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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

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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,

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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

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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

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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 constraincon