Volume 143, number 9 PHYSICS LETTERS A 5 February 1990

T H E L I N E A R I Z E D S P A C E T I M E OF S U P E R C O N D U C T I N G C O S M I C STRINGS

T.M. HELLIWELL Department of Physics, Harvey Mudd College, Claremont, CA 91711, USA

and

D.A. K O N K O W S K I Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402, USA

Received 17 May 1989; revised manuscript received 29 November 1989; accepted for publication 30 November 1989 Communicated by J.P. Vigier

The external metric of straight superconducting cosmic strings is derived in linear approximation in terms of the current and integrals over string fields. Three parameters in an axial magnetic field geometry of L. Winen are expressed in terms of string parameters. The effect of current on the deflection of light is also derived.

In 1985, E. Witten showed that some sponta- neously broken gauge theories give rise to supercon- ducting cosmic strings which carry currents o f up to ~ l02° A or even more [ 1 ]. In the case o f bosonic charge carriers considered here, the theories have two pairs of scalar and gauge fields. A vortex string is produced by spontaneous symmetric breaking of one pair; symmetry breaking of the other pair inside the vortex produces the electric current and an external magnetic field.

The spacetime surrounding superconducting strings has been derived previously by analytic in- tegration of Einstein's equations over the string's stress-energy tensor [ 2 ], by using Dirac distribution sources in the linearized Einstein equations [ 3 ], and by numerical integration of Einstein's equations to- gether with the scalar and gauge field equations [ 4 ].

The Lagrangian density o f a bosonic supercon- ducting cosmic string is

~ = -- ¼Fab F a b - ½ Daa(Dat7) *

- - I F a b l ~ a b - - ½D~0(Da0) * - V(¢, a) ( 1 )

containing charged and electromagnetic fields a and Aa, and Higgs scalar and gauge fields q~ and Xa, where

Fab m V~Ab -- VbA~, Fab = Va,'tb -- Vb-4a ,

438

D a a = (Va + ieda)a , DaO= (V~ +i~X~)O.

The potential is

v(o, ~ ) = ~ G ( I Or ~ - ¢ ) ~+-~,l~ I al ~

+-~¢121t~12-½m21aL2. (2)

For an infinite straight string we can use the general cylindrically symmetric metric [2,5 ]

d s2=exp [2 ( ~ - ~) ] ( - d r 2 + dr 2)

+ o ? e x p ( - 2 ~ ) d 0 2 + e x p ( 2 ~ ) dz 2 , (3)

where the functions o~, y, ~r depend only upon r. In the vortex string model one assumes 0 = ¢ (r)e ie and A a = ~ - ' [ W ( r ) - 1 ]Va0 [6]. By cylindrical symme- try we can also let o-=o-(r)e iq~z) and Aa=A(r )Vaz .

For an infinitely long string we can further simplify by making the gauge choice q ( z ) = 0 .

Variation of ~ with respect to 0, a, Aa, and da yields the scalar and gauge field equations. In the flat spacetime limit (i.e. as y--,0, ~J--.0, c~-,r) the equa- tions are

(rO') '=r(b[W2/r2+½J.o(~2--rl2)+f62 ] , (4)

( r o ' ) ' = r a ( e2A2 + 2 o a 2 - m = + f02) , (5)

( W ' / r ) ' = (~2/r)OZ W , (6)

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Volume 143, number 9 PHYSICS LETTERS A 5 February 1990

and

( rA')'=re2a2A- rJ , (7)

where primes denote d/dr. The current is

I ( r ) = i dr2r~rJ, 0

so the integral of the Maxwell equation (7) gives A'=I /2nr outside the string, where I is the total current.

The stress-energy tensor may be computed from

Tab = -- 2 ~qg / ~gab'q- gab ~ . (8)

In the flat-space limit, the results are given in table I.

Numerical solutions of the scalar and gauge field equations ( 4 ) - (7) have been calculated for non- conducting a = 0, A = 0 strings and for more general cases [4,7,8 ]. Results show that if one begins with an A = 0 solution, the fields 0 and W are essentially unchanged as A is increased modestly. This is not surprising, since an iterative procedure in which so- lutions ¢~o, Wo, ao computed with A = 0 are inserted on the right-hand sides of eqs. ( 4 ) - (6) does affect a, but neither q~ nor W, at the first iteration.

The effect of current on the mass density and ten- sion of cosmic strings can be computed using a model in which the string has radius ro, beyond which fields have essentially their asymptotic forms O-,t/, W-,0, a-,O, A'-,I/27tr. Define the linear densities

r0

r a= f dr 2nrT a 0

for each a = 0, 1, 2, 3.

