J A G Vickers- Generalised cosmic strings

8/3/2019 J A G Vickers- Generalised cosmic strings

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Class . Quantum G rav . 4 (1987) 1-9. Printed in the U K

Generalised cosmic stringsJ A G VickersDepartment of Mathematics , Universi ty of Southampton, Southampton SO9 5 N H , U K

Received 30 April 1986, in final form 2 June 1986

Abstract. It is shown that two-dimensional and t imelike elementary quasi-regular s in -gularit ies provide a general ised version of the usual cosmic s t r ings. The propert ies of suchgeneral ised cosm ic s tr ings are invest igated and i t is shown in part icula r that they are total lygeodesic submanifolds on which the mass densi ty p is constan t .

1. IntroductionRecent ly there has been in teres t in cosmic s t r ings produced by phase t rans i t ions inthe early universe [ l , 21 an d the cosmological consequences of the exis tence of suchstr ings have been considered by a n um ber of authors [3,4] . Most of the at tent ion h asfocused o n cosmic strings which have a metric of the form

d s 2= d t 2- r 2-A 2 r 2dt12- z 2 (1)where 0s 0 < 27r. Such spacetimes may be considered as the l imit of spacetimes witha source consisting of stressed filaments lying along a tub e parallel to th e z axis [ S ,61or in terms of the distribution-valued curvature formalism [7] as a spacetime whosenon-vanishing energy-momentum tensor comp onents are

an d w h ere

a n d .S2 is a two-d imens iona l de l ta func t ion wi th sup por t on t h e r = 0 axis . (N ote thatwith a s l ight abuse of nota t ion som e authors wri te 2v&= p t j ( r ) / r w h er e 6 ( r ) is t h eordinar y Di rac del ta funct ion . )The met r ic ds2= d r 2 + A 2 r2d e 2 i s ju s t th e met ric o f an o rd inary two-dimensiona lcone a nd we therefore refer to th e spacet ime with metr ic g iven by (1) as a conicalspacetime. By changin g to the angular coordin ate e = A0 we may w rite (1) in the form

d s 2= d t 2 - d r 2 - r2 d 0 - d z 2 (3 )where 0s e < 27rA = a, say.This form of the metr ic shows that th e spacet ime is local ly Minkowskian bu t th isis not t rue g lobal ly du e to the presence of a s ingulari ty at r = O w hich has the effectof focusing geodesics for 0


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2 J A G Vickers

Figure 1. Geodesics o n an opened o u t cone.space t ime one has n o warning that one is about to hit it since the curvature is wellbehaved in the region of the singularity ( in fact in this case it vanishes) . Suchsingular i t ies have a topological nature and are known as quasi- regular s ingular i t ies[8]. A quasi- regular s ingular i ty is one which may not be removed s imply by extendingthe space t ime ( a r egu la r s ingu la ri ty ) bu t h as the p roperty th a t the cu rva tu re tenso rc o m p o n e n t s Rahcds) when m easured in a paral le l ly propag ated f rame { E (s)} tend toa well-defined limit along all curves y ( ) end ing at the s ingular ity . The cosmic s tringgiven by ( 1) is therefore jus t a par t icu la r exam ple of a quasi-regular singularity.The cosmic s tr ing descr ibed by a conical spacet ime is an example of a spacet imein which al l the curvature is contained in the s ingular i ty . However if one wan ts tocons ide r cosmic s t r ings in an expan d ing un iver se or the in teract ion of str ings with onea n o th e r a n d t h e s u r r o u n d in g m a t te r , including the effects of the gravitatio nal radia tion,i t becomes necessary to consider cosmic s tr ings in curved spacet imes. In th is paperwe will sh ow how elementary quasi- regular s ingular it ies provide a su i tab le general isa-tion of cosmic s tr ings to cer ta in curved spacet imes and invest igate the proper t ies ofsuch s ingular it ies . In par t icu lar i t will be sh ow n that such general ised cosmic s tr ingsare to tal ly geodesic subm anif o lds on which the mass densi ty p i s cons tan t . The samemeth ods ma y be u sed to s how tha t a general two-d imens iona l a nd t imel ike quas i - r egu la rsingularity is a generalise d cosmi c str ing. In a subs equ ent pa pe r we will consider th isgenera l case an d sh ow tha t they too a re to ta l ly geodes ic subman i fo lds on which themass density is constant.

