Black hole formation in c=1 string field theory

Physics Letters B 300 (1993) 336-342 North-Holland

PHYSICS LETTERS B

Black hole formation in c = 1 string field theory Jorge G. Russo t Theory Group. Department of Physws. Umversttyof I~'xas, Austm. TX 78712. USA Recewed 23 November 1992

A suggestion on how black holes may appear m Das-Jevlckl collecuve field theory is given We study the behavlour of a "test'" parhcle when energy ~s sent into the system A perturbauon movmg near the potentml barrier can create a large-d~stance black hole geometry where the seeming curvature singularity ~s at the posmon of the b a m e r In the s~mplest "'static" case the exact D = 2 black hole metric emerges

The double scahng c o n t m u u m hmit o f the c= 1 matrtx model [ 1-12] is expected to describe crittcal strings moving in two s p a c e - t i m e d i m e n s i o n s (see e.g. refs. [ 3,13 ] ). Smce the discovery o f a black hole background in this c o n t i n u u m two-dtmenstonal theory [ 14-16] there have been a n u m b e r o f attempts to find the counterpart m the matrix model formulation [ 17-19] "~ In the last two years there was considerable progress tn understanding another D = 2 string background, the Llouvdle theory coupled to one scalar matter field [ 7, I 1,20-23 ]. In particular, this continuum theory was shown to have the same woo s y m m e t r y that appears m the c = 1 matrix model [ 9-12,23 ], and a good agreement betwecn the tree-level S-matrices was found (see e.g. refs. [ 20,21 ] ). The matrix model is equivalent to a ( 1 + 1 ) - & m e n s l o n a l field theory' where the collective field ts the density o f clgenvalues o f the matrix [2]. Thts theory is expected to be the field theory" of the only p r o p a g a t m g m o d e o f D = 2 string theory, where there are no transversal exotat~ons. At classical level, one expects a relation w~th the string effectwe action after all other modes o f the spectrum have been gauged away or integrated out by the equations of motion. The absence o f a metric in a resulting field theory obscures the geometrical interpretation o f physical processes. To get a rough idea o f what are the signs which would m & c a t c the presence of a black holc m the DJ field theory, ~t ~s instructive to perform the mtegrat~on o f metric and dilaton in the D = 2 string theory, even ~f this will be done only m a very a p p r o x l m a t t v e way. To leading or' order, the tree-level string effecuve a c u o n for the metric, d d a t o n and tachyon ~s g~ven by

S= f d2xx~-G[e-2~( R +4 Ouq)Ou~+c)-O,,qOl'rl-(Ou(~Ou(j- V20+ 2/ot')rl2 +O(rl3) ] ,

(l)

where q ts related to the usual tachyon by q = e - ¢ 7 , and c = - 8/a'. In the conformal gauge Gu~ =c2Vrluv (r/oo = -~h~ = l ), the equations o f motion take the form 0÷0_ ( p - O ) = -

(2)

~ O+ (c~/)O_ ( e ~ ) ,

Work supported m pan by NSF grant PHY 9009850 and R A. Welch Foundauon E-mad address russo@utaphy ph utexas edu "J The present approach is &fferent from other interesting, alternatwe proposals (see rcfs [ 17,19 ] ) 336

0370-2693/93/$ 06.00 © 1993 Elsevier Socnce Pubhshers B V All rights reserved

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2 4 O+ O_ e - 2 O = c e 2p- 2¢_ _ e2?q2, Or'

(3)

02_+0 - - 2 0±p 0 : 0 = ~ 0± (eoq)0_+ (eOq),

(4)

O+O_q=(O+OO_O-O+O_O+2--l, e2p)~l+O(q2).

