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PHYSICAL REVIEW D, VOLUME 64, 065027

Scalar field and the anti–de Sitter spaceÕconformal-field theory conjecture

Saurya Das* Department of , The University of Winnipeg, 515 Portage Avenue, Winnipeg, Manitoba, Canada R3B 2E9

J. Gegenberg† and V. Husain‡ Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3 ͑Received 6 February 2001; published 28 August 2001͒ We describe a class of asymptotically AdS field spacetimes, and calculate the associated conserved charges for three, four and five dimensions using the conformal and counterterm prescriptions. The energy associated with the solutions in each case is proportional to ͱM 2Ϫk2, where M is a constant and k is a scalar . In five spacetime dimensions, the counterterm prescription gives an additional vacuum ͑Ca- simir͒ energy, which agrees with that found in the context of AdS conformal-field theory ͑CFT͒ correspon- dence. We find a surprising degeneracy: the energy of the ‘‘extremal’’ scalar field solution Mϭk equals the energy of global AdS. This result is discussed in light of the AdS/CFT conjecture.

DOI: 10.1103/PhysRevD.64.065027 PACS number͑s͒: 11.10.Kk, 11.25.Mj

The nonlinear coupling of to matter in general With this motivation we study the Einstein-scalar field relativity presents difficult technical problems in attempts to system, with minimally coupled massless scalar field and understand gravitational interactions of elementary particles negative cosmological constant. We present static spherically and strings, as well as questions such as the details of gravi- symmetric AAdS solutions of these equations. For spacetime tational collapse. Progress in the former area has come dimension dϭ3, the equations can be solved exactly. For d mainly from treating quantum fields as propagating on fixed у4, the corresponding equations can be solved analytically background geometries ͓1͔, whereas much of the progress in for large radial , i.e., asymptotically. We calculate the latter has come from detailed numerical ͓2͔. conserved charges associated to these spacetimes using the Exact solutions of the relevant matter-gravity equations conformal ͓6͔ and the counterterm ͓7͔ prescriptions. Finally can play an important role by shedding light on questions of we discuss some consequences of our results, in particular interest in both general relativity and string theory. One is the surprising energy degeneracies associated with the solu- often interested in certain classes of solutions, with specified tions: The ‘‘extremal’’ limit of our solutions have the same asymptotic properties, the most common of which are the energy as the corresponding global AdSd spacetime. asymptotically flat spacetimes. Recent work in string theory The solutions we discuss are singular at the origin. This however, has highlighted the importance of another class of raises the question of whether these are ‘‘admissable’’ in the spacetimes via the AdS/conformal-field theory ͑CFT͒ conjec- context of the AdS/CFT conjecture. The fact that the energy ture ͓3͔. These are the asymptotically anti–de Sitter space- of this class of solutions turns out to be finite is a hint that times ͑AAdS͒. the singularity may be resolved by quantum effects. Indeed, The AdS/CFT conjecture is a duality between string there has been a suggestion ͓8͔ that singularities are valuable ϫ 5 theory on AdS5 S and the large N limit of conformally in that they are concomitant with the absence of an energy invariant Nϭ4 SU(N) Yang-Mills ͑YM͒ theory on the lower bound, which is another criteria for excluding solu- boundary of AdS5. This conjecture proposes a direct corre- tions. This is not the case for the solutions we present, since spondence between physical effects associated with fields the corresponding energies are well defined and bounded be- propagating in AdS spacetime and those of a conformal low by the energy of global AdS. quantum field theory on the boundary of AdS spacetime. The equality of the vacuum energy of AdS ͑calculated via Significant evidence for the conjecture has come from study- the counter-term method͒, and the Casimir energy of the ing free scalar or other fields on a fixed AdS background ͓4͔. boundary YM theory is considered to be a piece of the evi- Another important aspect of this conjecture is that it connects dence for the AdS/CFT conjecture. However, our degeneracy infrared effects in AdS space to ultraviolet effects in the result demonstrates that there is another spacetime with the boundary theory ͓5͔. This in turn implies a connection be- same Yang-Mills Casimir energy. This raises an ambiguity tween low and high energy physics in the respective theories. concerning this dictionary entry of the AdS/CFT correspon- A fully non-linear gravity-scalar field solution can provide a dence. window into this aspect of the conjecture as well, as is the The dϭ3 matter-gravity model considered in this paper case for scalar fields on a fixed background. has been studied numerically in the full time dependent con- text with circular symmetry in Refs. ͓9,10͔, where critical behavior at the onset of black hole formation is observed. *Email address: [email protected] More recently the critical exponent for the apparent horizon †Email address: [email protected] radius found in ͓10͔ has been verified by an analytical per- ‡ Email address: [email protected] turbation theory calculation ͓11͔.

