Gauge Theory Duals of Cosmological Backgrounds and their Energy Momentum Tensors
aa r X i v : . [ h e p - t h ] N ov UK/07-11
Gauge Theory Duals of CosmologicalBackgrounds and their Energy Momentum Tensors
Adel Awad a,b , Sumit R. Das a , K. Narayan c and Sandip P. Trivedi d Department of Physics and Astronomy,University of Kentucky, Lexington, KY 40506
USA a Center for Theoretical Physics, British University of EgyptSherouk City 11837, P.O. Box 43, EGYPT b Chennai Mathematical Institute,Padur PO, Siruseri 603103,
INDIA c Tata Institute of Fundamental Research,Mumbai 400005,
INDIA d [email protected], [email protected]@cmi.ac.in, [email protected] Abstract
We revisit Type IIB supergravity backgrounds with null and spacelike singularities withnatural gauge theory duals proposed in hep-th/0602107 and hep-th/0610053 . We show thatfor these backgrounds there are always choices of the boundaries of these deformed
AdS × S space-times, such that the dual gauge theories live on flat metrics and have space-timedependent couplings. We present a new time dependent solution of this kind where the effectivestring coupling is always bounded and vanishes at a spacelike singularity in the bulk, and thespace-time becomes AdS × S at early and late times. The holographic energy momentumtensor calculated with a choice of flat boundary is shown to vanish for null backgrounds andto be generically non-zero for time dependent backgrounds. On leave of absence from Ain Shams University, Cairo, EGYPT
Introduction
In two previous papers [1, 2] three of us proposed gauge theory duals to a class of time dependentand null backgrounds of IIB supergravity. These solutions are deformations of
AdS × S backgrounds with non-normalizable modes of the metric and the dilaton. The null solutionsand their duals were also proposed in [3]. It is thus natural to conjecture that the dual gaugetheory is deformed by corresponding sources. Generally, the supergravity solutions are singularwith spacelike or null singularities where of course supergravity breaks down. The idea is toinvestigate whether the dual gauge theory remains well behaved in this region and possiblyprovides a way to continue the time evolution beyond the time where the supergravity issingular. Other discussions of similar solutions include [4]. For other approaches to the use ofAdS/CFT correspondence to study time dependent backgrounds, see [5].These solutions have an Einstein frame metric of the form (with the AdS scale R AdS = 1) ds = 1 z h dz + ˜ g µν ( x ) dx µ dx ν i + d Ω (1)and a dilaton Φ( x ) together with a 5-form field strength F = ω + ⋆ω (2)This is a solution if ˜ R µν = 12 ∂ µ Φ ∂ ν Φ , ∇ Φ = 0 (3)where ˜ R µν is the Ricci tensor for the four dimensional metric ˜ g µν ( x ).In this paper, the S part of the metric will remain unaltered and we will not write this outexplicitly.In these coordinates the boundary is at z = 0, and as argued in [1, 2], it is reasonable toassume that the dual gauge theory lives on a 3+1 dimensional spacetime with metric ˜ g µν ( x )and has a spacetime dependent coupling g Y M ( x ) = e Φ( x )2 . Of particular interest are solutionswhere ˜ g µν ( x ) dx µ dx ν = e f ( x ) h − dx + dx − + d~x i (4)Now, because of the spacetime dependence of the coupling constant, the Yang-Mills theoryis not conformally invariant in the sense of conformal coordinate transformations. However,the theory is still Weyl invariant under Weyl transformations of the metric and correspondingtransformations of the fields. One might therefore hope that the overall factor in (4) can beremoved by a Weyl transformation leaving us with a gauge theory on flat space with spacetimedependent coupling. Such a step would be, however, subtle in the quantum theory because ofa possible Weyl anomaly. 