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Ко 42Лп*£ Central Research Ко 42лП*£ KFKÍ-Л-ИЧ -чг/з QY. FODOR Z. PERJÉS CANONICAL GRAVITY IN THE PARAMETRIC MANIFOLD PICTURE Hungarian Academy of Sciences CENTRAL RESEARCH INSTITUTE FOR PHYSICS BUDAPEST CANONICAL GRAVITY IN THE PARAMETRIC MANIFOLD PICTURE Gyulafbdor and Zoltán Perjés Central Research Institute for Physics H-1525 Budapest 114, P.O.Box 49, Hungary ABSTRACT Canonical structures in the parametric picture of gravitation in general relativity are investigated. One of the fields, called the redshift factor, emerges as a secondary entity that interacts with a system consisting of the three-metric g and the connection form w. We are employing the (c,w) system for developing a canonical approach. While a naive attempt would appear to founder, we do find an extension of the phase-space of the system which is computationally viable. 1. INTRODUCTION This paper has the modest aim to investigate how a canonical analysis of rela- tivistic gravitation can be pursued in the parametric picture introduced in the preceding paper1, to be referred to as I. A Hamiltonian treatment will break the reparametrisation- invariance of the system, as is expected from the familiar behaviour of gauge-invariant systems under a canonical treatment. This breakdown of the higher symmetry can be taken care of by a decomposition in terms of Riemannian structures. It has become the conventional path to canonical gravity to choose the ADM decomposition3 of space-time. The many merits of this scheme are well-known by now. One may praise, for instance, the property of the ADM approach that the primary constraints are all first-class. Nevertheless, in lack of a conclusive theory, a search for alternatives is justifiable, even though initial complications are sometimes brought in. 1 The main point of our departure from the ADM approach is that the 3+1 decompo­ sition of space-time variables is induced in the present scheme, instead of the space-like foliation of the ADM formalism, by a congruence of time-like curves. In section 2, we shall carry out the necessary decompositions. We obtain the Ricci scalar of genera! rel­ ativity expressed in terms of the 3-metric #*, the redshift potential / and the vector potential u>i in the manifold of curves of a timelike congruence. There is a unique choice of the conformal gauge of the 3-space in which the Hubert action has the form of a dynamical system consisting of the fields gik and w< plus terms that describe the interaction of this system with the field /. The term describing the dynamics of the free (ar,w) system in the action is Jdt(Pxy/gR where Л is a curvature scalar of the parametric 3-space. This is precisely the conformal gauge regularly chosen in the theory of stationary space-times. In Sec. 3, we explore the canonical structures contained in the full Hubert action. Altough we obtain the canonical momenta, Hamiltonian and primary constraints, we shall not attempt a deep exploration of the canonical structures of this system. We consider rather the main features of our approach by exercising first on a simplified model. Such a model is naturally available by the property of the action that the interaction with the / field can be separated. In Sec. 4, we shall elaborate on a Lagrangian approach to the free (д,ш) system. We obtain two reparamctrisation-invariant sets of field equations. One set of the field equations has the simple form V*#'* = 0 where Нг> = gl* + (gklgki)g*'- The question naturally arises if there is any dynamical degree of freedom whatsoever left in the (g,u>) system. Or, possibly, is the field u> left undetermined by the dynamics? To soothe such worries, in Sec. 5 we set out to get a simple solution of the field equations under the assumption of spherical symmetry. The set of solutions turns out to contain a parameter A/, in addition to the time parameter appearing in u. In Sec. 6 we venture a naive canonical analysis of the (g,w) system, the full develop­ ment of which is hampered, however, by king-size constraints. To avert such difficulties, in Sec. 7 we enlarge the configuration space of the system by including #** among the 2 canonical coordinates and simultaneously freeing it from the metric by a constraint term in the Lagrangian. Both the Hamiltonian and the canonical equations of evolution take simple forms. Each of the momenta provides a primary constraint. In this first investigation of parametric structures, we do not wish to commit our­ selves to any particular choice of gauge. Hence the natural course for us is to follow Dirac's gauge invariant approach3. The first task is to evaluate the constraint-preservation de­ mand which may yield both secondary