using the metric Eq.~2.20! on the brane, we find

    N~r !usin z515111

    3S T2pM P2 ln rr 0D 1OS T2

    M P4 D ~2.25!

    A~r !usin z515112

    3

    T

    2pM P2 1OS T2M P4 D , ~2.26!

    which represents a conical space with deficit angle23 T/M P2 .

    Recall that for pure four-dimensional Einstein gravity, thmetric is N(r )51 and A(r )5(12T/2pM P

    2 )21, whichagain represents a flat conical space with deficit angleT/M P

    2 .Thus in the weak brane limit, we expect not only an exlight scalar field generating the Newtonian potential foundN(r ), but also a discrepancy in the deficit angle with respto the Einstein solution. Moreover, since Eqs.~2.25!, ~2.26!deviate from the four-dimensional Einstein solution, tbrane boundary conditions Eq.~2.8! imply that the braneitself possess an extrinsic curvature whose magnitudeO(T/M P2 ).

    We can ask the domain of validity of the solution Eq~2.25!, ~2.26!. Examining the boundary conditions Eqs.~2.8!and the bulk Einstein equationsGAB50 and comparing thesize of terms neglected with respect to those included, wethat on the brane this solution is only valid when

    r 0A TM P2 !r !r 0 . ~2.27!The left-hand inequality of Eq.~2.27! is the one of interestFor values ofr violating this condition, nonlinear contributions to the Einstein tensor become important and the wbrane approximation breaks down. But this is preciselyregime we are interested in, since we wish to understwhat happens whenr and T/M P

    2 are fixed andr 0`. Weneed to find a solution in this regime.

    E. The r r 0\0 limit

    The weak brane approximation breaks down whencondition Eq.~2.27! does not apply. Outside this domainvalidity, nonlinear contributions to the Einstein equations bcome important. However, by relaxing the condition thatbraneworld be located at sinz51, a perturbative solution tothe Einstein equations Eqs.~2.5!, ~2.6! with the boundaryconditions Eq.~2.8! can be found in the limit of intereswhen r !r 0 with T/M P

    2 held fixed. We are still interested ithe limit of a weak source, i.e.,T/M P

    2 !1, so that using thelinearized equations establish in Sec. II C is still applicab

    Recall that the coordinatez into the bulk acts as a polaangle, but where the space it parametrizes has a deficit pangle. The bulk is characterized by that part of space whsinz,b @i.e., 0

  • -int

    d

    nmo

    f

    nglf:

    r-

    in.e.,heen-

    e

    cal

    q.

    t-

    se,

    s-ngth

    ekesion,the

    s

    the

    ARTHUR LUE PHYSICAL REVIEW D 66, 043509 ~2002!n~z!ur 5R51

    3 S T2pM P2 D lnF Rr 0 ~11cosz!G , ~2.30!for someR such thatr 0AT/M P2 !R!r 0. We choose this specific asymptotic form for reasons that will become clearthe next section. This boundary value problem for a stasolution to Laplaces equation Eq.~2.12! is well posed, albeitin an unusual geometry and with unusual boundary contions.

    Unlike when sinz51, this solution to Laplaces equatiodoes not possess a simple closed form. However, onearticulate the dominant contributions to the Newtonian ptential in several key limits. Whenr !r 0AT/M P2

    n~r ,z!5F12 coszG rr 02F12 Pq~cosz!G S rr 0Dq

    1,

    ~2.31!

    with q511AT/pM P2 andPq(x) is the Legendre function othe first kind of orderq. Whenr @r 0AT/M P2 ,

    n~r ,z!51

    3 S T2pM P2 D lnF 32 rr 0A T2pM P2 ~11cosz!G1.~2.32!

    Indeed, one can arrive at an explicit form for the leadicontribution to the Newtonian potential on the brane itse

    n~r !51

    3 S T2pM P2 D lnF 11 32 rr 0A T2pM P2G1,~2.33!

    when r !r 0. One may ascertaina(r ,z) and b(r ,z) by di-rectly using Eqs.~2.13!~2.16!. By comparing the full Ein-stein equations and brane boundary conditions, Eqs.~2.6!~2.8!, with the linearized equations, Eqs.~2.11!~2.17!, onecan confirm that the Newtonian potentialn(r ,z) is valid intheentire regionr !r 0, while the expressions fora(r ,z) andb(r ,z) using Eqs. ~2.13!~2.16! are valid when r!r 0AT/M P2 and whenr @r 0AT/M P2 .

