Negative energy in string theory and cosmic censorship violation

PHYSICAL REVIEW D 69, 105001 ~2004!

Negative energy in string theory and cosmic censorship violation

Thomas Hertog* and Gary T. Horowitz†

Department of Physics, UCSB, Santa Barbara, California 93106, USA

Kengo Maeda‡

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan~Received 30 October 2003; published 5 May 2004!

We find asymptotically anti–de Sitter solutions inN58 supergravity which have a negative total energy.This is possible since the boundary conditions required for the positive energy theorem are stronger than thoserequired for a finite mass~and allowed by string theory!. But the stability of the anti–de Sitter vacuum is stillensured by the positivity of a modified energy, which includes an extra surface term. Some of the negativeenergy solutions describe the classical evolution of nonsingular initial data to naked singularities. Since thereis an open set of such solutions, cosmic censorship is violated generically in supergravity. Using the dual fieldtheory description, we argue that these naked singularities will be resolved in the full string theory.

DOI: 10.1103/PhysRevD.69.105001 PACS number~s!: 04.65.1e

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I. INTRODUCTION

Anti–de Sitter space~AdS! has the property that a scalafield with a negative mass squared does not cause an ibility provided that m2>mBF

2 where mBF2 ,0 is a certain

lower bound known as the Breitenlohner-Freedman~BF!bound@1,2#. This is important since many supergravity theries arising in the low energy limit of string theory contafields with negativem2, but they all satisfy this bound. It iscommonly believed that there is a positive energy theor@3–6# which ensures that the total energy cannot be negawhenever this condition is satisfied.

We will show that, while this is indeed true form2

.mBF2 , the positive energy theoremcan be violated for

fields which saturate the BF bound. This will be demostrated by explicitly constructing nonsingular initial dawith a negative total energy. In fact, the energy can be atrarily negative. This is because the positive energy theorequires boundary conditions which are too strong for fiewhich saturate the bound. In particular, it requires stronboundary conditions than those required for a finite maWith natural boundary conditions, it is only the sum of tusual AdS energy and another~finite! contribution from theasymptotic scalar field which is required to be positive.

The negative energy solutions we find are quite differfrom previous examples of negative energy in AdS. The Asoliton @7# has negative energy, but asymptotically aproaches AdS only locally. One must introduce a perioidentification asymptotically to construct it. The solutions wpresent do not require any identification. The counterteapproach can result in a negative energy for AdS itself@8#.We will work with the standard energy relative to AdS,the energy of AdS is zero by definition.

We also consider the evolution of our negative eneconfigurations and show that an open subset of them lea

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]

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naked singularities. This shows that cosmic censorship@9# isviolated generically in a low energy limit of string theorWe had previously shown that cosmic censorship can belated in spacetimes which asymptotically approach AdS@10#.But in that work, we considered gravity coupled to a scafield with a general potential~satisfying a positive energytheorem!, and found that for some potentials cosmic censship was violated. We did not require that the potential cobe derived from string theory. The present work shows tthe same phenomena can happen even in string theory.

N58 gauged supergravity theories in both four@11# andfive @12,13# dimensions have scalar fields which saturateBF bound. This is of particular interest since these theoare believed to arise as the low energy limit of string theo~or M theory! with boundary conditions AdS43S7 @14# orAdS53S5. For these boundary conditions, we have the poerful AdS conformal field theory~CFT! correspondencewhich relates string theory to a dual field theory@15#. Thisraises another puzzle since the field theory Hamiltonianbounded from below, so how can it describe negative enestates in the bulk? The answer seems to be that thetheory Hamiltonian should not be identified with the usuAdS energy, but rather a modified energy which is indealways positive.

It is of great interest to ask what happens in a full quatum theory of gravity, when the semiclassical solution devops a naked singularity. One can now begin to addressissue using the dual field theory description. Althoughhave not yet explored this in detail, there appears to bereason for the field theory evolution to break down. Itlikely that the full quantum theory resolves the naked singlarity. This provides a new approach for studying cosmolocal singularities since the naked singularities we findspacelike inside some region. They are like big crunch sgularities embedded in an asymptotically anti–de Sitspace.

Since boundary conditions will play an important roleour analysis, recall that if we write AdSd ~with unit radius ofcurvature! in global coordinates

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HERTOG, HOROWITZ, AND MAEDA PHYSICAL REVIEW D69, 105001 ~2004!

ds252~11r 2!dt21dr2

11r 2 1r 2dVd22 , ~1.1!

then solutions to¹2f2m2f50 with harmonic time dependencee2 ivt all fall off asymptotically like 1/r l6 where

l65d216A~d21!214m2

2. ~1.2!

The BF bound is

mBF2 52

~d21!2

4. ~1.3!

For fields which saturate this bound,l15l2[l and thesecond solution asymptotically behaves like lnr/rl. For defi-niteness, we will focus ond55 since the dual field theory isimply four dimensionalN54 super Yang-Mills theory. Inthis case, the fastest that a mode withm25mBF

2 can fall off is1/r 2.