(5)

Ignoring the r/back reactmn the general solutmn to eqs. ( 2 ) - ( 4 ) is the black hole background discussed m refs. [ 1 4 - 1 6 ] . Inserting thts background into eq. ( 5 ) one can see that q becomes massless far away. By making perturbation theory around the hnear dilaton background, the q dependence in eqs. ( 2 ) , ( 3 ) can be neglected m the first order a p p r o x i m a t i o n . So let us assume that far away we have 0+0_ ( P - 0 ) ~ 0 and hence we can fix the " K r u s k a l " gauge p ~ 0. Then eq. ( 3 ) becomes, 4 0+ 0_ e - 2~ ~ c, i.e. e - 2~= ~cx +x - + h + ( x + ) + h _ ( x - ). The leading pmce o f the constraint equatmns ( 4 ) takes the form (we assume 0 ± r/>> q 0 ± 0 ~ - q/2x +-for x -+--, + oo )

O~e-2~~-O±qO+~. Therefore h"± ~ - 0 ± q 0 ± q, or

h±(x+-)~- I ~O+_qO+_q.

(6)

Inserting these solutions for p and O into eq. ( 5 ) , what remains ~s a nonlinear integro-dlfferenual equation expressed purely tn terms o f q. F r o m this equatmns one can obtain scattering a m p l i t u d e s corresponding to this scalar field effective theory. To see gravitational collapse we imagine that an energetic wave r/o, 0+0_r/o~ 0 is sent m, expand r/= qo+e, and study the b e h a v i o u r o f the small fluctuation or "'test" particle e m the incoming background r/o. Then the conformal factor and dilaton become e - 2p= e - 20 = ~cx + x - + h + ( 0 + qo) + h_ ( 0_ qo) + O (e), which for a large class ofqo represents a black hole. Then eq. ( 5 ) takes the form

O+O_e=m(qo)~+j(rlo)+O(e 2) ,

(7)

where both m, j ~ 0 asymptotically. The black hole geometry has to be read out o f m (r/o) in eq. ( 7 ) , since in the conformai gauge It does not show up in the lapaclan. However, by repeating the above procedure, e.g., in the linear dflaton gauge 0 = - x one finds a laplaclan o f the f o r m f -tO2,e,-fO~e+O(e2),f- 1 - M ( q o ) e -2x, where the underlying large-d~stance geometry is exh~bxted in a manifest way. The bosonlc h a m i l t o m a n for the double scaled c = 1 matrix model is given by [2] (we follow the notation o f ref. [71 ).

1t=

Hc(0x~)Hc+ 6g~(0x~)3+~(l-x2)(0~)

'

(8)

where x denotes the (rescaled) space o f mgenvalues o f the original matrix model, /-/¢(x) is the m o m e n t u m conjugate to ~(x). The equations o f m o t m n which follow from the above h a m i l t o m a n are 0,~=g2/TcOx~,

0,/-/'¢='~gs2 0 x / / ~ +

rr2 1 x. ~g20~(O~)2--g~

(9)

The general solution to these equatmns ~s [ 7 ] 17¢=

P+ + P 2g 2

0:,~=

P+-P2~z

'

(lO)

where p± (x, t) = a ( a ± ) s m h ( t - a + ), a(er) is an arbitrary funclmn ofcr and a± = a ± (x, t) are the two solutmns

of x=a( a)cosh( t-a). 337

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The momenta p_. obey the transport equation

Otp+ =x-p+_ OxP*_.

(11)

The ground state is g~ven by the static solution He=0,

O x ( o = l ( x 2 - 1 ) '/2,

Ixl>l.

(12)

Now we introduce the scalar field ~Vand ~ts momentum conjugate H as

~=~o(x)+ -gs : - ~v, x/n

n~= "/~H

(13)

gs

After introducing a new coordinate q as x = cosh (q) (we consider the right hand s~de of the bamer and 0 ~
tl=~ ~ dq(H2+(Oq~)2+ slnh2(q~ gsxFn [/--/2 0q t/t+ ~ (0q ~u)3 ] ) . The equations of motion for ~ a n d / / a r e ( [ ~V(q, t ) , / / ( q ' ,

" //2 + (0q t/'/)2

0,//=0q2~V+tgOq

s-~-nhT~-~ , 0,~v=

/](

g

l+sm ~ h2(q

(14)

t) ] =ld(q--q') )

OqtP)

(15)

,

where g = g~x/'n. Ehminatmg / / from these equations one easdy obtains 1

A( ~ 02~ - A ( ~ ) 02 ~=F(tp), 2g(q) ( ~ ) (0t IP) 2 0q[g(q) 0q~] F(~J) -~- A(~J)~2Ot~JOqOt ~ - A3g(q)

g(q) ( A ~1 tanh(q)

(0'~)2+(0q~V)2 ) ,

(16)

where

A(~) =l+g(q)Oqtp, g(q)=

g slnh2(q).