1ull solutions with the conformal factor depending on a single null coordinate are of specialinterest from several points of view. First, for such solutions, with f = f ( x + ) and Φ = Φ( x + )the equation (3) becomes 12 ( f ′ ) − f ′′ = 12 (Φ ′ ) (5)This means that we have a one function-worth of solutions. We can pick any f ( x + ) and solvefor Φ( x + ). In particular, we may look for solutions where the dilaton is bounded everywhere.Indeed an interesting solution is given by e f ( x + ) = (tanh x + ) e Φ( x + ) = g s | tanh( x + | √ (6)This solution asymptotes to AdS × S with string coupling g s at x + → ±∞ . Both themetric and the effective string coupling e Φ drops to zero at x + = 0 which is the location ofthe null singularity. This is good : things appears to be controlled. What makes this kindof background appealing is the fact that the Weyl anomaly in the gauge theory living in themetric (4) identically vanishes for such null backgrounds. Therefore, we could perform a Weyltransformation without bothering about the anomaly, and obtain a dual theory which is on flatspace. In this dual theory, the coupling is always bounded, vanishing at x + = 0. One wouldexpect that the Yang-Mills theory is well behaved for such profiles. A careful argument alongthe above lines was given in [2].This provides a clean formulation of the problem in terms of light front evolution. Considerthe strong ’t Hooft coupling regime of the gauge theory g Y M = g s → N → ∞ g Y M N = finite and large (7)Then at x + → −∞ we start with a gauge theory in its ground state. Since the ’t Hooft couplingis large, supergravity in AdS × S provides an accurate description in this regime. Now turnon a source leading to x + dependence of the effective coupling, g Y M → g Y M e Φ( x + ) . In thedual supergravity this means that we have turned on a non-normalizable dilaton mode Φ( x + ).As x + increases, the system evolves. In the gauge theory side the effective coupling decreases,but remains large so long as | x + | is large enough. In this regime, therefore, the supergravitydescription remains good and that is what is described by the supergravity solution. As weapproach x + = 0, the gauge theory coupling becomes weak, becoming exactly zero at x + = 0.A gauge theory with weak coupling is not well described by dual supergravity - stringy effectsare important. Indeed, if we look at the evolution on the supergravity side, and extrapolatethe solution to small x + - where supergravity should no longer be valid - we encounter a singu-larity. However, because the gauge coupling is weak, one would imagine that the gauge theorydescription remains good - maybe even well approximated by perturbation theory. This gaugetheory then describes the region which appears to be singular if the supergravity solution is2xtrapolated to the regime where it shouldnt have been in the first place. Because of null depen-dence of the coupling, the gauge theory has several properties which indicate that the theoryis actually well behaved at x + = 0. A null isometry ensures absence of particle production,and in a light cone gauge the kinetic terms are standard so that only positive powers of the x + dependent coupling appear in the nonlinear terms [2]. Since the coupling in fact vanishesat x + = 0 one might hope that perturbation theory is reliable in this region and may be evenused to extend the time evolution through x + = 0 to positive x + .In contrast to the null solutions, the time dependent solutions found in [1] do not have suchnice features. These solutions have boundary metrics which are Kasner cosmologies, e.g. ds = 1 z " dz − dt + X i =1 t p i dx i dx i i p i = 1 e Φ( t ) = | t |√ − P p i ) (8)The string coupling - and therefore the Yang-Mills coupling - still goes to zero at the spacelikesingularity at t = 0, but diverges at early or late times. As we shall see below it still turns outthat the Weyl anomaly of the boundary theory in these coordinates vanishes. However, becauseof the divergence of the Yang Mills coupling, it is unclear whether the gauge theory makes sense.Furthermore, as discussed below, time dependent backgrounds (unlike null backgrounds) havecurvature singularities at z = ∞ for any time. For the solution (8), at any fixed time t theRicci scalar diverges as z t . At early and late times the divergence, however, goes away.While the results stated above are suggestive, there are several causes for serious concern.The first issue concerns the choice of boundary described above. In this choice, the null or space-like singularity