Two-axion string wormholes

Volume 233, number 1,2

PHYSICSLETTERSB

21 December 1989

TWO-AXION STRING W O R M H O L E S K TAMVAKIS Dlvtston of Theoretwal Phystcs, PhysicsDepartment, Umverstty of loannma, GR-451 10 loanmna, Greece

Received 9 September 1989

Wederive and analyzewormholelnstanton solutionsgeneratedby both ax~onsof the effectivefour&menslonaltheoryobtained from a compactlficahonof the ten &menslonal superstrmg

Although quantum gravitational phenomena are expected to be relevant only at scales of the order of the Planck length, there exist, possibly, exceptions that have an influence on physics at much larger scales. Such effects are the ones associated with topology changing configurations (wormholes) [ 1-3 ] A wormhole is a euclidean solution to the equations of motion of gravitation and matter fields, which corresponds to a geometry of two asymptotically fiat regions of space-time connected by a tube A number of interesting consequences have been attributed to the existence of wormholes such as a quantum indeterminacy of the constants of nature and the proposal that the universal wave function has a delta-function peak at vamshmg cosmological constant [ 4-6 ] Although superstrmgs provide a general framework for a quantum theory of grawty we are still quite far from having understood how to consistently obtain such a theory All investigations related to topology change and wormholes have been derived within the poorly understood euclidean path integral formulation of quantum gravity Wormhole solutions come out as stauonary points of the euchdean acuon and one expects that they should make a significant contribution to the euchdean path integral in the semlclasslcal regime ( h ~ 0 ) The wormhole contribution to the euchdean path integral ~s interpreted as the amplitude for a tunnehng event corresponding to the nucleation of a closed "baby universe". The first exphclt wormhole solution was derived by Glddlngs and Strommger [ 3 ] in a superstrmg-motlvated simple system of Einstein gravity minimally coupled to a second rank antisymmetnc tensor field (axlon) [7] Wormhole solutions were subsequently derived by Lee [8 ] m the theory of a canomcal complex scalar field mmlmaly coupled to gravity for the case of a spontaneously broken U ( 1 ) global symmetry Wormholes arise as extrema of the action that enters m the transition amphtude between states of defimte charge Abbott and Wise [ 9 ], Coleman and Lee [ 10 ] and also Grmsteln [ 11 ] derived wormhole solutions for a canonical complex scalar field coupled to gravity in more general cases The effective low-energy field theory resulting from the ten &mens~onal superstrlng with six compactlfied internal dimensions [12,13] possesses two axlon fields a, a and two dllatons f, ~, combinations of the ten &menslonal ddaton and the breathing mode that determines the size of the compact manifold. These scalars are often expressed in the form of the complex fields T = v / 3 exp(f/v/3) + l a and S=exp(tp) + i a In their original paper Glddmgs and Stromlnger [ 3 ] considered the wormhole solution generated by S alone and found that it exhibits a singular behawour, giving an mfimte value for ~ and the action, for a finite value of the scale factor. It has been shown [ 14], however, that the coupled system of T with grawty, including the ra&atlvely corrected scalar potential for T, leads to wormholes that can be interpreted as giving rise to the nucleation of expanding universes m Mmkowskl space [ 15 ] Very recently Glddmgs and Strommger [ 16 ] have reexamined the axlondilaton system and analyzed the non-singular wormholes generated by T alone. 0370-2693/89/$ 03 50 © ElsevxerSciencePubhshers B V ( North-Holland )

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Volume 233, number 1,2

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21 December 1989

In the present article we work out the general case of wormholes generated by both T and S We find that wormholes with finite action and non-singular behavlour exist for a range of parameters The two extremal cases have infinite action due to the asymptotic growth of the dllatons with the scale factor In these cases the wormhole effects in the low-energy theory can be represented with local operators determined in terms of a cut-off action A peculiar asymptoUc decompacUficatlon of the internal dimensions appears m these cases The effective low-energy action for the two axion-dilaton fields S and T that results from the ten dimensional superstrmg by compacUfylng six internal dimensions on some manifold can be put in the form [ 12 ] (in euclidean space) SE= f d 4 x x / / g ( - R 4 - 6

10~rl = (T+/~)~ + 2

10~SI2 ~

(S+~)2 J ,

(1)

neglecting any scalar potential terms. SE can equivalently be expressed In terms of real scalars as BE ~_ ;

d4x N//g [ __~.~_1 (Ouf)2+ ½(0Ua)2 exp( - 2f/x/~ ) + 1 (Ou~)2+ ½(Oua)2 exp(-2¢0)