(17)

Here we do not assume that ~Vis small and thus we shall keep all higher powers in ~vm eq. (16). For large q the equation simplifies to

(a,~- a2)~~o.

(18)

with solution ~u= ~v+ (t+q) + ~v_(t-q). Along similar lines as in the first part of this note, now we assume a physical s~tuation in which there is an incoming wave ~vo and study the dynamics of small fluctuations, ~ = ~o + e ,

e << ~o •

(19)

We shall be interested only m large distance physics, where one expects to find analogous results as those coming from the a ' expansion of the D = 2 continuum string theory. Inserting the expansion (19) in ( 16 ) and retaining only the linear terms in c we find

1 a2e_A(,/.,o)[ l _

A(~o----~

(g(q) 0a,~o'~2-]~2 ~,-~o) ) J°qe-

=A (~Po)O 2 ~ o - A - ' (tPo) 02 tPo+ F(~o), 338

2g(q) O,q-'oo,,O:+QO,:+TO, e

A(~Vo)2

(20)

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where Q - = - 0 q [ g ( q ) 0a~o]

g(q) 02~Vo+0 { g2(q) '~'0 ~ ,2 4g2(q) q~kA3--~"~ojt t o) + A 3 ~ o Ot~lJoOqOt~Vo,

A2(t//o)

q ) )Ot ~Po_ A2(~%) T_ _ O,( Ag( 2~__~o 2g(q) OqOt ~10 . It is remarkable that the 0q~ and 0,e terms in eq. (20) are such that th~s equation can be written in the form (21)

Ggr~ Vu0~e=Jdr+O(e 2) , where

g(q) A - ' (~V°)

GeU~-=

4.

"~

A 2 ( ~ ) °'~°

g(q) _A(~o)[- I A2( o) O,'o L

/

(g(q) 0,~Vo,~2-]| ,

(22)

) JJ

Jen,=A (~Vo) 02 ~Vo-A - ' ( % ) 02 ~Po+F(~Vo).

(23)

An effective, large-distance geometry has emerged. The geometrical interpretation may break down in the v~cinity of the wall (q = 0 ) where the O (e 2) terms can no longer be ignored. A curious fact is that in these coordinates we have det Gefr= - I automatically. From eq. (22) we get ~, 2 ~[, (g(q) O, o'~ -1 2 ~ g(q) ds2=A(~Vo,L,-~,. ~ - ~ o ) ) ] d t - A - t ( ~ o ) d q 2 - z ~ O t ~ V o d q d t . (24) Eqs. (20) or (21) can be interpreted as the propagation of the scalar field e in a nontrivial geometry. The source term J~rf m the right hand side of eqs. (20), (21) vanishes asymptotically because we demand ~Po to satisfy the free field equation ( 18 ). Now we would like to show explicitly that for a large class of incoming waves this geometry corresponds to a black hole. The simplest case ~s that m which ~o ~s a static solution. This case ~s no less unphys~cal than the D = 2 stattc black hole solution, but it illustrates some points. From 02 ~Vo= 0 it follows 0q~o=const. Th~s can also be obtained from the exact solution M gs Ox~=Ox(M--Ox(O-- l_lt[ ( X 2 - 2 M - I )'/2-(X2-1)1/2] ~ - Itlx~' or 0q~V~ 0q~o- -

(25)

M/g. Thus we choose M g

(26)

Inserting eq. (26) into eqs. (20), (24) we obtain

(A~ 102-Ao O~)e=Jo +O(e 2) , Jo=O(M2e -2q) , I

dsE=Aodl2-Aff Idq 2, Ao-l-Msmh2(q-----~

(27)

I - 4 M e -2q.