Then using integration by parts,

ro

- _I dr2nr(A'2+a2eZA2) r t _ Tz_~.

o

= -A( ro ) I . (9)

Also

zt=-Ix-½A(ro)I and

zz=_u+½A(ro) i '

where

nO

/ t= ½ J dr 27tr[ W'Z/e2r2+ Wz(~2/F 2 0

+O'2+a'2+2V(O, a) ] . (10)

I f we begin with an I = 0 solution and then turn on the current, the first three terms in the expression for /t are unaffected in first iteration, since neither W nor ~ is affected. We can show that the sum of the last two terms is likewise unaffected in first iteration. Let a = ao + 8a where a and ao are the charge field with and without current, and 8a is assumed to be small. Then

i dr2~r(a'2-a'°z) 0

~- f dr 2nra'oSa'=- f dr 2~(ra3) 'Sa 0 o

ro

= - ~ dr 2~rao().~a~-m2+fO~)Sa ( 11 ) 0

integrating by parts and using eq. (5) in first iter- ation. Also,

T a b l e 1

A a/2 a2e2A2/2 W'2/2e2r 2 W2O2/2r2 ¢~'2/2 t r ' 2 / 2 V

T~ - 1 - 1 - 1 --1 - 1 - 1 - 1 T~r 1 - 1 1 - 1 1 1 - 1 T~ - 1 - 1 1 1 - 1 - 1 - 1 T~ 1 1 --1 - 1 - 1 - 1 - 1

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Volume 143, number 9 PHYSICS LETTERS A 5 February 1990

V - Vo= ¼2.[ ( ao + 8 ~ ) 4 - a a ]

+ ~o2[ (~o + 8a )~ - Go 2 ]

- ½rn2[ (CZo + 8rz)z- cr 2 ]

~_ao(k¢~2-mZ + fOz)sa, (12)

so the sum

i dr2xr[~'2+2V(O, o')] 0

is independent o f current in first iteration. There- fore , / t as a whole is insensitive to current, and

IrZl = l - l t + ½A(ro)ll

decreases as I increases, giving rise to the phenom- enon o f " c o s m i c springs" [ 9 ].

The linear stress densities r ~ and r ° can also be de- composed into current-dependent and current-in- dependent terms for small currents. From table 1 we can write

r0

rr+ T°= t dr 2nr( - a2e2A 2 + W ' 2 / e 2 o ' 2 - - 2 V) 0

= a + f l ( I ) , (13)

separated into a current- independent part

or= i dr 2 x r ( W ' Z / ~ Z r 2 - 2 V o ) (14) 0

and a current-dependent part

f l ( I ) = i dr 2~r[2( Vo- V)--~72e2A 2 ] , ( 1 5 )

0

where V= V(~, a) and Vo= V(0, ao). We will show below from Einstein's equations that T'+r°=I2/4x. Therefore c~=0 and /~ ( I ) =I2/4~. From table 1 we can also write

ro t~

½ j dr 2nr( W 2 0 2 / r 2 - - O '2 T 0 =

0

- - O'~2 + W ' 2 / ~ 2 r 2 _ 2 V o )

+ l r i dr 2 x r [ - A ' a - ~ r g e 2 A 2 0

- (rT'2+2V-a{~2-2Vo) ] . (16)

We have shown above that the last two terms of the first integral sum to zero and the last four terms of the second integral sum to zero, so

To=fi-- ½A(ro)l (17)

using eq. (9), where

r0

f i= ½ j dr 2~r( W2~2/r2--(Y2--O"o 2) . (18) 0

An analogous procedure shows also that

Tr= --fi+ ½A(ro)I+ I2/4n . (19)

The external geometry of straight superconducting cosmic strings may be found in terms of the stress tensor Tg by integration of Einstein's equations [2,5 ]

a " - - - - - 8 x G ( T t t + T~)v/g , (20)

(ot~")' = 4~G( T~ + T r + To o - T§ )x /~ , (21 )

017"= -- 0l~ll '2 + 8xGT°x/Cg , (22)

(c~7') '= 8nG( T; + T°)x/g, (23)

in terms of the metric coefficients oe, 7, ~t of eq. ( 3 ). We will retain only linear terms in the Tg and in 7, ¢/and ~ where c~= ( 1 + / / ) r . Then we can set v / g = r and c~=r (in all but eq . (20 ) ) , and we can neglect the first term on the right in eq. (22). Furthermore, only the magnetic field contributes to T~, outside the string, so

Tg(r>ro)=(I2/8xZrZ)diag(-1, 1 , - 1 , 1), (24)

as given by the first column of table 1. For r>~ ro, first integrals yield

c ( = 1 + 4 G ( r t + r r) , (25)

q/'= (2G/r) ( z t+ z ' + r ° - r z)

