Annalen der Physik. 7. Folge, Band 47, Heft 2/3, 1990, S. 93-100 J. A. Barth, Leipzig

On the Vacuum-Interaction of Two Pairallel Cosmic Strings

By M. BORDAG

Sektion Physik Karl-Marx-Universitat Leipzig, Leipzig, GD R

Dedicated to Prof. Dr. A . Uhlmunn on the Occusion to his 60th Birthday

Abst rac t . Cosmic. strings are well known solutions of the Einstein equations. In classical physics there is no interaction between such strings. I n quantum physics there is an interaction due to vacuum fluctuations like the well known Casimir effect. The interaction energy is calculated in the case of two parallel cosmic strings and shows an attractive force between them.

Zur Vakuumwochselwirkurig zweier paralleler kosmiischer Strings

Tnhal tsubersicht) . Kosmische Strings sind wohlbekamnte Losungen der Einsteinschen Ulei- chungen. Im Rahmen der klassischen Physik gibt es keine Wechselwirkungen zwischen den Strings. In der Quantenphysik erhalten wir eine Wechselwirkung infolge Vakuumfluktuationen wie im Fall des Casimir-Effekt,s. Wir berechnen die Wechselwirkungsenergie fur den Fall zweier pralleler kos- rnischer Strings und zeigen, daB eine anziehende Kraft zwirichen ihnen besteht.

1. Introduction and Summary

of parallel strings the corresponding metric has the form [2] Cosmic stfrings are well known solutions to the Einstein equations [ 11. T n the case

where the Newtonian potential of the strings is given by

V ( s ) = 2 21, I n r a , a

r,-heing the distance from the point .r to the i-th string in the plane perpendicular to the strings and A, is the linear mass density.

In classical physics there is no interaction between this strings [ a ] . Jn quantum physics, however, there is a interaction due to vac:uum-flnctuations. This situation is completely similar to that with the well known Catsimir effect [3] in eleclxodynamics. In order to investigate the interaction as a simple example the vacuum-expectation value E, of the energy of a scalar field in the presence of two strings will he calculated using field theoretical methods This quantity is infinit and one has to introduce a intermediate regularization E:, + EFP. Then one remarks that the contribution Eiif responsible for the vacuum-interaction of the two strings should depend on the mass densities A and A' of both strings. All other terms, resulting from the vacuum energy of one string alone and from that of the free space, carry the divergences and are keeped abvay from the total vacuiirn energy E';t'" to get E:ff,f. After that the regularization can be removed and

94 Ann. Physik Leipzig 47 (1990) 2/3

we have Ei;: ---f Ei,,t which is finit. We consider two strings parallel to the z-axis, located a t x1 = 0, x2 = 0 resp. x1 = b, x2 = 0. Because the wave equation in the metric (1) of two stings cannot be solved exactly some kind of perturbation theory must be used. We assume the mass density of the second string (denoted by A‘) to be small. This is meaningful having in mind that a typical value considered in cosmology is [I].

2

A- (:p;) - -10-6. (2)

The first string can be treated exactly up to the final result. The quantity Eint to be really calculated for straight strings is the energy density

per unit string length. In the case of a massless field it is calculated explicitly (formulae (31)). For a small mass density A of the first string the result is simply

and shows a attractive force between the strings.

8. Field Theoretical Formulation It is useful to start with the action of a scalar field @(z) in the metric (1)

4% @(x) + a;2 + r-8%’--A’ - % - m2) 1 @ ( 4 Y (4) s=-Ja 1 2

where

r = ix f + xi, r t = i(zl - b)2 + x i .

By the change of variables x + 5 r‘ = ae ( r = fx: -1- xi, p = f f : + pl =w

a Y cp = arctan - , y = arctaii [ X

X, = tz (a = 0 , 3 ~ with the notation

4 A - 1 - a ( O < C L < l )

the action (4) transforms into

(ag, - - ?4] @ ( 5 ) , (7 ) -t r‘-81.’

