PHYSICAL REVIEW D VOLUME 35, NUMBER 2 15 JANUARY 1987

Quantum field theory in the space-time of a cosmic string

B. Linet Unite Associee au Centre National de la Recherche Scient~fique No. 769, Uniuersite Pierre er Marie Curie,

Institut Henri Poincare. 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France (Received 16 September 1986)

For a massive scalar field in the static cylindrically symmetric space-time describing a cosmic string, we determine explicitly the Euclidean Green's function. We obtain also an alternative local form which allows us to calculate the vacuum energy-momentum tensor. In the case of a conformal scalar field, we carry out completely the calculations.

I . INTRODUCTION

The metric of the space-time in which a conical singu- larity describes a static, cylindrically symmetric cosmic string have recently been found.' The cosmic strings have some interesting implications for cosmology.2 In particu- lar, such a space-time is locally flat but of course it is not globally flat. Therefore it presents some distinctive global gravitational effects: it can act as a gravitational lens" and it can induce a repulsive force on an electric ~ h a r g e . ~ The theory of a free quantum field in a curved back- ground space-time is also a global problem and conse- quently vacuum expectation values of some physical quantities should not vanish.

To study quantum field theory, there are certain advan- tages to working within the Euclidean where techniques exist for obtaining well-defined physical quan- tities, especially the vacuum expectation values of the energy-momentum tensor operator. It is the principal purpose of the present paper to determine explicitly the Green's function GE (x ,xo ) (i.e., the Feynman propagator) for a massive scalar field in the Riemannian manifold re- sulting from the complexification of the time coordinate (i.e., a Wick rotation) in the space-time describing a static, cylindrically symmetric cosmic string. In order to give the expressions of the vacuum energy-momentum tensor, it will be more convenient to determine also GE(x,x0) as a sum of the usual Green's function in Euclidean space and a regular term, assuming that point x o is near point x .

The plan of the present work is as follows. In Sec. 11, we will recall some results, which will be needed, on the massive scalar field in the space-time of a cosmic string. In the corresponding Riemannian manifold, we will give in Sec. I11 the different expressions of the Green's func- tion for a massive scalar field. As an example we will cal- culate in Sec. IV the vacuum energy-momentum tensor of the conformal scalar field. We will add in Sec. V some concluding remarks.

11. PRELIMINARIES

The metric of the space-time describing a static, cylindrically symmetric cosmic string can be written as

ds2= - d t 2 + d p 2 + ~ 2 p 2 d d 2 + d ~ 2 with O < B 1 ( I )

in a coordinate system ( t,p,d,z) with p 2 0 and 0 5 d < 277. The constant B is related to the linear mass density p of the cosmic string located at p=O by

1-B P = 4 with 0 $ ; L < f

We will work in units for which G =c =Ti= I. In a previous work,' we investigated in detail the solu-

tion, denoted G (p,d,z;po,do,zo;m ), to the equation

( A - m 2 ) ~ = -6 '3 ' (p ,d ,z ;po,d0,~O) , (3)

where A is the Laplacian operator for metric (1) in the static case. We found that G can be expressed in an in- tegral form:

where is defined by the case B > which is physically justified (on the

p2+po2+(z -zo)* cosmological level p - Then, for all points (p ,b ,z) coshv = ( 7 7 2 0 ) . and (po,do,zo) of space-time (1) such that the coordinates

~ P P O d and do verify the inequalities

For certain calculations, we derived in our previous work7 -

a more convenient form for G. We restrict ourselves to

35 536 - 8 1987 The American Physical Society

35 - QUANTUM FIELD THEORY IN THE SPACE-TIME OF A . . . 537

G can be expressed as the sum of two terms biscalar GE(x,xO) on the manifold which obeys the co-

where the quantities r, R , and FB have the expressions

The first term in (6) is a solution to Eq. (3) which coin- cides with the usual solution in the Minkowskian space- time (by performing the change of coordinate f3=Bd) and the second term in ( 6 ) is regular.

In the Euclidean approach of the quantum field theory, the definition of the Green's function is simple: this is a

variant equation

( 0 - m 2 ) ~ E = -8(4)(X ; X o ) ,

where is the Laplacian operator, and which vanishes when the points x and x o are infinitely separated. In our metric ( I ) , we replace t by - i r and the Riemannian metric can be written

d s 2 = d ~ + d p 2 + ~ Z p 2 d d 2 + d z 2 with O < B < 1 (9 )

in the coordinate system (r ,p,d,z) with p 2 0 and 0 d < 2 ~ . Equation (8) takes the form

The solution to Eq. (10) can be written in the following in- tegral form:

1 + m GE(x,xo)=- S ~ ~ ~ , d , z ; ~ ~ , d ~ , z ~ ; ( m ~ + h ~ ) ~ ~ ~ ) c ~ ~ h ( ~ - - r ~ ~ d h .

2.rr - m

Result (1 1) can be easily verified taking into account the formula

1 + m 8(7-r0)=- 1- cosh( r - rO)dh .

