Rend. Sem. Mat. Univ. Poi. Torino Voi. 54, 3 (1996)

Geom. Struc. for Phys. Theories, I

G. Ferrarese

RIEMANNIAN GEOMETRY IN GENERALIZED FRAMES OF REFERENCE

To W. Tulczyjew for his 65-th birthday

Abstract. We summarize the main properties of a generalized frante of reference (in the polar sense) [1], and the related non-holonomic techniques; we then study the Riemann geometry of the space-time, in terms of non-orthogonal projections: longitudinal and transversai covariant derivatives, commutation formulae, decomposition of the Riemann tensor, transversai Bianchi identity, etc. - . Thus the general formulae contained in [2] and [3], Ch.4, are extended to the non-orthogonal case. For complete proofs and details see [4].

1. Introduction

In general Relativity, a standard frame of reference is generally a locai orthogonal 1 x 3 strutture (1-time and 3-space) (see f. i. [3], Ch. 4); it is characterized by a unit timelike vector field 7 or by the congruence of its flow lines t. From the kinematical point of view, T defìnes a standard Continuum, where the spacetime gradient of 7 summarizes its three fundamental characters: e2Ci (4-acceleration), u ^ {proper vortex) and kik {proper deformation rate).

The preceeding characterization can be generalized according to the theory of polar Continua [5]-[6], by considering a locai non-orthogonal 1 x 3 strutture', i.e. the Space -time is supposed provided with two geometrical ingredients:

i) a time-oriented congruence T, with an unitary tangent vector field 7:

(1.1) 7 - 7 = - i ;

ii) a spacelike distribution of 3-planes: È, non orthogonal to 7, namely a second timelike vector field 17, which defìnes the congruence normal to È and, without any loss of generality, can be choosen such that:

(1.2) 7 • « ? = - ! .

258 G. Ferrarese

Such a ( I \S ) non-orthogonal quasi produci structure can be introduced, more generally, in a differential Manifold, by means of two vector fields: ya and rja, contravariant and covariant respectively, both defined up to a scalar factor, and with the condition: 7aila "fi 0 [2]. However, the most interesting physical situations belong to the riemannian case.

Examples of this structure occur in many fields of the Physical-Mathematics; for instance:

- Superfluids (binary mixture), which we proved to be equivalent to Cosserat Continua [4];

- evolution of a surface (wave) with the associated rays, generally non orthogonal to the wave-fronts, also in the case of the light propagation: nuli congruences; - Dynamics of holonomic Systems, with tinte dependent constraints, wheré the Events space: En±i, strictly euclidean, is naturally provided with a non-orthogonal structure, defined by the temporal lines x° = var. and the rc-Manifolds x° = const. respectively [7]. Of course, in the last two cases, the S-distribution is integrable, and the framework is non-reiativistic.

In this paper, first of ali, we summarize the principal properties of a generalized frames of reference, and the associated non-holonomic techniques adapted to the structure; thus we study the riemannian geometry of the space-time, inside of the frame of reference. More precisely, by means of the non-orthogonal projection, we deduce longitudinal and transversai covariant derivatives, commutation formulae, Riemann tensor decomposition, transversai Bianchi identity, etc. Of course, if the coordinates are assumed to be adapted to the congruence Y, the internai group ofcovariance is the same as in orthogonal case.

2. Digression on generalized frames of reference

Let us suppose the space-time V4 to be provided with a generalized frame of reference (IV è), i.e. with two timelike vector fields (7, rf) satisfying conditions (1.1) and (1.2)1; as in the ordinary case, we assume the coordinates (xa) to be adapted to the congruence T, in the sense that

7 = 7 % , 7 ° > 0 ; such coordinates are defined up to transformations xa —• xa of the kind:

(2.1). x°' = x0'(x), / = z * V , z 2 , z 3 ) , (i = 1,2,3),

where the functions at right hand (invertible) satisfy the conditions:

dx0' dx1' (2.2) ^ > 0 , 0, which preserve the orientation in both time and space.

Transformations (2.1) constitute the fundamental (continuous) Group of the frame of reference, even if such a Group does not depends on the distribution E. Associated to

Condition (1.2) does not exclude the field n to be isotropie; therefore, the structure of generalized frames of reference can be used olso to study nuli congruences.

