koorio.com
漆講恅踱 恅紫蚳模
婝翑妀蟈諉
絞ヶ弇离ㄩ忑珜 >> >>

The Rest-Frame Instant Form of Relativistic Perfect Fluids and of Non-Dissipative Elastic M


The Rest-Frame Instant Form of Relativistic Perfect Fluids with Equation of State 老 = 老(n, s) and of Non-Dissipative Elastic Materials.
Luca Lusanna
Sezione INFN di Firenze L.go E.Fermi 2 (Arcetri) 50125 Firenze, Italy E-mail LUSANNA@FI.INFN.IT

arXiv:hep-th/0003095v1 13 Mar 2000

and Dobromila Nowak-Szczepaniak
Institute of Theoretical Physics University of Wroclaw pl. M.Borna 9 50-204 Wroclaw, Poland E-mail dobno@ift.uni.wroc.pl

Abstract

For perfect ?uids with equation of state 老 = 老(n, s), Brown [1] gave an action principle depending only on their Lagrange coordinates 汐i (x) without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski spacetime, the Wigner-covariant rest-frame instant form of these perfect ?uids is given. Their Hamiltonian invariant mass can be given in closed form for the dust and the photon gas. The action for the coupling to tetrad gravity is given. Dixon*s multipoles for the perfect ?uids are studied on the rest-frame Wigner hyperplane. It is also shown that the same formalism can be applied to non-dissipative relativistic elastic materials described in terms of Lagrangian coordinates.

February 1, 2008

Typeset using REVTEX 1

I. INTRODUCTION

Stability of stellar models for rotating stars, gravity-?uid models, neutron stars, accretion discs around compact objects, collapse of stars, merging of compact objects are only some of the many topics in astrophysics and cosmology in which relativistic hydrodynamics is the basic underlying theory. This theory is also needed in heavy-ions collisions. As shown in Ref. [1] there are many ways to describe relativistic perfect ?uids by means of action functionals both in special and general relativity. Usually, besides the thermodynamical variables n (particle number density), 老 (energy density), p (pressure), T (temperature), s (entropy per particle), which are spacetime scalar ?elds whose values represent measurements made in the rest frame of the ?uid (Eulerian observers), one characterizes the ?uid motion by its unit timelike 4-velocity vector ?eld U ? (see Appendix A for a review of the relations among the local thermodynamical variables and Appendix B for a review of covariant relativistic thermodynamics following Ref. [2]). However, these variables are constrained due to [we use a general relativistic notation: ※; ?§ denotes a covariant derivative] i) particle number conservation, (nU ? );? = 0; ii) absence of entropy exchange between neighbouring ?ow lines, (nsU ? );? = 0; iii) the requirement that the ?uid ?ow lines should be ?xed on the boundary. Therefore one needs Lagrange multipliers to incorporate i) anf ii) into the action and this leads to use Clebsch (or velocity-potential) representations of the 4-velocity and action functionals depending on many redundant variables, generating ?rst and second class constraints at the Hamiltonian level (see Appendix A). Following Ref. [3] the previous constraint iii) may be enforced by replacing the unit 4velocity U ? with a set of spacetime scalar ?elds 汐 ? i (z ), i = 1, 2, 3, interpreted as ※Lagrangian (or comoving) coordinates for the ?uid§ labelling the ?uid ?ow lines (physically determined by the average particle motions) passing through the points inside the boundary (on the boundary they are ?xed: either the 汐 ? i (z o , z )*s have a compact boundary V汐 (z o ) or they have assigned boundary conditions at spatial in?nity). This requires the choice of an arbitrary spacelike hypersurface on which the 汐i *s are the 3-coordinates. A similar point of view is contained in the concept of ※material space§ of Refs. [4,5], describing the collection of all the idealized points of the material; besides to non-dissipative isentropic ?uids the scheme can be applied to isotropic elastic media and anisotropic (crystalline) materials, namely to an arbitrary non-dissipative relativistic continuum [6]. See Ref. [7] for the study of the transformation from Eulerian to Lagrangian coordinates (in the non-relativistic framework of the Euler-Newton equations). Notice that the use of Lagrangian (comoving) coordinates in place of Eulerian quantities allows the use of standard Poisson brackets in the Hamiltonian description, avoiding the formulation with Lie-Poisson brackets of Ref. [8], which could be recovered by a so-called Lagrangian to Eulerian map. Let M 4 be a curved globally hyperbolic spacetime [with signature ?(+ ? ??), ? = ㊣] whose points have locally coordinates z ? . Let 4 g?糸 (z ) be its 4-metric with determinant 4 g = |det 4 g?糸 |. Given a perfect ?uid with Lagrangian coordinates 汐 ? (z ) = { 汐 ? i (z )}, unit 4-velocity vector ?eld U ? (z ) and particle number density n(z ), let us introduce the number ?ux vector

2

n(z )U ? (z ) =

J ? (汐 ? i (z ))
4 g (z )

,

(1.1)

﹟ and﹟ the densitized ?uid number ?ux vector or material current [?0123 = 1/ 4 g ; ? i ) describes the orientation of the volume in thematerial space] ?汐 ( 4 g??糸老考 ) = 0; 灰123 (汐 J ? (汐 ? i (z )) = ? ? ? ?
4 g??糸老考 灰 ?i (z ))?糸 汐 ? 1 (z )?老 汐 ? 2 (z )?考 汐 ? 3 (z ), 123 (汐

n(z ) =

?4 g?糸 (z )J ? (汐 ? i(z ))J 糸 (汐 ? i (z )) | J (汐 ? i(z ))| ﹟4 = 灰123 (汐 ? i(z )) , 4 g (z ) g
4g

?? J ? (汐 ? i (z )) =

[n(z )U ? (z )];? = 0,
i 4 gnU ? ](z )? 汐 ? ? (z )

J ? (汐 ? i (z ))?? 汐 ? i (z ) = [

= 0.

(1.2)

This shows that the ?uid ?ow lines, whose tangent vector ?eld is the ?uid 4-velocity timelike vector ?eld U ? , are identi?ed by 汐 ?i = const. and that the particle number conservation is automatic. Moreover, if the entropy for particle is a function only of the ?uid Lagrangian coordinates, s = s(汐 ?i ), the assumed form of J ? also implies automatically the absence of entropy exchange between neighbouring ?ow lines, (nsU ? ),? = 0. Since U ? ?? s(汐 ? i ) = 0, the perfect ?uid is locally adiabatic; instead for an isentropic ?uid we have ?? s = 0, namely s = const.. Even if in general the timelike vector ?eld U ? (z ) is not surface forming (namely has a non-vanishing vorticity, see for instance Ref. [9]), in each point z we can consider the spacelike hypersurface orthogonal to the ?uid ?ow line in that point (namely we split the tangent space Tz M 4 at z in the U ? (z ) direction and in the orthogonal complement) and 1 consider 3! [U ? ??糸老考 dz 糸 _ dz 老 _ dz 考 ](z ) as the in?nitesimal 3-volume on it at z . Then the 3-form 1 灰 [z ] = [灰123 (汐 ?)d汐 ? 1 _ d汐 ? 2 _ d汐 ? 3 ](z ) = n(z )[U ? ??糸老考 dz 糸 _ dz 老 _ dz 考 ](z ), (1.3) 3! may be interpreted as the number of particles in this 3-volume. If V is a volume around z on the spacelike hypersurface, then V 灰 is the number of particle in V and V s灰 is the total entropy contained in the ?ow lines included in the volume V. Note that locally 灰123 can be set to unity by an appropriate choice of coordinates. In Ref. [1] it is shown that the action functional S [4 g?糸 , 汐 ?] = ? d4 z
4 g (z ) 老(

| J (汐 ? i(z ))|
4 g (z )

, s (汐 ?i (z ))),

(1.4)

has a variation with respect to the 4-metric, which gives rise to the correct stress tensor ?老 |s ? 老 for a perfect ?uid [see Appendix A]. T ?糸 = (老 + p)U ? U 糸 ? ? p 4 g ?糸 with p = n ?n The Euler-Lagrange equations associated to the variation of the Lagrangian coordinates are [1] [V? = ?U? is the Taub current, see Appendix A] 1 汛S 1 ?s ﹟4 = ??糸老考 V?;糸 灰ijk ?老 汐 ? j ?考 汐 ?k ? n T = 0, i ? 2 ?汐 ?i g 汛汐 1 1 汛S ﹟4 ?? 汐 ? i = ?汐汕污汛 V汐;汕 U 糸 ?糸?污汛 ? T ?? s = 2V[?;糸 ] U 糸 ? T ?? s = 0. i ? 2 g 汛汐 3 (1.5)

As shown in Appendix A, these equations together with the entropy exchange constraint imply the Euler equations implied from the conservation of the stress-energy-momentum tensor. Therefore, with this description the conservation laws are automatically satis?ed and the Euler-Lagrange equations are equivalent to the Euler equations. In Minkowski spacetime ? i (z )), while the the conserved particle number is N = V汐 (z o ) d3 z n(z ) U o (z ) = V汐 (z o ) d3 z J o (汐 ? i ). Moreconserved entropy per particle is V汐 (z o ) d3 z s(z ) n(z ) U o (z ) = V汐 (z o ) d3 z s(z ) J o (汐 ?糸 over, the conservation laws T ,糸 = 0 will generate the conserved 4-momentum and angular momentum of the ?uid. However, in Ref. [1] there are only some comments on the Hamiltonian description implied by this particular action. This description of perfect ?uids ?ts naturally with parametrized Minkowski theories [10] for arbitrary isolated relativistic systems [see Ref. [11] for a review] on arbitrary spacelike hypersurfaces, leaves of the foliation of Minkowski spacetime M 4 associated with one of its 3+1 splittings. Therefore, the aim of this paper is to ?nd the Wigner covariant rest-frame instant form of the dynamics of a perfect ?uid, on the special Wigner hyperplanes orthogonal to the total 4-momentum of the ?uid. In this way we will get the description of the global rest frame of the ?uid as a whole; instead, the 4-velocity vector ?eld U ? de?nes the local rest frame in each point of the ?uid by means of the projector 4 g ?糸 ? ?U ? U 糸 . This approach will also produce automatically the coupling of the ?uid to ADM metric and tetrad gravity with the extra property of allowing a well de?ned deparametrization of the theory leading to the rest-frame instant form in Minkowski spacetime with Cartesian coordinates when we put equal to zero the Newton constant G [11]. In this paper we will consider the perfect ?uid only in Minkowski spacetime, except for some comments on its coupling to gravity. The starting point is the foliation of Minkowski spacetime M 4 , which is de?ned by an embedding R ℅ 曳 ↙ M 4 , (而, 考 ) ↙ z ? (而, 考 ) ﹋ 曳而 and with 曳 an abstract 3-surface diffeomorphic to R3 , with 曳而 its copy embedded in M 4 labelled by the value 而 (the scalar mathematical ※time§ parameter 而 labels the leaves of the foliation, 考 are curvilinear coor’ ’ ) are 曳而 -adapted holonomic coordinates for M 4 ). See dinates on 曳而 and 考 A = (考 而 = 而, 考 r Appendix C for the notations on spacelike hypersurfaces. In this way one gets a parametrized ?eld theory with a covariant 3+1 splitting of Minkowski spacetime and already in a form suited to the transition to general relativity in its ADM canonical formulation (see also Ref. [12], where a theoretical study of this problem is done in curved spacetimes). The price is that one has to add as new independent con?guration variables the embedding coordinates z ? (而, 考) of the points of the spacelike hypersurface 曳而 [the only ones carrying Lorentz indices] and then to de?ne the ?elds on 曳而 so that they know the hypersurface 曳而 of 而 -simultaneity [for a Klein-Gordon ?eld 耳(x), this ?(而, 考 ) = 耳(z (而, 考 )): it contains the non-local information about the embedding]. new ?eld is 耳 Then one rewrites the Lagrangian of the given isolated system in the form required by the coupling to an external gravitational ?eld, makes the previous 3+1 splitting of Minkowski spacetime and interpretes all the ?elds of the system as the new ?elds on 曳而 (they are Lorentz scalars, having only surface indices). Instead of considering the 4-metric as describing a gravitational ?eld (and therefore as an independent ?eld as it is done in metric gravity, where one adds the Hilbert action to the action for the matter ?elds), here one 4

