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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 fromth e 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 firsttas k 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.
(%*-5#*W+*w-*w> (2i) and the parametric three dimensional connections
Гу* - g^Wj + 9ij*i - 9ij.l) (2-2) have the difference
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 - 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. We can divide the Lagrangian (3.3) into two parts, M c = c + c б ia independent of /, but every term in £(/) contains /. The corresponding split of the actioa suggests the picture that the space-time in general relativity consists of the system of fields («**,«<) in interaction with the scalar field /. The Lagrangian of the free (gik, иц) system (the parametric manifold) is CS^, and the interaction with the / field is described by CS^. ш the rest of this paper, we shall investigate the dynamics of a free (gik,u>i) system. 4. FIELD EQUATIONS OF THE ($,w) SYSTEM In this section, we derive the field equations of the dynamical system described by the action S=lbdt (4.1) where the Lagrangian L = [см<Рх (4.2) ia an integral of the Lagrangian density £(") = —y/gR of the infinite-dimensional dy namical system (gtk,Ui) over the 3-dimcnsional differentiable manifold M. Using Eq.(2.8) for A, dropping divergence and time derivative terms, and expand ing the covariant derivative V« in terms of the Riemannian three dimensional covariant derivative £><, we get: к £<«> . y/g\-lt» - PDiUj - дЧдцВ »к + u/W*i + »wF 1 . - 1 1 1 .. I (43) - 2и*^9Ц9Ы9ы - 2Ui9*'9jkf»k + jwfy" Ái)a + ^29tJ9ij Variation of C^ with respect to o>< and дц yields the Euler-Lagrange equations, respec tively, (Cf. Appendix) VjR* m 0 (4.4) where к, Я" « f* + (д дк1)д» (4.5) 7 and V W Rii + j^Öw^il + *4 1*>H) - 9ij9 (V* - w*)wi + WiWj - У(|»я = 0 . (4.6) The field equations (4.4) and (4.6) are reparametrisation-invariant. Elimination of the second time derivatives firom the field equations by taking algebraic combinations yields the constraint C\ su>7D*ú>i — WV-DJW,- + (uf'úti)2 — wV'wjiiy — -J*g''ufjiuj — -ш2ыкиьд**дц 1 .. . . 1 1 . . <47) + \ь>79й'9цВкь>к + w^ <3>RÍJ - L2Л*3) = 0 . We have not found any other combination of the field equations containing no higher than first time derivatives. The time derivative of this constraint is a new equation containing second time derivatives. We can add to it an algebraic combination of the field equations once again to eliminate these second-order terms. Lack of patience however prevents us from writing out the resulting lengthy constraint Ct = 0 in detail. 5. THE SPHERICALLY SYMMETRIC SOLUTION We now obtain the spherically symmetric solution of the field equations of the (g*k,Wi) system. We assume that the metric d»2 = gn,dx*dxk has the spherically symmetric form d»2 s Ä(r,t)dr2 + r2(d$2 + ат2$аф2) (5.1) and that the field w has only a radial component: w 8 The field equation (4.4) yields immediately that the only undetermined function h in the 3-metric does not depend on i and #'* = 0. Thus we have the Theorem: The metric of a spherically symmetrical parametric manifold does not depend on the time parameter. This is not unlike the Birkhoff theorem of general relativity. Computation of the Ricci tensor yields all other components vanishing. Here a prime denotes partial derivation with respect to the coordinate r. Using this form of the Ricci tensor, the (r,r) and (9,6) components of the field equation (4.6) are, respectively: 2w2-2w'+2ww+v*--^-r- = 0 (5.3) h r hr rW - rV + rluw + ~r2df -3rw + l-/i-í£ = 0 (5.4) 2/t 2ft and all the other field equations arc identically satisfied. Subtracting we get: . 