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by Helliwell and Konkowski [17] who found (see also Lillet [18]. Dowker. [19.20] and Smith (21)). Note that since \* > 1 for positive n, the Casimir energy-density ...
NO9200024

UNIVERSITY OF OSLO

Gravitational effect of the quantum vacuum outside a cosmic string Øyvind Grun

x i

and Harald H. Soleng

Report 92-05 ISSN-0332-5571

2

Received: 1992-02-12

DEPARTMENT OF PHYSICS REPORT SERIES

^•.••JMMSH&M

iir ,, ifn

;nr''nrvni''!n!'^ ' ' " 1

1

,

i

• •r.-'fJ.'ffv .' *!/ yW\ -- *Z - 0\ Report 92-05 ISSN-0332-5571

Received: 1992-02-12

'Oslo College of [engineering Con Adeleres gt.W. \-' w h o - r t e s > , ,( t h a t

>t >•> m>t fit

twn.thl,

to ignore the strongly curved interior of the string (..) when assessing the integrated effect of vacuum polarisation. Hence, to complete Hiscock's model one should determine (T ) on the interior background metric and calculate the gravitational perturbations caused by the Casimir energy in the interior of the string also. Here we will, however, take a less ambitious approach: Wc will derive an exact so­ lution of Einstein's field equations with a source of the form (1). Then we match this exterior solution to the interior Gott-Linet-Hicock solution. To obtain a self-consistent solution, this matching requires a singular surface layer [23], Now, since the exterior solution is unique, given the energymomentum tensor and the geometrical symmetries of the problem, the Gott-Linet-Hisrock .source plus the singular layer must correspond to the integrated gravitational effect of a cosmic string and its interior Casimir energy. Although this method does not provide any information about the internal microstructure of the string, it provides a self-consistent model which explains the gravitational attraction in Hiscock's model [13]. M

V

3

Exact solutions of Einstein's field equa­ tions

: In* geometry of the string is static and cyliiidricaiiy symmetric, with Lorentz symmetry ;tlong the axis. Hence, the metric is of the form tlj = -A'ltlt*

2

2

2

- th } r B' i]*nt // enrrespond ti» • //

2: a KaMiei type solution Willi vanishing n^it;. 7", :

,

•'"'./*-'

,/.-'

(» fli' Middirtv-k»

:

:

* \7>/\l< \r d,-

llirln*-

* -••,/*•

-

/\"' '• '

Hence, we must consider the case of variable H to get a solution with negative energy density (cf. cqs. (1) and (2)). Using equation (11) to replace r with H as an integration variable in the expressions (12) and (13) gives d

A



=

A

t l 4

) ;

( # + 2)(3ff + l )

and dB _ j3H + 4)dH D (H + 2){3H + 1) ' Integration of (11) gives r* = / " 2 ( 3 f f + lf *(H + 2 ) - ' ' i / I/5

/

/ r

1

(1G)

where / is a constant. Here we assume that H £ {(). oo) so that /• G jol/n». 3»/Y_ y^y in this range r is a monotonously decreasing function of H. Xow we may use Eq.( 11) and (16) to express dr in terms of If and -

>••

II

, -IC./il i

1 i •

, i » fliri

1 ,I r\;

i mi

r -

To i;ef the tol a I t;ta\ national ma-- we uiu^t aKo add the coul I ihution fioui t lie -,f I Ulli I t s e l f /",

• i

I"'.

f i l e Ci Mil n l i l t t loll f t o l l l t h e U l ' e i t ' i r I".

II

ol III loll 1^ / e n i. I M V . H I ^

I'lle l o h i i a i l l u i l — |MT l e i m l l l of t h e - M l l l i J I-

t m ii l»\ / ' • i

I

i

v

./...;

. M

i l l ,

Using that the surface energy-momentum is traceless and assuming// < < 1, we get jx, = 2a-l*r

2

= - ^

0

+ 0( )



M

(42)

Defining the total Tolman mass as the sum of the Tolmnn mass of the string (42) and the Tolman mass of the exterior quantum vacuum (40), we find that the string with vacuum polarisation has a positive gravitational mass l

fi(r) = \.(r) +ft

= ^h*L + OUi ) • (-53) 4o;rH Thi.s .should he compared witli the Newtonian approximation. The New­ tonian gravitational potential o{r) is given by y = - 1 + 2o. Expanding ijtt to first order in H gives f

3

tt

l

l

ffu = - ( 1 - # ) = - d -i }r )

.

(44)

which then may be identifie rylindnrally syiuuw'ric i;»oiii«f r\ with an «•nncj iia-m* tt'um o n*nt "f