? 糸 replaces the 4-metric with the the induced metric gA ’B ’ [z ] = zA ’ on 曳而 [a functional ’ 灰?糸 zB ’ ? ? A ? ? are ?at inverse tetrad ?elds on Minkowski spacetime with the zr of z ; zA ’ *s ’ = ?z /?考 tangent to 曳而 ] and considers the embedding coordinates z ? (而, 考 ) as independent ?elds [this ’ ? A ? is not possible in metric gravity, because in curved spacetimes zA ’ = ?z /?考 are not tetrad ?elds so that holonomic coordinates z ? (而, 考 ) do not exist]. From this Lagrangian, besides a Lorentz-scalar form of the constraints of the given system, we get four extra primary ?rst class constraints 而而 r ’而 H? (而, 考 ) = 老? (而, 考) ? l? (而, 考)Tsys (而, 考 ) ? zr ’? (而, 考 )Tsys (而, 考 ) > 0,

(1.6)

而而 r ’而 ’ [ here Tsys (而, 考 ) = M(而, 考), Tsys (而, 考) = Mr (而, 考 ), are the components of the energymomentum tensor in the holonomic coordinate system, corresponding to the energy- and ∩ momentum-density of the isolated system; one has {H(?) (而, 考 ), H(糸 ) (而, 考 )} = 0] implying the independence of the description from the choice of the 3+1 splitting, i.e. from the choice of the foliation with spacelike hypersufaces. As shown in Appendix C the evolution vector ? ? ’ ? ? is given by z而 = N[z ](f lat) l? + N[r ’ , where l (而, 考 ) is the normal to 曳而 in z (而, 考 ) and z ](f lat) zr ’ N[z ](f lat) (而, 考 ), N[r z ](f lat) (而, 考 ) are the ?at lapse and shift functions de?ned through the metric like in general relativity: however, now they are not independent variables but functionals of z ? (而, 考 ). The Dirac Hamiltonian contains the piece d3 考竹? (而, 考 )H? (而, 考 ) with 竹? (而, 考 ) Dirac mul’s ’ ’s ’ tipliers. It is possible to rewrite the integrand in the form [污 r = ?? 3 g r is the inverse of 4 3 3 the spatial metric gr ’s ’ = gr ’s ’ = ?? gr ’s ’, with gr ’s ’ of positive signature (+ + +)] ? ’s ’ 糸 竹? (而, 考 )H? (而, 考 ) = [(竹? l? )(l糸 H糸 ) ? (竹? zr )(污 r zs ’糸 H )](而, 考 ) = def def

r ’s ’ 糸 = N(f lat) (而, 考 )(l? H? )(而, 考 ) ? N(f lat)’ r (而, 考 )(污 zs ’糸 H )(而, 考 ),

(1.7)

’s ’ ? with the (non-holonomic form of the) constraints (l? H? )(而, 考) > 0, (污 r zs ’? H )(而, 考 ) > 0, satisfying the universal Dirac algebra of the ADM constraints. In this way we have de?ned new ?at lapse and shift functions

N(f lat) (而, 考 ) = 竹? (而, 考 )l? (而, 考 ), ? N(f lat)’ r (而, 考 ) = 竹? (而, 考 )zr ’ (而, 考 ).

(1.8)

which have the same content of the arbitrary Dirac multipliers 竹? (而, 考 ), namely they multiply primary ?rst class constraints satisfying the Dirac algebra. In Minkowski spacetime they are quite distinct from the previous lapse and shift functions N[z ](f lat) , N[z ](f lat)’ r , de?ned starting ? from the metric. Since the Hamilton equations imply z而 (而, 考 ) = 竹? (而, 考 ), it is only through the equations of motion that the two types of functions are identi?ed. Instead in general relativity the lapse and shift functions de?ned starting from the 4-metric are the coe?cients (in the canonical part Hc of the Hamiltonian) of secondary ?rst class constraints satisfying the Dirac algebra independently from the equations of motion. For the relativistic perfect ?uid with equation of state 老 = 老(n, s) in Minkowski space4 time, we have only to replace the external 4-metric 4 g?糸 with gA ’B ’ (而, 考 ) and ’B ’ (而, 考 ) = gA i i the scalar ?elds for the Lagrangian coordinates with 汐 (而, 考) = 汐 ? (z (而, 考 )); now either the 汐i (而, 考 )*s have a compact boundary V汐 (而 ) ? 曳而 or have boundary conditions at spatial in?nity. For each value of 而 , one could invert 汐i = 汐i (而, 考 ) to 考 = 考 (而, 汐i ) and use the 汐i *s as a special coordinate system on 曳而 inside the support V汐 (而 ) ? 曳而 : z ? (而, 考 (而, 汐i)) = z ’? (而, 汐i). 5

By going to 曳而 -adapted coordinates such that 灰123 (汐) = 1 we get [污 = |det gr ’s ’|; ﹟4 ﹟ g = |det gA ’B ’ | = N 污] ﹟ ’ ’ J A (汐i (而, 考 )) = [N 污nU A ](而, 考 ),



g=

’u ’v ’ 1 2 3 i J 而 (汐i (而, 考 )) = [??r ?r ’汐 ?u ’ 汐 ?v ’ 汐 ](而, 考 ) = ?det (?r ’汐 )(而, 考 ), 3 ’ Jr (汐i (而, 考 )) = [ i=1;i,j,k cyclic ’u ’v ’ j k ?而 汐i ?r ?u ’ 汐 ?v ’ 汐 ](而, 考 ) =

1 ’u ’v ’ j k ?ijk [?而 汐i ?u = ?r ’ 汐 ?v ’ 汐 ](而, 考 ), 2 ? |J | n(而, 考 ) = ﹟ (而, 考 ) = N 污
’ B ’ A ?gA ’B ’J J (而, 考), ﹟ N 污 V汐 (而 )

(1.9)

with N = the conserved particle number and particle. The action becomes S= =? =? 老(

d3 考J 而 (汐i (而, 考 )) giving 3 而 V汐 (而 ) d 考 (s J )(而, 考 ) giving the conserved entropy per

d而 d3 考L(z ? (而, 考 ), 汐i(而, 考 )) = |J (汐i(而, 考 ))| ﹟ , s(汐i (而, 考))) = d而 d3 考 (N 污 )(而, 考 )老( ﹟ (N 污 )(而 考 ) ﹟ d而 d3 考 (N 污 )(而, 考 ) 1 污 (而, 考) (J 而 )2 ? 3 g u ’v ’
’ + Nu ’J 而 J v ’ + Nv ’J 而 Ju (而, 考 ; 汐i(而, 考 )), s(汐i (而, 考 ))), N N

(1.10)
’ ’ with N = N[z ](f lat) , N r = N[r z ](f lat) . This is the form of the action whose Hamiltonian formulation will be studied in this paper. We shall begin in Section II with the simple case of dust, whose equation of state is 老 = ?n. In Section III we will de?ne the ※external§ and ※internal§ centers of mass of the dust. In Section IV we will study Dixon*s multipoles of a perfect ?uid on the Wigner hyperplane in Minkowski spacetime using the dust as an example. Then in Section V we will consider some equations of state for isentropic ?uids and we will make some comments on non-isentropic ?uids. In Section VI we will de?ne the coupling to ADM metric and tetrad gravity. In Section VII we will describe with the same technology isentropic elastic media. In the Conclusions, after some general remarks, we will delineate the treatment of perfect ?uids in tetrad gravity (this will be the subject of a future paper). In Appendix A there is a review of some of the results of Ref. [1] for relativistic perfect ?uids.

6

In Appendix B there is a review of covariant relativistic thermodynamics of equilibrium and non-equilibrium. In Appendix C there is some notation on spacelike hypersurfaces. In Appendix D there is the de?nition of other types of Dixon*s multipoles.

7

II. DUST.

Let us consider ?rst the simplest case of an isentropic perfect ?uid, a dust with p = 0, s = const., and equation of state 老 = ?n. In this case the chemical potential ? is the rest mass-energy of a ?uid particle: ? = m (see Appendix A). 4 Eq.(1.10) implies that the Lagrangian density is [we shall use the notation gA ’B ’ = gA ’B ’ with signature ?(+ ? ??), ? = ㊣1; by using the notation with lapse and shift functions r ’ s ’ 3 s ’ 3 given in Appendix C we get: g而 而 = ?(N 2 ? 3 gr ’s ’N N ), g而 r ’ = ?? gr ’s ’N , gr ’s ’ = ?? gr ’s ’ with r ’ r ’ s ’ ? N N N 3 而而 而r ’ r ’s ’ 3 r ’s ’ gr = N 2 , g = ?? N 2 , g = ??( g ? N 2 ); the inverse ’s ’ of positive signature (+++), g ’s ’ ’s ’ ’s ’ ’s ’ of the spatial 4-metric 4 gr is denoted 污r = 4污r = ?? 3 g r , where 3 g r is the inverse of the ’s ’ ﹟ ﹟ 3 3 gr 3-metric gr 污= ’s ’] ’s ’ and we use ﹟ ﹟ ’ B ’ A L(汐i , z ? ) = ? g老 = ?? gn = ?? ?gA ’B ’J J =
r ’ s ’ = ??N (J 而 )2 ? 3 gr ’s ’Y Y = ??NX, ’ Yr =

1 r ’ 而 (J ’ + N r J ), N (J 而 )2 ?
3g r ’s ’Y r ’Y s ’

X=

=



g ﹟ n = 污n, N

(2.1)

’ with J 而 , J r given in Eqs.(1.9). The momentum conjugate to 汐i is ’3 r ’u ’v ’ j k ?u gt Yt ?L ’r ’ 汐 ?v ’汐 ’? = ? |i,j,k cyclic = ??而 汐i X ’ ’ Yr Yt r ’u ’v ’ j k 3 ? ? ? 汐 ? 汐 = ? gt Tr =? ’r ijk u ’ v ’ ’i , ’ 2X X

旭i =

,

Tt ’i =

def

1 ’u ’v ’ j k ?ijk ?u g’’ ?r ’r ’), ’ 汐 ?v ’ 汐 = gt ’ (ad Jir 2 tr

(2.2)

?1 i where ad Jir ’ = (det J )Jir ’ = ?r ’汐 ) of the ’ is the adjoint matrix of the Jacobian J = (Jir i transformation from the Lagrangian coordinates 汐 (而, 考 ) to the Eulerian ones 考 on 曳而 . The momentum conjugate to z ? is r (J 而 )2 ?L 而 Y (而, 考 ). + ? zr? J 老? (而, 考) = ? ? (而, 考 ) = ? l? ?z而 X X

(2.3)

The following Poisson brackets are assumed
? 3 {z ? (而, 考), 老糸 (而, 考 } = ?灰糸 汛 ( 考 ? 考 ), ∩ ∩ i i 3 {汐 (而, 考 ), 旭j (而, 考 )} = 汛j 汛 (考 ? 考 ).
∩ ∩

(2.4)

’ ’i We can express Y r /X in terms of 旭i with the help of the inverse (T ?1 )r of the matrix

Tt ’i
’ 1 Yr ’i = (T ? 1 )r 旭i , X ?

