Strings in strong gravitational fields

Physics Letters B 295 (1992) 214-218 North-Holland

PHYSICS LETTERS B

Strings in strong gravitational fields H.J. De Vega LPTHE 1, Universit~ Paris VI, Tour 16, ler ~tage, 4 place Jussieu, F- 75252 Paris Cedex 05, France

and A. Nicolaidis Theoretical Physics Department, University of Thessaloniki, GR-54006 Thessaloniki, Greece

Received 11 September 1992

We study string propagation in curved space-time. In such a problem, the equations of motion and the string constraints are nonlinear and difficult to solve. We propose here a systematic expansion in c, the world-sheet speed of light, to solve the string dynamics. Since c is proportional to the string tension, this amounts to a large tx' expansion. To zeroth order each point of the string moves along a null geodesic (null string), while the first order correction incorporates the string dynamics. As an illustration we apply our formalism to the Robertson-Walker geometry. In this case, it turns out that the string expands or contracts at the same rate as the whole universe.

As a first step on the understanding of q u a n t u m gravitational phenomena in a string framework, a program was started in 1987 on string quantization in curved space-times [ 1 - 7 ] . For some relevant backgrounds (gravitational shock-waves [4,8] and plane waves [ 9 ], conical space-time [ 5 ] ) the string dynamics was solved in closed form. Generally the nonlinear string dynamics does not allow an exact solution and one has to resort to either numerical methods or an appropriate approximative scheme. In ref. [ l ] a systematic expansion has been proposed around the geodesic motion o f the center of mass of the string, with the string oscillations treated as perturbations. This expansion corresponds to low excitations of the string as compared with the energy scales of the background metric G ~ and it remains valid whenever the metric G ~ does not change appreciably over distances of the order of the string length. Since the length of the string scales with x / ~ 7 ( a ' is the inverse string tension) the expansion parameter is the dimensionless constant ~ = (x/-a 7)/Rc where Rc characterizes the curvature o f the spacet Laboratoire Associ6au CNRS UA280. 214

time under consideration. In the limit a ' --, 0 the string shrinks to a point and therefore it is not sensitive to possible variations of the gravitational field. The breaking down of the above approximation is signaled by the appearance of imaginary frequencies for the string oscillators [ 1,10-12 ]. The instabilities thus developed have been already studied [ 10,11 ]. Clearly it is desirable to obtain a systematic description o f the opposite limit, where the gravitational field varies significantly over distances of the size of the string length and where the relevant small parameter is g = Rc~o

.

Interest for strings with a small (or vanishing) tension arises also in a different context. String theory is a candidate theory for quantum gravity. The approximation scheme used so far is an expansion in powers o f a ' , thus obtaining an effective theory containing the massless modes. If we measure energy in string units (E/x//Too), then it is clear that the limit To-~0 corresponds to the high energy limit of the string theory (like the m - . 0 limit of particle theories is equivalent to the high energy limit). At energies larger than the Planck mass we might clarify the fundamental structure of q u a n t u m strings and consequently the Elsevier Science Publishers B.V.

Volume 295, number 3,4

PHYSICS LETTERS B

short-distance structure of space-time. The action for a closed bosonic string in an arbitrary D-dimensional riemannian manifold with metric Gu. is given by

3 December 1992

Defining c=22To, the string equation of motion reads

• p-c2X"P+FPr.x(J(xJ(X-c2X'KX 'x) = 0 , S = - T o f dzda

-~,

(1)

(10)

( 11 )

where gab is the induced metric

where F is the Christoffel connection for the metric Gu,. The constraints now become

gab = OaXu ObX" G.,,,, ,

j(ux'"Gu. = O,

(12)

(2)

J(uJ("Gu. + cEX'UX'"Gu. = O .

(13)

(3)

Identifying the conjugate momenta obtained from the two lagrangians, eqs. ( 1 ) and (9), gives 22 = 1/ To and we recognize that a disturbance propagates along the string with the speed of light (c= 1 ) in world sheet coordinates. We may now proceed into a systematic expansion in powers of c. Since c is proportional to the string tension To, this amount to a small To expansion (large a ' expansion). Writing

O<~a,b
O<~#,v<~D-1.

the conjugate momentum P~ to X u is

pu= OL

02---~= ~

To

[(x'~x',c.,)::.o,.

- ( X u X '"Gu, ) X " G , u ] ,

from which we identify the constraints

~'l = PuX'"Gu. = O,

(4)

~2 =Pup'Gu. + T2 X'uX'~Gu~ = 0

(5)

and that the hamiltonian density is zero (a dot and a prime denote, respectively, differentiation with respect to the world-sheet time and space variables, and a). The action eq. ( 1 ), does not allow us to reach the limit To--,0. Following the analogous massless particle case we introduce the hamiltonian density (6)

H=2~2 +P~l ,

where 2 and p are Lagrange multipliers. This leads to a phase space lagrangian

L = PuJ("Gu~ -

,~[I,I 2 - p ~ l l

I .