2 7 ~ m

111. THE GREEN'S FUNCTION

We are now in a position to determine the Green's function GE(x,xO ). TO do this, we first integrate by parts integral (4):

where

d l sinh({/B) g,(<,*)= -

d < sinh<[cosh(</B) -cos$]

Then we apply formula ( 1 1) with G given by ( 13); we may interchange the order of integrations. We have thereby

With the aid of the Fourier cosine transformation

where Jo denotes the Bessel function, we can perform the h integration in (14) and we obtain finally the integral form

B. LINET

where y is defined by

(r-ro)2+p2+po2+ ( Z -z0 )' coshy = ( ~ 2 0 ) .

~ P P O

Result (16) is the desired Green's function which can be integrated for B = 1 to give the familiar Green's function

m G E ( x , x ~ ) = ,/, ~ l ( m [r2+(r- ro)2]"2) for B = 1 ,

4 ~ ~ [ r ~ + ( r - r ~ ) ] (1 7 )

where K 1 is the modified Bessel function. In order to calculate in the next section the vacuum energy-momentum tensor, it is more convenient to use an alterna-

tive form for the Green's function G E ( x , x O ) It can be obtained by applying formula (1 1) with G given by (6) for B > t. With the aid of the Fourier cosine transformation

Som e ~ ~ [ - b ( a ~ + x ~ ) ' / ~ ] c o s x ~ dx = a b ( b 2 + Y 2 ) 1 / 2 ~ l ( a ( b 2 + y 2 ) 1 / 2 ) , ( 1 8)

we obtain immediately GE(x,xo) as the sum of two terms:

where

Result (19) is valid for all points x and xo of the manifold (9) such that the coordinates d and do verify inequalities ( 5 ) . The first term in (19) is a solution to Eq. (10) which coincides with the usual Green's function in Euclidean space (by per- forming the change of coordinate O=Bd). The second term G;(x,x,) and its derivatives are regular in the coincidence limit x =xo

It may be of some interest to give the limit of GE(x ,xo) given by (16) when the mass m goes to zero. As J o ( 0 ) = 1, we obtain obviously

Expression (21) corresponds to a result of ~ o w k e r ' on the Green's function, which is periodic in the imaginary Rindler time, for a massless scalar field in a Rindler space-time.

IV. THE VACUUM ENERGY-MOMENTUM TENSOR

To regularize the vacuum expectation values of the energy-momentum tensor operator within the Euclidean theory of quantum field, a standard prescription is the f-function regularization. However, in our present problem we point out that there is no trace anomaly because the Riemann tensor of metric (9) vanishes identically. Moreover, expression (19) of the Green's function GE(x,xo) shows off the singular part which coincides with the usual Green's function. We can ignore infinities arising from it because we are locally in a Euclidean space and these divergences are familiar. On the other hand, the second term and its derivatives are regular in the coincidence limit x =x,; therefore, there is no problem for well defining the vacuum expectation values.

We consider a general form of the energy-momentum tensor parametrized by a constant 6. A procedure similar to the one of ~ a w k i n g ~ yields

where V, (VPo) means covariant differentiation with respect to the coordinates xp ( x i ) . We check that

VPpG;(x,x) =m ' G ~ ( x , x ). It should be remarked that ( pPv(x) ) will not depend on the coordinate r . Consequently, ex- pression (22) gives the vacuum energy-momentum tensor in the space-time since the change of time coordinate is r=it. To determine ( p P v ( x ) ) given by (22), the calculations are straightforward but give complicated expressions. We have simple results for a conformal scalar field (i.e., <= and m =O).

The limit of G;(x,xo) when the mass m goes to zero is

3 5 - QUANTUM FIELD THEORY I N THE SPACE-TIME OF A . . .

By making use of the identity

we find that ( FPv(x) ) can be written

for B > t. The vacuum energy-momentum tensor in the case of a

conformal scalar field has been already computed by Hel- liwell and ~ o n k o w s k i " using another method. They have found

A numerical analysis shows that expressions (25) and (26) coincide for B > +. (Unfortunately, we have not a formal proof of this result.)

V. CONCLUSION

The static, cylindrically symmetric space-time describ- ing a cosmic string gives an example of a curved space- time in which the scalar Green's function in the corre- sponding Riemannian manifold has been explicitly deter- mined. The origin of the nonvanishing vacuum energy- momentum tensor is purely topological since the space- time is locally flat. We remark that the energy density of the vacuum energy-momentum tensor (25) or (26) diverges negatively as p+O.

'B. Linet, Gen. Relativ. Gravit. 17, 1109 (1985); and also for an extended cosmic string see J. R. Gott 111, Astrophys. J. 288, 422 (1985); W. A. Hiscock, Phys. Rev. D 31, 3288 (1985).

'A. Vilenkin, Phys. Rep. 121, 263 (19851, and references therein. 3A. Vilenkin, Astrophys. J. Lett. 282, 51 (1984); C. J. Hogan

and R. Narayan, Mon. Not. R. Astron. Soc. 211, 575 (1984). 4B. Linet, Phys. Rev. D 33, 1833 (1986). 5G. W. Gibbons, in General Relativity: An Einstein Centenary

Survey, edited by S . W. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 19791, p. 639.

6R. M. Wald, Commun. Math. Phys. 70, 221 (1979). 'B. Linet, Ann. Inst. Henri Poincare 45, 249 (1986). 8J. S. Dowker, Phys. Rev. D 18, 1856 (1978). 9S. W. Hawking, Commun. Math. Phys. 55, 133 (1977). l q . M. Helliweil and D. A. Konkowski, Phys. Rev. D 34, 1918

(1986).