(2.5)

8

where
’i (T ? 1 )r = i g ?s ’汐 . k det (?u ’汐 ) 3 r ’s ’

(2.6)

From the de?nition of X we ?nd X= ?J 而
? 1 )u ’ i (T ? 1 )v ’j 旭 旭 ?2 + 3 gu ’v ’ (T i j

.

(2.7)

Consequently, we can get the expression of the velocities of the Lagrangian coordinates in terms of the momenta ?而 汐i = ? namely
r ’ ?1 r i ) ’i 旭i ?而 汐i = ?r ’汐 N ? N (T ? 1 )u ’i (T ?1 )v ’j 旭 旭 . ?2 + 3 gu ’v ’ (T i j ’ i ’ 而 ’ i Jr ?r (N r J ? NY r ) ?r ’汐 ’汐 = , J而 J而

(2.8)

(2.9)

Now 老? can be expressed as a function of the z *s, 汐*s and 旭*s: 老? = l? J 而
’i ? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 + z J 而 (T ?1 )r ?2 + 3 gu 旭i . ’v ’ (T i j r ’?

(2.10)

Since the Lagrangian is homogenous in the velocities, the Hamiltonian is only HD = d3 考竹? (而, 考 )H? (而, 考), (2.11)

where the H? are the primary constraints
R H? = 老? ? l? M + zr ’? M > 0, ’

’ ’ ’i Mr = T 而r = J 而 (T ? 1 )r 旭i .

M = T 而而 = J而

? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 , ?2 + 3 gu ’v ’ (T i j

(2.12)

satisfying {H? (而, 考 ), H糸 (而, 考 )} = 0.


(2.13)

One ?nds that {H? (而, 考), HD } = 0. Therefore, there are only the four ?rst class constraints H? (而, 考) > 0. They describe the arbitrariness of the foliation: physical results do not depend on its choice. The conserved Poincar? e generators are (the su?x ※s§ denotes the hypersurface 曳而 ) p? s =
?糸 Js =

d3 考老? (而, 考 ), d3 考 [z ? (而, 考 )老糸 (而, 考 ) ? z 糸 (而, 考)老? (而, 考 )], (2.14)

and one has 9

?糸 {z ? (而, 考 ), p糸 s } = ?灰 ,

(2.15)

d3 考 H? (而, 考 ) = p? s ? +

d3 考 l? J 而

? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 (而, 考 ) + ?2 + 3 gu ’v ’ (T i j

而 ?1 r d3 考 zr ) ’i 旭i (而, 考 ) > 0. ’? J (T

(2.16)

Let us now restrict ourselves to spacelike hyperplanes 曳而 by imposing the gauge-?xings
? r ’ 汎 ?(而, 考 ) = z ? (而, 考) ? x? ’ (而 )考 > 0 , s (而 ) ? br ? 3 {汎 ? (而, 考), H糸 (而, 考 )} = ?灰糸 汛 ( 考 ? 考 ),
∩ ∩

(2.17)

? r ’ where x? ’ (而 ), s (而 ) is an arbitrary point of 曳而 , chosen as origin of the coordinates 考 , and br r ’ = 1, 2, 3, are three orthonormal vectors such that the constant (future pointing) normal to the hyperplane is 汕 ? 汐 l? (而, 考 ) > l? = b? (而 )b污 (而 ). ’ 而 = ? 汐汕污 b’ 1 (而 )b’ 2 3

(2.18)

Therefore, we get
? ? zr ’ (而, 考 ) > br ’ (而 ), ? r ’ 步? z而 (而, 考 ) > x 步? ’ (而 )考 , s (而 ) + br ’s ’ ’s ’ gr 污r (而, 考) > ??汛 r , ’s ’(而, 考 ) > ??汛r ’s ’,

污 (而, 考 ) > 1.

(2.19)

By introducing the Dirac brackets for the resulting second class constraints {A, B }? = {A, B } ? d3 考 [{A, 汎 ? (而, 考 )}{H? (而, 考 ), B } ? {A, H? (而, 考 )}{汎 ?(而, 考 ), B }], (2.20)
? ? r ’ ? we ?nd that, by using Eq.(2.15) and (2.16) [with x? ’ (而 )考 ? 汎 (而, 考 ) and s (而 ) = z (而, 考 ) ? br ? 糸 with the assumption {br ’ (而 ), ps } = 0], we get 糸 ? ?糸 {x? s (而 ), ps (而 )} = ?灰 .

(2.21)

The ten degrees of freedom describing the hyperplane are x? s (而 ) with conjugate momen? tum ps and six variables 耳竹 (而 ), 竹 = 1, .., 6, which parametrize the orthonormal tetrad b? ’ (而 ), A with their conjugate momenta T竹 (而 ). The preservation of the gauge-?xings 汎 ?(而, 考 ) > 0 in time implies d ? r ’ 步? 汎 (而, 考 ) = {汎 ?(而, 考 ), HD } = ?竹? (而, 考) ? x 步? ’ (而 )考 > 0 , s (而 ) ? br d而
? ? 糸 步 ? = 0 and b 步r 步 糸 (而 )] so that one has [by using b ’ (而 )br ’ (而 )br 而 ’ (而 ) = ?br ’ ’ ? ? (而 ) + 竹 ? ? 糸 (而 )b糸 (而 )考 r 竹? (而, 考 ) > 竹 , r ’ ? ? ? (而 ) = ?x 竹 步 s (而 ), ? 糸 步? 步 糸 (而 )]. ? ?糸 (而 ) = ?竹 ? 糸? (而 ) = 1 [b 竹 ’ (而 )br ’ (而 )br ’ (而 ) ? br ’ 2 r

(2.22)

(2.23)

10

Thus, the Dirac Hamiltonian becomes ? ? (而 )H ? ?糸 (而 )H ? ? (而 ) ? 1 竹 ? ?糸 (而 ), HD = 竹 2 (2.24)

and this shows that the gauge ?xings 汎 ?(而, 考 ) > 0 do not transform completely the constraints H? (而, 考 ) > 0 in their second class partners; still the following ten ?rst class constraints are left ? ? (而 ) = H ? d3 考 H? (而, 考) = p? s ?
’l ? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 + b? (T ?1 )r 旭l d3 考 J 而 l? ?2 + 汛u ’v ’ (T i j r ’ ’ d3 考考 r H糸 (而, 考 ) ? b糸 r ’ (而 ) ’ d3 考考 r H? (而, 考 ) =

(而, 考) > 0,

? ?糸 (而 ) = b? H r ’ (而 )

? 糸 糸 ? ? [br ’ (而 )b而 ? br ’ (而 )b而 ]

?糸 = Ss (而 ) ?

’ ? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 (而, 考 ) + J 而 ?2 + 汛u d3 考考 r ’v ’ (T i j ’l ’ J 而 (T ? 1 )s 旭l (而, 考) > 0. d3 考考 r

? ? 糸 糸 + [br ’ (而 )bs ’ (而 )] ’ (而 ) ? br ’ (而 )bs

(2.25)

?糸 Here Ss is the spin part of the Lorentz generators ?糸 ?糸 糸 糸 ? = x? Js s ps ? xs ps + Ss , ?糸 Ss = b? r ’ (而 )

’ 糸 糸 d3 考考 r 老 (而, 考 ) ? br ’ (而 )

’ ? d3 考考 r 老 (而, 考 ).

(2.26)

As shown in Ref. [10] instead of ?nding 耳竹 (而 ), T竹 (而 ), one can use the redundant variables ?糸 ( 而 b? ’ ), Ss (而 ), with the following Dirac brackets assuring the validity of the orthonormality A ’’ ?糸汐汕 A b 糸 糸 汐 ?汕 ? 汕 糸汐 糸 汕 ?汐 ? 汐 糸汕 condition 灰 ?糸 ? b? = 灰污 灰汛 灰 + 灰污 灰汛 灰 ? 灰污 灰汛 灰 ? 灰污 灰汛 灰 are the ’ = 0 [C污汛 ’ 灰 bB A structure constants of the Lorentz group]
?糸 老 ? 老糸 ? 老? 糸 { Ss , bA ’ ’ } = 灰 bA ’ ? 灰 bA ?糸汐汕 污汛 ?糸 汐汕 ? { Ss , Ss } = C污汛 Ss ,

(2.27)

? ? (而 ) > 0 has zero Dirac bracket with itself and with H ? ?糸 (而 ) > 0, these last so that, while H six constraints have the Dirac brackets ? ?糸 (而 ), H ? 汐汕 (而 )}? = C ?糸汐汕 H ? 污汛 (而 ) > 0. {H 污汛
糸 ? ?糸 {x? s (而 ), ps (而 )} = ?灰 , ?糸 ? 老糸 ? 老? 糸 { Ss (而 ), b老 ’ (而 ), ’ (而 )} = 灰 bA ’ (而 ) ? 灰 bA A
∩ ∩

(2.28)

? ? ?糸 i We have now only the variables: x? ’ , Ss , 汐 , 旭i with the following Dirac brackets: s , ps , bA

i 3 {汐i (而, 考 ), 旭j (而, 考 )}? = 汛j 汛 ( 考 ? 考 ).