(28)

This has the same form as the expression for the Witten black hole. More exactly, after a change of coordinates cosh (q) = x/i-St-M cosh (r) we obtain ds2=-dr2+fl2(r)dt

2, f l 2 ( r ) = ( M + l ) [ M c o t h 2 ( r ) + l ]

-t,

(29) 339

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which is the " e x a c t " metric found in ref. [ 16 ] (an exact solution to the tree-level sigma-model//-function equations, see also ref. [ 24 ] ) ~2 In ref. [ 14] the p a r a m e t e r M was identified with the A D M mass, which is usually assumed to be posmve. This geometry has a horizon at smh 2 (qh) = M and a singularity at q = 0 or x = 1, i.e. the position o f the Polchlnskl wall ( o f course, the present " l i n e a r " a p p r o x i m a t i o n breaks down much beforc getting to the wall, so the singularity may just be an illusion for distant observers). Considering this particular form for ~o is cq_uivalent to the expansion if= (,w + (gs/n)e (see eqs. ( 13 ), ( 19 ) ). If we now introduce a coordlnatc qM as x = v ' l + 2 M c o s h (qM) then we would obtain cq. ( 1 6 ) with e instead of ~u, i.e. an equation o f the form (0qZM- 0 , 2 ) e = O ( e 2) This o f course agrees with eq. (27) after changing q--*qM, • eft whtch takes the metric Gu, to the conformal gauge In this gauge the black hole geometry is not manifest, but the coordinate qM is not geodesically complete smce it does not cover the region , / l + 2 . M > x > I. In particular tt does not include the horizon at .x= ~ . . " ~ 1. As a second example we consider a htgh pulse ~ , = T+ ( x + t ) coming from .x'=oo which extends above the line p = Ixl. The elgenvalues above the line will be on trajectories which carry them over the barrier to x < 0. For a very high pulse the reflected part can be ignored. Consider for example the case m which tp+ is o f the form, i)qT+ ( x + t ) = - E e -~x+'~2` E < 0 . Then A ( ~ ) = 1 - [ g E / ( x 2 - 1 ) ]e - ~'+'~2. This geometry has naked singularltles at x = _+ 1. The pulse may be interpreted as a wormhole connecting the two asymptotically flat sides o f the barrier. O u r thtrd example is a low energy density pulse (by "low energy density pulse" we mean a pulse whtch does not represent a large deviation from the static solution, where some degenerate behaviours can occur, leading to multlvalued functions p :. [7] ). The exact tree-level S-matrix for these pulses m the bosomc formalism was found in ref. [ 7 ] (for discussions in the fermlonlc formulation see e.g. ref. [ 5 ] ). In the terms proportional to e we can make use o f the Polchlnskl exact solutions replacing To by tPsince this does not affect the equation to linear o r d e r in e. Replacing To by ~ in eqs. ( 2 2 ) , (24) and using eqs. ( 10 ), ( 13 ) we obtain

~l,~ G ~,r =

"2 s i n h ( q )

p+ + p _

P+ - P p+ + p _ p+ - p _

P+ - P 2p+ p_ (p+ - p _ ) s l n h ( q )

( 30 )

or

ds2=-

2p+ P+ -+pP_- dq d t . (p+ - p _ ) , vP/-xr s y -- -Id t 2 - ~-Y-":~;-2--!-p~ - p _ " d q 2 + 2 p+

(31)