- (4Gk/r) ln(r/ro), (26)

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Volume 143, number 9 PHYSICS LETTERS A 5 February 1990

y ' = 4 G k / r + 4 G ( 2 n i d r T ° - k / r o ) 0

= 4 G ( r r + r ° ) / r , (27)

where k - - I 2/4~. Consistency of the two solutions for y' requires that

r o

zr+r°=2nro ~ dr T o 0

and that r r+z°=k=IE /4 l t , as claimed above. Second integrals yield the linearized external met-

ric. After a linear coordinate transformation of t, r, and z to remove unneeded constants, the external metric of superconducting cosmic strings becomes

ds2= [ 1 + 4 G ( k - U + z z) ln(r /ro)

+ 4 G k ln2( r/ro) ] ( - d t Z +dr 2)

+r2[ 1 + 8 G r t - 4 G ( k + U - z z) ln( r/ro)

+4Gkln2(r / ro ) ] d02

+ [ l + 4 G ( k + r t - r ~) ln(r /ro)

- 4 G k l n 2 ( r / r o ) ] dz 2 . (28)

Note that radial and azimuthal stresses appear only in the combination zr-l-z°=I2/4g, so the geometry to this order is independent of/i , the current-inde- pendent part of rr+r °. As I - ,0 , k and z t - z z both vanish, so the metric reduces to the usual metric

dsE=-d t2 -Fdr2Wr2(1 - 8G/t) d02+dz 2 (29)

for a nonconducting string, whose (r, 0) sections form a cone with deficit angle 8nG/t. For I S 0 the four-geometry is no longer a two-cone set in flat spacetime, but in this linear approximation the (r, 0) sections remain conical, with deficit angle 8nG(l~+ I2 /8n ).

At distances such that all fields have their asymp- totic forms, the exact metric surrounding a super- conducting cosmic string should be equivalent to that found by L. Witten from the Einstein-Maxwell equations for static, cylindrically symmetric systems with axial magnetic fields [ 10 ]. Therefore, the met- ric of eq. (28) should agree with a linearization of Witten's metric. Babul, Piran, and Spergel have writ- ten Witten's metric in the convenient form [ 11 ]

ds2= (r/ro) -2mA 2(r)

X [ (r/ro)2m2( - d t 2 + d r 2) +D2r 2 d02 ]

-I- (r/ro)2mA -2 ( r ) dz 2 , (30)

where A = [ (r/ro) 2,, + x ] / ( 1 + x) . The metric de- pends upon the three parameters m, x, and D; the spacetime becomes fiat as m--,0, except for a conical singularity along r = 0 if D # 1. Therefore, we com- pare the metric of eq. (28) with Witten's metric for small m, expanded to order m 2. It is straightforward to show that they agree if we set

m2=4Gk , x = l - 4 G ( k + z t - r z ) / V / ~ ,

and

D = 1 + 4Gz t .

These equations express the three metric parameters of Witten in terms of three sources, z t, r ~, and k = I 2 / 4n, in the weak-source approximation.

Null geodesic equations in the linearized metric show that light trajectories in a constant-z plane obey

dr - +_ ( l + f l ) r [ 1 - y ( r ) ]

dO

× { ( r / g ) E [ l + 2 y ( R ) - 2 y ( r ) ] - l } ~/z (31)

for small f l - 4 G U and small 7 = - ( GIE/n) ln(r/ro); R is the radius of closest approach to the string. For currents in amperes,

y= ( G#oIEmp/ nc 4) ln(r/ro) ,

where # o = 4 n × l0 -7. For currents of 1021 A or less, y(r) remains small even over cosmological dis- tances. Integration of the trajectory equation yields the deflection angle

~=4nG{l t+ ½A(ro)I

+ (I2/4n) [1 + l n ( 2 R / r o ) ] } , (32)

which exceeds the 4riG# deflection o f / = 0 strings and which grows slowly as the radius of closest approach is increased. The current terms become comparable to the/t term only for the largest currents and R's of galactic scale.

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(1988) 877. "[ 5 ] K.S. Thorne, Phys. Rev. 138 ( 1965 ) B251. [6] H.B. Nielsen and P. Olesen, Nucl. Phys. B 61 (1973) 45.

[7] D. Garfinkle, Phys. Rev. D 32 (1985) 1323. [8] P. Laguna-Castillo and R. Matzner, Phys. Rev. D 36 (1987)

3663. [ 9 ] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett.

58 (1987) 1910. [ 10] L. Witten, ed., in: Gravitation (Wiley, New York, 1962) p.

382. [ 11 ] A. Babul, T. Piran and D. Spergel, Phys. Lett. B 209 ( 1988 )

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