1-Л (5.5) w = - 2r ' thus ü as 0. Substituting back in Eq. (5.3), ft' :l-f t (5.6) ft ~ r The solution is 1 ft= — (5.7) 1 - 2M/r where M is a constant. Integrating Eq. (5.5): 9 where a function of integration has been absorbed by using a gauge transformation. Our result is that the spherically symmetric metric of the parametric system (j**,Wi) has the familiar form of the space-like part of the Schwarzschild space-time. At infinity, Г-+00, the metric approaches the flat form and the potential w-»0. Although the potential u> increases with time linearly, u%, the only reparametrisation-invariant Quantity associated with u/ is stationary. 6. THE HAMILTONIAN The canonically conjugate momenta arc obtained from the Lagrangian (4.3) of the (j,u>) system by variation with respect to WJ and j,*: р' = уДОЧ+"У*!Ы , (6.1) (6.2) These momenta contain just the /-independent terms in the momenta (3.5) and (3.6) of the total Lagrangian C, written out in terms of the Ricmannian 3-connection Д instead of Vi. We have only one primary constraint: щырЧ = y/jjiw'u'DiUj - JD'uii) . (6.3) The Hamiltonian density is: + CC.V - srsV^ - >>W + **> (6.4) 10 Substituting the momenta (6.1) and (6.2) in the Lagrangian constraint Ci, we get the secondary constraint - Мт* • *'"*) Gr* • °**У -%iB> {^1 - °**) The looming size of this constraint shows that the Poisson brackets will prove too difficult to compute. We shall present a more viable approach to the Hamiltonian structure in the next section. T. EXTENDED PHASE SPACE In this section, we pursue an alternative approach to the Hamiltonian structure of the (o,u>) system which arises by enlarging the phase space with the canonical coordinates #** and with the corresponding momenta. The quantities H'k have been firstintroduce d in the field equation (4.4) for no other reason than economy of writing. Htk being a symmetric tensor, the suggested enlargement of the phase space is entirely different in character from Ashtekar's approach4. In the latter, the new degrees of freedom involve connections rather than tensors. We write the Lagrangian density £ CM = Vgl -#*> - Я" Di»; - *ф, + ifaf»! - \»3дц\ \ • (7.1) We can eliminate all derivatives of the fieldu> < by subtracting divergence terms: ik ik £<«"> = y/g[-№ +ukDiH - \н*шт - H С-щицГ*** +1 fcrW - jwV») \. (7.2) 11 Thai is to say, <*>< is the potential of an arbitrary external field. The form (7.2) of the Lagrangian is not suitable for launching a Hamiltonian treat ment since it contains second time derivatives of the metric components gik. Such terms do not appear explicitly in (7.2) but they are concealed by the notation employing Hik. We can achieve a first-order form of the Lagrangian by considering Hik independent variables and adding the definition (4.5) of #*k to the Lagrangian with a Lagrange multiplier function A«: Ч ik ik 1 2 £<»> =y/gl -# + «• Д Я* - \H Wi»k - H Qu/^Л.» + Igiwu, - \и дЛ ik ik ik + bék(H -g -g g"gr.)} (7.3) tk 1 As a result, the fields g , Я *, Л,* and щу are allowed to be treated as independent coordinates. The momenta are obtained by varying the Lagrangian with respect to the time th derivatives of the canonical coordinates. None of the momenta pn = (***, Пц, V , P'), corresponding to the coordinates q* = (?'*,#'*, Л^,ь>,-) contain any time derivative. Hence the momenta yield the primary Hamiltonian constraints; (7.4) where we introduce the tensor №M s «i*4^«» ~ Нци'даЬ - ur*ffj(|ft)< - Д(.и>4)) (7.6) with the symmetry property цяы = /*(«»)i< These constraints are all second class, since И the Poisson brackets {C',C\} and {С ,СШ} do not weakly vanish. The Hamiltonian ie # = /ftd3* where the Hamiltonian density H = y/g(№ - шк Di Я" - А^Я") (7.6) 12 does not contain any momentum variable. T>ie latter obeervatkm is of considerable value when computing the Poiason brackets. Mowing Dirac3, we introduce the total Hamiltonian #r = / Wr«** of the con strained system, к Нт = Н + и*9 Цк+и%С!1 + и}кС?