Riemannian geometry in generalized frames of reference 259

the coordinates (xa) there is the naturai basis {ea} and its dual {ea}; in order to have

bases adapted to the generalized frante of reference, we must choose a non-holonomic distribution {èa} ~ {è

a}. More precisely, without any loss of generality, we will assume the following quasi-natural distribution:

Ti' " • (2.3) é0 = 7, éi = e,-

Le0 G E, *7o

which is canonically associated to the coordinates (xa)2; for an arbitrary change of coordinates (2.1), the following transformation law holds:

0xl (2.4) èó' = è0 = inv., ét-/ = — -^è,-, c/a?1

which is typical of an holonomic basis.

In the following, we will unify the notation by adopting a """ for ali quantities relative to XJ; in particular we will write:

/ o r \ ~ def ~ „ def ~

(2.5) e,- = e?;, j{ = 7; = 7 • e»,

and denote by {è1} the dwfl/ fcuf'j of {è,-}.on S, which is characterized by the reciprocity conditions:

(2.6) è '-éi = n ;

moreover we denote by 7^ the induced metrics on è:

/ o rr\ - def ~ (2.7) T«ib = e* 'ek,

and by 7** /te dual, which is necessarily of the kind:

(2.8) 7'* = è'-è*.

In non-holonomic terms, the metrics ofV^ is characterized by the products gap = èa • è/?:

(2.9) flroo = — 1, 9ik = T.fci

where the components 7,-, given by (2.5)2, coincide with the difference ji — rji of the two vector fields 7 and rf:

(2.10) 7i=7i-r}i, 70-770 = 0.

/« contravariant form, (2.3) is equivalent to the following relations:

(2.11) é° = -i7, é* = e*;

then we have the dual metrics ga/3 = èa • è13:

(2.12) g™ = n-rì=-rì2, goi = -rj\ ~gik = f \

2An analogous distribution follows from the dual basis {éa}: é° = 7, è1 = e1 — ^s-e0.

260 G. Ferrarese

where the coefficients fj2:, rf and 7'* are given by the formulae:

(2.13) ; rf = (1 + 7^)~S rf = -7/2f, f* ' = fk " - 7?¥ 7V /

3. Commutation formulae and Jacobi's identities

The distribution (2.3) gives rise to the following pfaffian derivatives da'.

0 d Z _ ^ W # (3.1) a0 = a = 70ró> 5 « - » i /» e

w/i/c/i o/i/y depend on the field r)a, because: 70 == —I/70, 70 = ̂ o-

Of course, the non-holonomy tensor A^p of the distribution (5) does not vanish:

(3.2) [Sa,dp] = Àa/dp^0.

More precisely, the following commutation formulae hold, like in the ordinary case:

(3.3) [d,Si] = Cid, [di,Sk] = 2Ùikd;

they m\o\\etwo fundamental geometrie ingredients of (r, E): Q and Ùik (skewsymmetric), which have the usuai expressions( see f.i. [3], 117), except for the change of 7 with 17:

(3.4) QfSM-%) + ̂ , Ù^éUJd^-d^). rjo 2 \ r)0 r)0 )

Clearly, the usuai geometrie meaning disappears;indeed Q coincides with the spatial part (on E) of the Lie derivative of ì) along 7 ([2], p. 75), and the tensor Ùik characterizes, by the condition Ùik = 0, the distributions {E} which are holonomic, namely tangent to a oo1 family of spatial manifolds V3.

From the kinematical point of view, the tensor (c/r))Ùik (or better his adjoint on E), can be interpreted as the proper vortex of the generalized Continuum; however, such a vortex defines the free angular velocity, which is typical of the Cosserat Continua. In other words, Ùik is not determìned by the field of velocities of the Continuum: V = cy, which gives the constraint angular velocity. Anyway, in a generalized frame of reference, also the field % plays a fundamental role in fixing, by (12), the spatial platform E; of course 7; is independent on the relative potentials: rjo < 0, rji and 7^, which generate, by derivation, the fields (3.4) and the deformation tensor: Kik = ^djik respectively. In any case, for every non-holonomic distribution {é a }, we have also Jacobi's identities for the brackets [da,dp\; so the following differential conditions hold for the tensor Aap

p:

(3.5) %Aap]a ~ K\P°Àa^ = 0.