In the next section, we explicitly construct the negatienergy initial data. In Sec. III, we discuss the positive enetheorem, and the boundary conditions required for it to hoThe following section shows that the negative energy inidata evolve to naked singularities. We also find the cosponding solution of ten dimensional type IIB supergravFinally, Sec. V contains some further discussion, includthe implications for cosmological singularities.

II. NEGATIVE ENERGY SOLUTIONS IN SUPERGRAVITY

N58 gauged supergravity in five dimensions@12,13# isthought to be a consistent truncation of ten dimensional tIIB supergravity on S5. The spectrum of this compactification involves 42 scalars parametrizing the coE6(6) /USp(8). Thefields which saturate the BF bound corespond to a subset that parametrizes the cSL(6,R)/SO(6). From the higher dimensional viewpointhese arise from the,52 modes on S5 @16#. The relevantpart of the action for our discussion involves five scalarsa iand takes the form@17#

S5E A2gF1

2R2(

i 51

51

2~¹a i !

22V~a i !G , ~2.1!

where we have set 8pG51.1 The potential for the scalarsa iis given in terms of a superpotentialW(a i) via

V5g2

4 (i 51

5 S ]W

]a iD 2

2g2

3W2, ~2.2!

W is most simply expressed as

W521

2A2(i 51

6

e2b i, ~2.3!

1Our formulas differ slightly from@17#, since they use 4pG51.

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where theb i sum to zero, and are related to the fivea i ’s withstandard kinetic terms as follows@17#:

S b1

b2

b3

b4

b5

b6

D 5S 1/2 1/2 1/2 0 1/2A3

1/2 21/2 21/2 0 1/2A3

21/2 21/2 1/2 0 1/2A3

21/2 1/2 21/2 0 1/2A3

0 0 0 1/A2 21/A3

0 0 0 21/A2 21/A3

D3S a1

a2

a3

a4

a5

D . ~2.4!

The potential reaches a negative local maximum whenthe scalar fieldsa i vanish. This is the maximally supersymmetric AdS state, corresponding to the unperturbed S5 in thetype IIB theory. At linear order around the AdS solution, tfive scalars each obey the free wave equation with a msaturating the BF bound. Nonperturbatively, the fields couto each other and it is generally not consistent to set osome of them to zero. The exception isa5, which does notact as a source for any of the other fields.

We now find a class of negative energy solutions that oinvolve a5, so a i50, i 51, . . . ,4 in oursolutions. Writinga55f and settingg254 so that the AdS radius is equal tone, the action~2.1! further reduces to

S5E A2gF1

2R2

1

2~¹f!21~2e2f/A314e2f/A3!G .

~2.5!

We construct the solutions by first solving the constraequations on a spacelike surfaceS. We consider initial datawith all time derivatives set to zero. For time symmetrinitial data the constraint equations reduce to

(4)R5gi j f ,if , j12V~f!. ~2.6!

For spherically symmetric configurations the spatial mecan be written as

ds25S 12m~r !

3p2r 2 1r 2D 21

dr21r 2 dV32 . ~2.7!

The normalization is chosen so that the total mass is simthe asymptotic value ofm(r )

M5 limr→`

m~r !. ~2.8!

The constraint~2.6! yields the following equation form(r ):

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NEGATIVE ENERGY IN STRING THEORY AND COSMIC . . . PHYSICAL REVIEW D 69, 105001 ~2004!

m,r11

3mr~f ,r !

252p2r 3F „V~f!16…11

2~11r 2!~f ,r !

2G .~2.9!

The general solution for arbitraryf(r ) is

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r

e2*rr r (f ,r )

2/3dr

3F „V~f!16…11

2~11r 2!~f ,r !

2G r 3 dr.

~2.10!

We now specify initial data forf(r ) on S. We consider asimple class of configurations with a constant density insa sphere of radiusR0:

f~r !5A

R02 ~r<R0!, f~r !5

A

r 2 ~r .R0!. ~2.11!

The falloff of f is motivated as follows. Iff→0 slowly, wedecrease the contribution to the energy from the positivedient terms and increase the contribution from the negapotential term. Since we want to try to construct a solutwith negative energy, we clearly wantf to vanish as slowlyas possible. It is easy to verify that 1/r 2 is the slowest falloffthat yields finite total energy. In addition, this behavior is tsame as the falloff of the mode solutions~1.2! of the freewave equation which are going as;1/r l1. One can noweasily show that for these initial data, the negative contrition to the mass from the potential more than compensfor the positive contribution from the gradient terms. If wtake 0,A!R0

2 so that the field is everywhere small then E~2.10! gives

M'2p2A2. ~2.12!

Since we can makeR0 and thereforeA arbitrarily large, it isclear that the total energy can be arbitrarily negative. FoA.R0

2, f is not small inside the sphere, but by using the fthat V(f),2622f22(2/3A3)f3 for all f.0, one canobtain a general upper limit to the total mass,M,2p2A2/A3.