?糸汐汕 污汛 ?糸 汐汕 { Ss ( 而 ) , Ss (而 )}? = C污汛 Ss ( 而 ) ,

(2.29)

? ? ? After the restriction to spacelike hyperplanes we have zr ’ (而, 考 ) > br ’ (而 ), so that z而 (而, 考 ) > (?) ’ r ’ r ’ 步? ?? ? ?糸 N[z ](f lat) (而, 考 )l? (而, 考) + N[r 步? ’糸 (而 )考 . ’ (而, 考 ) > x ’ (而 )考 = ?竹 (而 ) ? 竹 (而 )br s (而 ) + br z ](f lat) (而, 考 ) br

11

As said in the Introduction only now we get the coincidence of the two de?nitions of ?at lapse and shift functions (this point was missed in the older treatments of parametrized Minkowski theories):
’ ? ? (而 )l ? ? l ? 竹 ? ?糸 (而 )b糸 (而 )考 s N[z ](f lat) (而, 考) > N(f lat) (而, 考 ) = ?竹 = N (而, 考 ), s ’ ? ? 糸 s ’ ? ? N[z ](f lat)’ r (而, 考 ) > N(f lat)’ r (而, 考 ) = ?竹? (而 )br ’(而, 考 ). ’ (而 ) ? br ’ (而 )竹?糸 (而 )b (而 )考 = Nr s ’

(2.30)

Let us now restrict ourselves to con?gurations with ?p2 s > 0 and let us use the Wigner ? ? ? ?糸 boost L 糸 (ps , ps ) to boost to rest the variables bA ’ , Ss of the following non-Darboux basis
? ? ?糸 i x? ’ , Ss , 汐 , 旭i s , ps , bA

of the Dirac brackets {., .}? . The following new non-Darboux basis is obtained [? x? s is no o more a fourvector; we choose the sign 灰 = sign ps positive]
? x ?? s = xs +

= x? s ? = x? s ?

??B 1 A 老 (u(ps )) 糸老 ?糸 (u(ps ))灰AB Ss = 2 ?ps? ? 1 o? o糸 ps糸 ps 糸? ( S ? S [ps糸 Ss + ?p2 )] = s s s o 2 ?p2 ?p2 s s (p s + 灰 p s ) ?p2 s
? ?o ?A [灰A ( Ss

1

?

?Ar pr S s s po s + ?p2 s

)+

2 ?o p? s + 2 ?ps 灰 o ?p2 s (p s

+

?p2 s)

?r r ?o S s ps ],

? p? s = ps ,

汐i = 汐i , 旭i = 旭i ,
? A bA ’, r ’ = ?? (u(ps ))br B B ??糸 = S ?糸 ? 1 ?A (u(ps ))灰AB ( ??考 (u(ps )) p糸 ? ??考 (u(ps )) p? )S 老考 = S s s s s s 老 2 ?ps? ?ps糸 1 汕? 糸 汕糸 ? o? 糸 o糸 ? ?糸 [ps汕 (Ss p s ? Ss ps ) + ?p2 = Ss + s (Ss ps ? Ss ps )], 2 o 2 ?ps (ps + ?ps ) ?糸 糸 ? ??糸 Js =x ?? ?糸 s ps ? x s p s + Ss .

(2.31)

We have
糸 ? {x ?? s , ps } = 0 , s s r 汛 is (pr ’ ? ps bA ’) s bA oi r ? ? , {Ss , bA ’} = po ?p2 s s + ?ij , br’}? = (汛 ir 汛 js ? 汛 is 汛 jr )bs’, {S s A A

??糸 , S ?汐汕 }? = C ?糸汐汕 S ?污汛 , {S s s s 污汛 12

(2.32)

and we can de?ne ?AB = ?A (u(ps ))?B (u(ps ))S ?糸 > [bA (而 )bB ? bB (而 )bA ] S s ? 糸 s r ’ 而 r ’ 而
’ ? 1 )u ’ i 旭 (T ? 1 )v ’j 旭 (而, 考 ) ? J 而 ?2 + 汛u d3 考 考 r ’v ’ (T i j B B A ? [bA r ’ (而 )bs ’ (而 ) ? br ’ (而 )bs ’ (而 )] ’l ’ J 而 (T ? 1 )s 旭l (而, 考 ). d3 考 考 r

(2.33)

Let us now add six more gauge-?xings by selecting the special family of spacelike hyper2 planes 曳而 W orthogonal to p? s (this is possible for ?ps > 0), which can be called the &Wigner foliation* of Minkowski spacetime. This can be done by requiring (only six conditions are independent)
? ? ? TA ’ (而 ) = bA ’ (而 ) ? ?A=A ’ (u(ps )) > 0

?

? A A bA ’ (而 ) = ?? (u(ps ))bA ’. ’ (而 ) > 灰A A

(2.34)

? Now the inverse tetrad b? ’ is equal to the polarization vectors ?A (u(ps )) [see Appendix A C] and the indices &’ r* are forced to coincide with the Wigner spin-1 indices &r*, while o ?=而 is a Lorentz-scalar index. One has

?AB > (灰 A 灰 B ? 灰 B 灰 A )S ?而 r ? S s r 而 r 而 s A B B A ?rs ? (灰r 灰s ? 灰r 灰s )Ss , ?rs > S s d3 考 J 而 [考 r (T ?1 )sl 旭l ? 考 s (T ?1 )rl 旭l ] (而, 考 ), d3 考 J 而 考 r ?2 + 汛uv (T ?1 )ui 旭i (T ?1 )vj 旭j (而, 考 ). (2.35)

而r r而 ?s ?s S > ?S =?

?AB with S ??糸 yields The comparison of S s s ?uv = 汛 ur 汛 vt S ?rt S s s vr ?rt 汛 S s ps t ?ov = ? S . s 2 p0 + ?p s s

(2.36)

? The time constancy of TA ’ > 0 with respect to the Dirac Hamiltonian of Eq.(2.24) gives

d ? ? ? ? [b’ (而 ) ? ?? ’ (而 ) ? ?r (u(ps )), HD } = r (u(ps ))] = {br d而 r 1 ? 汐汕 ? ? ? ?汐 (而 ){br = 竹 ’汐 (而 ) > 0 ’ (而 ), Ss汐汕 (而 )} = 竹 (而 )br 2 ? ?糸 (而 ) > 0, ?竹

(2.37)

? ?糸 (而 ) > 0 so that the independent gauge-?xings contained in Eqs.(2.34) and the constraints H form six pairs of second class constraints. ?? Besides Eqs.(2.19), now we have [remember that x 步? s (而 ) = ?竹 (而 )]

13

? l? = b? 而 = u (p s ), ? z而 (而 ) = x 步? s (而 ) = 糸 g (而 )u ? (p s ) ? x 步 s糸 (而 )?? r (u(ps ))?r (u(ps )),

N (而 ) = x 步2 s,

g (而 ) = [x 步 s? (而 )u? (ps ))],
3



污 = 1,

g而 而 = grs = ?? grs = ??汛rs , ? g而 r = ??x 步 s? ?r (u(ps )) = ??汛rs N s , N r = 汛 ru x 步 s? ?? u (u(ps )), ? ? Nr 1 g而 r = ? x 步 s? 汛 ru ?? ( u ( p )) = ? ? , g而 而 = = 2 , s r g N g N2 N rN s x 步 s? ?? 步 s糸 ?糸 u (u(ps ))x v (u(ps )) rs ) = ? ? ( 汛 ? ). g rs = ??(汛 rs ? 汛 ru 汛 sv [x 步 s ﹞ u(ps )]2 N2

(2.38)

On the hyperplane 曳而 W all the degrees of freedom z ? (而, 考 ) are reduced to the four degrees ? ?? ? of freedom x ?? s (而 ), which replace xs . The Dirac Hamiltonian is now HD = 竹 (而 )H? (而 ) with ? ? (而 ) = p ? ? H s ? d3 考 J 而 u? (ps ) ?2 + 汛uv (T ?1 )ui 旭i (T ?1 )vj 旭j ? (2.39)

?1 rl ? ?? ) 旭l (而, 考 ) > 0. r (u(ps )) ?(T

To ?nd the new Dirac brackets, one needs to evaluate the matrix of the old Dirac brackets of the second class constraints (without extracting the independent ones)
? ? 汐汕 , T 考 {H ’} = B ? ? 考汕 汐 考汐 汕 ? = 汛BB ’ [灰 ?B (u(ps )) ? 灰 ?B (u(ps ))] ? ? ? C=? 老 ? 污汛 ? 老 ? 考 ? {TA , H } = { T , T } = 0 ? ? ’ ’ ’ B A 污 老汛 老污 汛 . = 汛AA ’ [灰 ?A (u(ps )) ? 灰 ?A (u(ps ))]

?

? 汐汕 , H ? 污汛 }? > 0 {H

?

(2.40)

Since the constraints are redundant, this matrix has the following left and right null a汐汕 = a汕汐 0 eigenvectors: [a汐汕 arbitrary], . Therefore, one has to ?nd a left B 0 ?考 (u(ps )) ? , CC ? = CC ? = D , such that C ? and D have the same left and right and right quasi-inverse C null eigenvectors. One ?nds ?= C ? = CC ?=D= CC 0污汛?糸 1 B [灰 ? (u(ps )) ? 灰考? ?B 糸 (u(ps ))] 4 考糸 ?
1 汐 汕 (灰 汐 灰 汕 ? 灰糸 灰? ) 2 ? 糸 老 1 0A?糸 (灰 老 灰 D 2 而 A 1 [灰 ?D (u(ps )) ? 灰汛而 ?D 污 (u(ps ))] 4 污而 汛 BD 0考而

0汐汕D 而 ? ?D老 (u(ps ))?A而 (u(ps ))

(2.41)

and the new Dirac brackets are 1 ? 污汛 }? [灰污而 ?D (u(ps )) ? 灰汛而 ?D (u(ps ))]{T 而 , B }? + {A, B }?? = {A, B }? ? [{A, H 汛 污 D 4 考 ? B ? ? ?糸 + {A, TB } [灰考糸 ?B ? (u(ps )) ? 灰考? ?糸 (u(ps ))]{H , B } ].