I

2. Cosmic strings and isometriesThe con ica l spacet ime ( 1 ) may be though t of as ar is ing f rom identify ing poin ts whichare related by a ro tat ion about the z ax is th rough a n ang le cu rather than through theusua l ang le 2 r , the f ixed poin t set of the ro tat ion forming a cosmic s tr ing . Moreprecisely we s tar t with Minkow ski space in cy l indrical polar co ordi nate s , delete the{ r = 0 ) tw o p lane, pass to th e covering space an d then identify poin ts of the form( t , r, q5 + kcu, z ) with ( t , r, 4, z ) fo r k E Z. This proc edu re is readi ly generalised to a nyspacet ime ( M ,g) which ad mi ts an i somet ry f : M -+ M ( f * g = g ) with some fixed pointse t F = {x E M : ( x )= x}. On e p roceeds a s fo llows:

( i ) s ta rt w i th the s pace t im e ( M ,g),( i i ) remove the f ixed poin t set F to o b ta in ( M \ F , g),


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Generalised cosmic strings 3(i i i ) pass to the univecsalAcoveTing pace ( k ,)(iv) l i f t f : M + M t o f : M + M ,(v) ident i fy points in ( k ,*) related by(vi ) if necessary remove som e points to m ake the resul t ing space Hausdorff ,(vi i) end wi th (A?, E) a space with a quasi-regulari ty singulari ty on F.This techn ique is just a specia l case of a meth od d ue to El li s an d Schmidt [8] whichprovides an easy way of constructing spacetimes containing quasi-regular singulari t iesfrom spacetimes with symmetries. We call su ch singulari ties 'elementary' q uasi-regular

singularit ies. In the next section we will sho w tha t if the or iginal spacet ime ( M , g )was a vacuu m solut ion to Einste in 's equat ions, then the result ing spacet ime (G, ) salso a vacuum so lu tion to Einste in 's equa t ions but wi th a del ta fu nct ion l ike energy-m o m e n t u m t e ns o r on F.

3. Singularities and holonomyT h e e x a m p l e of the conica l spacet ime (1 ) shows that the presence of curvature in asingularity m ay be d etec ted by the effect i t has o n geodesic congruences. An equivalentway of measuring this effect is in terms of parallel t ranslat ion round a closed loopsurro und ing the s ingular ity , i .e . in te rms of holon om y. We star t wi th a two-dimensionalexample . Let ( M , g) b e a two-dimensional Riemannian manifold wi th rota t ionalsymmet ry abou t some po in t xo.The n there exists a one- param eter family of i sometr iesfa :M + M ( wh e r e f a represents a rota t ion through a a b o u t Xg ) with fixed point set{x~}. et O < (Y < 27r and let (6,) be the sp ace obta ined by ident i fying points in theuniversa l cover ing space equivalent unde r f a , This space has a qua si-regular singulari tya t xg which we show may be thought of as a point where the curvature has adelta-function-l ike singulari ty.For a regu la r two-d imens iona l Rieman nian mani fo ld on e has by the Gauss-B onne t ttheorem th e fol lowing expression for the integra l of Gaus sian curv ature K over someregion U homeomorphic to a d i sc :1"K d a = 2 7 r - ~ ~ d s ( 4 )I"where K~ i s the geodesic curvature . The term on the lef t is kno wn a s the integratedcurvature . Wh en the sp ace conta in s quasi- regular s ingular it ies we will regard eq uat ion(4 ) as defining the total integrated curvature within aU, inc luding the curvature dueto the singulari t ies. Let ( r , e ) be geodesic polar coordinates for ( M ,g) with xg as origina n d U, be the d i sc rad ius r centre xo. Then , s ince xu s a regular point of ( M ,g ) ,