In our case, according to (3.3), the only non trivial components of the non-holonomy tensor are: À^ = Q and ÀQik = 2Q«A,; SO (3.5) is equivalent to the following conditions for the fields Q and Ùik.:

(3.6) dÙik = S[iCk], d[iÙkh] - C[iÙkh) - 0.

Riemannian geometry in generalized frames of reference 261

Of course (3.6) reduces to trivial identities if both the tensors Q and Ùik are expressed in function of the potentials rja, by means of relations (3.4).

4. Fundamental derivatives and Ricci rotation coefficients

Let us now consider the derivatives of the fundamental fields é a , in order to extend the pfaffìan operator da to any tensor fìeld; first of ali we have

(4.1) daèp = 'Jtaf3pèp,

where 1Za/3p are the Ricci rotation coefficients associated to the assigned distribution

{éo} ~ {èa}- Such coefficients can be expressed in a compact form, by means of the Christoffel symbols and the non-holonomy tensor:

(4.2) r a / 3 ) < T = -(da

262 G. Ferrarese

As for the field Hik, the two parts: skewsymmetric H[ik] and symmetric H^k), have the meaning of total (and proper) angular velocity and deformation rate respectively. Indeed we have the following expressions:

(4.9) H[ik] = Ùik + V[i7fc], H(ik) = K%k ~ C(iTJb),

which are more general than the usuai ones\ in particular, we see that the deformation of the metric tensor jik and the curvature vector ofT are mixed, according to (4.8). As regards the geometrie meaning ofthe spatial rotation coefficients H-ikK they can be related to the Christoffel symbols relative either to the metric tensor 7^., or the metrics 7^ defìned by (4.7)2. In the first case, we have the following expression:

(4.10) nikj = 7jh[tik,h - (Hik - di%)% + bhikl

where 7 j / l is the reciprocai of jjh and Dhik is defìned as follows:

(4.11) Dhik = &hàk + Ùik7h - &hhli-

We see that the difference Hi^ — i W is not invariant for transformations (2.1), for the presence of the derivatives dijk', a more significant expression is the following:

(4.12) * , V = r $ v + A V , •

by means of the Christoffel symbols associated to the metric tensor 7^ :

(4.13) r^^^^dijkh + dkjhi-dhjik},

and the field:

.(4.14) DJ dM-^\H{ik)% + H[kh]ji - H[hi]jk).

The last (4.14), which is a well determinedfunction ofthe metric tensor jik, Hik and 7,;, has now tensor character.

In conclusion, the Ricci rotation coefficients are:

HOÌ° = Q + Hik%, nik° = Hik - Vijk, Kikj =Tik

j + Dikj,

where Hik is given by (4.7) , and the Ricci spatial coefficients 0,^ can be alternatively expressed by (4.10).

Of course, as for an ordinary frame of reference, where 7* = 0, also in the general case, the transformation law (2.4) influences ali the successive non-holonomic developments. Therefore, the non-holonomic formalism, and the relative geometric-kinematical ingredients, are ali invariant for internai transformations (2.1), namely for arbitrary changes of spatial coordinates and temporal flow.

(4.15)

Riemannian geometry in generalized frames of reference 263

5. Riemann tensor decomposition

As in the orthogonal case (ffi — 0 ), the distribution (2.3) directly gives the decomposition of ali tensor fields of V4, along 7 and E; in particular, from the differential point of view, such distribution induces, in the frame of reference (T, È), two fundamental derivatives: d and-V,-, both invariant for transformations (2.1). The covariant derivation V, which appears in naturai way, as in the orthogonal case, does not satisfy the Ricci theorem; in other words, the spatial metrics 7^. £ E is not invariant for V«:

(5.1) V i 7,- t=* i t -7*+^i*7."#0.

Of course, by a little modifìcation of the connection coefficients, we can defìne a spatial hat derivative: V, which satisfies the Ricci theorem; indeed, if we introduce the coefficients ili^ as follows:

(5.2) Kikj d~ t{k + j

jh(Ùhi% + Ùik% - Ùkhji),

according to (5.1), we find Vjjik = 0.