Far away, p± = + x, the dq dt term vamshes and the metric asymptotically approaches to the Minkowskl metric r/u~. Eq. ( 3 1 ) provides a geometrical interpretation o f the scattering process for all t, and large x. The metric (31 ) has a potential singularity at x 2 = I which may be absent in some specific cases. For example, if at x = 1 the m o m e n t a p_+ take its stattc values ~ + ,,."~¥~/- I then the potential smgulartty cancels out. But if at some time t a pulse is moving near the wall, p,_ ( x - 1, t) wtll take values very dlffcrcnt from the static case and the metric can have a singularity at x = 1, presumably a curvature singularity (as viewed by a distant observcr). Thus the picture is the following: whenever a pulse is sent in, an observer at large X=Xo will see a t i m e - d e p e n d e n t geometry G~'~(x= Xo, t) given by eq. (30), which is curved in the regions where th pulsc differs from the stattc solution, and nearly fiat elsewhere. When the pulse is close to the wall, a singularity may develop and the observer at xo may measure a large-dlstancc black hole geometry. To be more specific, let us consider the case o f a " s t e p " pulse coming from x = ~ and travelling anti-clockwise ,2 The fact that the exact DVV metrtc a p p e a r e d should actually be regarded as a fortunate " i m p r o v e m e n t " of the present a p p r o x i m a t i o n For example, a change oflJq~tto from the constant value 0q~o= - M/g to <'~qtP'o = - M/g+O(e- 2q) would modify the metric by O( e -4q) A generic feature o f all these DVV type metrics ~s the s m g u l a r u y at q = 0

340

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along the Ferm~ surface m m o m e n t u m space. W h e n the pulse passes by, p+ sw~tches from the static value x / x 5-~ i to x / x 2 - 2 M - 1, M > 0. After the step pulse has reached the wall, p+ = x / x 2 - 2 M - 1 for all x. T h e n p _ starts switching from its static value - x/.'x ~5~- 1 to - x / : x ~ - 2 M - 1 as the step pulse travels from x = 1 to x = ~ ,uv along the lower b r a n c h o f the Ferm~ surface. The t i m e - d e p e n d i n g metric Gcfr(x, t) is equal to r/u, at the p o i n t s x which have not yet b e e n reached by the pulse. It xs M

ds2=A~dtZ-,4? ~dq2-slnhz(q~dqdt,

..It-1

M 1 2 smh2(q)'

(32)

at the x which have been only reached by the m g o m g step pulse (we have d r o p p e d s u b l e a d i n g terms in e -2q, the exact form can be read from eq. (31 ) ), a n d

ds2=

p+ slnh(q)

dt 2

slnh(q)dq2~Aodt2_Aff p+

~dq 2

at the x which have also been reached by the o u t g o m g step pulse (Ao has been d e f i n e d m eq. ( 2 8 ) ). T h i s is the m e t r i c everywhere for t - - , ~ ; it has a h o r i z o n at p+ = 0 , i.e. x~ = 2 M + 1, a n d a singularity at x = 1. Now o n e is led to s o m e speculation. C o m p a r i n g with D = 2 c o n t i n u u m string theory, black holes, we see that in this scenario a n d in the large-d~stance a p p r o x x m a t i o n -0qtPo plays the role o f the integral o f an e n e r g y m o m e n t u m t e n s o r (cf. e.g. eq. ( 6 ) ) . If this e n e r g y - m o m e n t u m t e n s o r is posttive-definile then th~s integral ~s a m o n o t o m c a l l y n o n d e c r e a s i n g quantxty, which ~s the basic reason why classical black holes can only increase in size. O n e could be t e m p t e d to d e m a n d i n g p o s l u v l t y on some o f the derivatives o f -0qt/"o to garantee that only positive "energy d e n s i t y " ~s e n t e r i n g into the system. Thts ad hoc restriction o f the i n c o m i n g pulses would always lead - m the classical theory - to metrics of DVV type ( 2 8 ) as final state. In the case of the step pulse c o n s i d e r e d above -Oqt/'o is m fact m o n o t o m c a l l y n o n d e c r c a s m g , but, in particular, m the case o f l o c a h z e d pulses it Is not. After a localized pulse is reflected a n d gets off from the wall, p_+ takes again values n e a r the static solution a n d a n y s e e m i n g black hole geometry evanesces. W h e n all higher powers o f t are i n c o r p o r a t e d the exact scattering a m p h t u d c o f low-energy pulses is u n i t a r y a n d reveals n o singularity or a n o m a l o u s b e h a v l o u r . Perhaps this is a clue o f a secret reconcdmt~on between black hole physics a n d q u a n t u m mechanics. I wish to t h a n k W. Ftschler, L. Sussklnd a n d A. T s e y t h n for useful &scusslons.

