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"Behaviour of the Gravitational Field Around the Initial Singularity"

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"BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY" Gerson Francisco. t ? õ INSTIT0T0 01 flSICA TEÓRICA "BEHAVIOUR OF THE GRAVITATIONAL FIELD AROUND THE INITIAL SINGULARITY" Gerson Francisco Instituto de Física Teórica Rua Pamplona, 145 01405 - São Paulo-SP BRASIL. Abstract The ultralocal representation of the canonical^ ly quantized gravitational field is used to obtain the evolution of coherent states in the immediate neighborhood of the singulari- ty. It is shown that smearing functions play the role of classical fields since they correspond to cosrnological solutions around the singularity. A special class of ultralocal coherent states is shown to contain the essential aspects of the dynamics of the system when we choose a simple representation for the field operators of the theory. When the ultralocality condition is broken a conjec- ture will be made about the quantum evolution of coherent states in the classical limit. i-í . Intiüd^': I ion Cosmologica.1 solutions to Einstein equations a-o'jnd the Initial singularity have been thouroughly investigated _y b^Lin^kii, KhaiaLnikov and Lifshitz |1—3| (or BKL t\u short). [n fiLs connection ue also mention the u»ork of Eardley et al. |4| inn thu Hami ltoninr' tiaatment of Liang |5|. The main point of theso st'ciRs is ttip far:t thac a large class of solutions to Einstein rtn,iaf,ions \n r.ne asymptotic region close to the singularity exhi- ;•>: ' :. kifnj cr ?\.>antaneously decoupled dynamics in the sense that field ewoJution at a given initial point on a Cauchy hypersur- 5^ is totally independent from what occurs in its neighbor- hood. This oxtrerpeJy localized behaviour of the dynamics is often •T>rrPr! to as ul tr alocality |6|. The gravitational field Hamlltonian constraint Q., ic; tho ^ridtidl iisLric on 2Ü > Qa dttu'i » r\ is the scalar Ri_L L .i;/,i ., - it O,, , JllJ is the momentum canonically cunjuga- :" •' Í» ar"! X c l6TtGI& uihere (7 is Newton's constant. In na'1 / '-;on~in i o'jiral ^ociwis it happens that the kinetic term , nr jii.iuloL.3l '' i '• 11 tonian conatrainti in (l) containing the momen domiratas |l-5| over the potential term |ll| o Since the momentum variables can be expressed in terms of the "tinia" derivative of 4JJ (see (9)), cosmologists have called those cosmologicel solutions, that evolve according to (3) veloci- ty dominated. On the other hand all spatial derivatives responsi- ble for the dynamical correlation between spatial points ara con- tained in the potential (4) and thus the velocity dominated Hamil- tonian Jfc generates an ultralocal dynamics. In this paper we will quantize the velocity d£ minated Hamiltonian (3) using the ultralocal techniques as develo£ ed by KlauderJ4,l2|. Liang |5| remarked that only three components of the metric tensor are necessary to specify the dynamics around the singularity and it is these components, together with their canon^ cally conjugate momenta, that we propose to quantize* Having obtain ed the field operators of the theory we are able to introduce cohe rent states and to discuss dynamical evolution. Briefly the paper 1B organized as follows. In §2 we outline the Hamiltonian approach to velocity dominatod cosmological models. In §3 the ultralocal quan- tization of the model is presented and coherent states introduced. Dynamical evolution and the role of smearing functions is the ob- ject of §4 while §5 contains more discussion about classical models and a conjecture about what we should expect the evolution to be when ultralocality is violated; around the singularity. Throughout the paper we assume a 3*1 split of spacetimo P\mTj*ff{ » where) T~ is compact and without boundary; the four dimensional metric &(£ m 0,1,2,3 , has signature -•++. All four dinensional objects have a superscript * while three dimensional objects on have no superscript. §2. /elocity Dominated Cosmologies! Models The variational principle leading to the sour- ce free Einstein field equations is based on the classical action where f^cfet^flU* » f\ is tnB scalar Ricci curvature Arnouiitt, Deser and Pilsner |0 |, or ADM for short (see also Oirac |7 I)» have rewritten (5) in terms of three dimensional objects d£ fined on where fll*(m $„)" * and N • i^X) J* . Observe that the G#* components are sssentially Lagrange multiplier and only nas the spatial tensor on 2Ü, 9/i • f</ l,J*/fi,5 » dynamical con_ tent. The dot means derivative with respect to the parameter band is given by (1). The momentum constraint J£i nas tnB form •**» where the bar means couariant derivative u/ith respact to Qy , Fxom the ADM action we can see that the quantity uftp (H^H^ f plays the rola of tha Hamiltonian of the system» Varying (6) with respect to Qu and K* gives Hamilton equationa