The affine connection (5.2) has a precise geometrical meaning in E, because the coefficients H.^ coincide with the products djèk • è

J:

(5.3) Kikj =dièk'è>;

anyway, in the following, we will adopt the derivation V,-, which is more immediate, and univocally connected with V», because of the relation:

(5.4) nikj ='Rikj -Kikf. ~

Let us consider, now, the Riemann tensor of the space time; in non-holonomic terms, such a tensor can be given in the following vectorial form:

(5.5) Rapp = [dp,da]èp -Àpaad„èp ,

where does appear the commutator of the second derivatives of the vectòrs éa ; derivatives

which are determined by (4.4), and the non-holonomic tensor, the triple system Ra/?p summarizes the riemann tensor components, according to the following decompositions:

So, by means of (5.5)-(5.6), taking into account (4.4), we directly have the non-holonomic components of the Riemann tensor:

(5.7) Rapp" — àpUap — dallpp -VR-ap "&/?«/ ~^pp ^av ~ Apa %up \

of course, also in the presence of non-holonomy tensor, ali algebraic and differential properties of the Riemann tensor stili hold, because of their tensoriat character. Anyway, in both cases, either using (5.5) and (4.4), or (5.7)-(4.15) dyrectly, we find the following

(5.6)

264 G. Ferrarese

decomposition:

(5.8) Rijk — Pìjk + Qijk,Rijk -• Bìjk + Rijk 7l,

RQÌO ' = Ci , ROÌO = Ci % »

where we have posed, for the sake of brevity:

(5.9)

Pìjk — àj'Rik — dilZjk -f fai* l/R-jh — ^ Ì A l^«7i / def

/ def TT fT i Qijk •= HikHj — HjkHi —2QjiHk — 2i7[j Vi]7*;

^ Ci* d= (Vi + Ci)Ck - dHik - ÈéÈjk + VyjHpCih

from here the covariant form:

(5.10) Ì2ÌJ*A = lMRijk +7hBijk, Rijko — Bjik,Roiok = Ca-,

with

(5.11) Ci, d= d^kj =ViChkj + CiCt - M * + ^ W i i - # ; * ) +

T i C ^ ^ ' - ^ ' C f c ) .

From (5.8) and (5.10) we see that, in every generalized frame of reference (with characteristic tensors ji , Q and Hik), the Riemann tensor components are determined, as in orthogonal case («y,- = 0), by three spatial tensorfields: Cik,Bj,kj and Pa- / ; the last, in turn, has the meaning of covariant derivative Vi commutator, and therefore it is called spatial Riemann tensor.

6. Algebraic properties of decomposition tensors

The tensors (5.11) and (5.9)3 satisfy the following conditions:

(6.1) C[ik] = 0,B(ij)fc = 0, B[ijk] - 0;

except for the second, which is immediate, the other properties follow from the Jacobi ìdentity (3.6) which, for the symmetry of the coeffìcients iti^ on i and k, can be written in the form:

(6.2) 0Ùik = VpCjfc], %Ùkj] = C[iÙkj].

More precisely, we have the identities:

Ro[m] = C[ik] = V[iCk] - dÙik

(6.3)

R[ijk]o = Byik] = 2(V[jQjjb] - Ù[ijCk]),

Riemannian geometry in generalized frames of reference 265

which are true in general, i.e. for every distrìbution {éa} ([9], 213), and moreover they define precisely the geometrie meaning of the Jacobi identities, in connexion with the Riemann tensor.

As for the algebraic properties of the spatial Riemann tensor (5.9)i, first of ali we have the covariant farm:

(6.4) Pijkh = dj%iuth — diUjkji - Hik Ìtjh,i + ì^jk fcih,i — 4&k[i. Hj](i7h))

which satisfies the following properties:

' P(ij)kh = 0, Pij(kh) = 2ÙjiI

266 G. Ferrarese

Both the expressions (7.3) can be transformed, starting with the first and second term respectively of the right members. Finally, by introducing the two parts of Hik'. symmetric kik, and skewsymmetric a;,-* respectively:

(7-5) kik = Kik - C(iJk)>Uik = Ùik + V[i7fc],

as well as the tensor Hjk>i, which generalizes the analogous of the orthogonal case:

(7.6) Hjk>i d= (Vj + Q)kkl + (Vfc + Ck)ku - (V, + tt)kjk,

we have the following general formulae:

(7.7) . &&JJ = Hjk\dtjk = dHjk - (V,- + Q)d% + &„%,

where Hjk* (symmetric on j and k) is defined as follows:

Hjk' =Hjki + 2 c 0 « t / - 2C0n t ) ' ' + C'V07*) - kiky

udy,+

T"(Cjk-ÈjhHkhyy,.

