On a model of an unconstrained hyperfluid

8

January

1996

PHYSICS

ELSEVIER

Physics Letters A 210 (1996)

LETTERS

A

163-167

On a model of an unconstrained hyperfluid Yuri N. Obukhov Instituteftir

Theoretical

Physics,

University

’

qf’ Cologne, D-50923

Received 24 October 1995; accepted for publication 14

Cologne,

Germany

November I995

Communicated by P.R. Holland

Abstract

A hyperfluid is a classical continuous medium carrying hypermomentum. We modify the earlier developed variational approach to a hyperfluid in such a way that the Frenkel-type constraints imposed on the hypermomentum current are eliminated. The resulting self-consistent model is different from the Weyssenhoff-type model. The essential point is a conservation of the hypermomentum current such that the final metrical and canonical energy-momentum forms coincide. PAC.? 04.4O.+c;

04SO.+h

1. Introduction Recently [ I] a variational model of a hyperfluid has been developed. This is a classical continuous medium, of which the elements possess both polarizability and elasticity properties. Such a kind of matter might be a non-quantum source of metric-affine gravity [ 21 within the framework of the gauge theory for the general affine space-time symmetry group. The hyperfluid model in Ref. [ I] is constructed as a natural generalization of the phenomenological Weyssenhoff spin fluid [ 31, the variational theory of which was elaborated in Ref. [4] (in the Lagrange approach) and in Ref. [ 51 (in the Euler approach). Although the Weyssenhoff model, in our opinion, provides a reasonable description for a spin fluid, it is well known that there exist different models of continuous media with spin [6-81. Likewise, the properties of matter with hypermomentum are not sufficiently well known

’ On leave from Department State University, I 17234

0375.9601/96/$12.00 SSDIO375-9601

@

of Theoretical

Moscow,

Physics, Moscow

Russian Federation.

at present. Thus a study of possible different hyperfluid models appears to be a necessary step in the development of this subject. In the present paper we discuss a possibility of removing the standard Frenkel-type constraints usually imposed on the hypermomentum current.

2. Preliminaries: gravity

Our basic notations and conventions are those of Ref. [ 21, except for the metric signature which we assumed here to be ( +, -, -, -) . The field equations of the metric-affine gravity theory are derived from the general Lagrangian four-form L = V + I,,,,, with V = V(g,p,P,Q,p,F, Rap) and L,,, = L,,, (g,p, 6”) rap, P’, DW ) as the pure gravitational and matter Lagrangians, respectively. Here the metric zero-form g,p, the coframe one-form P, and the affine connection one-form Tap are the gravitational potentials, the respective field strengths are the non-

I996 Elsevier Science B.V. All rights reserved

(95)00903-5

matter currents in metric-affine

Y.N. Obukhou/Physics

164

metricity one-form Qap := -Dgap and the two-forms of torsion T” and curvature Rap [ 21. The general field equations read

6” sr/

= -Asp,

(2.1)

where the matter currents are defined as the variational derivatives

(2.2) It is worthwhile to mention that the standard EinsteinHilbert Lagrangian V = -4 Rap A * (19” A 19,) (considered, e.g., in Ref. [ 91) is a bad choice for the metric-affine gravity. Due to an auxiliary (the so-called projective, see Ref. [2]) invariance, this Lagrangian is only compatible with the dilaton-free sources for which A = Aa’, = 0, and the Weyl oneform Q = g”pQnp is not determined by the gravitational field equations (2.1). The Weyssenhoff-type hyperfluid [ 1 ] is described by the following currents: the metric stress-energy four-form @‘p = 77](E+P)u0uP

- &@I

+ 2uyu’*DAp)

Y’

(2.3) the canonical

energy-momentum

-% = up, -P(rla

three-form

-U&),

and the hypermomentum

(2.4) three-form

Asp = uJap,

(2.5)

where E is the energy density, p the pressure, u the flow three-form, J”p the hypermomentum density, and Pa the four-momentum, P, = *(EU A 6,

- 2uPgyl,DAypl).

(2.6)

As usual, rla = *P and ~7 is the volume four-form. The components of the four-velocity are defined by includes as irreu = llayla. The hypermomentum ducible parts the spin density S,, = J~,pl and the dilaton charge J = J”,.

Letters A 210 (1996) 163-167

The hypermomentum Jafiup = Japu,

density satisfies the constraint

= 0.

(2.7)

This is a natural generalization of the Frenkel condition .&pup = 0. Recently [ lo], it has been suggested that the generalised Frenkel condition could be weakened and, in particular, reduced just to the ordinary Frenkel constraint imposed on the spin part of the hypermomentum density. Below we study the possibility of constructing a completely unconstrained hyperfluid model.

3. Variational hyperfluid

principle

for the unconstrained

We start with the Lagrangian perfluid, cf. Ref. [ 11, La, = 4p>s,p*,)rl-puAdA,

four-form

;pp*,b,Bu

for the hy-

A Db:

+hzuAdX+AsuAds

+Ao(*uAu-rl)+A~(bSAbA-r16~).