extends all the way to the boundary. While a Weyl transformation removes thisand brings us to a flat boundary metric, the conformal factor required is clearly singular :this is certainly an uncomfortable situation. Furthermore, both for the null and the spacelikecases, the behavior of the Yang-Mills coupling is non-analytic [7], casting serious doubts abouta smooth time evolution through this point. In fact, it turns out that if the conformal factor e f ( x + ) is chosen to vanish at x + = 0 in an analytic fashion, e Φ( x + ) has this non-analytic behavior.On the other hand if e Φ( x + ) is analytic, e f ( x + ) becomes non-analytic. Finally, the Kasner likesolutions with space-like singularities do not appear to lead to a controlled dual theory.In this paper we take some steps in solving some of these problems. We show that for thesolutions with brane metrics conformal to flat space, one can always choose a foliation such thatthe boundary is flat. To show this, we use the well known fact that in asymptotically AdS space-times, a Weyl transformation of the boundary theory corresponds to a special class of coordinatetransformations in the bulk - the Penrose-Brown-Henneaux transformations. We find theseexact transformations for any null solution of this kind and for the Kasner solutions describedabove. For other time dependent solutions, we find these transformations in a systematic3xpansion around the (new) boundary. Thus in these new coordinates the boundary theory isexplicitly defined on flat space and the nontrivial feature is in the time (or null time) dependenceof the coupling. Furthermore, for the null solutions of the type (4), we do not have to worryabout the function f ( x + ) any more and choose the function Φ( x + ) to have a nice analyticbehavior at x + = 0.We then describe new supergravity solutions with spacelike singularities which asymptoteto AdS × S at early and late times. The couplings are bounded everywhere and vanishat the spacelike singularity. This opens up the possibility of posing questions about space-like singularities in the gauge theory dual in a fashion analogous to our formulation of nullsingularities. At this time, however, we are still unable to arrive at such a clean formulationfor spacelike singularities.We then compute the energy momentum tensors of these solutions using standard techniquesof holographic RG[11, 17, 18, 20, 21]. We find that the energy momentum tensor vanishes forall null backgrounds for both foliations. For time dependent backgrounds, the trace anomalyvanishes in the coordinates of (1), while the energy momentum tensor in the foliation leadingto a flat boundary metric is non-zero. In fact the answer diverges at the time when thecoupling constant vanishes. While a nonzero energy momentum may be interpreted as particleproduction, one cannot attach any significance to the divergence since this happens at the placewhere the supergravity description is invalid.While this paper was being written, [13] appeared on the archive, which has some overlapwith our Section 2. In this section we will rewrite the supergravity solutions of [1] in new coordinates leading to achoice of the boundary with a flat metric.
It is well known that Weyl transformations in the boundary theory correspond to special co-ordinate transformations in the bulk - the Penrose-Brown-Henneaux (PBH) transformations[8, 9]. Any asymptotically AdS space-time may be written in a standard coordinate system ofthe Feffermann-Graham form ds = 1¯ ρ d ¯ ρ + 1¯ ρ ˜ g µν ( x, ¯ ρ ) dx µ dx ν (9)Now consider the coordinate transformations [10, 11, 12]¯ ρ → ¯ ρ e − σ ( x, ¯ ρ ) x µ → x µ + a µ ( x, ¯ ρ ) (10)4hich keeps this form of metric invariant. For infinitesimal transformations this is ensured byrequiring σ to be a function of x alone, and1¯ ρ ∂ ¯ ρ a µ = − ˜ g µν ∂ ν σ (11)The transformation of the metric ˜ g µν is given by δ ˜ g µν ( x, ¯ ρ ) = 2 σ ( x, ¯ ρ )(1 −