0556-2821/2001/64͑6͒/065027͑5͒/$20.0064 065027-1 ©2001 The American Physical Society SAURYA DAS, J. GEGENBERG, AND V. HUSAIN PHYSICAL REVIEW D 64 065027

We consider the massless scalar field minimally coupled x2Ϫb2 h͑x͒ϭ ͑xϪb͒Ϫ(1Ϫa)/2͑xϩb͒Ϫ(1ϩa)/2, to gravity in d spacetime dimensions. With the parametriza- xϪab tion of the negative cosmological constant as ⌳ϭ Ϫ Ϫ Ϫ 2 (d 1)(d 2)/2l , the field equations are 1 1Ϫa2 xϪb ␾͑x͒ϭ ͱ lnͩ ͪ . ͑dϪ1͒ 2 2 xϩb ͒ ͑ ␾ ץ␾ ץϩ ϭ Rab 2 gab a b , 1 l Thus the scalar field is real for a2р1. The metric may be written in the new radial coordinate x as where factors of 8␲G are suppressed. We assume the static spherically symmetric form l2 2ϭϪ͑ Ϫ ͒(1ϩa)/2͑ ϩ ͒(1Ϫa)/2 2ϩ 2 ds x b x b dt ͑ 2Ϫ 2͒ dx h͑r͒ 4 x b 2ϭϪ ͑ ͒ ͑ ͒ 2ϩ 2ϩ 2 ⍀2 ͑ ͒ ds g r h r dt ͑ ͒ dr r d dϪ2 , 2 ϩ 2 ͑ Ϫ ͒(1Ϫa)/2͑ ϩ ͒(1ϩa)/2 ⍀2 ͑ ͒ g r l x b x b d dϪ2 . 8 of the metric in d spacetime dimensions, which give the The Ricci scalar of the metric is coupled equations 3x2Ϫ4b2ϩa2b2 rl2gЈϩ͑dϪ3͒͑gϪh͒l2Ϫ͑dϪ1͒r2hϭ0, RϭϪ2 , ͑9͒ l2͑x2Ϫb2͒ r ϭ ͑lnh͒Јϭ ͑␾Ј͒2, which shows that there are curvature singularities at x dϪ2 Ϯb, corresponding to the origin rϭ0. Also, it confirms that ͑3͒ the spacetime is AAdS, since R(x→ϱ)ϭϪ6/l2. Since the solution contains no horizons for non-vanishing scalar field, k the singularity at rϭ0 is naked. ␾Јϭ Ϫ , There are two special cases of this metric which are fa- grd 2 miliar, both of which correspond to vanishing scalar field k ϭ 2ϭ where the prime denotes the derivative with respect to r and 0. The first is a 1 for which the metric reduces to k is an integration constant obtained by integrating the Klein- r2 dr2 Gordon equation. ds2ϭϪͩ ϯ2b ͪ dt2ϩ ϩr2d␪2, ͑10͒ ␺ dϪ3 2 2 2ϯ By a field redefinition g(r)ª (r)/r , the