constraints and equations for the multiplier func­ tions of the constraint functional». In Sec. 7 we find, not without surprise, that the resulting relations are linear differential equations in the multiplier functions. This is in contrast with systems with finite degrees of freedom3 where the equations are algebraic in the multiplier functions. Fortunately, however, these equations can still be solved by algebraic methods. At this juncture we notice, as a further advantage of the extended phase-space approach that the primary constraints do not contain any derivative terms. As a consequence, no derivatives of the multiplier functions enter the equations of mo­ tion. This is not the case in the naive approach of Sec. 6, where the primary constraint (6.3) does contain a derivative term. 2. DECOMPOSITION OF PARAMETRIC STRUCTURES In this section, we expose the details of gravitational field quantities contained in the reparametrisation-invariant notation. We first break down the generalized connection V in Riemannian terms. The Christoffel symbols (%*-5#*W+*w-*w> (2i) and the parametric three dimensional connections Гу* - g^Wj + 9ij*i - 9ij.l) (2-2) have the difference <V -(%* - V - 5**W* + »iiu - »lit,) . (2.3) 3 Using the definition (2.10) in I, we get a formula similar to that of the Riemann tensor: r Z iik = W - WS + W - r/IV • (2.4) Expressing the starry derivative in terms of partial derivatives, and Г^* by ***Гу* and Cijk, we get the Riemann tensor and some additional terms. Introducing the (starry) covariant derivative instead of the partial derivatives we have: 3 2V 2C C 2 5 z\ik^wiik+V V+ i*<¥+ ÍU Ú • < - > Substituting <3>f,/ from /*(V.&* + Vjfa-Vtga) = 2 (3,V-^V-»*+^«-«*í«) + 2í?y*Wr , (26) r (3 г we can express Z ijk with & ^* and covariant terms. Using that Krijk = Z|ri];* » Rrijt = (Krijk + Kjkri)/1 and RÍJ = Л*^ , we get for the starry Ricci tensor: (9) к1 к k Rij = Л„ - -^hj - 2*Щ9 9к1+"09])ки -u Vk9ij + w*V(ij,)ft U -9 b>(i4j)9U + w<iV*ji)Jk + -зРдщЧ,у*1 + r^V*«,) - r$,>V*wk 2 2 2 (2.7) 1 */• TT l •*/• . * 2 *!• • . -kl - J» 9kiV(iUj) - -piUjg gki + -ш'д* Sék9ji + ^(i9j)k9 Щ . 1 kl- nt 1 * I • • i 1 • к I • 1 2 • */ • w + J» Sr« «0j)mw - -u/ u> 0jfc0)i + -géiw w gu - -w ^p 5« . This e starry curvature scalar Л, expressed with the three dimensional Riemann scalar Л*3* Я » fíW + 2w* V jy - 2д"и>кЧк9Ц " 9***Щ - f%**w* + ^9» -uTg^gij - -wV'fy + ^>i9J9jk^K + ш%ш'д^д"ды - -jurtf* дцу . The decomposition (9.8) of I of the four dimensional curvature scalar Л is written out in terms of / u>„ yy and R alone, R=-fR- jj+уЛ« + 2^/'+^"fo + ^<Л« Г - 57?/*"fo 4 There are time-derivatives hidden in the starry curvature scalar A, and in all (starry) covariant derivatives V,. 3. CANONICAL STRUCTURE The Lagrangian density of gravitation in general relativity is , £ = >/=iÄ = /- ^Ä , (3.1) where g and g are the determinants of the four and the three dimensional metric, respec­ tively. For any vector X\ the parametric divergence can be written i i i i y/gViX^y/gDiX-(y/g^iXy + y/gűiX , (3.2) where D% is the Riemannian three-dimensional covariant derivative. Using this and drop­ ping divergence and time derivative terms, we can eliminate the second derivatives from the Lagrangian. For convenience, however, we retain the second partial derivatives in + \*Pki»t + l^^9n9k,9ki + jih^sa + jfcW/ (3.3) к - g'tVvi - д*>дцЧ »к - 2jí(V7)(V4/) + Л*Ц*Л X*V)] . By functional derivation we can calculate the momentum variables canonicaly conjugate to /, utj and gij. But before doing this, we have to express the terms containing (starry) covariant derivatives V,-, using the three dimensional Riemannian derivative operator Di, to expose the time derivative hidden in V,-. But after the functional derivation the momenta conjugate to /, иц and g^ can be written in a simpler form using again the covariant derivative Vi : г = у/я(-р- у**** + jsMu + j*"' V*/) , (3.4) У = ^(íS + ^ V*í;* + Jvi/ + 2/4 VMJ , (3.5) (3.6) ТЪе primary constraints are: (3.7) and щжЧ - -ufip* + fP ж -2у/д1Ущ (3.8) Fbr the calculation of the Hamiltonian, we need: **%+fa + P/ - £ « ^[—f7 + £^1^яц-+ ^-^fV'#«J* 1 . • 1 • • 2>- 1 • .. 3 + 2/VtVM + iVV,/ - /»(УИ>, )(V\^) + 2ji(V7)(V,/) + Л<>] . Substituting here P, p' and JT,J, for the time derivatives, we get the Hamiltonian density: + + р2 + 1 -т*'9 Ч + г/* £<"^ - 7*""' £ ЗР*' - á?^w+%jpiD^ - ír*"»'+5рЛч+iPuiDif 2 . - *№МРр,л) - ^ А/) + 272 (/V + 3)(Х?7)(Д/)] + y/i*F> . (3.10) We shall explore this constrained system by removing first the complications due to the presence of the / field. Instead of making here a direct attempt at a complete canonical analyst*, momentarily we return to the structure of the Hilbert action.
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