l i m l K d a = O a n d th u s v2 IC?",~ d s = 27r.r - n U ,I f one deno tes the cor respo nding ob jec ts fo r (G, ) by k, & etc , one has

cylim J Kg d f = limr - o act K~- s = (Y1 - 0 Jau , 2 7

a n d t h u s by e q u a t i o n ( 4 )r


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4 J A G VickersO n e a ls o ha s d 5 = a /2 7 rr d r d e + O ( r * ) , o that k, he 's ingular ' part of I?, is given by

wherefor any set V containing x0otherwise.lV8, d a = { (xO)

Thus the curvature of (& g) is given by

I n orde r to generalise this result to a fo ur-dime nsional spacetime we give a geometricinterpretation to the right-hand side of equation (4). et y : [0, 11+ M be a parametr isa-tion of dU with y ( 0 )= y ( 1 ) . Let { E } a = l , 2e an orthonormal basis at y ( 0 ) an d E ( t )the result of parallel propagation along y . Then E ( 1 ) = L:E(O) wherea aa h

a n dp = 2%-- a K~ d s

i.e. the right-hand side of equation (4) s just the angle the frame is rotated throughas a result of parallel propagation around dU. Thus by using equation (6 ) we maywri te eq uat ion (4) n the equivalent formL = e x p I,;

where s2 is the curvature 2-form with components( 7 )

a=( K ") 'In th is way we may regard the Gauss-Bonnet t theorem as giving a relationship betweenan element of the holonomy group and the in tegrated curvature. For a loo p contain inga regu la r poin t L t as the loop sh r inks down on t? the po int , bu t fo r a loop con ta in ingthe point x0 in ( M ,g) t follows from the way ( M ,g) was constructed that L + L o , arotat ion throu gh 257 - y, as the loop shrinks down . Thus from equat ion ( 7 )

lim J E d 5 = 2 % - - a6,- 0

and aga inI? = K + ( 7 ) 2 ~ 8 , .n - a


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Generalised cosmic strings 5In the four-dimensional case we will use th e appro pria te genera l i sa t ion of ( 7 ) , ratherthan the usua l ve rs ion o f a Gauss-Bonne t t t heorem (4), in ord er to relate the integralsof the ful l four-dim ensio nal curvature to e lements of the holonomy grou p. We willi l lustrate the ideas by considering the case of a spacetime with rotat ional symmetryabo ut some axis. Let ( M ,g) be a n axisymm etr ic spacet ime wi th axis F, so that thereexists a on e-par ame ter family of i sometr iesf l, :M + M leaving F invariant . Let (G,)be the spacet ime P b ta in ed by ident i fying points in th e universal covering spaceequivalent under Ll where a E (0 ,277) is some constant . The n this spacet ime has aquasi-regular singulari ty on the surface F which we now show may be regarded as agenera l i sed cosm ic s tr ing. (The se are just th e type 11, singularities of Ellis an d Schmidt[8].) Let xo be a point o n F. Then we may int roduce 'geodesic cyl indr ica l polarcoord ina te s ' (t , r, 0, z ) in a n e i g h b o u r h o o d o f xowith the fol lowing proper t ies:

(i) the rotat ional symmetry is generated by a l a 0 and the fixed point set is givenby r = 0 ,( i i ) ( t , r , 6 + 2 k 7 r , z ) = ( t , r , 8 , z ) k E Z ,(i i i ) the curves t = 0 , z =O, O= con stan t ar e geodesics wi th affine param eter r,ending a t xo , and wi th tangents or thogonal to F.Let y : [ 1 , O ) +M be the curve y( I ) = (0 , r, 0,O). By choosing a p se u d o - o r t h o n o r m a lf r a m e a t y(1) and para l le l ly propagat ing i t a long y we may define a frame fieldr( ) = { E ( )}", , a long y. Fur therm ore by choosing r( ) appropr i a t e ly and by impos ing

addi t ional condi t ions on t h e t a n d i' coord ina te s we may a r range tha ta

a a aa n d E ( r ) + - as r + O .p,+, f i r '+ ; 4 d ZWe now para l le l ly propagate E( r ) a r o u n d t h e l o o p s r = constant , t = 0, z = 0 i n o rde rto def ine E( r, e ) . This in turn enables us to define a Lorentz transformation L $ ( ) , th ee lement o f ho lonom y genera ted by the lo op r = cons tan t , by

a

a

E ( r , 2 7 r ) = L " , r ) E ( r , O ) .U a

I t fol lows from the resul ts in the appendix tha t the appropria te genera l i sa t ion ofequa t ion (7 ) is

L ( r )= P, e x p ( h d 6 )6,

( 9 )where fi' h - k u h ' d X c Y d ( a , 6, c, etc , be ing f rame indices) , f i r , ,={ x EG: = 0 , t = 0, r < ro}, X a n d Y are uni t vectors in the r a n d 6' directions respec-t ively a n d P, exp denotes an r -pa th o rde red exponen t i a l . O n the o the r hand , it fo llowsfrom the way (A?, g') has been const ructed tha t as r + 0, L ( r ) -+ L where L describes arota t ion through p = 27r - 0: a b o u t t h e z axis . Thus

1 00 c o s p s i n p = e x po - s i n p c o s p o 0 -p 0 0 .


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6 J A G VickersIn the l imit a s r + O the pa th o rde red exponenti a l becomes an o rd ina ry exponenti a la n d we o b t a in

lim C l: d 6 = lim R,,, = ,8 =27r - y.1- 0 r - 0 IS0, 0,

c c c c

Iim J Jd 6 = lim R,,, = p = 27r - y.

r - 0 r - 0 J J0, 0, LO n e a ls o h a s d 6 = c ~ / 2 r r r r dB + O ( r 2 ) a n d t hu s R2,zi, the s ingular par t of k 2 3 2 , ,sgiven by Rz, , , = (+)ZTS2.7-cyPara ll el t r ansp or t a ro und any lo op no t con ta in ing F gives the ide nti ty L orentz transfor-mat ion as the s ize of the lo op is shrunk to zero an d thus E,,, , and compo nent s re la tedby the obvious symmetr ies , a re the only par ts of the curvature with a singular part .The regular par t of the curvature i s of course unchanged under the ident i f ica t ion bythe i sometry. I f one no w calcula tes the Einste in ten sor an d appl ies the f ie ld equat ionse a -- Sn-GfE, on e f inds tha t the only non-zero par ts of the energy-momentum tensora re

where1 297-cyP=,(-a).