We have found that there exist nonsingular configuratiin N58 supergravity with negative total mass. In Sec. IV wstudy the evolution of our initial data, but first we explawhy this result is not in conflict with the positive energtheorem@4,6#.

III. POSITIVE ENERGY THEOREM

How are our negative energy solutions compatible wthe fact that there is a positive energy theorem for supergity? How are they compatible with the AdS-CFT correspodence since the gauge theory Hamiltonian is bounded fbelow? To answer these questions, we first review the ament for positive energy of test fields originally given@1,2#, and then discuss the full nonlinear proof of the positenergy theorem.

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A. Positive energy for test fields

Consider a test field of massm2524 which saturates theBF bound in AdS5. We start with the action

S51

2E @2~¹f!214f2#r 3dtdrdV3

51

2E F f2

~11r 2!2~Df!214f2G r 3 dt dr dV3 ,

~3.1!

where D is the spatial derivative on a constantt surface.Since the background is static, one can compute the Hatonian in the usual way and obtain

E51

2E F f2

~11r 2!1~Df!224f2G r 3 dr dV3 . ~3.2!

This energy density is not positive definite due to the netive m2. However, if we writef5c/(11r 2), substitute intoEq. ~3.2! and integrate by parts we obtain

E51

2E @~ c !21~11r 2!~Dc!214c2#

r 3

~11r 2!3 dr dV3

2 R c2 dV3 . ~3.3!

The volume term is now manifestly positive. The surfaterm vanishes providedc goes to zero asymptotically, whicmeans thatf falls off fasterthan 1/r 2. But we are interestedin solutions that fall off precisely as 1/r 2. In this case, thesurface term is nonzero and manifestly negative. So thneed not be a positive energy theorem and indeed, ashave seen, negative energy solutions can occur. Noticethis is possible only for fields which saturate the BF bounIf m2.24, then the total energy of any configuration thfalls-off as 1/r 2 diverges. Finite energy configurations mufall off faster, so the surface term vanishes and the energalways positive. It is the delicate cancellation betweenm2f2 term and the gradient term in Eq.~3.2! which allowsfields with m2524 to have 1/r 2 falloff and finite energy.

One might have thought that the reaction to this wouldto claim that one has positive energy only form2.mBF

2 .Instead Breitenlohner and Freedman@1# proposed to includethe limiting casem25mBF

2 and modify the definition of theenergy.2 In the original papers from the early 1980s, this wdescribed in terms of an ‘‘improved stress tensor’’ whi

2In @2#, Mezincescu and Townsend note that a perturbative ansis is not sufficient to prove stability if the bound is saturated. LaTownsend@6# performed a nonperturbative analysis in spacetimof arbitrary dimension, following the approach of Boucher@5#, inwhich he claims to establish a positive energy theorem~and stabil-ity! even when the bound is saturated. However, as we will discin Sec. III B, the proof given in@6# does not apply to the usual AdSenergy ifm25mBF

2 .

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corresponds to adding abRf2 term to the Lagrangian. InAdS, R is a constant, so this indeed looks like a mass tefor a test field. But as soon as one goes beyond the linearapproximation, adding a term like this changes the theorythe context we have been considering,N58 gauged supergravity in five dimensions, there is nobRf2 term in theaction, so this is not an option.

However, one still has the possibility of adding a surfaterm to the action~3.1! to get

S51

2E @f¹2f14f2#r 3 dt drdV3

5S11

2 R f¹mf dSm. ~3.4!

Now if one derives the Hamiltonian, one finds an extra sface term in the expression for the energy which exaccancels the surface term in Eq.~3.3!. This is possible since ifn is the unit radial normal to the sphere at infinity,fn•¹f522f2. So starting with this modified action, the energyindeed positive.

B. Nonlinear positive energy theorem

We now turn to the full positive energy theorem for AdThis is a generalization of the spinorial proof for asymptocally flat spacetimes given by Witten@18#. We will follow theapproach in@6#. The boundary conditions needed to appthis proof do not seem to have been clearly spelled outAdS, there are no covariantly constant spinors, but there‘‘supercovariantly’’ constant spinorse0 satisfying

¹me011

2gme050, ~3.5!

wheregm are the five dimensional gamma matrices. Fortheory we are considering Eq.~2.5!, the scalar potential isderivable from a superpotentialW(f) via V5W82

2(4/3)W2 ~2.2!. One now defines a modified derivative¹m

[¹m2(1/3A2)W(f)gm and the Nester two-form@19#

Emn[egmns¹se1H.c., ~3.6!

wheregmns[g [mgngs] ande is an arbitrary spinor that asymptotically approachese0.