(8)

1 2 s = - j- a45 @([) [a;, +

where the angle y is restricted to

0 5 y 5 2na. Thus the theory with one string only (in our case A‘ = 0) is a free theory with a range of angles restricted by (8) and the parameter a is called “angle-deficit”. For later conve- nience we perform the transformation

fi + f i . P’ (i = 1, 2) , (9)

which does not change the action. Using the smallness of A‘ we divide the action as follows

s = so $. Sint (10)

M. UORDAG, Cosmic Strings 95

where So is the “free” part of the action

(11) 1

S o = a $ d 4 6 ~ ( o (a,,ap - m2) @ ( E L

describing a scalar field @ ( E ) in the metric of one cosmic string. The interaction with the second string in lowest order of its mass density A’ is given by

x (a;, - a;, - ?n2) di(t), (12) (note /I = ba/a).

ponding to the action (1) is given by 1

In the standard formulation of quantum field theory [4] the energy operator corres-

(13) YPy = -a,,@ . a,& + g,,,@(a,,af - m2) di

and the vacuum-expectation value of the energy per unit string length by

Eo = J d t l d52 (01 1’00 10) * (1 4) Denote by G(E, 7 ) the Green function of the theory. As usual in quantum field theory it is given in perturbation theory by

is the kernel of Sillt (12). The to the transformation (9) it is now “normalized” so as t o vanish for p r -+ 0. P(&, 11) is the Green function corresponding to the action So (1 I ) , obeying the equation

(atPat -k m2) w, 7) = &E - 7 ) . (18) Vor thc vacuum-energy Eo we get the perturbative contribution

Here the intermediate regularization as desired in the introduction is introduced as a point splitting with respect to these angles y, + yT.

For the Green function IY’(5, q ) with one string, defined by eq. (18) within the reduced angle region (S), several representations are known. The problem is essentially the same as that of a conducting wedge in electrodynamics and can be solved by the re- flexion principle. In the context of vacuum-energy it was studied for example in ref. [5]. Within the massless case there is a explicit expression through traiiscendent functions (see also ref. [GI, for example). The representation used here is known from the diffrac- tion of light on a wedge (a standard textbook is 173). It loolss

9G Ann. Physik Leipzig 47 (1990) 213

Here K i p @ ) is the Bessel function of imaginary argument and index, obeying the equa- tion

(+az% a, + p w - 1 K&) = 0 )

J

(21)

and the completeness relation 00

K&T) K&r') = y d(r - T ' ) . -- 00

The angle dependent part G,(y, y ' ) satisfies the equation

(-a:, + pu2) Gp(y , y ' ) = S(y - y ' ) and can be written as

where the restriction to the angles 0 5 y 5 2na is taken into account. In the following we use the representation

1 ,-P(I-vJ')-~w e~(u-u ' ) - -ac l

4p sh n(up + i ~ ) + sh n(ap - i ~ ) + 2e-fi"I~-~'I ) , ( (23) G ( - y ' ) =- P W

which is valid for I y - y' 1 5 2na. The third term in the rhs produces the singularities mentioned in the introduction. Remark, that it is independent on the angle deficit a of the string.

3. Calrulation of the Vacuum-energy

inserted into ErS (19). Before doing this i t is useful to rewrite eq. (19) in the form For the calculation of the vacuum-energy the Green function P(5, q) (20) has to he

(24)

1 E:;:;: = - j - dirld52 %',It (5) J d45J9e(5,5)

= J d5, a, =.%It (0 ( I , -t 1,).

z

with

where we have used the translation invariance in the 0, 3-directions. In formulae (24) a3 regularization we use the pointsplitting in the angular variables yc $. ypt which is the same as that in eq. (19) because the Green function G([, 6) (formulae (23) resp. (24)) depends on the difference yc - yc of the angles. Consider firstly the contribution I , to (24)

1 I ~ ~ (a? - a 2 to __ m2) J d45 D'(f, 5) (-a;o) w5, ('1. (25)

L I 1--

ltiserting expression (20) for the Green function we get

M. BORDAC, Cosmic Strings 97

The following step is the integration over re = it: + ti. Using formulae (6.5764) in [S] we have

--

- n2 p2 - v2 7 dr rKi,(yr) Kiy(yr) = - 0 4712 ch np - ch nv

Next we consider the integration over ye in the angle Green functions

We insert the representation (23) for the Green functions G,(yc - y;,) resp. G,(yc - yC,) into the last line of eq. (12). Hereby we keep away the contribution from the third term in the rhs of eq. (26). In this way we remove just the divergent, but a-independent contributions. After that all integrals converge and the intermediate regularization can be removed: y;

It remains to consider the integration over k . In the case m + 0 it cannot be carried out explicitly and one is left with numerical methods. So we set m = 0, perform the Wick rotation k, -+ ik,, and use again formulae (6.5764) in [S] to perform the k-integra- tion

ye, -+ 0.