(

7)

Integrating out the momenta we arrive at

L = 1 [ (j(uj(.Gu~) _2p(J(ax,~Gu.) + (P 2-422T2) (X'uX'"Guv) ] •

XU=AU+cZB~'+ ....

and substituting the above expression into eqs. ( 11 ), (12), ( 13 ) we find to zeroth order in c

A>+r~AxA~=0,

(15)

AuJI"Gu,,=O ,

(16)

AUA"Gu,,=O.

(17)

We observe that AU(a, z) describes a null string [ 13 ], i.e., every point of the string moves independently along a null geodesic. The only reminiscence from the string is the constraint, eq. (17), which requires the velocity to be perpendicular to the string. To first order in c we obtain the equation of motion for BP(a, z):

(8) =A "p+ F~A '~A'a,

Reparametrization symmetry permits to set p = 0. In this gauge, the lagrangian becomes

L = -~2 [ (J(uJ(~Gu") -422T2(X'uX"Gu~ ) ] '

(14)

(9)

and the string appears as a collection of points with the string tension providing nearest-neighbor interactions (harmonic forces).

( 18 )

supplied with the constraints

2.4UB"Gu.+.4UAWRGu.:+A'UA'"Gu.=O,

(19)

BUA'"Gu,, +A UB'"Gu,, +A UA'"BaGu,,o = 0,

(20)

1-'~,,~(Gu,,,.) indicates the derivative of F~(Gu,,) with respect to X". For vanishing string tension To, the length of the 215

Volume 295, number 3,4

PHYSICSLETTERSB

string becomes infinite with respect to a typical fixed length. Since any disturbance propagates along the massless string with a finite speed, it takes an infinite amount of time for the disturbance to reach from one end to the other of the string. Equivalently we can state that the speed of propagation is infinitesimally small. As an application of our formalism we consider the Robertson-Walker metric

dS2=dX02 --R2(Xo) (dX2~ +dX~ + d X ~ ) .

(22)

(23)

where the momenta Pi may depend upon a but they are independent of z. the constraint, eq. (16), takes the form "2 2 "2 Ao=R (A, +A~+.4~).

(24)

(25)

3

(26)

i=1

with

Fo(cr, z ) = A ~ - 2

(28)

P, lo2 Ai(a, z) =I~ + ~oo-2- [exp(2Por) - 1 ] ,

A'o

(29)

(30)

For the given metric, the equations of motion for 216

Ai •

AoJ~

(34)

It is evident that the general solution for B u is very sensitive (through the functions F u) upon the initial shape and momentum of the string. To simplify the calculation we assume that at z = 0 the string forms a circle of radius a in the X1-X2 plane and furthermore all the points of the string move with the same momentum P1 parallel to the X1 axis, i.e.,

P2 =P3 = 0 ,

I2=asine,

(35)

Po =P1 - P = c o n s t .

(36)

Io(a) = (2a cos or+ 2~)1/2,

(37)

Ao = (2a cos e + 2 ~ ) 1/2 e x p ( P z ) ,

A2 =a sin a , A3 = 0 .

(38)

It is instructive to evaluate the spatial length element of the string [between (or, z) and ( a + d a , z)]. We find a 2 cos2~

where I~ and Pa are functions of a only. The constraint, eq. (17), provides a relation between the initial four-momentum and the shape of the string:

P°l'°= l (, , = i P,I,).

,

(33)

i = 1, 2, 3.

dS 2 = -

i = 1 , 2 , 3,

23i ffi l /A,~2 A~2 l,,,-x i ] A3 + Ao '

A~ = (a cos e + ~ ) exp(2Pz) - ~ , (27)

In this case the solution for A u is given by Ao(a, z) =Io exp(Poz) ,

(32)

i = 1, 2, 3,

with ~> a. the overall solution for the A's reads now

Let us assume that the cosmological scale factor R behaves like

R= 1/Xo.

E - 2 P o B i =F~(tr, z ) ,

From the constraint, eq. (30), we find that

and the constraint; eq. ( 16 ), implies p 2 = ~ p2.

(31)

11(a)=acose,

Eqs. (22) and (24) lead to RAo=Po,

Bo - 2PoBo +P~Bo =Fo(a, r) ,

r,(a, z) =-A~'-2 A---~A,+ 2 ~oo Bo - 2 - ~ o Bo,

where R' -- OR/OAo. We obtain also the first integrals

R2Ai=P~, i = 1 , 2 , 3,

B u, eq. (18), using also the constraints eqs. (19), (20), read

(21)

The equation of motion of Ao is

Ao +RR' (,~2 + ~ +.~2) = 0 ,

3 December 1992

(2a cos a + 2 g )

exp( -

2Pz) d o "2

= - (Ra cos a) 2 da 2 .