(fi, ) may thu s be regarded as a spacetime with a distribution valued curv ature whichsat isf ies the vacuum Einste in equat ions apar t f rom a l ine source which has the sameenergy-momentum com pon ents as the usual cosmic st r ing, an d we therefore regardthe s ingulari ty as a genera l i sed cosmic s t r ing. I t is impo rtant to note tha t the onlyfeature of th e (fi, ) which was important in the above argument was the existenceof a non-t r ivia l holo nom y gro up associa ted wi th the s ingular ity , namely on e genera tedby a rotat ion through 297- y ab ou t the direction of the singulari ty. In fact it is possiblet o sh o w [9] tha t a genera l two-dimensional an d t imelike quasi - regular s ingular i ty h assuch a holonomy group associa ted to i t and therefore a lso has energy-momentumtensor given by equat ion (1 1 ) . Conversely o ne can sh ow that any dis t r ibut ional solut ionof Einsteins equa t ions wi th energy-momentum tenso r a l ine source given by eq uat ion(1 ) , an d the regular par t of the curvature suff ic iently well behaved, mu st be a spacet imeconta ining a quasi - regular s ingular ity . The deta i l s of th is ar e som ew hat compl ica tedan d will be deal t wi th separa te ly in a future pap er [ lo ] . So far we have dealt with aho lonom y group gene rated by a rotat ion, but by making identifications under a boosto r null rota t ion on e can obta in e lementary quasi - regular s ingular it ies w i th holonomygro up genera ted by a boost o r null rotat ion. These are again spacetimes with distribu-t ional va lued curvature but the corre spon ding dis t r ibut ional energy-momentum tenso ris unph ysical in these cases since i t fai ls to sat isfy the weak energy condit ion. Th ustwo-dimensional t imelike quasi-regular singulari t ies in general , and the elementaryexamples const ructed in 9 2 in part icular, provid e the required gen eralisat ions of cosmicstrings. I n the next section we will examine some of the propert ies of elementaryquasi-regular singulari t ies, regarded as generalised cosmic strings.


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Generalised cosmic strings 74. Properties of generalised cosmic stringsTh e most im po rtan t pro per ty of an elem entary quas i-regular singulari ty is that i t is thef ixed point se t of an i sometry an d this in turn m eans tha t i t forms a tota l ly geodesicsubman i fo ld a s shown by th e p ropos it ion be low [111.Proposition. Let f : +M be an isometry with fixed point set F, t hen F is a totallygeodesic subm ani fo ld o f M .Proof: (W e give a sl ight ly ada pte d version of the proof in [111.)Le t x E F a n d c o n s i d e r D,f: ,M + T, , , ,M = T,M. We define D F ={ V E T , M : D , f( V ) V } o be the f ixed point se t of the der ived mapping. Now le tD U b e a n e i g h b o u r h o o d of the or igin in the T,M such tha t exp , :DU + M is one- to-on eand l et U = exp , ( D U ) . Th en i t is easy to see that U n F = e x p , ( D U n D F ) . Let y ( )be s ome curve in M with y ( 0 ) = x a n d X ( ) a parallel vector field along y , then sincef i s an i sometry f * ( X ) is parallel along f ( y ) . Now le t y be some curve in F a n dc h o o se X ( 0 )E DF, t h e n f ( y ) = y a n d ( f * ( X ) ) ( O ) X ( 0 ) o by the un iqueness of paralleltranslat ion f , ( X ) = X . T h u s X ( t ) s in the f ixed point se t off* for any point on y.However, the fixed point set is tangent to F so that a vector ini t ial ly tangent to Fremains tangent to F under para l le l t ransla t ion a long y. F i s thus an autopara l le lsu b m a n i f o l d of M , an d hence a to t al ly geodesic subman i fo ld o f M .T h e i m p o r t a n c e of the ab ove resul t is tha t i t imposes considerable const ra ints o n thedynamics of a genera l i sed cosmic s t r ing. A re la ted const ra int involves the holo nom yof an elementary quasi-regular singulari ty which is generated by a rotat ion throughth e same an gle 2 7 ~c y for a l l points o n the s ingulari ty an d hence th e mass densi ty pi s constant . Wh en on e considers possible interac t ions of cosmic s t rings one can extendthis result to prove a ' conse rva t ion o f ho lonom y ' theorem. In the (2 + 1) -d imens ion a lsett ing this is equivalent to th e Bianchi identi t ies of Regge calculus or the conse rvationof the mom ent um of a system of gravi ta ting poin t par tic les [12] . In the ( 3+ 1 )-dimensional case where the h olon omy g rou p is no longer Abel ian i t is hop ed to interpre tthis resul t as a conservat ion of mom entu m for interact ing cosmic s t rings (see [ 101 fo rfur the r de ta i l s ) .