(2.42)

? 汐汕 , B }?? = 0 is immediate, we must use the relation b ’ T ? ?D老 = ?T 老 While the check of {H ’ A? D A ? ? 老 ?? [at this level we have TA = T ] to check { T , B } = 0. ’ A A 14

? i Then, we ?nd the following brackets for the remaining variables x ?? s , ps , 汐 , 旭i 糸 ?? {x ?? = ?灰 ?糸 , s , ps } ∩ ∩ i 3 {汐i (而, 考 ), 旭j (而, 考 )}?? = 汛j 汛 ( 考 ? 考 ),

(2.43)

and the following form of the generators of the ※external§ Poincar? e group p? s, ? 糸 ?糸 ??糸 ?糸 Js =x ?? s p s + Ss , s ps ? x ir ?rs s ?oi = ? 汛 Ss ps , S s po ?p2 s s + ij ir js rs ? =汛 汛 S ? . S
s s

(2.44)

? ? (而 ) > 0, {H ? ?, H ? 糸 }?? = 0, of Let us come back to the four ?rst class constraints H Eq.(2.25). They can be rewritten in the following form [from Eqs.(1.9), (2.6) we have J 而 = ?det (?r 汐i ), (T ?1)ri = 汛 rs ?s 汐i /det (?u 汐k )] ? ? (而 ) = ?s ? Msys > 0, H (而 ) = u ? (p s )H Msys = = =?
def

d3 考 M(而, 考) =

d3 考 J 而 ?2 + 汛uv (T ?1 )ui 旭i (T ?1 )vj 旭j (而, 考 ) = d3 考 det (?r 汐k ) ?2 + 汛 uv ?u 汐i ?v 汐j 旭i 旭j (而, 考), [det (?r 汐k )]2

Hp (而 ) = Psys = =

d3 考 Mr (而, 考 ) = d3 考 ? 汛 rs ?s 汐i 旭i (而, 考) > 0, (2.45)

d3 考? J 而 (T ?1 )rl 旭l (而, 考) = ?

where Msys is the invariant mass of the ?uid. The ?rst one gives the mass spectrum of the isolated system, while the other three say that the total 3-momentum of the N particles on the hyperplane 曳而 W vanishes. ? ? 2 There is no more a restriction on p? s in this special gauge, because u (ps ) = ps / ?ps gives the orientation of the Wigner hyperplanes containing the isolated system with respect to an arbitrary given external observer. Now the lapse and shift functions are N = N[z ](f lat) = N(f lat) = ?竹(而 ) = x 步? s (而 )u ? (p s ), Nr = N[z ](f lat)r = N(f lat)r = ?竹r (而 ) = ?x 步? s (而 )?r? (u(ps )), so that the velocity of the origin of the coordinates on the Wigner hyperplane is
? ? x 步? s (而 ) = ?[?竹(而 )u (ps ) + 竹r (而 )?r (u(ps )),

(2.46)

[u2 (ps ) = ?, ?2 r (u(ps )) = ??].

(2.47)

The Dirac Hamiltonian is now HD = 竹(而 )H(而 ) ? 竹(而 ) ﹞ Hp (而 ), 15 (2.48)

?? 步 s = {x ?? = ?竹(而 )u? (ps ). Therefore, while the old x? 步? and we have x ? s , HD } s had a velocity x s not parallel to the normal l? = u? (ps ) to the hyperplane as shown by Eqs.(2.47), the new ? 步? x ?? ? ?s = l ﹞ xs is s has x s l and no classical zitterbewegung. Moreover, we have that Ts = l ﹞ x the Lorentz-invariant rest frame time. ? p2 The canonical variables x ?? s , Ts = s , ps , may be replaced by the canonical pairs ?s = ps o ps ﹞ x ?s /?s [to be gauge ?xed with Ts ? 而 > 0]; ks = ps /?s = u(ps ), zs = ?s (x ?s ? p ?s ) √ ?s qs . ox s One obtains in this way a new kind of instant form of the dynamics, the ※Wignercovariant 1-time rest-frame instant form§ with a universal breaking of Lorentz covariance. It is the special relativistic generalization of the non-relativistic separation of the center of P2 + Hrel ]. The role of the ※external§ center of mass mass from the relative motion [H = 2 M ? is taken by the Wigner hyperplane, identi?ed by the point x ?? s (而 ) and by its normal ps . The invariant mass Msys of the system replaces the non-relativistic Hamiltonian Hrel for the relative degrees of freedom, after the addition of the gauge-?xing Ts ? 而 > 0 [identifying the time parameter 而 , labelling the leaves of the foliation, with the Lorentz scalar time of the ※external§ center of mass in the rest frame, Ts = ps ﹞ x ?s /Msys and implying 竹(而 ) = ??]. After this gauge ?xing the Dirac Hamiltonian would be pure gauge: HD = ?竹(而 ) ﹞ Hp (而 ). However, if we wish to reintroduce the evolution in the time 而 √ Ts in this frozen phase space we must use the Hamiltonian [in it the time evolution is generated by Msys : it is like in the frozen Hamilton-Jacobi theory, in which the evolution can be reintroduced by using the energy generator of the Poincar? e group as Hamiltonian]

?

The Hamilton equations for 汐i(而, 考 ) in the Wigner covariant rest-frame instant form are equivalent to the hydrodynamical Euler equations: ?而 汐i (而, 考 ) = {汐i(而, 考 ), HD } = 汛 uv ?u 汐i ?v 汐j 旭j =? (而, 考 ) + ?u 汐m ?v 汐n det (?r 汐k ) ?2 + 汛 uv [det 旭 旭 (?r 汐k )]2 m n + 竹r (而 )?r 汐i (而 ), ?而 旭i (而, 考 ) = {旭i (而, 考 ), Hd} = (ijk cyclic) ?u 汐m ?v 汐n ? suv j k 2 + 汛 uv ? ? 汐 ? 汐 旭m 旭n + ? = u v ?考 s [det (?r 汐k )]2 ?u 汐n 汛uv ?v 汐m slt m u ? ?ipq ?l 汐p ?t 汐q ) det 旭 旭 (汛i 汛s ? det (?r 汐l ) (?r 汐l ) m n + (而, 考) + ?u 汐m ?v 汐n 旭 旭 ?2 + 汛 uv [det (?r 汐k )]2 m n ?r 汐m ? suv j k ? ? 汐 ? 汐 旭m + u v ?考 s det (?t 汐l ) ?r 汐m slt m + (汛i 汛rs ? ? ?ipq ?l 汐p ?t 汐q )旭m (而, 考 ). det (?t 汐l ) + 竹 r (而 )
?

HD = Msys ? 竹(而 ) ﹞ Hp (而 ).

(2.49)

(2.50)

? In this special gauge we have b? A √ L A (ps , ps ) (the standard Wigner boost for timelike ?糸 ?糸 r Poincar? e orbits), Ss √ Ssys [Ssys = ?ruv d3 考 考 u J 而 (T ?1 )vs 旭s (而, 考 )], and the only remaining canonical variables are the non-covariant Newton-Wigner-like canonical ※external§

16

3-center-of-mass coordinate zs (living on the Wigner hyperplanes) and ks . Now 3 degrees of freedom of the isolated system [an ※internal§ center-of-mass 3-variable qsys de?ned inside the Wigner hyperplane and conjugate to Psys ] become gauge variables [the natural gauge ?xing to the rest-frame condition Psys > 0 is Xsys > 0, implying 竹r (而 ) = 0, so that it coincides ? with the origin x? ?? s (而 ) = z (而, 考 = 0) of the Wigner hyperplane]. The variable x s is playing the role of a kinematical ※external§ center of mass for the isolated system and may be interpreted as a decoupled observer with his parametrized clock (point particle clock). All the ?elds living on the Wigner hyperplane are now either Lorentz scalar or with their 3-indices transformaing under Wigner rotations (induced by Lorentz transformations in Minkowski spacetime) as any Wigner spin 1 index.

17

III. EXTERNAL AND INTERNAL CANONICAL CENTER OF MASS, MOLLER*S CENTER OF ENERGY AND FOKKER-PRYCE CENTER OF INERTIA

Let us now consider the problem of the de?nition of the relativistic center of mass of a perfect ?uid con?guration, using the dust as an example. Let us remark that in the approach leading to the rest-frame instant form of dynamics on Wigner*s hyperplanes there is a splitting of this concept in an ※external§ and an ※internal§ one. One can either look at the isolated system from an arbitrary Lorentz frame or put himself inside the Wigner hyperplane. From outside one ?nds after the canonical reduction to Wigner hyperplane that there is an origin x? s (而 ) for these hyperplanes (a covariant non-canonical centroid) and a noncovariant canonical coordinate x ?? s (而 ) describing an ※external§ decoupled point particle observer with a clock measuring the rest-frame time Ts . Associated with them there is the ※external§ realization (2.44) of the Poincar? e group. Instead, all the degrees of freedom of the isolated system (here the perfect ?uid con?guration) are described by canonical variables on the Wigner hyperplane restricted by the rest-frame condition Psys > 0, implying that an ※internal§ collective variable qsys is a gauge variable and that only relative variables are physical degrees of freedom (a form of weak Mach principle). Inside the Wigner hyperplane at 而 = 0 there is another realization of the Poincar? e group, the ※internal§ Poincar? e group. Its generators are built by using the invariant mass Msys and the 3-momentum Psys , determined by the constraints (2.25), as the generators of ?AB as the generator of the Lorentz subalgebra the translations and by using the spin tensor S s P 而 = Msys = P r = Psys = ? d3 考 J 而 ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考 ), d3 考 J 而 (T ?1 )ri 旭i (而, 考 ) > 0, d3 考考 r J 而 ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考), d3 考考 u J 而 (T ?1 )vi 旭i (而, 考 ). (3.1)

?而 r √ Kr = J而r = S s

1 r uv ?s J r = Ssys = ?ruv S √ ?ruv 2

By using the methods of Ref. [13] (where there is a complete discussion of many de?nitions of relativistic center-of-mass-like variables) we can build the three ※internal§ (that is inside the Wigner hyperplane) Wigner 3-vectors corresponding to the 3-vectors *canonical center of mass* qsys , *Moller center of energy* rsys and *Fokker-Pryce center of inertia* ysys [the analogous concepts for the Klein-Gordon ?eld are in Ref. [14] (based on Refs. [15]), while for the relativistic N-body problem see Ref. [16] and for the system of N charged scalar particles plus the electromagnetic ?eld Ref. [17]]. The non-canonical ※internal§ M?ller 3-center of energy and the associated spin 3-vector are rsys = ? K 1 = ? P而 2P 而 d3 考 考 J 而 ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考 ),

18

?sys = J ? rsys ℅ P ,
r {rsys , P s } = 汛 rs , r s {rsys , rsys }=? r {rsys , P而} =

1 rsu u ? ?sys , (P 而 )2 1 s rsu {?r (?u (?sys ﹞ P ) P u ), sys , ?sys } = ? sys ? (P 而 )2
r

Pr , P而

而 {?r sys , P } = 0.

(3.2)

r s r The canonical ※internal§ 3-center of mass qsys [{qsys , qsys } = 0, {qsys , P s } = 汛 rs , s rsu u {J , qsys } = ? qsys ] is

qsys = rsys ? =? + > rsys

K

(P 而 )2 ? P 2 (P 而 + + (K ﹞ P ) P P > 0;

J ℅ ?sys

= (P 而 )2 ? P 2 ) J ℅P + (P 而 )2 ? P 2 )

(P 而 )2 ? P 2

(P 而 )2 ? P 2 (P 而 + (P 而 )2 ? P 2

, {qsys , P 而 } = P > 0, P而

P 而 (P 而 )2 ? P 2 P 而 + f or

Sq sys = J ? qsys ℅ P = = > Ssys , P 而J + (P 而 )2 ? P 2 f or

(P 而 )2 ? P 2

K℅P

?

(P 而 )2 ? P 2 P 而 +

(J ﹞ P ) P

(P 而 )2 ? P 2



P > 0,

r Ssys = ?ruv

d3 考 考 u J 而 (T ?1 )vs 旭s (而, 考 ),
r s rsu u { Sq Sq sys . sys , Sq sys } = ?