12 ¯ ρ∂ ¯ ρ )˜ g µν ( x, ¯ ρ ) + ∇ ( µ a ν ) ( x, ¯ ρ ) (12)The expression (12) explicitly shows that this transformation includes a Weyl transformationof the metric ˜ g µν .Consider now a metric of the form ds = 1 z h dz + e f ( x ) η µν dx µ dx ν i (13)Our aim is to perform a PBH transformation to remove the conformal factor e f ( x ) in theboundary metric. However we need to do this for finite PBH transformations. When the conformal factor f ( x ) is a function of a single null coordinate x + , i.e. when theoriginal metric is of the form ds = 1 z h dz + e f ( x + ) ( − dx + dx − + d~x ) i (14)it turns out that it is easy to figure out the correct finite PBH transformations. These are givenby the following z = w e f ( y + ) / x − = y − − w ( ∂ + f ) x + = y + ~x = ~y (15)In these coordinates the metric becomes ds = 1 w (cid:20) dw − dy + dy − + d~y + 14 w [( f ′ ) − f ′′ ]( dy + ) (cid:21) = 1 w (cid:20) dw − dy + dy − + d~y + 14 w (Φ ′ ) ( dy + ) (cid:21) (16)where in the second line we have used (5).The new coordinates provide a new foliation of the space-time. The boundary w = 0 isnaively the same as the original boundary z = 0. However, it is well known that AdS/CFT5equires an infra-red cutoff in the bulk which corresponds to a ultraviolet cutoff in the dualgauge theory. For any such finite cutoff ǫ , the boundary w = ǫ is not the same as z = ǫ , andbecomes flat in the ǫ → x + dependent coupling.Notice that in these coordinates, there is only one function Φ( x + ) which we are free tochoose. In particular, e Φ can be chosen to bounded and vanishing at x + = 0 in an analytic fashion. The proposed dual will then have a coupling which is bounded and vanishes at x + = 0in a smooth fashion.For such solutions, ∂ + Φ will, however, diverge at x + = 0. This means that the bulk space-time will be as usual singular. This may be seen by looking at the behavior of geodesics as in[1, 2]. In the new coordinates, these geodesics are w = z F ( y + ) y − = y − − z ddy + ( F ( y + ) (17)where F ( y + ) = e − f ( y + ) / . The affine parameter along such geodesics is given by λ = Z y + F ( y + ) dy + (18)It is easy to find functions Φ( x + ) so that the singularity at y + = 0 is reached in finite affineparameter. For such colutions F ( y + ) must diverge at y + = 0 The magnitude of the tidalacceleration between two such geodesics separated along a transverse direction is given by (seeequation (2.10) of [2]) | a | = ( F ( y + )) F ′′ ( y + ) (19)and would diverge as well. The form of the metric (16), however, shows that this singularityweakens as we approach the boundary w = 0 leaving a flat boundary metric. Similar considerations apply to time dependent solutions of the form of (8). We will concentrateon solutions with p = p = p = . Redefining the time coordinate, this solution may bewritten in the form ds = 1 z (cid:20) dz + 2 t (cid:16) − dt + ( dx ) + · · · ( dx ) (cid:17)(cid:21) e Φ( t ) = | t | √ (20)This solution has a spacelike singularity at t = 0.6ince the boundary metric on z = 0 is conformally flat, there should be a PBH transforma-tion which leads to a foliation with a flat boundary. This is indeed true. The solution for t > ds = 1 ρ dρ − (16 T − ρ ) T dT + (16 T − ρ ) (16 T + 5 ρ ) T (cid:16) ( dx ) + · · · ( dx ) (cid:17) (21)where the new coordinates ( ρ, T ) are related to the coordinates ( z, t ) in the region ρ < T bythe transformations z = 32 ρT √ T − ρ t = T T + 5 ρ T − ρ ! (22)The dilaton may be written down in new coordinates by substituting (22) in (20).It is clear that in this new foliation defined by slices of constant ρ , the boundary ρ = 0 hasa flat metric. However these coordinate system has a coordinate singularity at ρ = 4 T , butmay be extended beyond this point.The arguments in the previous section then indicate that there is a dual field theory whichlives in a flat space-time, but with a time dependent coupling which vanishes at T = 0. Unikethe null solutions, the coupling diverges at early or late times - and we cannot make any carefulargument about the behavior of this dual theory.As noted in the introduction, these solutions have a curvature singularity at any finite time,though the singularity goes away at early and late times. The bulk Ricci scalar is given by R = − ( 9 z t + 20) (23)In the global geometry the Poincare horizon is a product of a null plane times a S . Thissingularity appears at one point on this null plane. The rest of the Poincare horizon is non-singular. A necessary condition for a well defined dual theory is that the coupling should be bounded atall times. This motivates us to search for new solutions which have space-like singularities ofthis type. We will present such solutions in this section.These solutions are special cases of a class of time dependent solutions whose boundary7etrics are FRW universes. The Einstein frame metric is given by ds = 1 z " dz + A ( t )[ − dt + dr − k r + r ( dθ + sin θ dφ )] (24)with k = 0 , ±