Appendix. The relationship between integrated curvature and holonom yIn this appe nd ix we re la te the effec t of para lle l pro paga t ion ar oun d so me c losed curvey to the integra l of the curvature over some spanning 2-surface U wi th boundaryaU = y. We parametr ise U by mean s of the fol lowing func t ion. Let 4 : [0 , l ] ?+ M b ea sm o o t h m a p su c h t h at

( i ) 4 , [0, 1 ] + M , defined by 4 , ( s )= +(s, ) is a closed loop,( i i ) 4 ( 1 ~ ) =, + , ( I ) = ~ u ,(i i i ) &( t ) = cons tan t = xo .Now le t U" be so me or thonorm al f rame a t 4 (0 , ) . We parallel ly propagate this frame

a long s = 0 a n d t h e n a r o u n d t = cons tan t to ob ta in a l if t & of 4. Tha t i s, & : [0, 112+ L Mis a sm o o t h m a p su c h t h a t( i ) &CO, 1) = U0( i i ) T O & ' = +


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8 J A G Vickers

( i i i ) +*($)I d(n . o) = o -(.*(:))=owhere w is the connec t ion fo rm on the bund le o f o r thon orm a l f ram es L M . We nowuse 4 t o d e t e r m i n e t h e e l e m en t of the ho lonomy gro up ob ta ined by pa ra l le l p ropaga t ionr o u n d y r= 4 r ( ) by defining L( ) E2: according toL ( t ) 4 ( 0 , )= . ( I , t )or

L ;( t ) E ( O , )= F(1, t )a

where 6 ( s , t ) = { E ( $ , ) > 2 = = , .We have f rom equa t ion ( A 6 ) in [6]

L r ( t , ) i 2 h ( s , ) d s d t ,where

Solving the in tegra l equat ion ( A l ) by repea ted i t e ra t ion we ob ta in

x . . . R:;(s, t , ) f l > ( s , t , ) d s d t , . . .d t ,or suppress ing indices

where : : ind ica te s tha t the expres sion shou ld be o rde red so tha t te rms with a smal lert va lue p recede those wi th a h ighe r t va lue . Wi th ou t the o rde r ing ( A 2 ) wou ld s implybe the exponen t ial o f

I' + ( I 2 )We therefore write ( A 2 ) formal ly as

where PI exp den o tes the t -o rde re d pa th exponen t ia l de f ined by ( A 2 ) . Note tha t a s aconsequence of the Bianchi ident i t ies and the genera l ised Stokes ' theorem the r ight-hand s ide o f (14) i s independen t o f the cho ice o f spann ing su r face U a n d M onlyd e p e n d s o n y = d U.


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Generalised cosmic strings 9References

[ I ] K i b b l e T W B 1976 J . Phys. A: Marh. Ge n . 9 1387[ 2 ] Vilenkin A 1981 Phpy. Reo. D 24 2082[3] Vilenkin A 1981 Phys. Reo. Letr. 46 1169[4] Zeldovich Y B 1980 Mon. Not. R. Astron. Soc. 192 663[ 5 ] M ar d e r L 1959 Proc. R. Soc. A 252 45[6] Vickers J A G 1985 Class. Qu a n t u m Gr a v . 2 75 5[7] Linet B 1985 Gen. Re/ . Grav . 17 1109[8] Ell is G F R a n d S c h m i d t B G 1977 Gen. Rel . Grav. 8 91 5[9 ] Vickers J A G 1980 DPhil thesis University of York

[ lo ] V icker s J A G Quasi-R egular singularities and C osm ic Strings in p repara t ion[ l l ] K o b ay s h i S a n d N o m i z u K 1969 Foundations ofDifSerentia1 Geomerry (N ew York : In ter sc ience) ch VI1[12] RoEek M an d Wi l l i am s R M 1985 Class. Q u a n f u m G r av . 2 70 1

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J A G Vickers- Generalised cosmic strings