The exact metric about global cosmic strings

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  • Volume 215~ number I PHYSICS LETTERS B 8 December 1988

    THE EXACT METRIC ABOUT GLOBAL COSMIC STRINGS

    Andrew G. COHEN Lpnan Laboratoo' of Physics, Harvard University, Cambridge. MA 02138, US.4

    and

    David B. KAPLAN Department of Physics, B-OI9, University of California at San Diego, La Jolla, CA 92093, LISA

    Received 6 September 1988

    We present the exact solution to Einstein's equations for the metric about a straight, infinite cosmic string resulting from the breakdown of a global symmetry. Our solution is general enough to account for the effects of matter and currents trapped on the string. The metric exhibits a singularity at finite distance from the core. We examine geodesics and find that for a vortex scale F< 6.3 X 10 ~6 GeV the deflection of light is small, agreeing with the linearized result; however for 6.3 X 1016 GeV

  • Volume 215, number I PHYSICS LETTERS B 8 December 1988

    no closed orbits exist about a global string: all matter is eventually repulsed to the outer singularity.

    We conclude by presenting the exact metrics for global strings with (i) trapped matter at rest, and (ii) trapped light-like matter currents along the core.

    2. General solution

    We consider the example of a scalar field O which carries a global U (1) charge and which develops a vacuum expectation value at a scale F. This theory admits a string solution of the form

    )=F(r) exp( i0) , (2.1)

    where F(0) =0 and F( r ) =F for r>ro, the core ra- dius. Typically ro -~ 1/F.

    The general form of the metric outside the core is constrained by translation invariance in the z, t, and 0 directions; it will exhibit boost invariance only if there are no currents or trapped matter at rest along the core. We will assume that the Killing vectors as- sociated with these symmetries all commute, except for 0: with 0,. (This exception will allow us to include the effects of light-like currents along the string. ) We may then chose coordinates such that the metric takes the form

    ds2=A2(dt -Cdz)2 -D2d2 2 -B2(du2-t -d0 2) ,

    (2.2)

    where A, B, C, D are all functions of the radial coor- dinate u. Note that a string without matter currents is boost invariant in the z direction, so that C= 0 and A = D; a string with bound matter at rest will simply have C=0; and a string with a purely light-like cur- rent will satisfy C2+_C+I=D2/A 2. These condi- tions follow from considering how the stress tensor within the core behaves under boosts.

    The stress tensor for the global string does not van- ish outside the core, as it does for a gauge string, but instead has nonzero components from the Goldstone boson field (i.e., the phase of ~). The stress tensor outside the core (where O=Fexp( iO) ) is given by

    r , ,= (O,c)*3,0-~g, , 100[2+h.c.) . (2.3)

    Given the metric (2.2) and the vortex solution (2.1), one finds the nonvanishing components

    8nGT~,= 8nGT :z = 8nGT u = _ 8nGT o=f 2/B2 , (2.4)

    where

    f2=_SnGF2 . (2.5)

    The Einstein equations then become

    Rt,=0

    =A" /AB 2 +A'D ' /ADB2 + K 2 ,

    R--==0

    =D" /DB2 +A'D ' /ADB 2 -1K2,

    R'==0

    = (K /B) (2A ' /A+K' /K+B' /B) ,

    R%=O

    = ( 1 /AB 2) (A" -A 'B ' /B ) - ( 1/B 3) (B t2 /B -B '' )

    + ( l /DB 2) (D" -D 'B ' /B ) - K 2 ,

    Ro = - 2 f 2/B2

    =A'B ' /AB 3 - ( 1 /B 3 ) (B, Z/B- B" ) + B' D' /B3D, (2.6)

    where K = C 'A /BD. The general solution to these equations is

    A 2 = (U/Uo) (g/co)sinh ((co~g) ( 1 +g ln u) ),

    D2= (1/A 2)(u/uo) 2,

    C= 1 - (co /g)u ' / s inh( (co~g) ( 1 +g in u) ),

    B2=y2(uo /u) ( l - J2 ) /2exp[ (u~-uZ) /uo] , (2.7)

    where

    Uo =-- 1 I f 2 , (2.8)

    and 7, co, and g are constants of integration that can- not be removed by coordinate transformations. They are related to the form of the stress energy inside the string core. Note that for a symmetry breaking scale (0 ) = F