1, and Φ( t ) = ±√ Z dtA ( t ) (25)where A ( t ) = C sin(2 √ k t ) + C cos(2 √ k t ) . (26)The solutions with k = − A ( t ) = | sinh(2 t ) | , thedilaton becomes e Φ( t ) = g s | tanh t | √ (27)so that the coupling is bounded and vanishes at t = 0. There is a spacelike singularity at t = 0.In the following we will restrict our attention to the “big crunch” part of the space-time, i.e.for t <
0. In this case we use A ( t ) = | sinh(2 t ) | .The boundary metric is in fact conformal to parts of Minkowski space. This is seen bydefining new coordinates (for t < r = R √ η − R e − t = q η − R (28)The solution now becomes ds = 1 z " dz + | − η − R ) | [ − dη + dR + R d Ω ] e Φ = | η − R − η − R + 1 | √ (29)The t > t in (28). In these coordinates it is clear that as t → −∞ , i.e. η − R → ∞ , the space-time is AdS and e Φ asymptotes to a constant.The coordinate transformation (28) is valid in the region η − R >
0, and η − R = 1 arethe two spacelike singularities. Even though we started with the form of the metric (24) wecould extend the solution beyond part this part of Minkowski space in the standard manner. Inthis extended solution, there are timelike singularities at R − η = 1. As is evident, the dilatonshows a singular behavior at the location of these singularities, even though the value of e Φ goes to zero. In the following we will be interested in the solution in the regions ( η − R ) > These solutions can be derived from a generic ansatz with diagonal metric, and imposing that the dilatonis a (spatially homogeneous) function Φ( t ) of time t alone. z = ∞ .In the big crunch region, The bulk Ricci scalar is given by R = − − z (sinh(2 t )) ! , (30)where t is as in (24) with A ( t ) = − sinh(2 t ) in this t < t → −∞ there is no such singularity.We can, therefore, view these backgrounds in the same way as the null backgrounds. For t < AdS × S in the infinite past. As time evolves one generatesa space-like singularity at t = 0 which extend to the boundary defined at z = 0. However,since the boundary metric is conformal to flat space, we can choose a different foliation byperforming a PBH transformation and choose a boundary which is completely flat. (In thiscase, we have not been able to find the exact PBH transformations, but - as detailed in theAppendix - the PBH transformation may be found in an expansion around the boundary). Thegauge theory defined on this latter boundary is on flat space with a time dependent couplingconstant which vanishes at the location of the bulk singularity. The source in the gauge theoryevolves the initial vaccum state. On the supergravity side, a (timelike) singularity develops at z = ∞ . While we not have a clear idea of the meaning of this singularity in the gauge theoryit is reasonable to presume - in view of the usual AdS/CFT duality - that this should manifestitself in the infrared behavior. Finally, as the time evolves, the gauge coupling goes to zero -this manifests itself as a spacelike singularity in the bulk in a region where supergravity itselfbreaks down.The analysis of this dual gauge theory appears to be more complicated than the dual gaugetheory for null backgrounds. One issue is related to the fact that the gauge theory lagrangianhas an overall factor of e − Φ . When Φ depends only on a null direction, it was shown in [2]that a choice of light cone gauge, together with a field redefinition converts the kinetic termsin the action into standard form for constant couplings. All factors of couplings then appearin the nonlinear terms as positive powers of e Φ( x + ) , which vanish at the location of the bulksingularity. This allowed us to arrive at some clean conclusions about the behavior of the gaugetheory. In [13] a different gauge choice was used - this again made analysis of the gauge theoryeasier. For time dependent backgrounds, we have not been able to find a gauge choice and afield redefition which leads to a similar simplification. Nevertheless we expect that the theoryis amenable to perturbative analysis near t = 0 where the gauge coupling becomes weak.9 The Holographic Stress Tensor