z p2 - v2 -- - - 24.2 ch np - ch nv

In doing so we get the expression I l (25) in the form

The integrals over p and v can be calculated explicitly. This is done in the appendix with the result (A.5)

(I - 2) (1 + 6a2) I , =

3 60 n2r!a4 . The second contribution I, to the energy (24) is calculated in an analougeous manner. It similifies by means of eq. (18) and we have to consider

98 Ann. Physik Leipzig 47 (1990) 213

Inserting now expression (23) without its last term into the last line of the foregoing formulae and remove the regularization (i.e. set [ = ['). After that in the massless case the k-integration can be carried out by means of

which is a consequence of formulae (6.5764) in [S]. We get

48n rt --oo sh np 1, = 24 -1 7 dp A 1 - p2) e w ( l - a )

1 (sh n(ap + ic) 4- sh n(ab - i ~ )

After obvious rearrangements we arrive a t

Using formulae (3.5522) in [S] for this integrals we have -(I - a ) (1 + lla2)

1440n2r?a4 - I , =

Summing the two contributions (27) and (28) we get (1 - a2) (3 + 1 3 ~ ~ ~ )

1440n2rfa4 ' I1 + I , = (29)

This has to be inserted into the energy (24). Because the regularization is already re- moved we get Eirlt

On carrying out the angle-integration by means of

7 c+ In (r2 -j- 1 - 2r cos p/a) = 4 n In r e (r - 1) 0

the remaining integration becomes trivial

and we get for the energy the expression

il'(1 - a2) (3 + 13a2) 9Onp2a5

Eint = -

In terms of the mass density il = (1 - a)/4 (6) of the first string the interaction ener- gy looks like

Ail' (1 - 22) (3 + 13(1 - 8, 45n( l - 4A)5

E. - _ _ int -

For small il the leading order becomes simply

11' 32 Eint - - -

b2 4 5 n '

M. BORDAG, Cosmic Strings 99

Acknowledgement . I thank V. P. Frolov who has directed m y attention to this problem and W. D. Skarshinsky for useful discussialns.

Appendix Here we calculate expression (26)

with the abbreviation

Changing p -+ v in the second contribution and introducing an intermediate regulariza- tion (c > 0) in the denominator for the singularity a t p = v we rewrite (A.l) by

with

x v(v2 - p2) (1 - y+)z) ( I - (qq). Using the formulaes (3.5334) and (9.622) in [S] one derives the relation

which is valid for -z < Re t < z. By setting t == z

,J dv ((ch zv - ch zp + i ~ ) 2

izp - E here we get

sh zv .,,2rn+ 1 03

+ - ch 7cp - ic)2

Here we can set e = 0 because the expression is regular for p = 0. Inserting now eq. (A.4) with the well known explicit expressioiis for the Bernoulli polynomials into eq. (A.3) we get

16(2pZ - l ) p I’ =

so that I, eq. (A.2) looks like ’ 3zsh7cp ’

100

After trivial rearrangements we get

Ann. Physik Leipzig 47 (1990) 213

Using formulae (3.5222) in [8] we get finally (1 - u') (1 + 6a2) Il =

3 60 n2a4

Litoraturverzeichnis [l] VILENKIN, A.: In: 300 Years of Gravitation. (Ed. S. W. HAWKING and W. ISRAEL). Cambridge:

[2] LETELIER, P. S.: Classical and Quantum Gravity 4,4 (1987) p. L75. [3] CASIMIR, H. B. G.: Kon. Ned. Akad. Wet. 1 (1948) 79. [4] ITZYKSON, C.; ZUBER, J.-B.: Quantum Field Theory. New York: McGraw-Hill, 1980. 161 LUKOSZ, W.: Physica 56 (1971) 109. [6] DOWKER, J. S.: J. Phys. A: Math. Gen. 10, 1 (1977) 115. [7] JACKSON, J. D.: Classical Electrodynamics, New York: Wiley, 1962. [8] GRADSTEIN, I. S.; RYSHIK, I. M.: Tablizy Integralov. 5th ed. Moskwa: Nauka, 1971.

University Press, 1987, p. 499.

Bei der Redilktion eingegangen am 29. Juni 1989.

Anschr. d. Verf.: Dr. M. BORDAG Sektion Physik der Karl-Marx-Universitat Leipzig DDR-7010