(39)

We thus see that the string follows the cosmological evolution, that is the string expands at the same rate as the whole universe ~1. We can compute also the energy-momentum tensor for the null string. The en~1 A n a l o g o u s p h e n o m e n a were f o u n d in refs. [ 1,10,11 ].

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PHYSICSLETTERSB

ergy-momentum tensor T u " ( A ) at the space-time point A, is written as [ 14]

T~'~(A) =-~

(40)

where AU(cr, z) are the string coordinates. It is convenient to integrate T u " ( A ) over a spatial volume, completely enclosing the string at fixed time A o, as was proposed in ref. [ 14]. Defining

-xf--~t G TU~(A) d3A,

(41)

we find that IOO= -~ n PAo,

(42)

= -~PA~.

(43)

1ol

We observe that the total energy (momentum) of the string grows linearly (quadratically) with the cosmic time Ao. This growing energy is clearly supplied by the contracting geometry. The solution to eqs. (31 ) - (34), using eq. (38) and keeping the leading terms in ~, is given by Bo = (½OJoZZ+Voz) e x p ( P z ) , B1 = [ ½colr2+ (2a cos a+28)l/~Vo-c ] e x p ( 2 P z ) , 0)2

B2=- ~z+

O92

~

[ e x p ( 2 P z ) - I 1,

B3 =0,

(44)

where 0)0=

0)1 = - a

o92=

a a+3cos a (2a cos a + 2 5 ) :/2 a cos a + d '

a + 6 cos a a cos a + ~ ' a(f sin a a cos a + 5 '

Xo= [ ½o9oz2+ Voz+ (2a cos or+ 2~) 1/2] exp(P~), Xl = [½0)t z2+ (2a cos a+2~)l/2Vor+ (2 cos a+(f) ] ×exp(2Pr)-3,

dad~fl~ft"ti(A-A(a, r)),

IU~(A°)= f

3 December 1992

(45)

and Vo is a function of a fixed by the initial conditions. The overall string solution (X u =A ~'+ B u) reads

0)2

0)2

k½ = - -~-fi ~ + - ~

[ exp ( 2P¢ ) - l ] + a sin a .

(46)

In the problem we studied there are two inherent distinct time scales. One time scale, ~ , is associated with the external geometry (rate of change of the metric G~,) while the other, ~ , is associated with the internal motion of the string (frequencies of the string oscillators). The approximation scheme we developed applies when ~n >> ~x. In this limit ("impulsive" limit [ t0] ) the string modes do not have enough time to follow the fast changes of the metric Gu, and they remain frozen. Indeed the answer we obtained, eq. (46), indicates that while the overall motion of the string is treated exactly, for the internal motion a short time (z) expansion is applied. In the opposite case .~x >> ~ , the approximation described in ref. [ 1 ] is valid. There ("adiabatic" limit) the string follows the slow changes of the external metric. Thus we have at our disposal two systematic expansions, complementing each other, which hopefully would allow to study the full string dynamics in curved space-times, work along these lines will be reported elsewhere. This work has been partially supported by an EIE ( Greece ) - CNRS (France) exchange program.

References [ 1 ] H.J, de Vegaand N. Sanchez,Phys,Lett. B 197 ( 1987) 320. [2] H.J. de Vegaand N. Sanchez, Nucl. Phys. B 309 (1988) 552. [3]HJ. de Vega and N. Sanchez,NucL Phys. B 309 (1988) 557. [4l HJ. de Vegaand N. Sanchez, Nucl. Phys. B 317 (1989) 706. [ 5 ] HJ. de Vegaand N. Sanchez,Phys.Rev.D 42 (1990) 3269. [6] H.J. de Vega, M. Ramon Medrano and N. Sanchez,Nucl. Phys. B 351 (199I) 277. [7] HJ. de Vegaand N. Sanchez,Phys.Rev.D 45 (t992) 2783. [ 8 ] D. Amati and C. Klimcik,Phys. Lett. B 210 (1988) 92. [ 9 ] M. Ademolloet al., Nuovo Cimento 21A (1974) 77. ll0] N. Sanchezand G. Veneziano,Nucl. Phys. B 333 (1990) 253.

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[ 11 ] M. Gasperini, N. Sanchez and G. Veneziano, Intern. J. Mod. Phys. A 6 (1991) 3853. [ 12] M. Gasperini, Phys. Lett. B 258 ( 1991 ) 70. [ 13] A. Schild, Phys. Rev. D 16 (1977) 1722; A. Karlhede and U. Lindstr6m, Class. Quantum Grav. 3 (1986) L73; F. Lizzi, B. Rai, G. Sparano and S. Srivastava, Phys. Lett. B 182 (1986) 326;

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R. Amorim and J. Barcelos-Neto, Z. Phys. C 38 (1988) 643; J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Nucl. Phys. B 338 (1990) 143; U. Lindstr6m, B. Sundborg and G. Tbeodoridis, Phys. Lett. B253 (1991) 319. [14] H.J. de Vega and N. Sanchez, Intern. J. Mod. Phys. A 7 (1992) 3043.