REND. SEM. MAT. UNIVERS. POLITECN. TORINO

Vol. 34° (1975-76)

M A U R O FRANCAVIGLIA

ON THE RIEMANNIAN STRUCTURE FOR THE SPACE OF METRICS OF GENERAL RELATIVITY

IN CANONICAL FORM (*)

RTASSUNTO - Dopo unintroduzione storico-critica delVapproccio dinamico alia Relativita Generale, si descrive una sua rigorosa sistemazione nel-I'ambito della geometria difjerenziale in dimensione injinita. La strut-tura riemanniana dello spazio delle metriche riemanniane su di una varieta (compatta) M viene studiata, descrivendone i concetti analoghi al tensore metrico, alia connessione ajfine, alia derivazione covariante. Si dimostra infine che le equazioni di Einstein nel vuoto si possono scri-vere in forma geodetica rispetto a tale metrica, provando cosi una con-gettura di D. CHRISTODOULOU.

1. INTRODUCTION.

Since the earlier works of P.A.M. DIRAC ([11]), the dynamical formulation of General Relativity played a central role into the developments and understanding of the theory itself, both at the classical and the quantum level.

New insights in this dynamical formulation were gained after the work of R. ARNOWITT, S. DESER and C. MISNER ([1]); here refer­red as ADM). They, making use of the lapse function N and of the shift vectorfleld N{ together with six pairs of canonical variables

Classificazione per soggetto A MS (MOS) 1970: 53B30, 83C05. (*) Work sponsored by «Gruppo Nazionale per la Fisica Matematica» (GNFM)

of Italian C.N.R.

30

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g{j and n11 (see after for details), were able to split the system of the ten Einstein eqns. 6ra(3 = 0 (a, ft = 0, . . . , 3) into two parts. The six eqns. G{j = 0 (i, j = 1, 2, 3) are evolution eqns. for g and n\ the remaining four Q\ = 0 are "constraints" imposed to the variables g and n themselves on a fixed hypersurface 20 of space-time F4 . In this dynamical formulation, N and N{ do not play a real dynamical role. (*)

According to the deep studies made in this subject by A. LICH-

NEROWICZ and Y. 0. BRUHAT ([22] and [3]), the constraints (r° are in fact the "Oauchy da ta" on H0 for the dynamical theory of G. E . itself; they, when satisfied on 20y are preserved in time out of H0 by virtue of the field eqns. G{j = 0. A great amount of lite­rature about the interdependence between the two sets of equations has been written by many Authors, and we refer to it the reader (see e.g. [16], [19], [20]).

In the dynamical formulation above, we are obliged to make use of a "slicing" of space-time, namely to describe F 4 as a sequence in time of 3-hyper surf aces j^) ieiR . As it is well known from many counterexamples, no reason a priori exists to assure that the evolution in time of a certain H0 will not develop into a 2t, of different topological structure. Hence, in general, F 4 has not the topology of a product manifold M x 1R, where M is a certain fixed 3-dimensional (7°°-manifold. However, simple continuity argu­ments show that if we cut a slice 20 in F4 , at least a finite part of F 4 nearby 2"0 will be represented by a "tubular neighborhood" ^o x ] ~ fi> + fc'[? fi being a real number (possibly also e = + oo.).

In order to avoid the difficulties involved with the changes of topology, we shall agree to consider only space-times of the form M X 1R, since locally any space-time has such a topological struc­ture (2). Furthermore, we recall tha t tins topological assump­tion is closely related to the causality conditions on F4 , through the "global hyperbolicity theory" of J . LERAY (see [17], [21]). Finally, we shall consider only the cases of compact without boun­dary or open and asymptotically flat spaces M.

(') We shall see later that they can in fact be interpreted as Lagrange multipliers. (") Detailed discussions about such topological problems can be found in [18J.

