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 < F < 2x 10 ~7 GeV we find deflection angles greater than 2n. For F> 2 X 10 j 7 GeV the singularity is too close for comfort.

1. Introduction

Cosmic strings are topological defects that arise from the spontaneous breakdown of symmetry at some scale F ~1. If a theory admits string solutions, then such strings would have been copiously pro- duced in the early universe. Residual strings may ex- ist today within our horizon and could perhaps be seen by their deflection of light. For this reason it is interesting to determine the gravitational metric about the string. Knowing the metric may also be of use in considering the evolution of strings in the early universe, as well as their possible effect on galaxy for- mat ion [2 ].

Cosmic strings come in two types: gauge and global, depending on the nature of the symmetry which is broken. From the point of view of gravity, these types of strings are very different. Gauge strings have van- ishing energy and m o m e n t u m outside a small radius (the core of the string). The space- t ime about gauge strings has been found to be flat, with a conical defect at the core and a deficit angle around the string [ 3 ].

Junior Fellow, Harvard Society of Fellows. ~ For a review of the properties of cosmic strings, see ref. [ 1 ].

However, global strings have a nontr ivial Goldstone boson field extending beyond the core, giving rise to a nonzero ene rgy-momentum tensor throughout space. Harari and Sikivie have found the metric in the linearized approximation for a straight infinite global string [4]. Since in flat space the energy den- sity of a global string falls offas 1/r 2, the total energy diverges logarithmically and one expects that not only will the linearized approximation break down suffi- ciently far from the string, but that eventually one will reach a singularity.

In this paper we calculate the exact solution to Einstein 's equations for the metric about a global cosmic string, As expected, the solution exhibits a physical singularity at a critical distance from the core of the string. In general, strings may have energy and m o m e n t u m trapped along the core (e.g, in the form of massive particles, zero-modes, or vibrational modes of the string itself), and we solve for the met- ric in this general case. We first consider the particu- lar case of a string with no trapped energy- m o m e n t u m along the core, making contact with the linearized solution in ref. [4]. We examine the exact geodesics of this metric, and display some typical ex- amples in the figures. We are also able to show that

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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 o f 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

d s 2 = A 2 ( d t - C d z ) 2 - D 2 d 2 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 C 2 + _ C + I = D 2 / 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 = F e x p ( i O ) ) 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~,= 8 n G T :z = 8nGT u = _ 8 n G T ° o = f 2/B2 , (2.4)

where

f 2 = _ S n G F 2 . (2.5)

The Einstein equations then become

Rt,=0

= A " / A B 2 + A ' D ' /ADB2 + ½K 2 ,

R--==0

=D" /DB2 + A ' D ' / A D B 2 - 1 K 2 ,

R'==0

= ½ ( K / B ) ( 2 A ' / A + K ' / K + B ' / B ) ,

R%=O

= ( 1 / A B 2) ( A " - A ' B ' / B ) - ( 1/B 3) ( B t 2 / B - B '' )

+ ( l / D B 2) (D" - D ' B ' / B ) - ½K 2 ,

R°o = - 2 f 2/B2

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

(2.6)

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

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

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

C = 1 - ( c o / g ) u ° ' / s i n h ( ( c o ~ g ) ( 1 + g i n u) ),

B 2 = y 2 ( u o / u ) ( l - ° J 2 ) / 2 e x p [ ( u ~ - u Z ) / u o ] , (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 << Mm, Uo is very large.

3. The unexcited string

The first case that we will consider is that of the bare string with no currents of any sort along the core.

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Volume 215, number 1 PHYSICS LETTERS B 8 December 1988

Then the form of the metr ic is const ra ined addi t ion- ally by boost invar iance 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 l imits co/g-,O and g ~ 0 in the expression for the general metr ic (2.7) . The resulting line e lement for this spe- cial case is

d s 2 = (u/uo) ( d t 2 - d z 2 )

_72(Uo/U) ,/2 exp[ (Uo - u2)/uo] ( d u 2 + d O 2) •

(3.1.)

In this coordina te basis it is not immedia te ly clear where the core o f the string is, and so we now show how our metr ic matches on to the solution in the li- nearized approximat ion presented in ref. [4]. In fact, we will see that the outer singulari ty, is at u = 0 , and the core is at u-~ uo. To this end, we first make a change of radial coordinate:

u = Uo - l n x . (3.2)

The line e lement then becomes

[1 - l n ( x ) / u o ] ( d t 2 - dz 2)

- 7 2 exp[ - l n2 ( x ) / uo] [ 1 - l n ( x ) / u o ] -1/2

X (dx2+x2dO 2) • (3.3)

To fix the constant , 7, we consider values o f x near 1, so that In x<< 1. Then keeping terms to first order in 1/Uo~ (F /MpI)a:

d s 2 - ~ [ 1 - l n ( x ) / u o ] ( d t 2 - d z 2)

-7211 + ½ ln (x) /uo - l n 2 ( x ) / u o ] (dxZ +x2dO 2 ) .