    When r !r 0AT/M P2 , the metric on the brane is detemined by the line element

    ds25N2~r !usin z5b~dt22dx2!2A2~r !usin z5bdr22b2r 2df2~2.34!

    with

    N~r !usin z5b511r

    2r 0A T

    2pM P2 1O~r 2/r 02! ~2.35!

    A~r !usin z5b512r

    2r 0A T

    2pM P2 1O~r 2/r 02!, ~2.36!04350ic

    i-

    ay-

    where b512AT/2pM P2 . The brane solution, Eqs.~2.35!,~2.36!, is distinct from the weak brane solution, Eqs.~2.25!,~2.26!. In particular, the extrinsic curvature of the branethe first case is nonperturbative in the string tension, iO(AT/M P2 ), whereas extrinsic curvature of the brane in tweak brane approximation is perturbative in the string tsion, i.e.,O(T/M P2 ).

    The deficit polar angle in the bulk isp(12a) whereagain sin(pa/2)5b, while the deficit azimuthal angle in thbrane itself is 2p(12b). In the limit when r 0`, thegraviton linewidth vanishes and we recover a flat conispace with a deficit angle 2p(12b)5T/M P

    2 , the solutionfor a cosmic string in four-dimensional Einstein gravity E~2.9!. Consequently, the cosmic string solution Eqs.~2.35!,~2.36! does not suffer from a VDVZ discontinuity, supporing the results found for cosmological solutions@14# in thisbraneworld theory with a metastable brane graviton.

    III. MATCHING BETWEEN PHASES

    We wish to address the matching of the Einstein phaEqs. ~2.35!, ~2.36! to the weak brane phase, Eqs.~2.25!,~2.26!. These two solutions are in distinct coordinate sytems. Nevertheless, when the cosmic string source streT/M P

    2 is small, there exists a region inr, namelyr 0AT/M P2!r !r 0, where both solutions are valid. In this section wshow that there exists a coordinate transformation that tathe Einstein phase into the weak brane phase in this regimplying that these phases are simply different parts ofsame solution.

    In the regionr 0AT/M P2 !r !r 0, the Einstein phase takethe form

    n~r ,z!51

    3 S T2pM P2 D lnF 32 rr 0A T2pM P2 ~11cosz!G1a~r ,z!5

    2

    3

    r 0r S T2pM P2 D cosz1 ~3.1!

    b~r ,z!5SA T2pM P2 2 2r 03r T2pM P2 D cosz1,where the brane is located at sinz512T/2pMP

    2 ~or cosz5AT/2pM P2 ), and neglecting contributions ofO(T/M P2 ) butkeeping leading order contributions, e.g.,;(T/2pM P

    2 )(r 0 /r ). The weak brane phase in the same region hasform

    n~r ,z!51

    3 S T2pM P2 D lnF rr 0 ~11cosz!G1 ~3.2!a~r ,z!5

    2

    3

    r 0r S T2pM P2 D cosz19-6

  • -

    do

    te

    ino

    .he

    ion.

    a-ng.

    h,t in

    e.icne.

    ur-

    -ianom

    the

    nof

    teme-

    -ofmicser-ns-

    fi-es

    m-

    glyous

    tion

    tiong

    orynd

    bles.

    COSMIC STRINGS IN A BRANEWORLD THEORY WITH . . . PHYSICAL REVIEW D66, 043509 ~2002!b~r ,z!522

    3

    r 0r S T2pM P2 D cosz1,

    where the brane is located at cosz50, and where again contributions ofO(T/M P2 ) are neglected.