— 467 —

In ADM's and DIRAG'S approach it is easily recognized that G00 plays the role of the Hamiltonian functional for the canonical formulation of General Relativity. The canonical field variables are the six components g{j (x

l, t) of the metric induced on It by the space-time metric 4#a(3 satisfying Einstein vacuum eqns.. The canonical momenta conjugated to g{j are a set of six tensor-field-densities of weight 1/2, namely:

(1) *»=i \'g [& ~ g« (trk)} ,

where ~k{j is the 2nd fundamental form of lt as embedded in F4. The "lapse function" N is given by (—g00)v2, while the "shift vec-torfield" N{ is simply 4g0i and transforms as a vectorfield on 2t

under changes of the spatial coordinates oo\ With these notations the metric of F4 can be written as follows:

(2) ds2 = - (N2 ~ N{N{) dt2 + 2Nidxidt + g{j (x, t) dx{dxj.

Since the beginning of this dynamical formulation, it was reco­gnized the central importance of a four index tensorfield defined on each St -slice, namely the tensorfield:

(3) Gijmn-•=— (g':,"gin4 gingj'n 2gljgm")

We can easily check that lc{j contracts with OtJmn to give just (Yg)~[ nm"\, furthermore, the Hamiltonian jf can be simply expressed as:

(4) )/g.G^%kmn-\/g-R ,

where B denotes the scalar curvature of 2t (see [9], [28]). The

tensorfield Gijmn enters also essentially into the Hamilton-Jacobi

equation for General Relativity, as e.g. introduced by J. A. WHE­

ELER ([30]). To G we associate the corresponding density:

(5) G/J'm" = fg- &im" .

— 468 —

ADM developed the theory on formal grounds. The rigorous setting of it was only recently achieved, by means of infinite dimen­sional differential geometrical techniques, after the extensive work of A. E. FISCHER and J. MARSDEN ([13], [14]).

We shall here shortly describe this rigorous approach. Space-time Vi with its "slicing" M x 1R can be interpreted as a smooth sequence of metrics on M, linked altogether in a 4-metric by Einstein equations. Configuration space is then the space Eiem (M) of all C°° Eiemannian metrics on M: each pair (2t£± ilf, g{j (ocl, t)), t being fixed, represents an istantaneous configuration of space. In the sequel the ^-dependence of the 3-metrics g^ (x\ t) which are linked together to form space-time metric will be generally omitted. Namely, we shall agree to work on a fixed t -slice, momentarily forgetting that the continuous sequence [g^ (xl, t)] te1R consitutes in fact a smooth curve in Eiem( M).

Eiem(if) is defined as the space of C°° -cross sections of the bundle X+, which has M as base space and whose fibre in x is the space of twice covariant positive-definite symmetric tensors on Tx M (namely, the inner products of Tx M). (3)

The space Eiem( M) is naturally endowed with a Frechet mani­fold structure, such that for any metric g the tangent space Tg Riem(ilf) is the space 82{M) of symmetric 2-covariant 0°°-tensorfields on M.

Let's for a moment abandon this description, to shortly intro­duce the concept of Superspace, which is due to original ideas of WHEELER'S ([28]). If we consider the space components g}j of space-time metric as the really dynamical important variables and we take into account the coordinate independence of Einstein equations, we are naturally led to identify on each 2t all the me­trics related to each other by a change of coordinates. We define Superspace on M the quotient space S(if) of Riem(if) obtained by such an identification, through the natural action on Riem( if) of the group Diff(M) of 0°°-diffeomorphisms of M (i. e., locally, by the changes of coordinates). Each equivalence class is called,

(;i) In this paper we shall sharply distinguish between the terms tensorfield and tensor.

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according to WHEELER ([29]), a "geometry" on M. A beautiful analysis of S(Jf) has been carried by A. E. FISCHER ([12]), who showed that %(M) is a stratified manifold, which unfortunately is not globally a smooth manifold. Each stratum corresponds to diffe­rent symmetries of the metrics of M. The strata corresponding to the so-called homogeneous universes, which constitute the finite dimensional part of S(Jf) (see [12], page 342), have been given by MISNER the name of "Minisuperspace" ([24]): we are returning after on this last concept.