+*? where the unknown coefficients um are functions of the coordinates x* and parameter t. The Eamiltonian equations of motion take the simple form The velocities are, from (7.8), jT»-«* Я*» = «# Á.»=tA ú>.=ur (7.9) and the time derivatives of the momenta are: *.» - v^| -(3) R.b + \l*3)9u* + \ыИаОш^ Лу№<•"»> - \H.WÍ) (7.10) - 2Ы*9к{шЩе^ - fmUbHib + 2<«>20í.#»fc + \tb9ik - К9хш9кЬ + 2^»f 2^*"We + A<* "xec9ik)\| W W Й.» • yfrl A.» - P(,u>») + uj* í -W.U;»Íjk + 2 ^K« ») - l^9i»9kkJI (7.11) P''=^(*#'-«;4«ÍW#») (7.12) P*mtf(DtH*-ufHÍ-iUub) . (7.13) n n The time derivative of an arbitrary dynamical quantity Y(q ,pn) = /y(f ,pn)d*z is given by the Poisson bracket х-«{У,Яг) - (714) 13 The condition that the primary constraints are preserved can be expressed that the Poisson brackets of these constraints with the total Hamiltonian vanish. Since none of the primary constraints contain any derivative term, the multiplying factors um can be moved out in fronto f the variations. We can employ a distributional notation where the multiplier functions of the constraints are not displayed: <3) {C1»**T} = VG[- Ä-» + I^J.» + \н#*П+>ы + D* (Я>(.и,»}) л в - «í*í £ W«ft"i + -jb>.utbHik + ^«.tfJwjW* - 2 ** 0 + <^+«.\-«?Л»] (7.15) 2 {СЦ,Нт} = V? А.» - Ö(.wk) + uf (-иддк{тыЬ) - -ш дя,дьк\ + u£wb) (7.16) {С?УНТ} = Jg{H« - < + и?***"') (7.17) e {CZ, Яг} - y/g(DhH* - «;Vi* - «#«•) . (7.18) The vanishing of these Poisson brackets provides us with relations between the so far undetermined multiplying functions wm and functions of the phase-space variables. Some of these equations can be solved algebraically for multiplier functions: uab = я.ь _ 1 щдпъ (7 19) 3 a;*«? = 4J{D^b) - Кь) - <*(Л* -\\- gu> tff) . (7.20) The commutators (7.15-7.13) also provide secondary constraints. Substituting the expressions (7.10) and (7.20) in (7.16), we get: Н 1 2 {C&*HT) =^9 Kb- A."») + J«'*«.**) - \^ « - \»*»ьЩ + -и> даьЩ c + ^ (0{вш\е\ + Dcu(„ - 2\е(й)шЬ) - -jWb{D *c - К) • (7.21) 14 The trace of this yields the secondary scalar constraint С s y/g{\% - Dew« -г Ju».w»tf*» + ^wV(l)»uic - A»e)) = 0 . (7.22) In addition, we get a tensor constraint by subtracting a multiple of C* from (7.21), 2 СД ж WV5(AC- - D&4 - \u Hcé + IwVrfffi) = 0 (7.23) e where Pl = S\ — £sbt wm is the projector into the tangent subspace orthogonal to w, with nonvanishing variations CDC 1 JCpc | These new constraints do not depend on the momenta. The Poisson bracket with the Hamiltonian of an observable у depending onh on the canonical coordinates has the form {V,Яг} = «f ^ + tiHjjp; + «, — + uik— . (7.25) Commuting С further with the total Hamiltonian, we have {CÍJÍT) = -? (!«*<* + £cW*) - £<С£Ч k k 7 26 - о^диЯекЩ - ^gaHik) - ukD(eu9 d) + -u Dku,ed ( - ) 2 + ^ufaipk + £>|к,цм) - 2Arf)* + -и> #,,)к) + -^Я;«>' 2 2 - -Яс*<и;' - -D(cu^ - -ut VHcd + gW0e<«HÍ + «erf } • The first three terms containing constraints can be dropped, and the last expression is projected in both indices, as is the constraint C£, into the orthogonal complement of u>.. All the remaining multiplier functions u'/j and u,\ can be obtained as functions of uH\ from Eqs. (7.15), (7.18) and (7.26). The Poisson bracket {C\HT} aoes not yield any information on u« and uA. 15 Up to this point, the extended phase space approach proved easier to pursue than the naive canonical method of section 6 (Coini>are, e.g., the forms (6.4) vs. (7.6) of the Hamiltonian density). If we next transvect (7.15) with ur*w* and subtract from {C",tfr}, we get a lengthy new constraint C\ involving the Einstein tensor. There is a remote possibility that the Poisson bracket of the constraint C\ with the total Hamiltonian will yield the missing information on un\. One can