In this section we use the standard techniques of covariant counterterms [11, 17, 18, 20, 21]to calculate the holographic stress tensor. The gravity-dilaton action in five dimensional space M , with boundary ∂ M is given by, I bulk + I surf = 116 πG Z M d x √− g (cid:18) R (5) + 12 −
12 ( ∇ Φ) (cid:19) − πG Z ∂ M d x √− h Θ . (31)Where the second term is the Gibbons-Hawking boundary term, h µν is the induced metric onthe boundary and Θ is the trace of the extrinsic curvature of the boundary ∂ M .The above action is divergent. Therefore, one might use one of the known techniques toregularize such action. Here we choose to work with the covariant counterterm method sincewe are interested in calculating the boundary energy momentum and its trace. To have a finiteaction one can add the following counterterms I ct = − πG Z ∂ M d x √− h " R −
18 ( ∇ Φ) − log( ρ ) a (4) (32)where ρ is a cutoff on the radial coordinate ρ which has to be taken to zero at the end ofthe calculation. R is the Ricci scalar for h . The term proportional to log( ρ ) is required tocancel a logarithmic divergence in the action (31). However this term does not contribute tothe renormalized energy momentum tensor.Now the total action is given by I = I bulk + I surf + I ct . Using this action one can construct adivergence free stress energy tensor [17]: T µν = 2 √− h δIδh µν = 18 πG " Θ µν − Θ h µν − h µν + 12 G µν − ∇ µ Φ ∇ ν Φ + 18 h µν ( ∇ Φ) (33)Here G µν and ∇ are the Einstein tensor and covariant derivative with respect to h. In theregime where the supergravity approximation is valid, the vev of the CFT’s energy momentumtensor < T µν > is related the above stress tensor by the following equation q − ˜ g ˜ g µν < T νσ > = lim z → √− h h µν T νσ . (34)where we have used the notation of equation (9).The energy momentum tensors calculated in the holographic RG approach correspond tooperators in the dual field theory which are regularized using the specific boundary metric usedto perform the bulk calculation. Θ ab = − ( ∇ a n b + ∇ b n a ), where n a in the unit normal vector to the surface z=constant and pointing tothe boundary ∂ M .1 Conformally Flat Boundary Let us first consider bulk metrics of the form (1). This means we use a cutoff defined in termsof the radial coordinate z . Using the above expression for the stress tensor, one can easily showthat for a any solution with conformally flat boundary (i.e. of the form of equation (4), thestress tensor vanishes. Let us see how this result is obtained. First, the extrinsic curvature fora solution with a conformally flat boundary isΘ µν = − h µν . (35)The extrinsic curvature terms in the expression then cancel with the term proportional withthe induced metric. Using (3) and its contraction, one can see directly that the last threeterms exactly cancel leading to the vanishing of the stress tensor. As a result, the traceanomaly vanishes. Comparing this result with the known results in the literature one findsthe following. Our result does not match with the field theory calculation of trace anomaly in[16]. The reason is that in this calculation only terms up to quadratic order in the dilaton wereincluded and all higher orders have been ignored. But this result agrees with the holographicanomaly expression calculated in [18] since their expression contains these terms which arecrucial to have a vanishing anomaly. We will now consider the energy momentum tensor which is defined by a choice of foliationwhich leads to a flat boundary metric. This is of course a different regularization and wouldlead to a different answer which would give us the energy momentum tensor of the gauge theorydefined on flat space in an appropriate regime.