(3.1)

Here the first two terms describe the internal energy density E and the kinetic energy density, respectively. As usual, we assume that the internal energy of the hyperfluid depends on the particle density p, the specific entropy s and the specific hypermomentum density ,u*,. It seems worthwhile to notice at this point that the indices A, B, . . ., which run from 1 to 3, are merely labels and they have no any geometrical meaning. The material frames b* (which is a one-form with an expansion b* = btSa) and bA (which is a threeform with analogous expansion bA = bzr],) are independent of the reference frames which an observer may introduce and change (i.e. rotate and deform) at his own choice. The material frames together with the specific hypermomentum are a sort of internal variables which describe the polarizability and elasticity properties of the fluid elements. Material frames are rigidly attached to elements of the continuum and their evolution is determined by the equations of motion of the fluid. From the geometrical point of view, pAB is a scalar and E is explicitly a general coordinate invariant quantity. The energy density cannot be a function of the hypermomentum density tensor, as is incorrectly

Y.N. Obukhov/Physics

assumed in Ref. [9], since this is incompatible with the general coordinate invariance of the fluid action. The remaining terms in (3.1) describe constraints. The first three express the conservation particle number, constancy of entropy, and identity of elements along the flow lines,

165

Letters A 210 (1996) 163-167

= ;pb,Bu A Db;, -;ppABbflDb;

- pdA,

= 0,

(3.10)

+A2dX+Aids-2A0*u=0, ;ppAB * (u A Db”,)va

d(pu)

(3.9)

+ AA,bA = 0,

(3.11)

(3.2) ; * D(ppABbfu)P

uAdX=O,

(3.3)

u A ds = 0.

(3.4)

The last two terms in (3.1) have the meaning fluid velocity is timelike and normalized, *l4 A u = 77,

that the

(3.5)

+ A;bB = 0.

(3.12)

These equations determine the Lagrange multipliers. While A,, AZ, A3 satisfy differential evolution equations, one can explicitly solve (3.8)-( 3.11) with respect to the other Lagrange multipliers, 2Ao=p

(

$

(3.13) )

and that material frame variables are dual in the sense A; = ;p,uCB * (u A b*,Db;). bB A bA = 6;~.

(3.6)

This is the crucial modification of the hypertluid model [ 11. Recall that the material triad is rigid in the spin fluid model (which can only rotate), while in accordance with the affine gauge approach we assumed that in a hyperfluid the material frame b: is elastic and can deform arbitrarily during the motion of the medium. However, in the Weyssenhoff-type model [ 1 ] the generalized material tetrad (u, bA) is still constrained as the material triad is assumed to be orthogonal to the velocity, *u A bA = 0. Now, developing the suggestion of Ref. [ lo], we remove the latter orthogonality condition. Hereafter such a medium is called the unconstrained hype@uid. The derivation of the Euler-Lagrange

equations is analogous to Ref. [ 11. As before, the independent variables are the metric-affine gravitational potentials g,p, 6”, ran, the material variables P = s, X}, and the Lagrange multiplier {bA,bB,p,,uA,, zero-forms Ao, A I , A2, A3, A$ Varying the action (3.1) with respect to the Lagrange multipliers, one finds (3.2)-( 3.6). The variations of s, X, p, p* B and u, bB, bA yield the equations, respectively, d& rl +d(A3u) ( as 1

Substituting (3.14) into (3.11) and (3.12), we get the system of equations of motion of the specific hypermomentum density and the material frames, u A (d/L*, + pACb”BDbz - pCBb;Db;)

=O,

- ;pAsb,Bu A Db”, -u

AdA, = 0,

(3.7)

(3.8)

= 0, (3.15)

(SE - bib$pABu

A Db; = 0,

(3.16)

(St - b&b$)pABu

A Db”, = 0.

(3.17)

While (3.15) coincides with the analogous equation of motion for the Weyssenhoff-type hypertluid [ I], Eqs. (3.16), (3.17) arenew.Onecanrewrite (3.15)(3.17) in terms of the tensor variables, demonstrating that these equations express the conservation of the hypermomentum current. Indeed, contracting (3.15) with b”,b: and using (3.16), (3.17), we find u A D(p’,b”,b;)

= 0.

(3.18)

The matter currents of the unconstrained

hyperfluid are derived by direct calculation of the variational derivatives (2.2). Denoting as usual [ 1 1, Jap = ;p,uABb”,b;,

d(A2u) = 0,

(3.14)

and introducing

a.5

p:=p

the pressure

(ap>--&,

(3.19) in a standard way, (3.20)

166

Y.N. Obukhou/Physics

one obtains (using (3.15)-(3.18))

-pp1,

a@ = rl[@+p)uauP

& = &WI

-

p(rla - uua)

9

(3.21) (3.22) (3.23)

Asp = uJap.

The most essential difference between the resulting unconstrained model and the Weyssenhoff-type hyperlluid [ 1 ] is the complete decoupling of the hypermomentum from both energy three-forms (3.21) and (3.22). This is compatible with the Noether identities [ 21 though, as Eq. (3.18) (multiply by p and use (3.4) ) actually describes the conservation of the hypermomentum current, (3.24)

DAap = 0.