On the basis of the tensorial character of Einstein equations, WHEELER suggested that Superspace S(ilf) has to be the most sui­table configuration space for the dynamical formulation of General Eelativity. We are here, on the contrary, supporting the approach given on Biem(ilf) by FISCHER and MARSDEN: as already analyzed in some detail in other papers (see e. g. [7] for a discussion in the quantum mechanical case), the passage to S(i)f) can be most fruit­fully made after a deeper study on Eiem( M).

In order to give to Einstein equations a dynamical Lagrangian or Hamiltonian form, we first need a Lagrangian functional on Eiem(ilf). The general theory of infinite dimensional Lagrangian systems can be found in the rigorous exposition [2], or in the intro­ductory parts of [13] and [15].

In general coordinates (2), it has been since a long time reco­gnized (see e. g. [9]) that the most suitable Hamiltonian functional is:

(6) H=ftt[Jiil+te(NlGit + NGt!)ld*x .

In (6), N and N{ can be interpreted alternatively as Lagrange's multipliers ([23], p. 57) or as canonical momenta ([13], p. 548). The reduction from general ADM coordinates to Gaussian ones (N = 1, N{ = 0) is deeply described in [13], § 8. (4)

When it is written in Gaussian coordinates, the Lagrangian for

("') The same problem has been Avidely discussed within a different formalism by CHRISTODOULOU and myself in [8], Sect. 4.

— 470 —

General Eelativity becomes simply:

(7) L^fM[k^-(trk)*+K\^,

where fig is the volume element of if, which in coordinates reads )'g dx1 dx2 dx*.

The terms containing 7̂ can be put into the form GlJmn fc,y hmn. Hence, (7) takes at least formally the usual aspect of "kinetic" plus "potential" terms: this analogy with classical mechanics is not only formal, when viewed in the light of the theory developed in [2].

By the reasons above, the kinetic part has been given also the name of "DeWitt's metric", since D E WITT was the first to under­stand its central importance. It can in fact be shown that De-Witt's metric is a weak Eiemannian metric (in the sense of [2]), which is (weakly) non-degenerate.

Writing down the infinite dimensional analogue of Lagrange equations for L, we get six equations governing the time evolution of hfj = g{j, which are in turn equivalent to the six equations Gjj — 0. They are:

(8) k = l4kti - \ (trk) kij + I [(trky-kJ^]gii-2Rit+ \ Rgii.

Eqns. (8) are derived in detail in [13]; however, their appearance dates back, to works by TAUB ([27]), although there derived within a different formalism.

According to the general theory of Lagrangian dynamics, eqns. (8) are the "geodesic equations" for the Eiemannian metric

(9) GB(k,k)=\ / ^ - • M w . A t . •

More details can be found in the Eefs. cited above.

By working with a different approach, also D. OHRISTODOULOU

was led, independently and at the same time, to study the geome-

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trical structure naturally associated to General Eelativity by DeWitt's metric ([4], [5]). He was so led to consider as the central object the tensorfield-density Gipn", together with its inverse:

(10) G&. = Qfg)-1 {g iniojn OinOJm o 'JO inn) ' \ I

From analogy with the finite dimensional geodesic equation in a Eiemannian space, he stated (without proof) that Einstein equa­tions can be reformulated in the following form:

( i i ) n &>'"»&,+a**- m"k«L=- is (Rmn - 2 smnR) '

where Ciirs'mn is formally defined by:

1 /dG''im" 8Grsmn 3Gijrs\ (12) t W " " - 2 ( l i - + ^ - - % - ) - <'>

Eqns. (ll)x are essentially the four constraints G0a = 0. Eqn. (11)JI, which replaces the six equations G{j = 0, has the form of a forced geodesic equation for the "metric" G'pnn, where Ciirs,mfi; as defined by (12), is the "afiine connection". The forcing term, namely the left hand side (abrv. l.h.s.) of (12) itself, is of course related to the fact that the Lagrangian (7) is not only kinetic, but it contains also a potential term \/g R.