It is easy to check by a direct calculation that for the solutions with null singularities, theenergy momentum tensor continues to vanish.
Now consider the Kasner type solution in new coordinates (21). Using the above expression forthe holographic stress tensor, one gets the following T νµ = ρ π G T diag ( 9 , , ,
13) + O ( ρ ) . (36)The energy momentum tensor of the CFT as in (34) is given by < T νµ > = N π T diag ( 9 , , , , (37)11hich has the following non-vanishing trace: < T µµ > = 3 N π T , (38)here we have used G = π N . (39)The trace computed here agrees with the holographic trace anomaly found in [18]. The nonzeroenergy momentum tensor can be possibly interpreted as particle production. To calculate the energy momentum tensor for the new FRW solution with k = − τ = η − R , r = η + Rη − R (40)This puts the metric in the form ds = dz z + 1 z (1 − τ ) [ − dτ + τ r dr + τ r − r ) d Ω ] (41)The dilaton in this coordinates is given byΦ( τ ) = √ " τ − τ + 1 (42)One can use this coordinates to do a PBH transformations as explained in the appendix andobtain another form of this solution with Minkowski boundary. In this form the stress energytensor is given by T νµ = ρ π G ( ¯ T − diag (cid:16) − T , T , T , T (cid:17) + O ( ρ ) . (43)where the coordinate ¯ T is defined in the Appendix. Using (34) and (39), the energy momentumtensor of the CFT is given by < T νµ > = N π ( ¯ T − diag (cid:16) − T , T , T , T (cid:17) , (44)which has the following non-vanishing trace: < T µµ > = 12 N π ( ¯ T + 1)( ¯ T − (45)12gain this trace agrees with the calculation in [18]. Note that the energy momentum tensorvanishes at early times. This reinforces our claim that at early times we have started with Thevacuum state of the dual gauge theory, with a source which vanishes at ¯ T → −∞ . At latertimes, the source produces a nonzero energy momentum tensor as well as a nonzero expectationvalue of the operator dual to the dilaton. In other words, the Heisenberg picture state is thevacuum of the CFT. It is tempting to interpret the nonzero stress tensor as an effect of particleproduction. Once again the stress tensor diverges at the singularity ¯ T = 1. However this isprecisely the place where the holographic calculation cannot be trusted.The real question is whether the gauge theory is well behaved in this region. For nullbackgrounds, this appears to be so [22, 13, 23] For time dependent backgrounds, this is notclear at the moment, particularly because of bulk singularities at z = ∞ which signify thatthere are infrared problems in the gauge theory. These issues are under investigation. We would like to thank Costas Bachas, Ian Ellwood, Ben Craps, David Gross, A. Harindranath,David Kutasov, Gautam Mandal, Shiraz Minwalla, Tristan McLoughlin and Alfred Shapere fordiscussions at various stages of this work and Steve Shenker for a correspondence. S.R.D. wouldlike to thank Tata Institute of Fundamental Research, Indian Association for the Cultivation ofScience and Bensque Center for Science for hospitality. Some of the results were presented byS.R.D. in a talk at Benasque String Workshop in July 2007. K.N. would like to thank TIFR forhospitality during various stages of this work, as well as the organizers of Strings 07, Madrid, andthe String Cosmology workshops at ICTP, Trieste and KITPC, Beijing, where some of this workwas done. S.T. thanks the Swarnajayanti Fellowship, DST. Govt. of India, for support. Theresearch reported here was supported in part by the United States National Science FoundationGrant Numbers PHY-0244811 and PHY-0555444 and Department of Energy contract No. DE-FG02-00ER45832, as well as the Project of Knowledge Innovation Program (PKIP) of theChinese Academy of Sciences. It was also supported by the DAE, Govt. of India, and especiallythe people of India, whom we thank.