above set of l r /l 2b equations give where the ϯ signs correspond to aϭϮ1 respectively. Thus ␺Љ ͑dϪ1͒͑dϪ2͒r2ϩ͑dϪ3͒͑dϪ4͒l2 k2 aϭ1 is the non-rotating BTZ black hole with 2b ␺2ͫ r Ϫ ͬϭ . ϭ ϭ ␺Ј ͑ Ϫ ͒ 2ϩ͑ Ϫ ͒ 2 dϪ2 C/2. The second is b 0 and a arbitrary, which gives the d 1 r d 3 l zero mass BTZ black hole, rather than global AdS space- ͑4͒ 3 time. у This basic equation determines the spacetime metrics of in- d 4. For spacetime dimensions greater than three, Eq. ͑ ͒ terest. This equation is especially simple for dϭ3 and we 4 cannot be solved analytically. However it is possible to deal with it separately, followed by the cases dу4. obtain an asymptotic expansion of the solution for large r. ӷ ͑ ͑ ͒ dϭ3. The differential equation ͑4͒ reduces to ͓12͔ For r l and fixed l), Eq. 4 can be approximated as ␺ 2 gЉ͑r͒ Љ k 2 2 ␺2ͫ Ϫ͑ Ϫ ͒ͬϭ ͑ ͒ g͑r͒ ͫr Ϫ1ͬϭk . ͑5͒ r d 2 Ϫ , 11 g͑r͒ ␺Ј d 2 ␺ The complete solution in this case can be expressed as r which has the exact solution 0(r) given implicitly by ϭr(g) with ␺ 2 1/2(dϪ1) A 0 k rϭBl2/(dϪ1)ͫ␺2ϩ Ϫ ͬ gϩA/4ϩC/4 A/4C 0 dϪ1 ͑dϪ1͒͑dϪ2͒ ϭ ͑ 2ϩ Ϫ 2 ͒1/4 ͑ ͒ r Bl g Ag/2 k /2 ͩ ϩ Ϫ ͪ , 6 g A/4 C/4 ␺ ϩA/2͑dϪ1͒ϩC/2͑dϪ1͒͑dϪ2͒ (A/2C)[(dϪ2)/(dϪ1)] ϫͫ 0 ͬ ␺ ϩ ͑ Ϫ ͒Ϫ ͑ Ϫ ͒͑ Ϫ ͒ . where A and BϾ0 are integration constants, and C 0 A/2 d 1 C/2 d 1 d 2 2 2 ϭͱ8k ϩA . Without loss of generality, we can set Bϭ1. It ͑12͒ is convenient to define the variables xϭgϩA/4,bϭC/4,a ϭA/C, in terms of which Eq. ͑6͒ becomes As before the metric for large r may be written using the ϭ␺ ϩ Ϫ ϭ Ϫ Ϫ variables x 0 A/2(d 1), b C/2(d 1)(d 2), and a (1Ϫa)/4 (1ϩa)/4 ϭ Ϫ ␺ ϳ (dϪ1) rϭl͑xϪb͒ ͑xϩb͒ . ͑7͒ (d 2)A/C. For large r, 0(r) r . The next term in the asymptotic expansion for large r is Using Eq. ͑3͒ gives obtained by writing