Equations which are very similar to eqns. (11) were also derived by MISNER ([24]) and EYAN ([25]), for the purposes of applying

(") It can be easily shown that C;,-,„„ Qmna coincides with the generalized Kro-

nccker symbol <5?- = - (<5"<\- +fla(5. ). These tensorfields on M have been widely used

in the literature, but without the notation (—1). This notation dates back to CHRISTO-

DOULOU ([4]), who introduced it to avoid the danger of confusing G^mn with the

lower index form of G '•""". (") Christodoulou's equation contains a factor —1/2 in front of the l.h.s.; this is due

to different choices of the numerical factor appearing in (3) and (10).

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them to the canonical quantization approach for General Relativity. However, the Authors above were only working in the finite dimen­sional case represented by Minisuperspace: we shall see later that such a restriction is not necessary and slightly misleading. Fur­thermore, in order to evaluate the connection terms (12), they

made use of a different metric, by substituting the tensorfield G to the density GlJm": we shall also see that this choice does not lead to Einstein equations.

The problem whether choosing Gl]m" or Gijmn (or any other geo­metric object on M) as the central quantity to study the geome­trical properties of this dynamical formulation of General Rela­tivity ("geometrodynamics"), has troubled for a long time the studies on this topic. Analysis and hypotheses were worked out by DeWitt and others, on the metric that must be chosen for geome­trodynamics (7); the main problem in this area was to decide whe­ther retaining or not the density-forming term )lg in the object assumed as "metric tensor". The present paper strongly supports the assumption of retaining |/#; in the next sections we shall inve­stigate the metric structure naturally associated with (7) and (9). We shall explicitly evaluate the "formal" connection (12), studying also its properties (8). Finally, we shall rigorously demonstrate that (11 ) n are in fact equivalent to Einstein equations, thus sho­wing that the natural metric for geometrodynamics is Gl]!"n rather

than Gijmn.

2. T H E METRIC OF RIEM(M).

We have already seen that a weak Riemannian metric G exists on Riem(ilf), associated to General Relativity, namely the metric:

(i3) Gg(k, A>~ fM e*-v-^» >

(7) See, for a discussion, [10]. (s) A direct application of this formal connection has been already useful in [6].

— 473 —

where k, k! e Tg Riem(ilf) = S2(M) and GiJmH is given by (3) for the fixed metric g e Riem (If).

It is clear that (13) gives us an intrinsic definition on M, in the sense that Gg (Jc, k') does not depend upon the particular choice of a coordinate system. In other words, we can say that the canonical action of Diff(M) on Riem(M) is an isometry for G (9).

The integrand of (13) may be easily written as follows:

(14) [(tr k) (tr k') - tr (k x k')] ,

where k x kf denotes the new tensorfield whose components are, in any fixed basis, /e-ft'y. Expression (14) shows us that also the integrand of Gg has an intrinsic meaning, being constructed by means of coordinate invariants on M (namely, by means of traces). Hence, Gg can be interpreted as the integral over M of a certain fa­mily of "metrics" Gg , each one on the corresponding fibre over x. (10) Each of the metrics Gg has an intrinsic meaning; however, we can explicitly give expressions for them only after the choice of a certain coordinate stystem in M. If such a system is given, we have in any point x eM a set of three coordinates xl (x); with respect to these coordinates, Gn takes the form

(15) &~ix)=1 fc£) (g^x) g»{x) + . . . ) ,

and the transformation from the given coordinate system to a new one \x1' (a?)) xeM, is accomplished by applying the usual tensor laws in M. (n)

In expression (15) we can consider x as a fixed point of Jf, and consequently think of the tensor indices as deriving from the choice of a coordinate system in a suitable neighborhood of x in M. We may therefore subintend the x -dependence and consider the entity

(") A more detailed discussion can be found in [8], Sect. 6. (10) The rigorous explanation has already been given in [8], Sect. 2. (u) The procedure here described is the rigorous setting of what has been often

called in the literature with the name of «metric representation of Superspace » (see e.g. [9] and [10]); that terminology is of course unprecise, since we are really dealing with Riem (M).

_ 474 —

(15) as a "metric" operating on the space whose "points" are, for any fixed coordinate system chosen, the components g^x). This formal way of dealing with Ol]m"(x) is allowed since we know that G is invariant, by the fact that the #-dependence and the "change of representation" induced by any coordinate transformation are related through tensorial laws in M.