The dynamics of all irreducible parts of the hypermomentum is contained in (3.24), and it is indeed unconstrained since neither the Frenkel-type conditions (2.7), nor any other are imposed on the hypermomenturn density. The complete decoupling of Amp from the energy currents of course does not mean that the unconstrained hypermomentum does not affect the gravitational field: it still enters as the source in the third metric-affine gravity field equation (2.1). At first sight, it may seem surprising that such a simple modification of the original hyperfluid model as the elimination of the orthogonality condition of the material triad and velocity, may yield the above described essential change of the equations of motion of the hypermomentum and reduce the energymomentum three-form to that of the ideal structureless fluid. However, everything becomes clear when one notices that the Lagrangian (3.1) of the unconstrained hyperfIuid possesses an extra gauge symmetry besides the usual coordinate and local gravitational frame GL( 4, R) invariances. Namely, (3.1) is invariant under the simultaneous transformations of the material frames and the affine connection components, b; ----+ L;bf, rap -

Lf(r,”

b,B -

b;L-‘pa,

+ X;d) L-lp,,

(3.25)

where L; E GL(4, R). Notice that this is different from the affine gauge gravity transformation, since the

Letters A 210 (1996) 163-167

gravitational frame 6” and metric g,p are not transformed. The total Lagrangian L = V + L,,, certainly does not possess the symmetry (3.25). Applying to the Lagrangian (3.1) the Noether machinery [ 21, one can derive a conservation law corresponding to (3.25). This is exactly (3.24).

4. Discussion

and conclusion

In the Weyssenhoff-type hyperfluid [ 1 ] the generalized Frenkel condition (2.7) plays an important role. It indeed restricts the possible motions of the medium, as well as the very structure of the hypermomentum current. In particular, (2.7) rules out the case of purely dilatonic Weyssenhoff matter. Using the standard decomposition [ 21 of the hypermomentum into its irreducible parts, Jap = rap + i JSZ + S”,, one finds in view of (2.7) that the dilaton charge is expressed in terms of the proper hypermomentum (shear) according to J = -~&&UP. Thus vanishing shear yields also J = 0. Unlike this, the unconstrained hyperfluid may be of purely dilatonic type. In this case our model gives a description of a physical source for the generalized Einstein-Weyl gravity theory considered recently in Ref. [ 111. In Ref. [ lo] an attempt was made to construct a sort of intermediate model in which the constraint (2.7) reduces to the original Frenkel condition, F, = Sapup = 0. Technically this can be achieved if one adds to (3.1) the term [“F, with a Lagrange multiplier (four-form) 5”. Such a term breaks the symmetry (3.25). However, the resulting equations of motion for the fluid and the gravitational currents (2.2) look unusual and their physical interpretation is unclear. Contrary to the expectations of the authors of Ref. [ lo] (who even failed to solve the highly nontrivial system of constraint equations with respect to [“), at the end one does not find a Weyssenhoff-type dynamics for the spin part of the hypermomentum. It should be noted that the above described unconstrained hyperfluid is not a subcase of the general Weyssenhoff hyperfluid model. They are close though, in the sense that the two theories may admit the same particular solutions for the gravitational and matter field equations. The next decisive step will be a comparison of different ideal hyperlluid models [ 1,9,10] with the real physical media of which the elements

Y.N. Obukhoo/Physics

display polarizability

and/or

elasticity properties.

Letters A 210 (19961 163-167

161

[2] F.W. Hehl, J.D. McCrea, Phys. Rep. 258 (1995)

Acknowledgement

[3]

J. Weyssenhoff

[4]

W. Kopczynski,

[ 51 YuN. (1987)

I would like to thank Friedrich W. Hehl for stimulating discussions and useful comments. This research was supported by the Deutsche Forschungsgemeinschaft under the project He-5281 17- 1.

and A. Raabe, Acta Phys. Pol. 9 Phys. Rev. D 34 (1986)

Obukhov

and V.A. Korotky,

M. Lavorgna,

Phys. Rev. D 28 (1983) [8]

HP. de Oliveira,

[9]

L.L. Smalley

( 1993)

[I I]

( 1982)

fluid

and B.N. with

LANL

10.59.

713.

Frolov,

intrinsic Preprint

e-archive

The

473.

Moscow

gr-qc/95090

( 1995)

variational

hypermomentum

R.W. Tucker and C. Wang, Class. Quantum 2587.

Grav. 4

and C. Stomaiolo,

and J.P Krisch, J. Math. Phys. 36

Babourova

University,

G. Platania

Gen. Rel. Grav. 25 (1993)

time with nonmetricity, Phys. Lett. A I84

Class. Quantum

R. De Ritis,

of perfect

and R. Tresguerres,

7.

1633.

J.R. Ray and L.L. Smalley, Phys. Rev. Lett. 49

O.A.

( 1947)

352.

[7]

References

17.

and Yu. Ne’eman,

[6]

[ IO]

[ I ] Yu.N. Obukhov

E.W. Mielke I.

in space-

Pedagogical I3

778.

theory State

( 1995). Grav. I2

( 1995)