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␺ ͒ϭ␺ ͒ ϩ␣ 2 2͒ ͑ ͒ TABLE I. Conserved in the two approaches. ͑r 0͑r ͑1 l /r , 13 where ␣ is a constant. For dϭ5 one finds ␣ϭ1 by substi- Dim Q1 Q2 ͑ ͒ tuting this expression into Eq. 4 , a result which is useful for 3– ab/4G calculating conserved charges. For AAdS geometries, the calculation of conserved 4 ab/Gab/G charges is complicated by the occurrence of divergent ex- ␲ ␲ ϩ ␲ 2 pressions. These occur essentially because the metric di- 59ab/16G 9 ab/16G 3 l /32G verges as r2 for large r. There exist two quite distinct proce- dures for obtaining ‘‘regularized’’ finite expressions for The counter-term method proposes that the Einstein- asymptotic conserved charges. These are the so-called ‘‘con- ͓ ͔ ͓ ͔ Hilbert SEH should be supplemented with additional formal’’ 6 and ‘‘counter-term’’ 7 methods. boundary terms dependent on the intrinsic metric ␥. Since In the conformal method, a conformal transformation the variational principle is defined with fixed boundary met- ϭ⍀2 ˆ gab gab is performed on the physical spacetime ric, this does not change the . The full ˆ ˆ ϩ (M,gab) under consideration, such that the asymptotic re- action S Sct is used to obtain an effective energy momen- gions get mapped to a finite distance in a new manifold tum associated with the boundary, ͓ ͔ (M,gab) 6 . A boundary is then added to this conformally transformed ͑and unphysical͒ manifold. This is especially ␦ ϩ ͒ 2 ͑SEH Sct useful for AAdS spacetimes, because many of the canonical T . ͑17͒ abªͱ ab metric components diverge asymptotically and limits such as Ϫ␥ ␦␥ r→ϱ become rather tricky. On the other hand, the trans- formed manifold has a completely well-behaved structure. The conserved charge associated with the symmetry gener- The procedure basically involves showing that the electric ated by a vector field ␰ is then defined by parts of the Weyl tensor satisfy, as a consequence of Ein- stein’s equations, a conservation law at null infinity, I,inthe 1 ͵ dϪ2ͱ␴ a␰b ͑ ͒ Q␰ d Tabu , 18 conformally transformed spacetime. The end result is that the ª8␲G ⌺ following equation holds on I: M, ua is the timelike unitץ where ⌺ is a spatial slice of ͑dϪ3͒ ⌺ ␴ ␥ ϩ D pE ϭϪ8␲G T nahb , ͑14͒ normal at , and abª ab uaub . mp l ab m For dϭ3, 4 and 5, the following expression suffices to yield finite charges for commonly encountered AAdS space- where D p is the intrinsic covariant derivative on I, compat- times, including the scalar solutions: Ϫ 2 ible with the induced metric habªgab l nanb (na ␥⍀), E is the electric part of the Weyl tensor at I dϪ2 lͱϪ ٌ ª a ab ϭϪ ͱϪ␥Ϫ ͑␥͒ ͑ ͒ 2 ⍀3Ϫd m n ⍀2Ϫd ˆ Lct ͑ Ϫ ͒ R . 19 defined as Eabªl Cambnn n and Tabª Tab on l 2 d 3 I. The above conservation equations dictate the following form of conserved charge associated with the conformal Kill- The resulting stress-energy tensor is ing vector field ͑KVF͒ ␰: dϪ2 1 l T ϭK Ϫ␥ KϪ ␥ ͓ ͔ Ϫ Ͷ ␰a b ͑ ͒ ab ab ab l ab Q␰ C Eab dS . 15 ª 8␲G dϪ3 C l 1 ϩ ͩ ͑␥͒Ϫ ␥ ͑␥͒ͪ ͑ ͒ Ϫ Rab abR . 20 ͑An ordinary KVF on Mˆ becomes the conformal KVF on d 3 2 M.͒ In the presence of matter fields, this charge satisfies the covariant balance equation The conserved charges for various dimensions are calcu- ͑ ͒ lated using Eq. 15 in the conformal method (Q1), and Eq. ͑ ͒ ␲ 18 in the counterterm method (Q2). Restoring the 8 G ͓ ͔Ϫ ͓ ͔ϭ ͵ ␰a b ͑ ͒ factors we obtain the expressions given in Table I. Q␰ C2 Q␰ C1 Tab dS 16 ⌬I Note that the conformal method does not apply for dϭ3, as the Weyl tensor is identically zero. The following com- I where C1 and C2 are two cross-sections on bounding the ments discuss additional aspects of the solutions: region ⌬I. Equations ͑15͒ and ͑16͒ are the fundamental re- ͑i͒ Although we have not been able to find an exact solu- lations which we will use to define conserved quantities. tion of Eq. ͑4͒ above three dimensions for all r, there is Thus apart from volume factors, the electric part of the Weyl strong evidence that the solutions in higher dimensions have tensor is the relevant to be calculated. For our met- curvature singularities at the origin. This may be seen ana- rics the scalar field vanishes too quickly to be captured on lytically and numerically. Analytically, Eq. ͑4͒ can be inte- the right hand side of Eq. ͑16͒. grated near rϭ0. For example, in dϭ5 this equation gives