We should of course have no trouble if the metric gVi of each t -slice was in fact independent on the "space variables" x\ This is the case of Misner's Minisuperspace, namely the case of the so-called "spatially homogeneous universes". (12) In this case the a?-dependence completely disappears from (15), thus enabling us to deal simultaneously with the whole of M, instead of working point-wise on it.

By these reasons, many Authors were led to restrict their con­siderations only to this restricted class of metrics: however, as it was already claimed in the Intrd., the same kind of analysis may be carried in the general situation, by making use of the family of invariant metrics Gn described above.

"X

The first extensive studies concerning the properties of Q'J'nn

date back to DEWITT'S famous paper on quantum gravity ([9]), where he described in a full and detailed analysis the essential geo­metrical structure induced by Gl]"" on the space of g{/a. In that paper, however, the exact mathematical hypotheses underlying the theory were slightly unclear; because of this reason, the subsequent investigations led to the erroneous idea that the "metric represen­tation" can be carried only in the homogeneous case. Furthermore, De Witt made use of a particular choice of coordinates in g^s space, which was based on the existence in it of timelike, null and spacelike directions (the metric Gis in fact non-definite). That way of handling the problem, on one side led to deep insights in the properties of g{/$ space (signature, foliation in a set of 5-dimensio-nal Einstein manifold, etc.; see [9], App. A); on the other side,

('") In these cases an isometry group F acts on space-time, denning a slicing where each £-slice is topologically equivalent to a fixed manifold M and constitutes an invariant submanifold of space-time for the action of r. The metric then takes, with respect to suitable coordinates, the form: ds~ — gaf) {t)dxaclxP; configuration space is then finite dimensional, since on each £-slice it is enough to specify the constant components gij. These topics are deeply described in [26], pp. 103-107.

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however, it led also to formulas and expressions not easily and immediately manageable. Therefore, the present work wants also to be a complement and not only an extension of De Witt's older analysis.

3. S T U D Y O F THE " M E T R I C " G'Jm".

Looking at the symmetries of the tensorfield G1pnn defined over the manifold M, we soon notice that the following symmetry rela­tions are satisfied:

(16) Qi?'""-- Qjimn __ ~Qijnm _~Qjinm .

(16)n Gl'}m" = Gmn,:i .

Thus, the independent components of G are only 21. Since the independent components of any symmetric tensorfield on M (among which we have also the metric tensorfield gi}-) are only six, we can represent our <7»'s space as a six dimensional one.

One of the possible ways is, for example, to label each geome­tric quantity with indices ranging from 1 to 6, according to the fol­lowing convention: each pair of "small" latin indices will be repre­sented by a "capital" latin index, with the assumption that, when A represents the symmetric pair (ij), its numerical value is given by i + j + 1 iff i ^ j and by i iff i — j . (13)

In the sequel, the Einstein summation convention will be adopted, in the obvious sense that a summation over one capital index will represent a summation over both the small indices cor­responding. The generalized Kroneclcer symbol 6%J

mn will be simply written d^.

With the above conventions, the "metric tensor" will be written as GAB (A, B = 1, . . . , 6): it will be symmetric in A and B.

Its inverse, akeady given by (10), will be denoted by GAB: we have omitted the notation (—1) since in this new context no con­fusion can arise (see ftn. (5)). In fact, it is extremely easy to prove

(l:{) This choice has firstly been suggested in my thesis. It is easy to verify that it is consistent.

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the following relations (14):

(17)x GABGBC=dC , hence:

(17)n GABGBCGAD=GaD ,

which simply say us that GAli and GBC are inverse.

Before going on, it will be important to emphasize a particular feature of this capital index formalism. Although it is certainly useful to handle the calculations with only half the number of indices, we can soon recognize that the explicit expression of G, B in terms of the variables gc involves very unlikely mixing of the small indices. In fact, if A represent (ij) and B represents (mn), GAB will also contain the terms gim, gjn, etc. , which correspond to generally different capital indices. This difficulty can be avoided by introdu­cing the following notion of "full symmetrization", which has been suggested to our attention by a careful analysis of the symmetries presented by GJmn and by the formulas which will be derived later.