{Sq sys , P } = {Sq sys , qsys } = 0, Ssys ℅ P

(3.3)

The ※internal§ non-canonical Fokker-Pryce 3-center of inertia* ysys is ysys = qsys + = rsys + Ssys ℅ P ,

(P 而 )2 ? P 2 (P 而 + P 而 (P 而 = Ssys ℅ P + (P 而 )2 1

(P 而 )2 ? P 2 ) = P而 ?
rsu u Ssys

P 而 (P 而 )2 ? P 2 , , (3.4) ? P2 (P 而 )2 ? P 2 )

qsys = rsys +

P 而 rsys + +

(P 而 )2 ? P 2ysys (P 而 )2

?

P 2)

r s {ysys , ysys }

+

P 而 (P 而 )2 ? P 2

(P 而 )2 ? P 2 (P 而 +

(Ssys ﹞ P ) P u

P > 0 ? qsys > rsys > ysys .

The Wigner 3-vector qsys is therefore the canonical 3-center of mass of the perfect ?uid r ? con?guration [since qsys > rsys , it also describe that point z ? (而, qsys ) = x? s (而 ) + qsys ?r (u(ps )) where the energy of the con?guration is concentrated]. 19

s 耳 o i oi i o ir 耳 s ?i Js =x ?o =x ?o ?i [for x ?o s ps ? p o s ps ? x s ps ? x s ps + 汛 s = 0 this is the Newton-Wigner po s +? s s +? s ?糸 decomposition of Js ] we can build three ※external§ collective 3-positions (all located on the Wigner hyperplane): i) the ※external canonical 3-center of mass Qs connected with the ※external§ canonical non-covariant center of mass x ?? s ; ii) the ※external§ M?ller 3-center of energy Rs connected with the ※external§ non-canonical and non-covariant M?ller center of ? energy Rs ; iii) the ※external§ Fokker-Pryce 3-center of inertia connected with the ※external§ covariant non-canonical Fokker-Price center of inertia Ys? (when there are the gauge ?xings 考sys > 0 it coincides with the origin x? s ). It turns out that the Wigner hyperplane is the natural setting for the study of the Dixon multipoles of extended relativistic systems [20] (see next Section) and for de?ning the canonical relative variables with respect to the center of mass. The three ※external§ 3-variables, the canonical Qs , the M?ller Rs and the Fokker-Pryce Ys built by using the rest-frame ※external§ realization of the Poincar? e algebra are

There should exist a canonical transformation from the canonical basis 汐i (而, 考 ), 旭i (而, 考), i i to a new basis qsys , P = P耳 , 汐rel (而, 考 ), 旭rel i (而, 考 ) containing relative variables 汐rel (而, 考), 旭rel i (而, 考 ) with respect to the true center of mass of the perfect ?uid con?guration. To identify this ?nal canonical basis one shall need the methods of Ref. [16]. The gauge ?xing qsys > 0 [it implies 竹(而 ) = 0] forces all three internal center-ofmass variables to coincide with the origin x? s of the Wigner hyperplane. We shall denote qsys )? ? ? x( ( 而 ) = x (0) + 而 u ( p ) the origin in this gauge (it is a special centroid among the many s s s ? possible ones; xs (0) is arbitrary). As we shall see in the next Section, by adding the gauge ?xings Xsys = qsys > 0 one can show that the origin x? s (而 ) becomes simultaneously the Dixon center of mass of an extended object and both the Pirani and Tulczyjew centroids (see Ref. [18] for a review of these concepts in relation with the Papapetrou-Dixon-Souriau pole-dipole approximation of qsys )? an extended body). The worldline x( is the unique center-of-mass worldline of special s relativity in the sense of Refs. [19]. With similar methods from the rest-frame instant form ※external§ realization of the ij j i ir js rs i Poincar? e algebra of Eq. (2.44) with the generators p? ?i ?j s , Js = x s ps ? x s p s + 汛 汛 S 耳 , Ks =
汛ir S rs ps (S ℅p )r

Rs = ?

ps o Ssys ℅ ps 1 K s = (x ?s ? o x ?s ) ? o o , o ps ps ps (ps + ?s ) ps o zs po Ssys ℅ ps s Rs + ?s Ys x ? = = R + , = s s o o o ps ?s ps (ps + ?s ) po s + ?s

Qs = x ?s ? Ys = Qs +

Ssys ℅ ps Ssys ℅ ps , = R + s ?s (po po s + ?s ) s ?s 1 r s {Rs , Rs } = ? o 2 ?rsu ?u ?s = Js ? Rs ℅ ps , s, (p s ) (Ssys ﹞ ps ) pu 1 rsu u s S + ? , sys o+? ) ?s po ? ( p s s s s

{Ysr , Yss } =

ps ﹞ Qs = ps ﹞ Rs = ps ﹞ Ys = ks ﹞ zs ,

20

ps = 0 ? Qs = Ys = Rs ,

(3.5)
??

with the same velocity and coinciding in the Lorentz rest frame where ps = ?s (1; 0) In Ref. [13] in a one-time framework without constraints and at a ?xed time, it is shown r that the 3-vector Ys [but not Qs and Rs ] satis?es the condition {Ks , Yss } = Ysr {Yss , po s } for ? being the space component of a 4-vector Ys . In the enlarged canonical treatment including time variables, it is not clear which are the time components to be added to Qs , Rs , Ys , to ? ? rebuild 4-dimesnional quantities x ?? s , Rs , Ys , in an arbitrary Lorentz frame 忙, in which the o origin of the Wigner hyperplane is the 4-vector x? s = (xs ; xs ). We have x ?? xo ?s (而 )) = x? s (而 ) = (? s (而 ); x s ? x ?o s =
2 (T + 1 + ks s ? 1 糸? o? o糸 ps糸 ps ), p S + ? ( S ? S s糸 s s s s ?s (po ?2 s + ?s ) s

p? s,

ks ﹞ zs 2 (T + k ﹞ q ) = x0 , 2 ) = 1 + ks po s s s s s = ?s 1 + ks , ?s ks ﹞ zs zs + (Ts + )ks = qs + (Ts + ks ﹞ qs )ks , ps = ?s ks . (3.6) x ?s = ?s ?s

for the non-covariant (frame-dependent) canonical center of mass and its conjugate momentum. Each Wigner hyperplane intersects the worldline of the arbitrary origin 4-vector x? s (而 ) = ? ? ? ? ? z (而, 0) in 考 = 0, the pseudo worldline of x ?s (而 ) = z (而, 考 ) in some 考 and the worldline of ? ? the Fokker-Pryce 4-vector Ys (而 ) = z (而, 考Y ) in some 考Y [on this worldline one can put the ※internal center of mass§ with the gauge ?xing q耳 > 0 (q耳 > r耳 > y耳 due to P耳 > 0)]; one ? also has Rs = z ? (而, 考R ). Since we have Ts = u(ps ) ﹞ xs = u(ps ) ﹞ x ?s √ 而 on the Wigner ? ? hyperplane labelled by 而 , we require that also Ys , Rs have time components such that they too satisfy u(ps ) ﹞ Ys = u(ps ) ﹞ Rs = Ts √ 而 . Therefore, it is reasonable to assume that x ?? s, ? ? Ys and Rs satisfy the following equations consistently with Eqs.(3.2), (3.3) when Ts √ 而 and qsys > 0 x ?? xo ?s ) = (? xo s = (? s; x s ; Qs + = (? xo s; Ys? ps o x ?s ) = po s

zs ks ﹞ zs qsys )? + (Ts + )ks ) = x( + ?? 考u, s u (u(ps ))? ?s ?s = (? xo s ; Ys ) = = (? xo s; Ssys ℅ ps ks ﹞ zs 1 [zs + ] + ( T + )k s ) = s ?s ?s [1 + uo (ps )] ?s

? =x ?? s + 灰r

(Ssys ℅ ps )r = ?s [1 + uo(ps )] qsys )? u = x( + ?? s u (u(ps ))考Y , = (? xo s; 1 ks ﹞ zs Ssys ℅ ps [zs ? )k s ) = ] + (Ts + o o ?s ?s u (ps )[1 + u (ps )] ?s
? 灰r

? Rs = (? xo s ; Rs ) =

=

x ?? s

?

(Ssys ℅ ps )r = ?s uo (ps )[1 + uo(ps )] 21

qsys )? u = x( + ?? s u (u(ps ))考R , (qsys ) Ts = u(ps ) ﹞ xs = u (p s ) ﹞ x ?s = u(ps ) ﹞ Ys = u(ps ) ﹞ Rs , 糸? o? ?r? (u(ps ))[u糸 (ps )Ss + Ss ] = 考 ? = ? = o [1 + u (ps )] rs s rs s Ssys ps Ssys u (p s ) 而r r = ?Ssys + = ? r + > s 耳 o ?s [1 + u (ps )] 1 + u o (p s ) rs s rs s Ssys u (p s ) Ssys u (p s ) r > ?s qsys + > , o 1 + u (p s ) 1 + u o (p s ) r qsys )? ?r? (u(ps ))[x( s

x ?? s]

r qsys )? 考Y = ?r? (u(ps ))[x( ? Ys? ] = 考 ? r ? ?ru (u(ps )) s

(Ssys ℅ ps )u = ?s [1 + uo (ps )]

=考 ?r +

rs s Ssys u (p s ) r r = ?s rsys > ?s qsys > 0, 1 + u o (p s )

r qsys )? ? 考R = ?r? (u(ps ))[x( ? Rs ]=考 ? r + ?ru (u(ps )) s

(Ssys ℅ ps )u = ?s uo (ps )[1 + uo(ps )] rs s S rs us (ps ) [1 ? uo (ps )]Ssys u (p s ) r =考 ? r ? o sys = ? r + > s sys o o o u (ps )[1 + u (ps )] u (ps )[1 + u (ps )] rs s [1 ? uo(ps )]Ssys u (p s ) > , o o u (ps )[1 + u (ps )] f or qsys > 0, (3.7)

qsys )? ? x( (而 ) = Ys? , s

namely in the gauge qsys > 0 the external Fokker-Pryce non-canonical center of inertia qsys )? coincides with the origin x( (而 ) carrying the ※internal§ center of mass (coinciding with s the ※internal§ M“ oller center of energy and with the ※internal§ Fokker-Pryce center of inertia) and also being the Pirani centroid and the Tulczyjew centroid. Therefore, if we would ?nd the center-of-mass canonical basis, then, in the gauge qsys > 0 and Ts > 而 , the perfect ?uid con?gurations would have the four-momentum density peaked qsys )? i on the worldline x( (Ts ); the canonical variables 汐rel (而, 考), 旭rel i (而, 考 ) would characterize s the relative motions with respect to the ※monopole§ con?guration describing the center of mass of the ?uid con?guration. The ※monopole§ con?gurations would be identi?ed by the vanishing of the relative variables. Remember that the canonical center of mass lies in between the Moller center of energy and the Fokker-Pryce center of inertia and that the non-covariance region around the FokkerPryce 4-vector extends to a worldtube with radius (the Moller radius) |Ssys |/P 而 .