think of three possible reasons for these complications. As has been stressed by Isham* in the context of the ADM approach, the particular choice of the canonical variables is by no means sacrosanct. For example, it has been suggested that a different choice of the confonnal factor of the three-metric may lead to a simpler formulation. However, this suggestion must tie rejected in the present context since our confornml gauge is the unique otic in which the the action (3.3) describes two distinct dynamical systems, (д,ш) and / in interaction. In any other conformal gauge, the Ricci-like scalar R would pick up a conformal factor. Other choice:« of the canonical variables are still possible. We have followed up the variant of the extended phase-space approach in which one of the canonical coordinates is H'k (rather than #'*). This has lead to certain improvements (stich as the disappearance of the tensor /'„4е), but the overall gain does not appear compelling. The third possible reason may lie that the canonical description, as we adopt it, does not fulfil the requirement of rcparainetrisation-invariance. This objection may, at first, apjiear to lie of little significance since a Hatniltonian approach is known to break gauge symmetries. Nevertheless, we are currently employing the insight gained in this work for developing a repartuiietrisntioii-invariaut canonical scheme which appears to offer more .structural economy. APPENDIX We would like to calculate the variation of Ош) with respect to g*' . If wc have two (starry) derivative operators, Vj and Vj , belonging to the metric g%i and gli respectively, but to the same form u>i, then for any scalar / and form 14 , (Vi - Vi)(/u>) = /(Vé - Vé)«i . (АЛ) 16 k Hence if both V« and V< is torsion free,ther e exists a symmetric tensor Ci} such that for any tensor ifcfc , v**fc£ - Wtä+£<*,* afcjíe - E^rfci» • M-2) The*; forth e two Zebnanov curvatures, *** « S'*,. + VA' - V^' + Cífc-C^ - C^-Cj . (A3) Using ÄVy* « Zw* »ÄHi* = (*rti» + KjkriW . Ä*i = Ä*y* M»d Л = А<е, the starry curvature scalar is Ы Л - fVJtaw =9**9 Ккц, « f«*^ . (A4) The variation of Л is * Л = Z\ii)k 6g* + f ««**(ад. . (A5) Using (2.15) fromI , *W - *« +|*"(*«vl^ + **iVpwn). (АЛ) The variation of 2*п,н» using (A.3) and dropping terms proportional to C^* , «W - *(**7>Л - v*(^*) • (AT) Since the covariant derivative of the metric is zero, и «V - \д (Ъ*9ц + Vjigu - ViS9ii). (A8) Substituting this into equation (A.7), multiplying by y/gg** , using (3.2) and dropping divergence and time derivative terms, we get к у/дд'Щц)к ш у/д[дцд \V*w, - w*w,) - V&j + *&,]Sg** . (A9) Using that 6y/g = -у/ддц6д**/2, the variation of £<"> with respect to gfi is: ^£(w) ill *' l 2 2 (АЛО) kl + 9ij9 {Vhűi - w*wi) - V(,) + (ОД . 17 Multiplying by 9%i , we caa eliminate Ä, and we get the Euler-Lagrange equation (4.6). REFERENCES [1] Perjés, Z.: The parametric manifold picture of space-time, paper I [2] Arnowitt, R., Deser, S. and Misner, C. W., in Gravitation, Ed. Witten, L. (Wiley, 1962) [3] Dirac, P. A. M.: Lectures on Quantum Mechanics, Yeshiva University (New York, 1964) (4) Ashtekar, A., Phys. Rev. D36,1587 (1987) [5] Isham, C, in: Quantum Gravity, Eds. Isham, C, Penrose, R. and Sciama, D., Oxford University Press, 1984 18 The issues of the KFKI preprint/report series are classified as follows: A. Particle and Nuclear Physics B. General Relativity and Gravitation С Cosmic Rays and Space Research D. Fusion and Plasma Physics E. Soad State Physics F. Semiconductor and Bubble Memory Physics and Technology Q. Nuclear Reactor Physics and Technology H. Laboratory. 