We shall say that two capital indices A and B are fully symme­trized if the usual tensorial symmetrization is performed on the four small indices corresponding to A and B; this operation shall be denoted, without danger of confusions, with the usual notation: namely, by enclosing the capital indices A and B into round brac­kets (example: GA{BGcw). (15)

With the aid of this notations, it is easy to put the metric GAB

into the following form:

(18) GAB=A-^g{AgB) _ 2gAgB] ,

where A denotes the square root of the determinant of {&>);,_/. Some other useful relations concerning GAB and its inverse are:

(19) t GABSA=~ &-'gn ,

(19)„ G**gA=-Ag° .

(w) Relation (17)i is the capital index translation of the relation mentioned in ftn. 3.

(T") In this paper the tensor symmetrization is defined by:

,. . . , 1* yl'i»2 •••'„) =—: (y ' l ' s . - ' n + syrara. terms).

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In the formulas above, gA is simply the capital index form of one contravariant coefficient, say giJ-, the relations between gA and the full set of gB (B = 1, . . . , 6) are given by the well known rule of inversion for a non-singular matrix. However, we may alterna­tively interpret the relations (19) as the implicit dependence rela­tions between the #,,'s and the #7?'s. Throughout this paper, the capital indices are not raised (lowered) by means of GAB (GAB): namely, if 1c is a twice covariant symmetric tensor, with compo­nents \ in a fixed basis, its capital index forms will be denoted by hA and JcA. However, kA does not mean GAnlcn, but it is the usual contravariant form of k {i.e.: JcA — W = gl1 gjm ftmZ).

Nevertheless, the "metric" GAli defines a raising action on sym­metric tensors; whenever we need this action, we shall follow the commonly accepted use and we shall write:

(20) nB = GABkA .

We easily recognize that nn is the capital index form of the canonical momentum (1), of course at the fixed point x. (16)

4. EVALUATION OF THE " A F F I N E CONNECTION".

The purpose of this section is to explicitly express, in terms of the gA 's, the "affine connection" (12), which in capital index nota­tion takes the following form:

(21) CAB>°=- (8AGIW+8BGAC - b°GAB) ,

where 3A denotes the derivative with respect to gA. (17) To evaluate each of the terms appearing in (21), we shall use

the following:

(22) dAGBC=~GnEGCF(dAGEF) ,

(in) The raising action (20) is the pointwise action induced by the canonical action defined by (13); see [13] for its discussion.

(l7) We denote such a derivative with an upper index A, because it involves always the appearance of a factor gA . See next ftn.

— 478 —

which in turn can be immediately obtained by taking the dA -diffe­rential of both sides of (17)P

From (18) we soon find:

(23) dAGw= - \ GBFg-< ^-l[6dfEgw) - 2digF - 2digE] ,

where the first term appears since GEF contains a factor ^ -1. (18) Hence, from (21), (22) and (23) we find the following:

(24) CAB>G=\ (gAGBG+gBGAC - g°GAB) ~- 3A-'[dfEgF]GCEGBF±

4 + »^)G

CEGAF - qgnW"} - 2g°G AB

In order to make expression (24) simpler, we must transform each of the terms appearing in square brackets, which shall be denoted for shortness as follows:

(25) &ABC = dc{EgF)G

AEGBF .

By putting pairs of small indices in place of each capital index in (25), we get after some tedious calculations:

(26) &**c = - A[— 2gAGBC — 2gBGAC+ WABa] ,

where ipAliC is given, in small index notation with A — (ij), B = {M) and C = (pt), by:

(26)' y/ABC=2 \Gmkgl)p + Gimksl)l + GklpUgiH + Gklt(igi)p] . tiQ)

As we can see, at this computational level a great mixing occurs among small indices coming from different capital ones; this

(ls) We have in fact, by the rule of differentation of a determinant:

(ln) In this expression, of course, round brackets denote the usual tensor symme-trization on two small indices. The same for (27).