As for dH.jk\ we see that the.symmetry on the indices j and k is conserved, as it's naturai; moreover, this tensor depends on fy and its first derìvatives, as well as on ali characteristics of the frame ofreference: Ci, u>ik and kik; in the orthogonal case (%• = 0), the dipendence on u;,-*. disappears ([3], 139).

8. Commutators of d and V2 derìvatives. Spatial Bianchi identity

The spatial covariant derivative V; does not commute, i.e. [V,-, VA?] ^ 0, likewise to (3.2). Anyway, the V,- commutator is essentially connected to the spatial Riemann tensor (5.9) i; more precisely, taking into account that the Ricci formula does not change in non-holonomic bases:

(8.1) fiptValSv^-RapSSp,

by means of the picture (4.15), we deduce the following commutation formulae:

(8.2) [Vj, Vk]èh = 2Ùikdsh + Pikh3èj, [d, Vjs*» = Cidsh - Hih

3Sj.

We note that, differently from (8.2)1,(8.2)2 is not true in general, but only for vectors s E E, i.e. such that so = 7*Sj\ anyway, it defines precisely the geometrie meaning of the tensor H^, given by (7.8).

Let us consider, now, the Bianchi identity which, in non-holonomic terms, is the following:

(8.3) VfoÉaflp, = 0;

for spatial indices, we have not the same form for the spatial Riemann tensor Pijku as happens also in orthogonal case [8]-[2]. More precisely, starting from (8.3), using

Riemannian geometry in generalized frames of reference 267

decomposition (5.10) and Jacobi identities (3.3), and finally taking into account that the field Ù[ijÙh]i, being skewsymmetric, vanishes in 3-dimensions:

(8.4) «[y«fc]/ = 0,

we deduce the following general spadai Bianchi identity.

(8.5) V[hPij]kl + 2 % / ^ - yìmHvmPjh]k

nyn + ÈbnPhi]knji = 0.

In the orthogonal case (ji = 0) we find the reduced form [2]:

V[fcPjj]fc/ -f 2Q[ijHh]kl = 0;

so we have the typical form of the Bianchi identity either for Ùik =. 0, or Kik = 0.

REFERENCES

[1] FERRARESE G., BINI D., Riferimenti fluidi polari in relatività generale, Suppl. Ricerche di Matematica 41 (1992), 159-172.

[2] FERRARESE G., Proprietà di 2° ordine di un generico riferimento fisico in relatività generale, Rend. Mat. Roma 24 (1965), 57-100.

[3] FERRARESE G., Lezioni di relatività generale, Pitagora Editrice, Bologna, 1994. [4] FERRARESE G., STAZI L., Proprietà di 2° ordine di un riferimento generalizzato, in corso

di stampa nella Riv. Mat. Univ. Parma. [5] FERRARESE G., Intrinsic formulation of Cosse rat continua dynamics, Trends Applications

Pure Mathematics to Mechanics 2 (1979), 97-113. [6] FERRARESE G., Relativistic polar continua, in: Gravitation, electromagnetism and

geometrical structures, Int. Conf. for the 80-th birthday of A. Lichnerowicz, Pitagora Editrice, Bologna, 1996, 37-53.

[7] FERRARESE G., Sulle equazioni di moto dì un sstema soggetto a un vincolo anolonomo mobile, Rend. Mat. Roma 22 (1963), 1-20.

[8] FERRARESE G., Contributi alla tecnica delle proiezioni su una varietà riemanniana a metrica iperbolica normale, Rend. Mat. Roma 22 (1963), 147-168.

[9] FERRARESE G., Introduzione alla dinamica riemanniana dei sistemi continui, Pitagora Editrice, Bologna, 1979.

Giorgio FERRARESE Dipartimento di Matematica "Guido Castelnuovo" Università "La Sapienza" Piazzale Aldo Moro, 2, 00185 Roma, Italy.

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