Let S be a nonsingular spacelike surface with boundary

infinity, and let e be a solution tog i¹ie50 ~where i runsonly over directions tangent toS) which asymptotically ap-proachese0. Then the integral of¹mEmn over S is non-negative, and vanishes if and only if the spacetime is Aeverywhere.~If there is matter in addition to the scalar fielits stress tensor must satisfy the dominant energy conditi!Hence

R EmndSmn>0. ~3.7!

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~Note that the volume element picks out the componeorthogonal to the three-sphere at infinity.! If W is constant,this reduces to the usual definition of mass in asymptoticAdS spacetimes. However, in our caseW is not constant, andfor smallf, W/3A2'21/22f2/6. So there is an additionasurface term

R f2~ e0gmne0! dSmn. ~3.8!

Since the area of theS3 at infinity grows liker 3, f;1/r 2,and e0 is supercovariantly constant at infinity, one mighave thought that this surface term would always vanish.it does not. Supercovariantly constant spinors grow liker 1/2

in AdS ~see, e.g.@20#!. In retrospect this is not surprisinsince the square of a supercovariantly constant spinorKilling vector, and a timelike Killing vector in AdS has normproportional tor. So in order for this surface term to vanisand recover the usual positive energy theorem, one needfto vanish faster than 1/r 2 at infinity. We have seen that thiboundary condition is too strong for fields which saturateBF bound. In general dimensiond, the required boundarycondition on f in order to apply the positive energtheorem3 is thatf must vanish faster thanr 2(d21)/2. A natu-ral way out of this conundrum is to modify the definition oenergy to include the extra surface term~3.8!. We have seenthat the combination of this with the usual energy cannotnegative and vanishes only for AdS.

Supersymmetry implies that the square of the superchashould be positive. Although we have not checked it,believe that the supercharge also has an extra contributiothis case, so that the positive quantity is indeed the ensurface term~3.7!. It would be interesting to verify this byextending the work of@21# to N58 supergravity. Since theHamiltonian of the dual field theory must be positive~or atleast bounded from below if one includes Casimir energy! itshould be identified with this modified energy.

IV. EVOLUTION AND NAKED SINGULARITIES

In this section, we consider the evolution of the negatenergy initial data constructed in Sec. II, and show that thevolve to naked singularities. But first, we point out anothinteresting difference between the usual energy andmodified energy, which arises in evolution.

A. Is the energy time dependent?

For fields behaving asf5A/r 21O(1/r 3) at larger, theusual energy is time dependent ifA is a function oft. Thereis a nonzero flux of energy at infinity. The modified energon the other hand, is always time independent. To see thsuffices to consider the linearized theory, sincef is verysmall asymptotically. In terms of the conserved stress ten

3In @4# it is incorrectly stated that ind54, one only needsf tovanish faster than 1/r . It is easy to see this is incorrect by construcing negative energy solutions analogous to those in the prevsection which vanish liker 23/2.

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Tmn5¹mf¹nf21

2gmn@~¹f!212V~f!# ~4.1!

the usual energy~3.2! is just the integral ofTmnjm over aspacelike surface, wherejm is the timelike Killing field. Thelocal flux of energy at infinity is thusTmnjmnn wherenn isan asymptotic unit radial vector. Integrating this flux betwet1 and t2 yields

E~ t2!2E~ t1!5 limr→`

Et1

t2r 4 dt dV3f~r ] r !f

52E @A2~ t2!2A2~ t1!#dV3 . ~4.2!

It is now clear that if we add to the definition of the energysurface term*A2 dV3, the modified energy will be timeindependent. This is precisely the same surface term wmakes the energy positive.

If one wants the usual energy to be time independent,can require thatA be independent of time. This can bachieved by imposing boundary conditions at a largefinite R and requiringf5A/R2 ~with fixed A) at this radius.~This is automatically implemented in most numerical evlution schemes.! The radiusR is like a cutoff, and in prin-ciple should be taken to infinity to obtain the true solutio

The fact that the total energy may be time dependholds only for fields which saturate the BF bound. Ifm2

.mBF2 , then finite energy requires fields to fall off fast

than 1/r 2 and then the flux always vanishes at infinity.

B. Cosmic censorship violation

Recall that our initial data consisted of a constant fif5A/R0

2 inside a sphere of radiusR0. The proper size ofthis sphere initially is

L'E0

R0 dr

@11~Hr !2#1/2'H21 ln R0 , ~4.3!

where H252V(A/R02)/6. So for largeR0 there is a large

region r ,R0 of constant energy density and we can mothe evolution inside its domain of dependence by ak521Robertson-Walker universe,

ds252dt21a2~ t !ds2, ~4.4!

whereds2 is the metric on the four dimensional unit hypeboloid. The field equations are

a

a5

1

6 FV~f!23

2f2G , ~4.5!

f14a

af1V,f50, ~4.6!

and the constraint equation is

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a22a2

6 F1

2f21V~f!G51. ~4.7!

It is well known that a homogeneous scalar field, rollindown a negative potential, produces a singularity in fintime @22,23#. The argument is the following. We start witf5A/R0

2!1 andf50, so initially we have

f~ t !5A

R02

cosh 2t, a~ t !5H21cosHt. ~4.8!