22

IV. DIXON*S MULTIPOLES IN MINKOWSKI SPACETIME.

Let us now look at other properties of a perfect ?uid con?guration on the Wigner hyperplanes, always using the dust as an explicit example. To identify which kind of collective variables describe the center of mass of a ?uid con?guration let us consider it as a relativistic extended body and let us study its energy-momentum tensor and its Dixon multipoles [20] in Minkowski spacetime. The Euler-Lagrange equations from the action (1.10) [(2.1) for the dust] are ?L ?L ﹟ A ’B ’ ? 糸 ? ?A [汐]zB ’ ’ [ gT ’ ](而, 考 ) = 0, ? (而, 考 ) = 灰?糸 ?A ? ?z ?zA ’ ?L ?L ? (而, 考 ) = 0, ? ?A ’ ?耳 ??A ’耳

(4.1)

where we introduced the energy-momentum tensor [with a di?erent sign with respect to the standard convention to conform with Ref. [1]] 2 汛S ’’ (而, 考 ) = T AB (而, 考 )[汐] = ﹟ g 汛gA ’B ’ = =
dust

? 老g AB + n
’ ’

’’

J AJ B ?老 ’’ |s (g AB ? ’ D ’ ) (而, 考 ) = C ?n gC ’D ’J J
’’ ’ ’





? ?老U A U B + p(g AB ? ?U A U B ) (而, 考 )
’ A ’ B ’ ’ ? 1 )u ’ i (T ? 1 )v ’j 旭 旭 . ?2 + 3 gu ’v ’ (T i j

J AJ B = ???nU U = ?? 2 ﹟ 而 N 污J

(4.2)

﹟ ? When ?A ’ [ gzB ’ ] = 0, as it happens on the Wigner hyperplanes in the gauge Ts ? 而 > 0, ’B ’ ? ’’ A = 0. 竹(而 ) = 0, we get the conservation of the energy-momentum tensor T AB , i.e. ?A ’T Otherwise, there is compensation coming from the dynamics of the surface. As shown in Eq.(A9) the conserved, manifestly Lorentz covariant energy-momentum ?老 |s ? 老)] is tensor of the perfect ?uid with equation of state 老 = 老(n, s) [so that p = n ?n T ?糸 (x)[? 汐] = = =
dust

? ?老 4 g ?糸 + n

J ?J 糸 ?老 4 ?糸 |s ( g ? 4 ) (x) = ?n g汐汕 J 汐 J 汕

? ?老U ? U 糸 + p(4 g ?糸 ? ?U ? U 糸 ) (x) = ? ?(老 + p)U ? U 糸 + p 4 g ?糸 (x)

= ??? n U ? U 糸 (x),


? A nU ? = J ? = ???糸老考 ?糸 汐 ? 1 ?老 汐 ? 2 ?考 汐 ? 3 = nzA ’U = ? r ? A ? 而 ’ = zA ’J . ’ J = z而 J + zr ’

(4.3)

Therefore, in 曳而 -adapted coordinates on each 曳而 we get

23

B A (而, 考 )T ?糸 (x = z (而, 考 ))[? 汐] = (而, 考 )z糸 T AB (而, 考 )[汐] = z? B A (而, 考 )T ?糸 (而, 考 )[汐 = 汐 ? ? z ], (而, 考 )z糸 = z? ’ ’

’’





(4.4)

On Wigner hyperplanes, where Eqs.(2.38) hold and where we have
? u z ? (而, 考 ) = x? s (而 ) + ?u (u(ps ))考 , 3 4 ?糸



= ? u ? (p s )u 糸 (p s ) ?

糸 ?? r (u(ps ))?r (u(ps )) , r =1

N =x 步 s ﹞ u (p s ), Yr = n=
dust

N r = 汛 ru x 步 s? ?? u (u(ps )),

而 Jr + N rJ而 J r + 汛 ru x 步 s? ?? u (u(ps ))J = , N x 步 s ﹞ u (p s )

=

?gAB J A J B = (J 而 )2 ? 汛rs Y r Y s N ?J 而 , ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j

(4.5)

’ = A] we get [A
汕 u T ?糸 [x汕 s (而 ) + ?u (u(ps ))考 ][汐] = 而 糸 s 糸 而 ? r ? x 步 s (而 ) + 汛B ?s (u(ps ))]T AB (而, 考) = x 步 s (而 ) + 汛A ?r (u(ps ))][汛B = [汛A 而而 ? 糸 rs =x 步? 步糸 s (而 )x s (而 )T (而, 考 ) + ?r (u(ps ))?s (u(ps ))T (而, 考 ) + 糸 ? r而 + [x 步? 步糸 s (而 )?r (u(ps )) + x s (而 )?r (u(ps ))]T (而, 考 ),

?老 + [x 步 s ﹞ u(ps )]2 ? (J 而 )2 ?老 ? ) (而, 考 ) + n |s ( ?n [x 步 s ﹞ u(ps )]2 [x 步 s ﹞ u(ps )]2 [(J 而 )2 ? 汛uv Y u Y v ] J而 dust = ?? ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考 ), [x 步 s ﹞ u(ps )]2 汛 ru x 步 s? ?? u (u(ps )) T r而 (而, 考 ) = T 而 r (而, 考 ) = ? ?老 ? [x 步 s ﹞ u(ps )]2 ?老 汛 ru x 步 s? ?? u (u(ps )) ? n | s (? + ?n [x 步 s ﹞ u(ps )]2 J rJ 而 ) (而, 考) + [x 步 s ﹞ u(ps )]2 [(J 而 )2 ? 汛uv Y u Y v ] Jr dust ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考 ), = ?? [x 步 s ﹞ u(ps )]2 步 s? ?? 步 s糸 ?糸 u (u(ps ))x v (u(ps )) rs rs ru sv x T (而, 考 ) = ?老(汛 ? 汛 汛 )? [x 步 s ﹞ u(ps )]2 T 而 而 (而 考 ) = ? 24

x 步 s? ?? 步 s糸 ?糸 ?老 u (u(ps ))x v (u(ps )) |s ?(汛 rs ? 汛 ru 汛 sv )+ ?n [x 步 s ﹞ u(ps )]2 JrJs + (而, 考 ) [x 步 s ﹞ u(ps )]2 [x 步 s ﹞ u(ps )]2 [(J 而 )2 ? 汛uv Y u Y v ] J rJ s dust ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (而, 考 ). = ?? [x 步 s ﹞ u(ps )]2 J 而 ?n Since we have
? ? 糸 ? 糸 x 步? 步 s糸 (而 ) = s (而 ) = ?竹 (而 ) = ?[u (ps )u (ps ) ? ?r (u(ps ))?r (u(ps ))]x 2 x 步2 s (而 ) = 竹 (而 ) ? 竹 (而 ) > 0 , ?竹(而 )u? (ps ) + 竹r (而 )?? x 步 ? (而 ) r (u(ps )) ? =? , Us (而 ) = s 2 (而 ) ? 竹 2 (而 ) x 步2 ( 而 ) 竹 s

(4.6)

= ? ? u? (ps )竹(而 ) + ?? r (u(ps ))竹r (而 ) ,

(4.7)

the timelike worldline described by the (i.e. gauge dependent): x? s (而 ) may be ready said the real ※external§ center of ? 1 糸? o? o糸 ps糸 ps x? ) s (Ts ) ? ?s (po +?s ) ps糸 Ss + ?s (Ss + Ss ?2
s s

origin of the Wigner hyperplane is arbitrary any covariant non-canonical centroid. As almass is the canonical non-covariant x ?? s (Ts ) = : it describes a decoupled point particle observer.

In the gauge Ts ? 而 > 0, Xsys = qsys > 0, implying 竹(而 ) = ?1, 竹(而 ) = 0 [g而 而 = ?, N = 1, ? N r = g而 r = 0], we get x 步? s (Ts ) = u (ps ). Therefore, in this gauge, we have the centroid
(qsys )? ? x? (Ts ) = x? s (Ts ) = xs s (0) + Ts u (ps ),

(4.8)

which carries the ?uid ※internal§ collective variable Xsys = qsys > 0. In this gauge we get the following form of the energy-momentum tensor [Y r = J r ]
(qsys )汕 u ? 糸 而而 T ?糸 [xs (Ts ) + ?汕 u (u(ps ))考 ][汐] = u (ps )u (ps )T (Ts , 考 ) + 糸 ? r而 + [u? (ps )?糸 r (u(ps )) + u (ps )?r (u(ps ))]T (Ts , 考 ) + 糸 rs + ?? r (u(ps ))?s (u(ps ))T (Ts , 考 ),

T 而 而 (Ts , 考 ) = ? ?老 + +n
dust

= ?? J 而 ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (Ts , 考 ), ?老 J rJ 而 |s 而 2 (Ts , 考 ) ?n (J ) ? 汛uv Y u Y v ?老 |s ?汛 rs + ?n

?老 (J 而 )2 | s (? ? 而 2 ) (Ts , 考) ?n (J ) ? 汛uv Y u Y v

T r而 (Ts , 考 ) = ? n
dust

= ?? J r ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (Ts , 考 ), JrJs + 而 2 (J ) ? 汛uv Y u Y v 25

T rs (Ts , 考 ) = ?老汛 rs ? n

(Ts , 考 )

dust

J rJ s = ?? ?2 + 汛uv (T ?1 )ui (T ?1 )vj 旭i 旭j (Ts , 考 ), J而 with total 4 ? momentum

? [汐] = PT

? u d3 考T ?糸 [x? s (Ts ) + ?u (u(ps ))考 ][汐]u糸 (ps ) =

而 ? = ?P 而 u? (ps ) ? P r ?? r (u(ps )) > ?P u (ps ) = = ?Msys u? (ps ) > ?p? s,

and total mass
? M [汐] = PT [汐]u? (ps ) = ?P 而 = ?Msys .