],

(2)

with the identification

T-v/3exp(f/x/~)+la,

S - e x p ( ~ ) +w~

(3)

A particular combination of f and tp defines the breathing mode that determines the size of the internal manifold In fact, we have

dsZ=gu,(x) dxU d x " + exp (~P+f/x/~) gmn(Y) dY 'n dy"

(4)

There are two conserved U ( 1 ) currents, namely J ~ = e x p ( - 2 ~ p ) 0 u a and J~r=exp(-2f/x/3)OUa, corresponding to global translations of a and a The charge densities j o = p and j o _ n coincide with the conjugate momenta of a and a Let us consider the transition amplitude between elgenstates ofdefimte charge density on given spacehke hypersurfaces Z"and Z" [ 10,11 ]

~=(n(Z),p(Z)ln'(X'),p'(S'))=

f

(d/z) exp(--SE)

(5)

The measure if integration refers to (dg)u.(dc~) (da) [dip exp(-~0) ] [ d f e x p ( - f / x / ~ ) ] The boundary conditions in (5) have given rise to the surface terms incorporated ~n~

The vectors n u and p U in (6) are conserved vectors with the property nu~/uiz= n(-Y) and pUqulz=p(X ) The vector ~/u is a future directed unit vector orthogonal to S By dZ"u we mean qu dZ'= t/ux/~ d3x The charges on 2" are defined as

Qr= ~ d~UPu, Qs= f cL~Unu

(7)

Variation of the action (6) leads to the equations of motion

Ru~ - ~gu.~= ½0ufOJ+ ½0u~oO~o+ ½0uaO~aexp( - 2f/x/~ ) + ½0ua O.a exp(-2~o) - ~gu.C , [Elf=- ~

1

(Oa)2 exp(- 2f/x/~).

O~,[v/ggU"(O.a)

[3tp=-(0a)2exp(-2~o),

e x p ( - 2 f / v / 3 ) ] =0.

0u[v/ggU"(0~a)exp(-2~o)] = 0 ,

#' In SEwe consider included the Gibbons-Hawkingsurfaceterm 108

(8)

Volume 233, number 1,2

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21 December 1989

and the boundary conditions [qu(0ua) e x p ( - 2 f / x / ~ ) lz, z, = i p ( Z , S' ),

[qu(Oua) e x p ( - 2 ( o ) ]z,z =17r(S,Z' ) .

(9)

It is evident in (9) that we have obtained imaginary saddle points for the axlons a, a [ 10 ]. In order to be able to obtain a solution to (8) and (9) we consider the case of homogeneous charge densities and p together with the usual ansatz for the metric ds2_- ( d z ) 2 + R 2 ( z ) (dg23) 2 and look for r-dependent solutions f ( z ) , ~0(t), a (z), ct (z) and R (z) According to (9) the relation between charges and densities of a hypershere of radius R (z) will be

QT

p = 27~2R3 ,

Qs

7z= 2~zZR3

(10)

With this ansatz the transition amplitude under calculation will be proportional to Aoc g ( Q r ( S ' ) --QT(~,) ) g ( Q s ( S ' ) -Qs(,Y,) ) e x p ( - S )

( 11 )

The delta functions have resulted from the integration of the zero modes a ( S ) , or(S) In ~ we substitute the solutions for the fields which we shall obtain from (8) and (9) The equations for the axions are

IQr & exp(-2tp) = iQs ap -2f/x/~= 27t2R3(z ) , 27r2R3(z) '

(12)

and can be substituted in the rest of the equations yielding

~ exp(2~)

q~" R - l = g : ,'oz{\ ~I'c2_l_ j - i vl~z " - ~--gexp(

--2flvi3)--

f + -~--f= 3R qZr 6 exp(2f/x/~), v/-~R

~0+ -~3R ~b= ~g q2 exp(Ztp)

(13)

We have introduced the effective charges qr.s- QT,s/27r 2. Taking the neck of the wormhole at z = 0 we must h a v e / ~ ( 0 ) = 0 For simplicity we can also take ) ' ( 0 ) = ~b(0) = 0 although wormholes exist for non-zero values o f f ( 0 ), ~b(0) under certain conditions. The wormhole size R (0) = L is related to the other two remaining constants of integration by

L4=~2{qZ-exp[2f(O)/x/~] + q2 exp(2~0(0) }.