By Eq. ~4.5!, a/a is always less than its initial value2H2

~which is close to one!. So the scale factor must vanish intime less thanp/2H. SincefÞ0, the vanishing of the scalefactor causes the energy in the scalar field to diverge, resing in a curvature singularity. More precisely, after a certatime T0 the potential term in Eq.~4.6! is unimportant and thefield behaves asf5c/a4, wherec is a constant. Matching aT0 givesc'A/R0

2. From Eq.~4.7! it follows that the changein f induces a change in the form of the scale factor wha2f2 is of order one, which occurs whena3'c. Assumingthe potential term is negligible compared to the kinetic te~which can be confirmed after the solution is found! Eq. ~4.7!reduces to a22c2/(12a6)'0, which implies a(t)}(Ts2t)1/4 and hencef(t)}2 ln(Ts2t). Actually, Eq. ~4.7! alsodetermines the coefficient so thatf52(A3/2)ln(Ts2t).Since the scalar field diverges, one has a curvature singity.

Before one can claim that our initial data evolve tosingularity, one must check that the homogeneous apprmation is valid all the way to the singularity. This is nocompletely obvious since the boundary of the domain ofpendence is a null surface, and in pure AdS, a radial lightcan travel an infinite distance in finite time. So we needcalculate the proper distance on the initial surface traveledthe inward going radial light ray from the border of the hmogeneous region atr 5R0 to the singularity. From theRobertson-Walker form of the metric, this isl5a(0)*0

Ts dt/a(t). In pure AdS the distancel diverges. But,as we have seen, in our casea(t) changes its form near thsingularity resulting in finitel. If f(R0)'c!1 then the cut-off on t where a(t) changes its form occurs close to thmaximum valuep/2H, yielding l}2 ln c1/3.1 ~for instancefor c5.01 one hasl'2.3). Since the proper distance is prportional to lnr this implies that the homogeneous appromation is good all the way to the singularity for radii lethane2 lR0. This is much smaller thanR0 but it can easily bemade arbitrarily large by increasingR0 keepingf(R0) fixed.If f(R0) is of order one, then the size of the singular regiis ;R0, for largeR0.

If the total mass could not increase, one could easily shthat a black hole could not enclose this singularity as flows. If this singularity lies inside a black hole, then we ctrace the null geodesic generators of the event horizon bto the initial surface, where they will form a sphere of radiat leastRs5e22lR0 @one factor ofl is for the reduced size o

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the domain of dependence at the singularity and the secis because the event horizon is an outgoing null surface~seeFig. 1!#. The area theorem for black holes only requiresnull convergence condition and hence still holds eventheories withV(f),0. Since the area of the event horizocannot decrease during evolution and the mass cannocrease, the initial massM must be large enough to supportstatic black hole of sizeRs . Clearly it is impossible to pro-duce a Schwarzschild AdS black hole, since this requirepositive massMBH}Rs

4 , and our mass is negative. But oncould imagine the formation of a black hole with scalar hawith f(r );r 22 at larger so that the hair renders the totmass finite and negative. However we have numerically vfied that with our potential all black hole solutions with sclar hair havef(r ); ln r/r2 at larger. Thus our finite massinitial data cannot evolve to a black hole with scalar hair

We have seen that the total mass is not conserved,might increase during evolution. If it increases enoughblack hole could form. To ensure that a naked singularityproduced, we can impose a large radius cutoff as mentioabove. This is discussed in more detail in the AppendSince the cutoff can be at arbitrarily large radius, we wcontinue the following discussion ignoring the cutoff.

Inside the domain of dependence of the homogeneousgion the singularity will be spacelike, like a big crunch. Tsingularity is likely to extend somewhat outside this domof dependence~so our estimate forRs is really a lower limit!,but not reach infinity. So the singularity will either endbecome timelike. In both cases, one has naked singulariIn fact, there is really no way to distinguish these two casince the evolution ends at the first moment that a nasingularity appears. To see a timelike singularity, one wohave to know the appropriate boundary conditions to impat the singularity, which is not possible classically. If tsingularity did reach infinity, it would cut off all space, producing a disaster much worse than naked singularities.this is unlikely since there would then be a radiusRc on theinitial surface such that the outgoing null surfaces forr.Rc expand indefinitely and reach infinity, while those wir ,Rc hit the singularity and~probably! contract to a point.This indicates that the surface withr 5Rc would reach a

FIG. 1. If an event horizon encloses the singularity, it must haan initial size greater thanRs .