(3.4)

Changing variables one last t ime,

7x=-r{1 + 2 ( 1 / U o ) [ I - I n ( r / y ) ]

+ ( l /2uo) ln2(r /7 )} , (3.5)

the line e lement becomes

d s 2 ~ [1 -- ( 1 / u o ) l n ( r / ? ) ] ( d l 2 - d z 2 ) - d r 2

- [ l+~_/Uo- (2 /uo) ln ( r /7 ) ] redO 2 . (3.6)

I f we knew the stress tensor inside the core we could perform a complete matching and de te rmine the pa- rameter 7 in terms oF the comple te energy and mo- mentum inside the string. However, since we only know the form of the stress tensor and metr ic outside

the string core, we have an arbi t rar iness in y which reflects the arbi t rar iness in what size ro we choose as the core radius. (Of course, we know that ro ~ F - ~. )

It is convenient to choose ro so that the metr ic coef- ficient g,o is jus t this radius squared at the surface. Then the circumference o f the string at this poin t is just 2nro. With this convention, the paramete r y is given by

7=ro exp( - ¼ ) ~ F -~ (3.7)

Then to leading order in 1/Uo the convent ional radial coordinate r is related to u by

u = u o - ~- ln ( r / ro ) . (3.8)

o r

r = r m a x e x p ( - u ) , rmax ~ e x p ( u o - - ~) . (3 .9)

The linearized approximat ion (3.6) requires r << r ... . . Note that when r is near the core radius, x-~ 1, and

therefore u~-Uo~- (Mj,I/F) 2. As we move away from the core, r increases, and therefore u decreases. Thus in the u coordinate system u = 0 is the point (r/ ro) -~ exp ( 1 / f 2), a m a x i m u m radius away from the string, while u'-- Uo is where the core begins, and u = oo is the center string. Our solution for the metr ic sug- gests that there may be physical singularit ies at u = 0 and u = o c . Here we see that the lat ter is at the center of the string, inside the core, where our solution is not val id (since 87rGT,, does not look like (2.4) in this region). The singularity at u = 0 on the other hand is a ~ingularity which occurs at finite coordinate dis- tance from the surface of the string. In fact it is also at finite proper distance, as we shall see when we dis- cuss geodesics in the next section. Since coordinate invar iants constructed from derivat ives of the stress tensor are singular at this point , u=O corresponds to a physical singularity.

4. Geodesics

The geodesics may be found for the general metric, (2 .7) , but for s implici ty we will look simply at the case (3.1) . We will focus on geodesics in the (u, 0) plane. The constants o f the mot ion associated with the Kill ing vectors O, and O= provide two conservat ion laws, conservation of energy and angular momentum:

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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 mot ion 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)=-- i d u B ( E a / A : - j 2 / B a - m 2 ) - 1 / 2 . (4.3) tel

If we now consider a null radial geodesic, J = 0, m = 0, then we m a y 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(½Uo) j dyy ~/4 exp( - ½uoy 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 <. For F<< Mpj, Uo is very large, and we can approximate the integral by extending it to y = oe (i. e., by integrating to the center of the core). We find

2rad ~-- (7/E)exp( 1 / 2 f 2 ) F ( 9 ) , (4.5)

expressed in terms of the symmetry breaking scale f by means of (2.8). For strings associated with low energy scales, f--+ 0 and the singularity moves off to infinite affine parameter.

The equation for the shape of geodesics may now be obtained by eliminating the affine parameter f rom the equations, giving 0 as a function o f u:

O= if du(J/B) ( E 2 / A 2 - j 2 / B 2 - m 2 ) -1/2 , (4.6)

Ui

where u~ is the position on the geodesic where 0= 0. In fact, there are no closed orbits about the string. This follows since there is only one value of u where the radical in (4.6) vanishes, and so there is only one

~-~ The Ricci scalar has a branch point at u=0, vanishing as .~.~.

turning point. Thus the only possible orbits are cir- cular; but the derivative of the radical is never zero at the radius of this orbit, showing that such an orbit does not exist.

Therefore all geodesics eventually reach the singu- larity at u = 0 . Then we may take u~=0, and follow the geodesic in towards the string.

The general geodesic may be expressed in terms of the distance of closest approach, u+, which satisfies the turning point equation:

E2/A2(u+ ) - j2/B2(u+ ) - m 2 = 0 . (4.7)

It is convenient to define a "velocity" by

v2= [E2-m2AZ(u+ ) ]/E 2 . (4.8)

(Note that O<~v2<~ 1, and v2= 1 corresponds to m = 0.)