(14)

The scalar field equations immediately yield

!2fZR6-- ½q2exp(2f/x/~) = -- ½qZrexp [ 2 f ( 0 ) / x / ~ ] ,

(15a)

½(oZR6_½q2exp(2~o)=

(15b)

-

, 2 exp [2tp(0) ] , ~qs

which substituted in the scale factor equation gives the wormhole differential equation with the known solution [31

R2

1= - (L/R) 4 .

(16)

Integrating (15a) and (15b) we get exp{ - I f - f ( 0 ) ]/x/~} = cos [7 c o s - l ( L 2 / R 2) l ,

(17a)

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Volume 233, number 1,2

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21 December 1989

exp{ - [ ~ - ~0(0) ]} = cos [x/3 ( 1 - 7 2) c o s - I ( L 2 / R 2 ) ] ,

(17b)

where we made use of (14) and have introduced the parameter (18)

qr 7 ~ 2 ~ 3 L 2 exp[f(O)/x/~]=x/q~+q~exp{2[~(O)_f(O)/x/3]~ In the case that the field values at the neck o f the wormhole are zero the parameter 7 is Y= ~

qr -

(19)

In any case 7 is a number between zero and one The ngbt band s~de of (17a) is the cosine o f an angle between zero and ½n, tbe latter value being acb~eved only for R = oo In the special case y = 1 [qs= 0 and tberefore ~0= ~p(0) ] ~t takes tbe sxmpler form exp{ - [ f - r i 0 ) ]/x/~} =L2/R 2

(20)

In contrast, eq (17b) can have singular bebavlour Tbe factor m front of tbe reverse cosine [3 ( 1 - 72)] 1/2 = [ 1 + 2 ( 1 - 372) ] 1/2 becomes greater tban one for y < Yc= v/I- Tben, tbe argument o f tbe cosine becomes ½n at a fimte value/~ o f the scale factor g~ven by /~=L{cos [ ½n/x/1 + 2 ( 1 - - 7 2 / 7 c ~ ] } -1/2 ,

(21)

and the corresponding value of the ddatlon becomes mfimte ~o(/~) = oo For 7 = 7¢= x/~, we get the simple soluUon exp{- [~0-~0(0)]}=L2/R 2

(22)

Values above the critical value are safe Tbus, we bave obtained non-singular wormboles for x/~<7...< 1 The non-singular radml ax~on wormbole obtained by G~ddmgs and Strommger [ 16 ] corresponds to 7= 1 As was observed by tbe same autbors for the case o f the 7= 1 soluuon, the internal dimensions dccompactlfy asymptoucally as R - . oo Tb~s is a property o f the two extreme cases ( 7 = 1 and 7= ~ ) only as can be sbown ~mmedmtely. According to (4), tbe relaUve factor m front of the internal metric is

exp(~o+ f/x/~)=exp[~o(O)+f(O)/x/~]

{cos[ycos-~(L2/R2) ] c o s [ x / ~ - 7 2 )

cos-l(L2/R2) ]} -~

In the case cases 7=1 and 7 = x / ~ , it becomes correspondingly (R2/L 2) e x p [ ~ o ( 0 ) + f ( 0 ) / x / 3 ] and (R2/ L 2) e x p [ ~ 0 ( 0 ) + f ( 0 ) / x / ~ ] [ c o s ( n / x / ~ ) ] -~ m the hmlt R--,oo Tbus, in botb tbese cases the internal dlmens~ons decompactffy asymptotically. For all otber values o f y, bowever, exp[~p(0) +f(O)/x/-3] exp(~°+f/x/~) ~ cos(½Yn) cos[ ½ x / ~ 1 - 7 2 ) n ] '

(23)

no sucb pbenomenon is exhibited and the internal space stays 1/R 2 Umes small m comparison to euchdean space. The wormhole action SE iS comprised of three different terms, Le the volume integral SE, the surface term due to the axlon boundary condlnons and the G i b b o n s - H a w k i n g [ 17] surface term - 2 f c l X ~ (k u - kou) which ~s the integral o f the extnnslc curvature of the boundary minus that of the boundary embedded in flat space The last object is - (0/'dr/) f d X - - - 2 0Z/0~/equal to the derivative o f the area o f the boundary along the outward normal In our case this amounts to - 2 ( d / d z ) 2hER3= - 12n2R2R For fiat space we have R = 1, and therefore the G i b b o n s - H a w k i n g term is -