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finite radius asymptotically, just like the stationary horizowhich are ruled out.4

A similar argument allows us to say something aboutgeometry near the naked singularity. Consider the area ofspherical cross sections on an outgoing null surface whhits the naked singularity. If the areas shrink to zero asreaches the singularity, then a nearby null surface startinslightly larger radius will have the areas decrease nearnaked singularity and then increase as the surface reainfinity. This contradicts the Raychaudhuri equation andnull convergence condition. We conclude that the areaspheres near the naked singularity remain of nonzero sThe naked singularity is metrically a sphere and not a poWe have seen that inside the domain of dependence ofhomogeneous region, the singularity is a strong curvatsingularity and all spatial distances shrink to zero. Tshows that as the singularity extends outside this regionbecomes weaker, and when it ends, the two sphere remafinite size. The curvature, however, still diverges.

The above arguments assumed spherical symmetry,that was not essential. In the central region, the collapsRobertson-Walker metric develops trapped surfaces. Weclearly perturb our initial data and construct nearby initdata ~which need not be time-symmetric! which will stillproduce trapped surfaces. The singularity theorem guatees that a singularity must form. On the other hand,energy will still remain negative, so the singularity cannotenclosed inside a black hole. Thus cosmic censorship isnerically violated in the theory~2.5!. In fact, cosmic censor-ship is generically violated inD55, N58 supergravity,since one can also perturb the other fields in the theorystill produce naked singularities.

The fact that the naked singularity is not a point holeven for general, nonspherical solutions. To see this, consthe boundary of the past of infinity in the maximal evolutioof our initial data.~We are assuming boundary conditionsinfinity, so the fact that infinity is timelike is not a problemfor evolution.! This is a null surface which ends on the naksingularity. Standard arguments show that this surface is gerated by null geodesics which cannot be converging. Soarea of any cross section increases into the future.5

C. Ten dimensional viewpoint

D55, N58 supergravity is believed to be a consistetruncation of ten dimensional type IIB supergravity on S5.This means that it should be possible to lift our five dimesional solution to ten dimensions. At the linearized level,fields which saturate the BF bound correspond to,52

4There is also the possibility that the singularity could beconull. If the null singularity reached infinity, it would again cut off aspace, and be worse than a naked singularity. If it remained insifinite region, it would be like a static black hole with singular hrizon. Numerical evidence suggests that even these solutions dexist when the energy is very negative.

5One can view this null surface as a type of event horizon sithe points inside cannot communicate with infinity. However thevent horizon becomes singular and does not correspond to adard black hole.

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modes on S5. Since the field diverges at the singularity, oexpects that the sphere will become highly squashed.

Even though it is not known how to lift a general solutioof D55, N58 supergravity to ten dimensions, this is knowfor solutions that only involve the metric and scalars thsaturate the BF bound@24#.6 So we can immediately writedown the ten dimensional analog of the solution that evolto naked singularities. The ten dimensional solution involvonly the metric and the self dual five form. To describe thewe first introduce coordinates onS6 so that the metric on theunit sphere takes the form (0<j<p/2)

dV55dj21sin2 j dw21cos2 j dV3 . ~4.9!

Letting f 5ef/2A3 andD25 f 22sin2j1f cos2j, the full ten di-mensional metric is

ds102 5D ds5

21 f Ddj21 f 2D21sin2jdw2

1~ f D!21cos2 j dV3 . ~4.10!

This metric preserves anSU(2)3U(1) symmetry of the fivesphere. The five form is given by

G552Ue523 sinj cosj f 21

* d f`dj, ~4.11!

where e5 and * are the volume form and dual in the fivdimensional solution and

U522~ f 2cos2j1 f 21sin2j1 f 21!. ~4.12!

In the homogeneous region of the asymptotic AdS5 space,the metric can be written in Robertson-Walker form~4.4!.Near the singularity we havea(t)}(Ts2t)1/4 and f(t)52(A3/2)ln(Ts2t). Therefore, over most of theS5 (jÞp/2) near the singularity, the metric approaches

ds102 5cosj@2~Ts2t !21/8dt21~Ts2t !3/8ds2

1~Ts2t !23/8dj21~Ts2t !23/8tan2j dw2

1~Ts2t !3/8dV3#. ~4.13!

Introducing a new time coordinateh5(Ts2t)15/16, thistakes a simple Kasner-like form

ds102 5cosj@2~16/15!2 dh21h2/5ds21h22/5dj2

1h22/5tan2j dw21h2/5dV3#. ~4.14!

Both the five sphere and the asymptotic anti–de Sitter spdevelop a singularity at the same time.

V. DISCUSSION

We have shown there are asymptotically anti–de Sisolutions toN58 supergravity that have negative total eergy. The reason such solutions can exist is that for fiesaturating the BF bound the positive energy theorem requstronger boundary conditions than those required for fin

6We thank Michael Haack for bringing this paper to our attentio

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mass. By contrast, in asymptotically flat space~and inanti–de Sitter space provided there are no fields saturathe BF bound!, the boundary conditions required for finitmass coincide with those required for the positive enetheorem to hold.