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2 4 ͑ ͒ k r vi The energy of Schwarzschild-AdS spacetime M SA ␺ϭͱ Ϫ , ͑21͒ ͑calculated using either method͒ can be matched by choosing 6 6l4 C and k of the scalar solutions such that from which the Ricci scalar is RϳrϪ6. Numerically, this equation can be integrated for all r and d ͑for example using 1 C2 ͑dϪ1͒ M ϭ ͱ Ϫ4k2 . ͑24͒ MAPLE͓͒13͔. For all initial conditions considered, ␾Ј shows SA 4 ͑dϪ2͒2 ͑dϪ2͒ a strong divergence at the origin, which is similar to the behavior seen analytically in three dimensions. Finally, an Thus the energy degeneracy pointed out above, concerning exact scalar field solution in five dimensions has been found global AdS, extends to the Schwarzschild-AdS metrics. ͓ ͔ recently by one of us 14 . Although this exact solution arises ͑vii͒ There exist certain intuitive criteria for the types of in different coordinates, it has the same symmetries as the gravitational singularities that might be permitted in the con- class considered in this paper. This provides further evidence text of the AdS/CFT conjecture ͓15͔. The central singularity that the spherically symmetric scalar field-AdS spacetimes in the present solutions does not violate these. Thus it ap- generically have naked singularties at the origin. pears that an explanation of the degeneracy is not yet avail- ͑ ͒ ii The two approaches for calculating conserved charges able. give positive energies for our scalar field solutions in spite of ͑viii͒ Some evidence that the energy degeneracy discussed the fact that the solutions contain a naked singularity at the above may not be manifested in other calculations in the origin. This is unlike the negative mass Schwarzschild naked AdS/CFT context comes from consideration of the full action singularity, where the associated conserved charge is nega- associated with our solutions. This computation can be done tive. exactly in three dimensions. Surprisingly, the divergence at ͑ ͒ ϭ iii Except in d 5, the two approaches yield identical the origin does not make the action infinite. ͑Recall that the ϭ results. In d 5, the counterterm prescription predicts an ad- divergence for large r is regulated by the subtraction proce- ditional Casimir energy, which is identical to that found in dure.͒ The key test is whether the scalar field parameter ap- the context of AdS/CFT correspondence, and is independent pears in the action. It does: SϭϪ␲k ͑multiplied by a factor of the scalar charge k. This is unlike AAdS spacetimes with coming from the t integration͒ for the ‘‘extremal’’ solution. rotation, where the vacuum energy depends on the rotation ͑For comparison the action of global AdS is Ϫ2␲.͒ This ͓ ͔ parameter 6 . Our results can be generalized to spacetime situation is analagous to the result for Schwarzschild-AdS, dimensions greater than five; the Casimir energy appears for where the action is a function of the black hole mass M. The all odd dimensional spacetimes. natural interpretation of the latter ͓16,3͔ is that it corresponds ͑ ͒ iv In terms of the scalar field strength k, the conserved to a CFT at finite . Since there is a scale in our charges are proportional to solution, determined by the scalar field strength, the corre- sponding CFT must have conformal invariance broken. The 1 C2 ͑dϪ1͒ exact nature of the breaking is apparently not due to tem- abϭ ͱ Ϫ4k2 . ͑22͒ 4 ͑dϪ2͒2 ͑dϪ2͒ perature since temperature cannot be associated with naked singularities. Nevertheless, these considerations provide a clear distinction between global AdS and this class of scalar ϭ Note that the k 0 case is the AdS-Schwarzschild solution field spacetimes, regardless of the energy degeneracy. ϭ Ϫ with mass M C/(d 2). Thus, surprisingly the presence of In summary we have described solutions of general rela- the scalar field effectively reduces the energy of the gravita- tivity with a cosmological constant coupled to a scalar field. tional solution. In three spacetime dimensions, the solution is exact for all r, ͑ ͒ v There is an unexpected and interesting result in the whereas for higher dimensions, the solution is an asymptotic ‘‘extremal’’ case one, for large r. The solutions have finite energy, although they do not possess an event horizon. The only nonzero C dϪ1 charges are those associated with the timelike KVF. Further- ϭ2kͱ . ͑23͒ dϪ2 dϪ2 more, these charges contain information about the strength of the scalar field k. As we have discussed above, these results lead to interesting questions concerning the AdS/CFT con- The conserved charge vanishes in the conformal calculation jecture. Among these are the issues of how a naked singular- for all dimensions and is therefore equal to the energy of ity on the gravity side translates to the field theory side, global AdS in this method. The same holds for the counter- given that the associated energy is finite, and the meaning of term method, except that the energies of the two solutions the energy degeneracy of the extremal solution and global now equal the Casimir energy in five dimensions. This sur- AdS. Finally, it would be interesting to see if the energy prising degeneracy of energy associated with two distinct degeneracies we have found can lead to the possibility of solutions on the gravity side raises an interesting question for phase transitions analagous to the Hawking-Page transition the AdS/CFT conjecture: What effects on the CFT side dis- ͓17͔ between Schwarzschild-AdS black holes and global tinguish global AdS from this extremal scalar field solution? AdS. It is possible that the answer lies in calculating other effects using the correspondence, such as n-point functions with This work was supported by the Natural Science and En- these spacetimes as background ͑see below͒. gineering Research Council of Canada.

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