for these quart titles («ore on this below). Varying H and 0? one obtains that JL and jh must vanish. ADM show that this corresponds respec- tively to the four constraint Einstein equations J|## =O and ^P ,S 0 . A geometric interpretation for JÇ, and fa is as follows |l9»15|t j£ is the generator of infinitesimal deforma tions in the normal direction away from JEL while Jfcg corresponds to the generator of infinitesimal coordinate transformations on£. Thus all possible variations of the metric and corresponding mnmen ta are encoded in the contraints In any physical theory the existence of cons - traints is a sign that there are redundant variables that can be eliminated without affecting the essential aspects of the dynamics (however in this elimination process one will surely loose explicit covariance). The canonical approach to gravity provides us with a clear insight into this problem since there are three constraint generators jj^i of coordinate transformations and the geometry des cribed by the six components Q.. should not depend on coordi- nate system's. The full ADM reduction process is a way of eliminat- ing all spurious variablas leaving only the "true" degrees of free dom. It consists in imposing four "coordinate conditions" on the metric tensor and in solving all the four constraints for the mo- menta reducing the system from six to two degrees of freedom* This prescription is rather vague but the process is considered success- ful if the action in terms of the two remaining pairs of variables with their conjugate momenta can be expressed in canonical form and if there is compatibility between the choice of variables and the Lagrange multipliers. Below u/e will use a remarkable property oF velocity dominated models to cast the ADM action into canonical form (see (17)) without actually solving the constraints jjj . In doing this we have two possibilities regarding the constraint V . The full reduction demands that JjT, be solved classically and this leads to a choice of a time variable, the intrinsic time |16| of the system. This time is "intrinsic" because it should depend only on the geometry of 4j» and not on coordinate systems (for example any monotonic function Of fls dcff» can in principle be usedte.g. the variable Qs-T'&ft in (17) conjugate to P ). Another possibly lity, the one we choose, is not to solve V • This partial ADM re duction process allows us to introduce the concept of proper time evolution in the quantum theory developed in §4. In this paper we will assume that a synchro- nous frame |l7| has been chosen that is, the Lagrangean multipli- ers H vanish. Also we impose that the remaining multiplier is constant and we choose flr «f ; this means that the parameter T can be identified with the proper time of the system henceforth denot- ed by 0 . Thus ths line element has the simple form ..' that (8) is time reversal invariant. In the frame described above the momenta canoni caJly conjugate to Qn are related to the extrinsic curvature % °f l spatial indices art» raisad or lowered by the rrtbtric 0.» ). Relationship (9) corresponds precisely to the equation of motion that would be obtained from (6) upon arbitrary variations of & * If tue combine (9) with the remaining equations of motion obtained upon variations with respect to A» then the six dynamical Einstein equations are reproduced (we use the mixed Ricci tensor for conve- nience |l?|)s (10) From (10) a very interesting property of velocity denominated mo- dels can be deduced. Suppose we are studying a cosmological situa- tion tuhuro the spatial derivatives play no significant role (e.g. around the initial spacelike singularity). This means that we can drop J^' from (10) since this is the only term where spatial deri vativiis occur. From (11) it follows that if K]\.; is diagonalized on the initial surface than it will remain diagonal when evolved out of ^L, . This will auraly be the case for Ú.. as well since 9u9'^%$\s\ ' The ccm elusion is thet only three components of the metric tensor will be necessary to specify the dynamics and we write wit»- XCLsO • Th» conjugate «ementa asauma tha form «ith 2. Q »O «The *£.{£) arc tine independent fields and serve to diagonallze tha ««trie tanaor on £ (BKL |l*3| and Liang |*| Meue)eX||Lf<|*Ho form an orthonormal coveetor triad frama on^[). Wo will eoncantrata our dlacuaaion on tho quantitiaa Ô^CL and Jti 9 Bk «inca thaaa are tha variablaa wo intand to quantize while (MJL. do not ovolva and-contain no dynamics. Although our approach to quantization in §4 uses different techniques fro» that of Liang (ha uses the Wheeler-DeWitt method |lO|) we fellow him in leaving aside the effects of the constraints J^j in trie quantum theory. Classically however these constraints impose some conditions on the fields Qt QÃt Jtí',& ,ü)êi and their spatial derivatives |5|.
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