Nevertheless supersymmetry guarantees there exismodified energy that is always positive. This preventsanti–de Sitter vacuum from decaying through quantum tneling to a different~regular! state with zero energy. So thexistence of solutions with negative energy does not imthe anti–de Sitter vacuum is unstable. Classical stabilitythe linearized level comes from the existence of a compset of mode solutions that are oscillating in time. It iscourse much harder to prove that small nonlinear fluctuatido not develop singularities. In light of our results this nobecomes a particularly important issue.

The modified energy consists of the usual energy plusextra surface term depending on the asymptotic value ofscalar field. To better understand this extra term, we briereview the situation for fields with masses slightly aboveBF bound@25#. For scalars in AdSd with mass2(d21)2/4,m2,2(d21)2/411, there are two complete sets of nomalizable modes with falloffr 2l6 ~1.2!. Usually, one as-sumes that the faster falloffl1 corresponds to nontriviastates in the dual CFT while the slower falloff correspondsmodifying the CFT by adding sources. This is supportedthe fact that solutions that behave asr 2l2 near the boundaryhave divergent total energy. But we know that the dimensof operators in the dual field theory is justl, and the fieldtheory contains operators of dimension less than (d21)/2,so if the AdS-CFT correspondence is correct then there mbe two ways to quantize this field so that the roles ofmodes are interchanged. This is indeed the case. The uaction S52 1

2 *(¹f)21m2f2 diverges for modes that faloff like r 2l2, so it is only appropriate for the faster fallofBut we can obtain a finite action by adding the surface te12 rf¹f. Furthermore, the energy derived from this actionnow finite ~and positive!, even for the modes which fall oflike r 2l2. For example, in AdS5, by writing f5c/(11r 2)l2/2 one can bring the usual energy to the form

E51

2E @~ c !21~11r 2!~Dc!21~4l21m2!c2#

3r 3

~11r 2!11l2dr dV32 lim

r→`

r 422l2 R c2 dV3 .

~5.1!

Adding the surface term to the action cancels the surfterm in this expression for the energy, leaving a manifespositive and finite result.

Even though the surface term one adds to the action his exactly analogous to the one we added in Eq.~3.4!, there isan important difference. When the BF bound is saturatthere is only one quantization of the field consistent withAdS symmetries@25#. Modes which fall off like 1/r 2 corre-spond to nontrivial states and modes which fall off likln r/r2 correspond to perturbing the theory. The surface te.

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we add is finite and does not change the quantization oftheory, but just the definition of the energy.7

Not surprisingly the existence of negative energy sotions has interesting physical consequences. We have shthat a subset of the negative energy configurations evolvnaked singularities. This means that cosmic censorshiviolated generically inD55, N58 supergravity, which isthe low energy limit of string theory with AdS53S5 bound-ary conditions. With AdS43S7 boundary conditions, one obtains D54, N58 supergravity. Although we have focuseon five dimensions, this theory also has fields which satuthe BF bound. So it will also have negative energy solutioand violate cosmic censorship.

It is an open question whether the cosmic censorshippothesis holds in string theory with asymptotically flboundary conditions. We have recently shown that a laclass of Calabi-Yau compactificationsM43K contain fourdimensional potentials which become negative@27#. In thiscase the positive energy theorem guarantees the positivithe ADM energy for all solutions that tend asymptoticallythe supersymmetric vacuum. However one might imagthere exist certain configurations that have a large cenregion with negative energy density which develops a sinlarity, but with a positive total mass that is too smallenclose the singular region by an event horizon. In@10# wehave shown this is possible in certain theories with a posienergy theorem in asymptotic AdS space, and we havelined a~numerical! program to test this possibility in asymptotically flat space.

Our results also provide a new approach for studying cmological singularities in string theory. The naked singulaties we find are spacelike inside some region. They arebig crunch singularities embedded in an asymptoticaanti–de Sitter space. Actually, since our initial data are tisymmetric, there are similar singularities in the past. Ththe central regions of our solutions look like~part of! uni-verses beginning in a big bang singularity and ending ibig crunch~see Fig. 2!. It is of great interest to ask whahappens to these singularities in a full quantum theorygravity. One can now begin to address this issue usingdual field theory description. Although we have not yet eplored this in detail, we can make the following preliminacomments.

The scalars which saturate the BF bound in AdS cospond in the gauge theory to the operators Tr@XiXj

2(1/6)d i j X2# where Xi are the six scalars inN54 superYang-Mills theory. Each quantum of the field in AdS corrsponds to the gauge theory state obtained by acting with

7If one considers a patch of anti–de Sitter space in Poincare´ co-ordinates then there is a continuum of oscillating modes with fal1/r 2 that form a complete set. However, if one considers the mowith falloff ln r/r2 the continuum is not complete. There is one exbound state that grows exponentially in time@26#. Therefore turningon this mode should correspond to an unstable deformation oboundary theory.