  • Volume 215, number 1 PHYSICS LETTERS B 8 December 1988

    Then the form of the metric is constrained addit ion- ally by boost invariance along the string axis, which implies that we may choose a coordinate system where A =D and C=O. This is achieved by taking the limits co/g-,O and g~0 in the expression for the general metric (2.7). The resulting line element for this spe- cial case is

    ds2= (u/uo) (dt2-dz 2 )

    _72(Uo/U) ,/2 exp[ (Uo - u2)/uo] (du2+dO 2) (3.1.)

    In this coordinate basis it is not immediately clear where the core of the string is, and so we now show how our metric matches on to the solution in the li- nearized approximation presented in ref. [4]. In fact, we will see that the outer singularity,is at u=0, and the core is at u-~ uo. To this end, we first make a change of radial coordinate:

    u= Uo - ln x . (3.2)

    The line element then becomes

    [1 - ln (x ) /uo] (d t2 - dz 2)

    -7 2 exp[ - ln2(x) /uo] [ 1 - ln (x ) /uo] -1/2

    X (dx2+x2dO 2) (3.3)

    To fix the constant, 7, we consider values o fx near 1, so that In x

  • Volume 2/5, number l PHYSICS LETTERS B 8 December 1988

    E=A2dt/d2, J=B2dO/d2, (4.1)

    where A = g, and B = goo, given in (3.1). The remaining equation of motion is

    A2( dt/d)t )2-B2( du/d~ ) 2 -B2( dO/dj.)2=m 2 , (4.2)

    where m is the mass of the particle. Combining these equations we can solve for the affine parameter as a function of the radial coordinate:

    )~(U)=-- iduB(Ea/A: - j2 /Ba-m2)-1/2. (4.3) tel

    If we now consider a null radial geodesic, J= 0, m = 0, then we may integrate from just outside the string core, u = uo, to the maximum distance away from the string, u = 0. This gives

    1

    2,.ad = (7/E)exp(Uo) j dyy ~/4 exp( - uoy 2) , 0

    (4.4)

    where we made the change of integration variable, y=-u/uo. 2,-aa is easily seen to be finite, and so u=0 represents a physical singularity

  • Volume 215, number ! PHYSICS LETTERS B 8 December 1988

    a b

    C d Fig. I, Light-like geodesics with no= 137, corresponding ~o F~ t0 ~ GeV, for various "impact parameters" u+: (a)u+ = 1 O, O,~r= -4,4~; (b) u+ = 4, 0~,~-= - 1.88~t. (c) ~f,~ = 3, O,~,r= - 1,23m (d) u+ = I, O,~r~ + 0,23~r~ The enclosing circle and central asterisk mark the outer singularity and string core, respectively,

    e= - ( ; z /uo) [ ln ( r . / ro ) +In 2 + t - 1/2~? 2 ]

    +O( ( l /u~) In ( r+/ r~) )2 , (4.12)

    which agrees with the expression lbr the deficit angle found in the linearized approximation ~3 [4], Note that (4. I t ) is valid over a much larger range of turn- ing points than (4.12). The latter always gives e m It behooves us to see when such peculiar behavior can arise,

    Expression (4.12 ) is valid to 10% for ( I /uo)In (r,~, / ~i)) ~< I. This is satisfied for any turning point within our horizon (r+ < 10 ~o ly) for F< 62 X t0 ~ 6 GeV. As long as F satisfies this bound (4.12) is accurate and the deflection angle e < 7r/10,

    If we require that the outer singularity occur out- side our horizon "~ we find an upper bound F

  • Volume 215, number I PHYSICS LETTERS B 8 December 1988

    In this case we find that light-like geodesics may wrap around the string more than 2zr.

    Consider a concrete example with uo=137, F~2 1017 GeV, corresponding to a string with a singularity at r ..... = 10 I ly. For a turning point satis- fying u+ >20, (4.11 ) is accurate to 10% - this cor- responds to r+=rmaxexp(-u+)