— 479 —

is in agreement with our previous remarks in Sect. 3. However, we can easily prove that:

(21\ — IJ*ABC+ lJfACB^_*[fBCA==__ _ VQvtk{ig1)l *Q<pU(i~1)K\

with the same indices correspondence as above.

Inserting (27) into (24) and taking into account the definition of full symmetrization, it is then easy, although again slightly long and tedious, to find finally the following:

(28) CAB-a=-^gaGAB+^gAGR0^g"G*G ~^G*Ag* .

This last equivalent form of the "connection" is more suitable for the next studies; as we can soon notice, it is manifestly symme­tric in A and B.

Expression (28) could also be derived by directly dealing with the small index notation; however, we emphasize that the calcula­tions would be, in that case, much more involved and obscure. Furthermore, we must stress that more complicated calculations are practically impossible in the small index form: as an example, the evaluation of the "Biemann tensor" associated to G (/>,, which will be the subject of a future paper, has revealed itself to be fea­sible only in the capital index notation, since in the other one the computations were extremely cumbersome.

Before going on, we need some useful relations concerning GAJi

and its <?c-derivative. A first expression for dcGA H has already been imphcitly given through expressions (22) and (23). Nevertheless, it will be interesting and useful to derive a different but equivalent form. Beforehand, we need the following:

(29) d°gF = - 2A~*GCF - gcgF ,

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which can be obtained by thinking gF as a function of all the gA 's.(-°) From direct computation (e.g. by making use of the usual small index notation) it is not difficult to prove the following identity:

(30) bcg(jgB)=_4A-ig(cGA)B5 g»g(AgC)i-l g°g(Agn) ;

Zi ZJ

making then use of the upper index analogue of (18), namely:

(31) GAlt=34\gAgJi-g<AgTt)] ,

and replacing (30) and (29) in oE GAB, we find, after some algebraic passages, the following:

(32) dcGAB = -^cGA)B+gcGAB-v\ gAGBC - gnGAC . (H)

By the symmetry of GA1i, we know that, for any C, we must have dc GAB = dc GBA. However, expression (32) is apparently not sym­metric in A and B\ this apparent lack of symmetry is only due to our particular choices, which led us to introduce the fully simme-trized term g(CGA)B. Hence, interchanging A and B in (32) and sub­tracting the new one from (32) itself, we get the following identity:

(33) 3\g<°GA>n - g<°GB)A] - ~ (gAGBC- gBGA0) ^ 0 . Zi

Let's now define the covariant derivatives induced by GAB to be formally given by the usual laws in which CAB' c plays the role

(20) Expression (29) can be deduced by (^-differentiating b ° t n sides of (19) and taking into account (23). It can alternatively be shown in small index notation, by noticing that the r.h.s. is (with E — {ij), F = (kl)):

-j («V'W),

which is the expression found by gl}-differentiating gatg1" = $a and solving for

dgij(f'). (21) (32) could also be derived directly from (28). In any case, long calculations are

needed; furthermore, it is more instructive to follow the preceding way of reasoning.

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of the connection term. As an example, the covariant derivative of a two capital indices object H, B is given by:

(34) F°HA1!=ccHAB+C™, BHB, ^ C«°, AHim ,

where GRO, B is obviously given by:

(35) C«°,B=GA1)CR0--< .

We can so prove the following:

PROPOSITION. GAB satisfies Bicci's theorem, namely GAn has zero covariant derivative.

Proof. From (35) we soon have:

(36) C*°, „=1 ^gnG«° + I (d°g*+d%°)- | ^ V ;

hence, transvecting with GRA we get:

(37) C™, BGRA=\ 4 " W 2 - \ WkS* + 1 3

+ 2 8 ^HA — 2 hRA°B g '

From the alternative form (32) of cc GAH, by applying the dual of (22), namely:

(38) ?cGAB=- GAEGBF(dcG™) ,

we finally obtain:

(39) dcGAn=ZGBRg^ + | A-^gjo _ goGAB _. j - i ^ c ,

which is of course equivalent to (23). Then, from (34), (37) and (39) we easily get:

(40) VCGABJ- ^ teJ* - gAA + | GBl!g«df- | GABSt°i* .