References [ I ] D Gross and N. Mllkovlc, Phys Lett. B 238 (1990) 217, E Br~zm, V. Kazakov and AI.B. Zamolodchlkov, Nucl. Phys. B 338 (1990) 673, G Panst, Phys. Left B 238 (1990) 209, P. Gmspargand J Zmn-Jusnn, Phys Lett B 240 (1990) 333 [2]S.R DasandA Jevlcki, Mod Phys Left A 5 (1990) 1639 [3] J. Polchmskl, Nucl. Phys. B 346 (1990) 253 [4] D. Gross and I Klebanov, Nucl. Phys. B 352 (1990) 671, A.M Senguptaand S R Wadta, Intern J Mod A 6 ( 1991 ) 1961. [5] D. Gross and I Klebanov, Nucl Phys. B 359 ( 1991 ) 3. [6] D. Gross, I. Klebanov and M Newmann, Nucl. Phys B 350 ( 1991 ) 621, U Damelsson and D. Gross, Prmceton preprmt PU PT- 1258 ( 1991 ) [7] J. Polchmskl, Nucl Phys B 362 ( 1991 ) 125 [ 8 ] K. Demeterfi, A Jevlekl and J P Rodngues, Nucl Phys B 362 ( 1991 ) 173, B 365 ( 1991 ) 199, G. Mandal, A.M Sengupta and S.R Wadla, Mod Phys Lett. A 6 ( 1991 ) 1465, G. Moore, Nucl Phys. B 368 (1992) 557, G. Moore, M.R. Plesser and S. Ramgoolam, Nucl. Phys B 377 (1992) 143 341

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[9] J. Avan and A Jevlck~, Phys. Lett. B 266 ( 1991 ) 35, B 272 (1990) 17, Mod. Phys Lett A 7 (1992) 357. [ 10] S.R. Das, A. Dhar, G. Mandal and S R Wadia, Intern. J Mod. Phys. A 7 ( 1992); Mod Phys. Lett. A 7 (1992) 71,937 [ 11 ] G. Moore and N. Selberg, Intern. J Mod. Phys A 7 (1992) 2601. [ 121 T Eguchi, H Kanno and S K. Yang, Newton Institute preprmt NT-92004 (1992). [ 13] A.A. Tseythn, Intern J Mod Phys. A 5 (1990) 1833. [ 14 ] E. Witten, Phys. Rev. D 44 ( 1991 ) 314 [ 15 ] G. Mandal, A Sengupta and S R Wadm, Mod. Phys. Lett. A 6 ( 1991 ) 1685. [ 16] R D0kgraaf, E. Verhnde and H. Verhnde, Nucl Phys. B 371 (1992) 269. [ 17] J Elhs, N E. Mavromatos and D. Nanopoulos, Phys Lett B 272 ( 1991 ) 261, B 267 ( 1991 ) 465, R. Brustem and S. de Alwls, Phys. Lett. B 272 ( 1991 ) 285 [ 18] E. Wltten, IAS prepnnt IASSNS-HEP-92/24. [ 19] S.R. Das, Tara Institute prepnnt TIFR-TH-92/62 ( 1992); A. Dhar, G. Mandal and S.R. Wadla, Tata Institute prepnnt TIFR-TH-92/63 ( 1992 ). [20] P Di Francesco and D. Kutasov, Phys. Left B 261 ( 1991 ) 385. [21 ] A Polyakov, Mod. Phys. Lett. A 6 (1991) 635 [221 E. Martmec and S. Shatashvlh, Nucl Phys. B 368 (1992) 338. [23] E Wltten, Nucl. Phys. B 373 (1992) 187, I. Klebanov and A. Polyakov, Mod Phys Lett. A 6 ( 1991 ) 3273. [24] A.A. Tseylhn, Phys Lett. B 268 ( 1991 ) 175

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