    We have chosen a region,r 0AT/M P2 !r !r 0, such that thesolution Eqs.~3.1! remains a valid solution to the linearizeEinstein equations while undergoing the following linear cordinate transformation. Take a new polar variable,Z, suchthat

    Z5z1A T2pM P

    2 sinz1OS T2pM P2 D . ~3.3!Since

    sinZ5sinzF11A T2pM P2 coszG1, ~3.4!the metric functions are still in the ansatz Eq.~2.4! with thenew polar coordinate,Z:

    ds25N2~r ,Z!~dt22dx2!2A2~r ,Z!dr22B2~r ,Z!3@dZ 21sin2Z df2#. ~3.5!

    Under this coordinate redefinition, when the brane is locaat cosz5AT/2pM P2 , it is now located at

    cosZ5cosz2A T2pM P

    2 1501OS T2pM P2 D ~3.6!and with a time renormalization

    t5t

    121

    6 S T2pM P2 D ln T2pM P2~3.7!

    the Einstein phase takes the form

    n~r ,Z!5 13 S T2pM P2 D lnF rr 0 ~11cosZ!G1

    a~r ,Z!5 23

    r 0r S T2pM P2 D cosZ1 ~3.8!

    b~r ,Z!52 23

    r 0r S T2pM P2 D cosZ1,

    with the brane located at cosZ50, which is identical to Eq.~3.2! up to O(T/M P2 ). Thus, one can match the Einstephase solution with the weak brane solution using the codinate transformation Eq.~3.3! and the time redefinition Eq~3.7!, implying that the Einstein phase in Sec. II E and t04350-

    d

    r-

    weak brane phase in Sec. II D are parts of the same solut4

    IV. DISCUSSION

    In different parametric regimes, we find different qualittive behaviors for the brane metric around a cosmic striFor an observer at a distancer @r 0 from the cosmic string,where r 0

    21 characterizes the gravitons effective linewidtthe cosmic string appears as a codimension-three objecthe full bulk. The metric is Schwarzschild-like in this regimWhenr !r 0, brane effects become important, and the cosmstring appears as a codimension-two object on the braWhen the source is weak~i.e., when the tension,T, of thestring is much smaller than the square of the fodimensional Planck scale,M P

    2 ), the Einstein solution with adeficit angle of T/M P

    2 holds on the brane whenr!r 0AT/M P2 . In the region on the brane whenr @r 0AT/M P2~but still wherer !r 0), the weak brane approximation prevails, the cosmic string exhibits a nonvanishing Newtonpotential and space suffers a deficit angle different frT/M P

    2 .We identified a coordinate transformation connecting

    weak brane phase, Eqs.~2.25!, ~2.26!, and the Einsteinphase, Eqs.~2.35!, ~2.36!. Each phase is a linear solutiowhich becomes strongly nonlinear outside of its domainvalidity simply because the corresponding coordinate sysin which each solution is linear differs from the other. Thfull nonlinear solution in this light is reminiscent of the ansatz introduced by Vainshtein@4# for the Schwarzschild solution in a massive gravity theory. Moreover, the presencethis weak brane phase at large distances from the cosstring may have non-negligible consequences for the obvational search for cosmic strings through gravitational leing techniques@18#. For a GUT scale (1016 GeV) cosmicstring, the Einstein deficit angle for the string is;1025. Thisimplies that the light deflection by the string differs signicantly from the predictions of general relativity at distanc

    r;r 0AMGUT2M P

    2;3 Mpc, ~4.1!

    wherer 0 is taken to be todays Hubble radius. Such a seeingly peculiar choice ofr 0 is intriguing cosmologically@13,19#. Detailed discussion on how such a correspondinsmall fundamental Planck scale is possible without seriphenomenological obstacles may be found in@11,12#.