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Ézsöletal IAEA SPE 3 Valuation of measurement results (in Hungarian) KFKM991-067G A Takács et al.: SMABRE as the dynamical model of the primary circuit for the code KARATE (in Hungarian) KFKM991-06/A J Balog et al: Locality problem for the Liouvflle field KFKM991-07/E N Éber et al Linear electromechanical effect in a polymeric ferroelectric liquid crystal KFKI-1991-ОвУС 8 Lukács Prices and sunspots KFKM991-097E M v Kozlovsky el al A chiral side chain polymer with layered structure KFKM991-10/G P. Vértes: Calculation of transmission and other functionals from evaluated data in ENDF format by means of personal computers KFKM991-11/E F Tóth: Non destructive testing methods (in Hungarian) KFKM901-12/H Vera Jánossy it al: Multichannel recordings and data analysis from spinal cord expiants and cell cultures KFKM991-13/0 Z Szatmóry: User's manual of program RFIT Part 1 General description KFKM991-14/Q Z Szalmáry User's manual ol program RFIT Pari 2 Parameter estimation KFKI-1991-16/Q Z Szalmáry: User's manuel of program RFIT Part 3. The data flies KFKM961-16/G Z. Szalmáry User's manual of program RFIT Part 4 Statistical analysis. Task PLOT KFKM891-17/A T Haugset et al: Is the breakup lime a free parameter when describing heavy Ion collisions? KFKMMM-18/G Margit Telbisz SNARE A response matrix based few group nonlinear reactor code in hexagonal geometry KFKM991-19/B Q Paál et al Inflation and compactification from galaxy redshlfts? KFKI-1991-2r>Q L Perneczky: ISP 27 Pre test calculations for BETHSY SB LOCA test with RELAP5/RMA code (in Hungarian) KFKI-1991-21/A V Sh Gogohla el al Phenomenological dynamics in OCD at large distances KFKI-1991-22/C Ml Verigin el al The dependence ol the Martian magnetopause and bow shock on solar wind ram pressure according to P hobos 2/T AUS ion spectrometer measure ments KFKI-1991-23/A V Sh GogoNa el al Dynamical chiral symmetry breaking and pion decay constant KFKI-1981-24/A Gy Bencze el al Coulomb screening In low energy nuclear reactions KFK1-1991-26VG E Vegh el al Compact simulators for WER 440 type nuclear power plants KFKM991-26/B I Rácz Spacetime extensions I KFKM991-27/K I Balásházy Calculaiion of flow fields and aerosol parties trajectories in bifurcating lubes KFKI-1991-28/A T Csörgő el al (eds ) Proceedings of Ihe Workshop on Relativislic Heavy Ion Physics al Present and Future Accelerators KFKM991-28/B LB Szabados Canonical pseudoiensors, Sparling's form and Noelher cur rents KFKM991-307B В Lukács el al: About the possibility of a generalized metric KFKI-1W1-31/L P Pelliontsz et al Development of an expert system for acoustic emission testing: "AE DATA EXPERT" KFKM691-32/C В Lukács et al. (eds ) Symmetry and topology in evolution KFKI-1991-33/G I. Vidovszky et al Study of Gd lattices (Calculations and Intercomparlson) KFKM981-34/J.K К Berelelal Bibliography of astatine chemistry and biomedical applicatlone KFKI-ieoi<3S/Q A. Keresztúri et al.: A nodal method (or solving the time depending diffusion equation In the 10$ approximation KRCMMI-ae/Q A. Récz: Detection of small leakage combining dedicated Kaiman filters and an extended version of the binary SPRT KFKMflei-37/A S Hegyi Analysis of the rapidity gap probability at CERN Collider energies KFKMWl-a8/Q G Hordósy: Calculation of collision probabilities и two dimensional geometry KFKM8ei~3tt/Q A. Péter: Acoustic detection of leakage phenomena KFKM9B1~40/B Agnes Holba et al. Cosmotogical parameters and redshift periodicity KFKMMM1/K *AfdrisUNénwlh,P.ZcinborW.Urb^^ laboratories for montortng envtronrnenljJ radiation KFKMM1-42/A+B Z Perjés: The perarnetric manifold picture of space-time Qy. Fodor and 2. Perjés: Canonical gravity In the parametric tnanaotd picture Kiadja a Központi Fizikai Kutató Intézet FaMós kiadó: Szegő Karoly Szakmai lektor: Szabadoa Uszló NyeM laktor: Dolintzky Tárnia PékJanyszám: 386 KétzűK a PROSPERITÁS Kft. nyomdaüzemében Budapest, 1982. mérek» hó