31

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It is now easy to prove that the r.h.s. of (40) is in fact vanishing. It is indeed the lower index form of identity (33), as it can be pro­ved by simply transvecting (33) with 0BE GAF. (Q.E.D.)

Going on this way we can aso evaluate the "Riemann tensor", the "Ricci tensor" and the "scalar curvature" associated to 0AH. This has in fact been made, and the details will be published else­where. We report here, as a result, that the scalar curvature turns out to be simply proportional to ^"]; this is in perfect agreement with results already obtained by De Witt ([9]).

5. EINSTEIN EQUATIONS IN GEODESIC FORM.

This last section is devoted to prove that eqns. ( l l ) n are equi­valent to the six dinamical equations G{j = 0. First of all, we rewrite them in their capital index form:

(41) GB°gB 4 C * %g:ll= - A (BP - \ Rg°) .

Throughout this section, the gAs are to be again considered as a smooth ^-sequence of points in Eiem(Jf), namely each gA is a g.j = g{. (xl, t). As usual, we shall write TeA instead of gA and TcA

instead of gA.

For any h e S2(M) we have the following:

(42) GABhA=\ *[hB -gB{trh)],

(43) GAhKK=\ Alh'h - (tr h)2] >

h . h being the scalar hA hA = h{j W (22).

H Cfr. [15], eqns. (21) and (25)I.

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As in [13] and above, we introduce the following tensor operation:

(44) (ftxfc)ff=fcfk„ .

In order to check the equivalence claimed, we shall also need an explicit evaluation oiOC{AgH) kAkH. Let's first pose G == (ij), A = (ab), B = (cd). We have then, by making use of simple calcu­lations:

(45) G°^gB)kAkB = i A {{tr k) [kc - gc(tr k)] \-2GCE(kx k)E} ,

where E is the new capital index corresponding to (ad). Thence, from (45) and (42), we have:

(46) Gc^AgB)kAkB = \ A[(trk) [kc-g°(trk)] -f 2[(kx k)°-gc(k.k)]).

Transvecting (28) with hA k}, by means of (46) we get then:

(47) CAB-ckAkB = i A{(trk)k°+^ (k • k)g° - 2 (kxk)° -

- I (trkyg°) .

We can now replace (47) in (41), easily finding the following equa­tions:

(48) - 4[kc -g°(tr'k)]+- A[(trk) k° +~(k - k)g°- 2(kxk)° -

-l(trkyg°] + J(R°-±Rgc) = 0 ,

which implicitly express the dependence of k° upon the Zc/;'s and the

Following now a method suggested in [13], we eliminate the <f's.

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(tr 7c) term of (48) by taking the trace of eqns. (48) itself. We find:

(49) ( « r f c ) - - i ( « r * ) » r | ( A - i ) - i / l , (»)

which replaced again in (48) transforms them into the following:

(50) ka = (kxk)°-\ (trk)kc + \ (trk)*gc-2 o

-l(k-k)g°-2R°^Bt(° ,

which are the capital index form of (8), apart the raising of indices. Hence, the claimed equivalence is proved. (24)

Let's again emphasize, to complete our discussions, that the presence of the zl-f actor in GAB is essential for the possibility of putting Einstein dynamical equations into the geodesic form (41).

If we try to replace Gl]mn by GtJm" in (12), in fact, we are no longer led to eqns. (8). This can be easily seen by observing that, as already

noticed in Sect. 4, the terms GEF gA would disappear from

(23), thus driving to different expressions for GAB'c which, in turn, can be proven to give different equations when replaced in (41).

All this is in perfect agreement with the symplectic approach of [13]: there, in fact, the central role is played by the metric G, which can be invariantly defined over M only by integrating a tensorfield-density.

(3:!) We remark that (48) and (49) correspond respectively to eqns. (3.3) and subs, of [13] complemented by the additional terms deriving from the potential part of (7).

("') The proof of the equivalence above was already announced in my preceding paper [16].

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MAURO FRANCAVIGLIA, Istituto di Fisica Matematica, Universita di Torino.