The geodesic may be expressed completely in terms of u+ and v:

u / t ¢ +

O(u)=u+ J dz(z- ' /2exp[(uZ+/uo)(1-z2)] 0

x{[ (l -z)lzllv2+ 1}- 1 )_,/2. (4.9)

This equation is exact. An explicit, approximate expression for the deflection angle may be derived when u2+/uo >> 1, for then the dominant contribu- tiofi to the integral comes from the region near the turning point (z_~ 1 ). Provided that

v2u+/Uo >> 1 (4.10)

(implying u2+/Uo >> 1 as well) we may expand the in- tegral in (4.9) for O(u+ ) to obtain

e=;c-zr(Uo/U+ ){1 - (uo/u+)[~ + 1/2v2-1n 2]

+O(uo/u~+ v2)}. (4.11)

Remembering that u varies from Uo = M;,JF 2 near the core o f the string to u = 0 at the outer singularity, we see that Uo> u+ > 0. Therefore the condition (4,10) is only satisfied for Uo<< 1 and u+ far from zero. In terms of the radial coordinate r in (3.8), the singu- larity at rm,x must be very far away in units of F - 1

(Uo >> 1 ), and the turning point r+ must be far away from the outer singularity (u+ >>0). In fact, if the turning point is very close to the core (i.e. ( u o - u+ ) / uo<< 1, or ln(r+/ro)<<Uo) then (4.11) may be fur- ther expanded to yield

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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 / u o ) [ l n ( r . / r o ) + I n 2 + t - 1/2~? 2 ]

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

which agrees with the expression lbr the deficit angle found in the linearized approximat ion ~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 << )z, while (4 .10) can yield 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 < 6 2 X t0 ~ 6 GeV. As

long as F satisfies this bound (4.12) is accurate and the deflection angle e < 7r/10,

I f we require that the outer singularity occur out- side our horizon "~ we find an upper bound F < ~ 2 . 0 X I0 ~v GeV. For symmetry breaking in the range 6,3X 10 ~' G e V < F < 2 . 0 X t0~7 GeV, the deficit angle may have to be calculated from (4.11 ) or (4.9),

~ Ourf ~ corresponds to v=/2 in ref, [4 t- ~4 By lhis we simply mean ti,~ > ! 0 m ly~ 5 × t 0 ~l GeV- ~,

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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+)<,%l ly . For such geodesics, (4.11 ) gives us deflection angles as large as 6~! Light has wrapped around three times before leaving the region near the string. For u+ < 20, the integral in (4.9) must be performed. Numerical re- sults are pictured in figs. 1 a - 1 d for u + + 10, 4, 3, and 1. Note that the deflection angle for u+ = 10 is - 13.8 rad, which is not apparent from the resolution of the figure. The linearized approximation (4.12) is only valid to 10% for l n ( r+ / ro ) < - 123, or r+ < 10 -26 cm from the string core!

5. Excited strings

We now consider strings with either time-like or light-like currents along the core. We must relax the condition of invariance under boosts along the string axis, and no longer assume A = D. This would be the case if there were additional matter density on the string. (The case of matter currents is handled trivi- ally by boosting the solution for matter at rest.) As long as a frame exists where the currents vanish (which is the case if the matter currents are not light- like) we may still take C=O. The solution (2.7) then becomes

A2= (U/Uo) ~+~,~ '

D 2 = (U/Uo) ~ -"~,

B2=TZ(uo /u ) l~ . . . . ) / 2 e x p [ ( u ~ - u Z ) / u o ] , (5.1)

(This can be obtained from (2.7) by setting g ~ 0 in conjunction with a boost along the z-axis and a re- scaling oft . ) Here o~ is related to the ratio of the extra mass per unit length to M ~ .

The preceding solution is inappropriate for a light- like matter current (which would be the case if the extra energy on the string were carried by massless modes) . I f the current is light-like and the matter density vanishes, we may take o)-~0 in (2.7) and get

A 2 = ( u / u o ) ( l + g i n u) ,

D 2 ~--- (U/b/o) ( 1 -~-g In u) - ~ ,

C = I - (i + g i n u) -1 ,

BZ= TZ( uo /u ) ~/2 exp[ ( u g - u 2 ) /uo] . (5.2)

In addit ion to these solutions, it is straightforward to include the effects of long-range electromagnetic fields from electrically charged currents on the string. These general solutions may be of interest in partic- ular models o f cosmic strings.

This work was supported in part by NSF contract PHY-87-14654 and by DOE contract DE-AS03- 81 ER40029. The authors would also like to acknowl- edge the Aspen Center for Physics where this work was completed. D.K. was supported by the Harvard Society of Fellows during part of this project.

References

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A. Vilenkin, Phys. Rev. Lett. 46 ( 1981 ) 1169, 1496(E). [3] A. Vilenkin, Phys. Rev. D 23 ( 1981 ) 852;

J. Gott Ill, Astrophys. J. Lett. 288 (1985) 422. [4] D. Harari and P. Sikivie, University of Florida preprint

FUTP-87-22 (1987).

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