    The solution presented here supports the Einstein solunear the cosmic string in the limit thatr 0`. This observa-tion suggests that the braneworld theory under consideradoes not suffer from a VDVZ discontinuity, corroboratinthe findings for cosmological solutions in the same the@14#. Far from the source, the gravitational field is weak, a

    4On the brane itself, the coordinate transformation is applicaincluding O(T/2pM P2 ). One can confirm this by comparing Eq~2.25!, ~2.26! with Eq. ~2.33! and Eqs.~2.13!, ~2.15!, and ~2.16!using the time renormalization Eq.~3.7!.9-7

  • llnrcsiss

    onrg

    corldtetha

    lf.re

    kee,ld

    om-

    .forSFrd

    ARTHUR LUE PHYSICAL REVIEW D 66, 043509 ~2002!the geometry of the brane within the bulk is not substantiaaltered by the presence of the cosmic string. Propagatiothe light scalar mode is permitted. However near the southe gravitational fields induce a nonperturbative extrincurvature in the brane. That extrinsic curvature suppresthe coupling of the scalar mode to matter and only the tenmode remains, thus Einstein gravity is recovered. Astakes r 0`, the region where the source induces a labrane extrinsic curvature grows withr 0, implying Einsteingravity is strictly recovered in this limit.

    In this paper, we investigate the spacetime around amic string on a brane in a five-dimensional branewotheory that supports a metastable brane graviton. This syshas the advantage of offering a semianalytic solution tometric around a compact object, while still providing a cle. B

    s

    04350yofe,cesoree

    s-

    mer

    example in which the VDVZ discontinuity manifests itseThe result may help shed light on the more difficult, moimmediately relevant problem of a Schwarzschild-lispherical source in this braneworld theory. At the same timthe cosmic string solution is itself interesting and, shouthese objects exist in nature, would have testable phenenological consequences.

    ACKNOWLEDGMENTS

    The author would like to thank C. Deffayet, G. Dvali, AGruzinov, M. Porrati, R. Scoccimarro and E. J. Weinberghelpful discussions. This work is sponsored in part by NGrant No. PHY-9996137 and the David and Lucille PackaFoundation.s.

    in,

    M.@1# Y. Iwasaki, Phys. Rev. D2, 2255~1970!.@2# H. van Dam and M.J.G. Veltman, Nucl. Phys.B22, 397~1970!.@3# V.I. Zakharov, JETP Lett.12, 312 ~1970!.@4# A.I. Vainshtein, Phys. Lett.39B, 393 ~1972!.@5# D.G. Boulware and S. Deser, Phys. Rev. D6, 3368~1972!.@6# I.I. Kogan, S. Mouslopoulos, and A. Papazoglou, Phys. Lett

    501, 140 ~2001!; 503, 173 ~2001!.@7# A. Karch and L. Randall, J. High Energy Phys.05, 008~2001!.@8# M. Porrati, Phys. Lett. B498, 92 ~2001!.@9# A. Karch, E. Katz, and L. Randall, J. High Energy Phys.12,

    016 ~2001!.@10# G. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B485,

    208 ~2000!.@11# G.R. Dvali, G. Gabadadze, M. Kolanovic, and F. Nitti, Phy

    Rev. D64, 084004~2001!.

    .

    @12# G.R. Dvali, G. Gabadadze, M. Kolanovic, and F. Nitti, PhyRev. D65, 024031~2002!.

    @13# C. Deffayet, Phys. Lett. B502, 199 ~2001!.@14# C. Deffayet, G.R. Dvali, G. Gabadadze, and A.I. Vainshte

    Phys. Rev. D65, 044026~2002!.@15# A. Vilenkin, Phys. Rev. D23, 852 ~1981!.@16# R. Gregory, Phys. Rev. Lett.59, 740 ~1987!.@17# A. Gruzinov, astro-ph/0112246.@18# See the following and references therein: A. de Laix, L.

    Krauss, and T. Vachaspati, Phys. Rev. Lett.79, 1968 ~1997!;A.A. de Laix, Phys. Rev. D56, 6193~1997!; J.P. Uzan and F.Bernardeau,ibid. 63, 023004~2001!.

    @19# C. Deffayet, G.R. Dvali, and G. Gabadadze, Phys. Rev. D65,044023~2002!.9-8