12n2R2(R - 1 )~6nZL4/R2--*O

The volume term of the a c n o n vanishes ldenncally This can be seen by taking the trace o f the Einstein equanon in (8) which gives ~ = ~ Substituting m (2) we get S E = 0 Thus, finally, we have llO

Volume 233, number 1,2

SE = -1Qs[ot(~) - a ( 0 )

PHYSICS LETTERS B

21 December 1989

] - 1QT[Ot ( ~ ) - o t ( 0 ) ]

The acUon for a full wormhole ( - ~ axlons, we get

(24)

< r < + ~ ) is twice SE Substituting in ( 2 4 ) the derived soluUons for the

Sw=(2nL) 2 { v / ~ 1 - y 2) t a n [ ½ n ~ / 3 ( 1 - 7 z) ] + 3~,tan(½n~)}

(25)

F o r the extremal values ? = 1 and y = v/~-~,the acUon becomes mfimte. It is useful to compute the action cut-off at some value ~"corresponding to a finite scale K = R ( t ) or, eqmvalently, to corresponding background values f ( / ~ ) , ~0(K) We get Sw[f, ~0] = Q s e x p ( ~ 0 ) n / 1 - e x p { - Z [ ~ o - ~ o ( O ) ] } + Q r n / ~ e x p ( f / v / 3 )

41-exp{

(2/w/3) [f-f(0) ] }

~Qs exp(~0) s m [ ½ n ~ / 3 ( 1 - 7 1 ) ] +Qrw/3 e x p ( f / n / 3 ) sm(½n~,)

(26)

In the effecuve low energy theory at scales >> L ~t should be possible to represent wormhole effects by the a c u o n o f local operators that have the wormhole q u a n t u m numbers A m p l i t u d e s hke (5) can be c o m p u t e d over t n v m l configuraUons by an insertion o f the o p e r a t o r V= e x p ( - S w )

exp[ -l(Qsot+Qra) ]

(27)

weighted by an a p p r o p r i a t e d e t e r m i n a n t and zero-mode factors W h e n the action is infinite one could use the cut-off action (26 ) m terms o f which the wormhole vertex o p e r a t o r takes the form V~ [ exp( - T ' ) ] Q r [ e x p ( - S ' ) ]Qs

(28)

with T' = x a + x/~ exp ( f ' / x / ~ ) = I a + x/~ exp ( f / x / ~ ) s i n ( ½zr~), S' = l o t + exp(~0' ) = l O t + exp(~0) s m [ ½ x / ~ v / ~ - r

2) n] .

C o n t r a r y to the case o f a canonical scalar field [ 10 ], here, the wormhole vertex operators d e p e n d non-trivially on the p a r a m e t e r s The author wishes to thank the G r e e k M l m s t r y o f Research a n d Technology for partial support and the C E R N Theory Division for partial support and hospltahty

References [ 1] S W Hawking, Phys Rev D 37 (1988) 904 [2] G V Lavrelashvdl, V Rubakov and P G Tmgyakov, JETP Lett 46 (1987) 167 [3] S B Glddmgs and A Strommger, Nucl Phys B 306 (1988)890 [4] S W Hawking, Phys Lett B 134 (1984) 403 [5]E Baum, Phys Lett B 133 (1983) 185 [6] S Coleman, Nucl Phys B 310 (1988) 643 17] R C Myers, Phys Rev D 38 (1988) 1327 [8] K Lee, Phys Rev Left 61 (1988)263 [9] L F Abbott and M B Wise, Caltech prepnnt CALT-68-1523 (1989) [ 10 ] S Coleman and K Lee, Harvard prepnnt HUPT-89/A022 (1989) [ 11 ] B Grmstem, Nucl Phys B 331 (1989) 439 [12] M Dine, R Bohm, N Selberg and E Wxtten, Phys Lett B 156 (1985)55 [ 13 ] P Candelas, G Horowltz, A Strommger and E Witten, Nucl Phys B 258 (1985) 45 [ 14 ] K Tamvakls and C Vayonakxs,Umverslty ofloannma prepnnt IOA-224 (May 1989 ), Nucl Phys B, to appear [ 15 ] V A Rubakov and P G Tmyakov, Phys Lett B 214 (1988) 334 [ 16] S Glddmgs and A Strommger, Harvard preprlnt HUPT-89/A031 (6/89) [ 17] G Gibbons and S W Hawking, Phys Rev D 15 (1976) 2752 111