A PBH transformations
In the coordinates displayed in (23), the spacelike singularity extends to the boundary. Howeverin the form (29) the boundary metric is conformal to flat space. This suggests that there shouldbe PBH transformations which leads to a flat boundary metric. In the case of FRW solutionshowever, we have not yet been able to find the exact PBH transformations. We will showbelow how to find these transformations systematically in the neighborhood of the boundary13nd obtain them to the order which is required for our analysis of the energy momentum tensorin the next section.Let us show how can we obtain such coordinate transformation for a solution with a con-formally flat boundary on the following form ds = 1 z [ dz + f ( t ) η µν dx µ dx ν ] (46)We chose the conformal factor to depend only on one coordinate since this will be sufficient todeal with the cases under consideration in this work. One can generalize such a procedure tocases with general conformal factor f ( x µ ) and boundaries other Minkowski. Define the followingcoordinate transformations t ( ρ, T ) = X n =0 , ,.. a i ( T ) ρ i z ( ρ, T ) = ρ X n =0 , ,.. s i ( T ) ρ i (47)One can choose a ( T ) = T , then expanding all metric components in ρ , they have the followingform in new coordinates g ρρ = 1 ρ + 4 s o [ s s − a f ] + 4 s [ 2 s s − a ˙ f s + 2 a f s − a f a s ] ρ + O ( ρ ) g ρT = 1 s [ ˙ s s − a f ] 1 ρ + 1 s [ ˙ s s + s ˙ s s − a f ˙ a s − a ˙ f s +4 f a s − f a s ] ρ + O ( ρ ) g T T = − fs ρ − s [ a s ˙ f − f s + 2 ˙ a f s − ˙ s s ] − s [2 s ˙ f a + s ¨ f a − f a s s +4 ˙ f a s ˙ a − f s s + 6 f s − f ˙ a s s + 4 f s ˙ a + 2 f s ˙ a +4 ˙ s s s − s ˙ s s ] ρ + O ( ρ ) g ii = fs ρ − s [2 s f − a s ˙ f ] + 12 s [2 s ˙ f a + s ¨ f a − f a s s − f s s + 6 f s ] ρ + O ( ρ ) (48)To keep the PG form of the metric and to get a Minkowski boundary, one imposes the followingconditions g ρρ = 1 ρ , g ρT = 0 , g µν = η µν ρ + O (1) , (49)this guarantee the existence of such a coordinate system, at least, close to the new boundary.These conditions lead to s ( T ) = f ( T ) , s ( T ) = ˙ f ( T ) f ( T ) s ( T ) = ˙ f ( T ) f ( T ) , ..a ( T ) = ˙ f ( T )4 f ( T ) , a ( T ) = 0 , .. (50)14pplying the above procedure to Kasner type solutions in (20) with f ( t ) = t , one can obtainthe following coordinate transformations z ( ρ, T ) = √ T ρ ρ T + ρ T ] + O ( ρ ) , t ( ρ, T ) = T + ρ T + O ( ρ ) (51)The metric in these coordinates has a Minkowski boundary and has the following form ds = [ 1 ρ + O ( ρ )] dρ − [ 1 ρ − T + 25256 T ρ + O ( ρ )] dT +[ 1 ρ + 18 T − T ρ + O ( ρ )] d ¯ x (52)which agrees with the exact coordinate transformation in (22) and the metric (21) upon ex-panding it in powers of ρ .Before applying this procedure to calculate the FRW solution with k = − τ = η − R , r = η + Rη − R (53)This puts the metric in the form ds = dz z + 1 z (1 − τ ) [ − dτ + t r dr + τ r − r ) d Ω ] (54)Following the above procedure one can obtain the PG form of this solution with Minkowskiboundary. The coordinate transformations and the metric are z ( ρ, ¯ T ) = √ ¯ T − ρ ¯ T [1 + ρ ¯ T ( ¯ T − + ρ ¯ T ( ¯ T − ] + O ( ρ ) ,τ ( ρ, ¯ T ) = ¯ T + ρ ( ¯ T −
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