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of these operators on the vacuum.8 To reproduce our classicaconfigurationf(r ), one could take the corresponding coheent state and map it to the gauge theory. Since this is a fienergy state, there appears to be no reason for the gtheory evolution to break down. This means that the AdCFT correspondence implies that string theory must resothe naked singularities. In fact one expects even the largNgauge theory to have a well defined evolution for all timindicating thatclassical string theory should resolve thessingularities.9 This can be viewed as saying that thea8 cor-rections prevent the curvature from diverging, but anotinterpretation is simply that the spacetime metric is not wdefined near the singularity.

Since the singularities are spacelike in the central regthe resolution of the singularity in the gauge theory shodetermine whether the universe can ‘‘bounce.’’ There are tpossibilities. After the formation of the singularity in AdSthe field theory state could correspond to a bulk metric whis semiclassically well defined only outside a finite region,it could correspond to a metric which is well defined evewhere. In the first case, the classical naked singularity wocontinue for a while.~It might continue for all time, or even-tually the metric could become well defined again andnaked singularity would disappear.! In the second case, thnaked singularity would only last an instant. This wouldanalogous to passing through cosmological singularitiesquantum gravity. There has been considerable debatecently about this possibility. We now have a new concrapproach for settling this issue.fs

he

8For a detailed discussion of the relation between the bulk fiand the Yang-Mills operator, see@28#.

9We thank D. Gross for pointing this out.

FIG. 2. Our solutions are like homogeneous universes beginnin a big bang singularity and ending in a big crunch, embeddedan asymptotically anti–de Sitter space. The dual field descripcan be used to study how the singularities are resolved.

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ACKNOWLEDGMENTS

It is a pleasure to thank the participants at the KITP pgram on String Cosmology for discussions, especiallyBanks, D. Gross, R. Kallosh, P. Krauss, D. Marolf, andWarner. This work was supported in part by NSF grant PH0244764.

APPENDIX: NAKED SINGULARITIESIN THE THEORY WITH A CUTOFF

Our argument for the existence of naked singularitiesSec. IV B is incomplete since we had to assume that the tenergy did not grow. One can ensure that the usual energconserved by picking a large but finite radiusR1 and requir-ing f(R1) to be constant in time. This is automaticalimplemented in most numerical evolution schemes, andstandard regulator in discussions of AdS/CFT. Howevesubtlety now arises since modes that behave like lnr/r2 nearthe cutoff cannot be excluded if they have a sufficiensmall coefficient. Since these modes fall off more slowthan the usual 1/r 2 modes we have been considering, thenergy is even more negative. The easiest way to shownaked singularities can be produced is to modify our inidata. We consider the following class of configurations:

f~r !5f05A

lnR1

lnR0

R02 ~r < R0!,

f~r !5A

lnR1

lnr

r 2 ~R0,r ,R1!. ~A1!

Note that at the cutoff,f(R1)5A/R12. For f0!1 and

lnR0/ lnR1!1 the total mass of the initial data is given by

Mi522p2A21p2A2

lnR1

1 p2A2S lnR0

lnR1D 2

. ~A2!

As we showed earlier, since the density is constant insidesphere of radiusR0, the central region will evolve as a colapsingk521 homogeneous universe and produce a curture singularity.

If the singularity lies inside a black hole then the sizeits event horizon on the initial surface must be at leastRs

10500

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aa

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he

-

f

'f02/3R0. In addition, the boundary condition on the field

R1 implies that the black hole cannot be a SchwarzschAdS black hole. It must have some scalar hair. At large rathe hair behaves as

f~r !5a/r 21b lnr /r 2. ~A3!

For any valuef(Rs)Þ0, both coefficientsa andb are non-zero, but to compute a lower bound on the mass of the blhole solution we consider the case in which all the hair isthe logarithmic mode.10 The boundary condition then requiresb5A/ lnR1, and assumingRs.1, we obtain the fol-lowing estimate for the total mass of the black hole:11

MBH'22p2A2 1p2A2

lnR1

13p2f08/3R0

412p2A2S lnR0

lnR1D 2

.

~A4!

Comparison with Eq.~A2! yields, for smallf0,

MBH2Mi ' p2f02R0

4 . 0. ~A5!

Since the mass of the black hole is greater than the inmass~and the mass is now conserved!, the singular regioncannot be enclosed by an event horizon. Cosmic censoris violated.

This example is not ideal since the limitR1→` is notstraightforward. Also the lnr/r2 behavior of the configurationsuggests that it should be described by a small perturbaof the gauge theory, and not just a nontrivial state. Howevthe large radius cutoff is believed to be equivalent to a Ucutoff in the gauge theory, and everything that happens inbulk should have an analog in the dual theory. So onestill hope to use the gauge theory to resolve naked singuties in the bulk.

10Here we assumea . 0 in the exact black hole solutions.11Near the horizon the precise radial profile forf(r ) deviates

from Eq. ~A3!, but numerical computations show that this estimais accurate, provided of course that lnR0/ lnR1!1 and thatf(Rs)!1.

s.

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Negative energy in string theory and cosmic censorship violation