(4.9)

The stress tensor of the perfect ?uid con?guration on the Wigner hyperplanes is T rs (Ts , 考). We can rewrite the energy-momentum tensor in such a way that it acquires a form reminiscent of the energy-momentum tensor of an ideal relativistic ?uid as seen from a local observer at rest (see the Eckart decomposition in Appendix B): T ?糸 [? 汐] = 老[汐, 旭] u? (ps )u糸 (ps ) + + P [汐, 旭] [灰 ?糸 ? u? (ps )u糸 (ps )] + + u? (ps )q 糸 [汐, 旭] + u糸 (ps )q ? [汐, 旭] +
rs 糸 + Tan [汐, 旭] ?? r (u(ps ))?s (u(ps )) (Ts , 考 ),

老[汐, 旭] = T 而 而 , 1 P [汐, 旭] = T uu , 3 u r而 q ? [汐, 旭] = ?? r (u(ps ))T , 1 rs Tan [汐, 旭] = T rs ? 汛 rs T uu , 3 u

uv 汛uv Tan [汐, 旭] = 0,

(4.10)

where i) the constant normal u? (ps ) to the Wigner hyperplanes replaces the hydrodynamic velocity ?eld of the ?uid; ii) 老[汐, 旭](Ts , 考 ) is the energy density; iii) P [汐, 旭](Ts , 考) is the analogue of the pressure (sum of the thermodynamical pressure and of the non-equilibrium bulk stress or viscous pressure); iv) q ? [汐, 旭](Ts , 考 ) is the analogue of the heat ?ow; rs v) Tan [汐, 旭](Ts , 考 ) is the shear (or anisotropic) stress tensor. We can now study the manifestly Lorentz covariant Dixon multipoles [20] for the perfect ?uid con?gurationon the Wigner hyperplanes in the gauge 竹(而 ) = ?1, 竹(而 ) = 0 [so ? ? (qsys )? ? that x 步? “? (Ts ) = x? s (Ts ) = u (ps ), x s (Ts ) = 0, xs (Ts ) = xs s (0) + u (ps )Ts ] with re? ? qsys )? spect to the origin an arbitrary timelike worldline w (Ts ) = z (Ts , 灰(Ts )) = x( (Ts ) + s 26

r ? (qsys )? r ? ? r ?? (Ts )+?? r (u(ps ))灰 (Ts ). Since we have z (Ts , 考 ) = xs r (u(ps ))考 = w (Ts )+?r (u(ps ))[考 ? def 灰 r (Ts )] = w ? (Ts ) + 汛z ? (Ts , 考 ) [for 灰 (Ts ) = 0 we get the multipoles with respect to the origin of coordinates], we obtain [ (?1 ..?n ) means symmetrization, while [?1 ..?n ] means antisym1 ..?n ?糸 1 ..?n ?糸 metrization; t? (Ts , 灰 = 0) = t? (Ts )] T T ?1 ...?n ?糸 tT (Ts , 灰) = tT 1 (? ...?n )(?糸 )

(Ts , 灰) =

=

qsys )汕 u d3 考 汛z ?1 (Ts , 考 )...汛z ?n (Ts , 考 ) T ?糸 [x( (Ts ) + ?汕 s u (u(ps ))考 ][汐] =

? r1 ..rn AB ?n 糸 1 = ?? (Ts , 灰 ) = r1 (u(ps ))...?rn (u(ps ))?A (u(ps ))?B (u(ps )) IT r1 ...rn 而 而 ?n ? 糸 1 = ?? (Ts , 灰) + r1 (u(ps ))...?rn (u(ps )) u (ps )u (ps )IT r1 ...rn rs 糸 + ?? (Ts , 灰 ) + r (u(ps ))?s (u(ps ))IT r1 ...rn r而 糸 ? (Ts , 灰) , + [u? (ps )?糸 r (u(ps )) + u (ps )?r (u(ps ))]IT r1 ..rn AB IT (Ts , 灰) =

d3 考 [考 r1 ? 灰 r1 (Ts )]...[考 rn ? 灰 rn (Ts )]T AB (Ts , 考)[汐],
?1 ...?n ?糸 tT (Ts , 灰 ) = 0, 而而 n = 0 (monopole) IT (Ts ) = P 而 , r而 IT (Ts ) = P r ,

u?1 (ps ) F or灰 = 0
?1 ...?n ? tT ? (Ts ) =

?n ? (qsys )汕 u 1 d3 考汛x? (Ts ) + ?汕 s (考 )...汛xs (考 )T ? [xs u (u(ps ))考 ][汐] =

def ?1 r1 ...rn A n = ?r1 (u(ps ))...?? A (Ts ) rn (u(ps ))IT r1 r2 A ’r1 r2 A A (Ts ) ? 1 汛 r1 r2 汛uv I uvA A (Ts ) = ir1 r2 (Ts ) ? 1 汛 r1 r2 汛uv iuv (Ts ), IT A (Ts ) = IT T T T 3 2 ’uvA A (Ts ) = 0, ’r1 r2 A A (Ts ) = ir1 r2 (Ts ) ? 1 汛 r1 r2 汛uv iuv (Ts ), 汛uv I I T T T T 3 r1 r2 r1 r2 A r1 r2 uvA iT (Ts ) = IT 汛uv IT A (Ts ) ? 汛 A (Ts ), ?1 ...?n ?1 ...?n ?糸 ?T t (Ts ) = tT (Ts )u? (ps )u糸 (ps ) = r1 ..rn 而 而 ?1 n = ?r1 (u(ps ))...?? (Ts ), rn (u(ps ))IT r1 而 而 ?1 r1 1 1 ?? t (Ts ) = ?P 而 ?? T (Ts ) = ?r1 (u(ps ))IT r1 (u(ps ))rsys , ?1 ?2 r1 r2 而 而 ?2 1 ?T t (Ts ) = ?? (Ts ), r1 (u(ps ))?r2 (u(ps ))IT

r1 ..rn 而 而 r1 ..rn rs r1 ..rn r而 The Wigner covariant multipoles IT (Ts ), IT (Ts ), IT (Ts ) are the mass,

1 1 r1 r2 而 而 r1 r2 而 而 r1 r2 uv而 而 ?T IT (Ts ) = I (Ts ) ? 汛 r1 r2 汛uv IT (Ts ) = ? iT (Ts ) ? 汛 r1 r2 汛uv? iuv T (Ts ), 3 2 1 r1 r2 r1 r2 而 而 ?uv而 而 (Ts ) = 0, ?T iuv 汛uv I (Ts ) = ? iT (Ts ) ? 汛 r1 r2 汛uv? I T (Ts ), T 3 r1 r2 r1 r2 而 而 uv而 而 ? (Ts ) ? 汛 r1 r2 汛uv IT (Ts ). iT (Ts ) = IT

(4.11)

27

stress and momentum multipoles respectively. ’r1 r2 A A (Ts ) and ir1 r2 (Ts ) are the traceless quadrupole moment and the The quantities I T T inertia tensor de?ned by Thorne in Ref. [21]. r1 r2 而 而 r1 r2 The quantities IT (Ts ) and ? iT (Ts ) are Dixon*s de?nitions of quadrupole moment and of tensor of inertia respectively. 1 ?? Moreover, Dixon*s de?nition of ※center of mass§ of an extended object is t T (Ts ) = 0 r而 而 r or IT (Ts ) = ?P 而 rsys = 0: therefore the quantity rsys de?ned in the previous equation is r s rs a non-canonical [{rsys , rsys } = Ssys ] candidate for the ※internal§ center of mass of the ?eld (qsys )? con?guration: its vanishing is a gauge ?xing for P > 0 and implies x? (Ts ) = s (Ts ) = xs ? x? (0) + u ( p ) T . As we have seen in the previous Section r is the ※internal§ M?ller s s sys s 3-center of energy and we have rsys > qsys > ysys . r drsys dI r而 而 (T ) ? r而 而 When IT (Ts ) = 0, the equations 0 = TdTs s = ?P 而 dT = = ? P r implies the correct s
drsys P momentum-velocity relation P > 0. 而 = dT s Then there are the related Dixon multipoles ?

?1 ...?n ? ?1 ...?n ?糸 pT (Ts ) = tT (Ts )u糸 (ps ) = pT 1 n (Ts ) = ? r1 ..rn A而 ?n 1 = ?? (Ts ), r1 (u(ps ))...?rn (u(ps ))?A (u(ps ))IT ?1 ...?n ? pT (Ts ) = 0, ? ? A ? ? p? T (Ts ) = PT [汐] = ???A (u(ps ))P > ??ps ,

(? ...? )?

u?1 (ps ) n=0

?1 ...?n r1 ..rn 而 而 ?n 1 ..?n ? 1 ?T p? (Ts )u? (ps ) = t (Ts ) = ?? (Ts ). T r1 (u(ps ))...?rn (u(ps ))IT

(4.12)

The spin dipole is de?ned as
?糸 rA而 ST (Ts )[汐] = 2pT (Ts ) = 2?[r? (u(ps ))?A (u(ps ))IT (Ts ) = ?糸 = Ss = 糸 rs ? 糸 糸 ? 而r = ?? r (u(ps ))?s (u(ps ))Ssys + [?r (u(ps ))u (ps ) ? ?r (u(ps ))u (ps )]Ssys , ?糸 而r 而 糸 r ?糸 u ? ( p s ) ST (Ts )[汐] = ??糸 r (u(ps ))Ssys = ?tT (Ts ) = P ?r (u(ps ))rsys , [?糸 ] 糸]

(4.13)

?糸 1 ?? with u? (ps )ST (Ts )[汐] = 0 when t T (Ts ) = 0 and this condition can be taken as a definition of center of mass equivalent to Dixon*s one. When this condition holds, the [rs]而 ?糸 ] rs而 barycentric spin dipole is ST (Ts )[汐] = 2?[r? (u(ps ))?糸 (Ts ) = s (u(ps ))IT (Ts ), so that IT ?糸 r s ?? (u(ps ))?糸 (u(ps ))ST (Ts )[汐]. As shown in Ref. [20], if the ?uid con?guration has a compact support W on the Wigner hyperplanes 曳W 而 and if f (x) is a C ﹢ complex-valued scalar function on Minkowski spacetime ?(k ) = d4 xf (x)eik﹞x is a slowly increaswith compact support [so that its Fourier transform f ing entire analytic function on Minkowski spacetime (|(xo + iy o)qo ...(x3 + iy 3)q3 f (x? + iy ?)| < o 3 Cqo ...q3 eao |y |+...+a3|y | , a? > 0, q? positive integers for every ? and Cqo ...q3 > 0), whose inverse d4 k ? f (k )e?ik﹞x ], we have [we consider 灰 = 0 with 汛z ? = 汛x? is f (x) = (2 s] 羽 )4

28

< T ?糸 , f > = = = =

d4 xT ?糸 (x)f (x) = dTs dTs dTs


d3 考f (xs + 汛xs )T ?糸 [xs (Ts ) + 汛xs (考 )][汐] = d3 考 d4 k ? f (k )e?ik﹞[xs(Ts )+汛xs (考 )] T ?糸 [xs (Ts ) + 汛xs (考 )][汐] = (2羽 )4 d3 考T ?糸 [xs (Ts ) + 汛xs (考 )][汐]

d4 k ? f (k )e?ik﹞xs (Ts ) (2羽 )4

(?i)n u n [k? ?? u (u(ps ))考 ] = n ! n=0 = dTs
﹢ d4 k ? (?i)n ?1 ...?n ?糸 ?ik ﹞xs (Ts ) (Ts ), f ( k ) e k?1 ...k?n tT 4 (2羽 ) n! n=0

(4.14)

and, but only for f (x) analytic in W [20], we get <T
?糸

,f > = ?

1 ?1 ...?n?糸 ? n f (x) dTs |x=xs(Ts ) , tT (Ts ) ?1 ?x ...?x?n n=0 n! ?n (?1)n n! ?x?1 ...?x?n n=0
﹢ ?1 ...?n ?糸 (Ts ) = dTs 汛 4 (x ? xs (Ts ))tT



T ?糸 (x)[汐] =

糸 AB (Ts , 考 )[汐] = = ?? A (u(ps ))?B (u(ps ))T ﹢ (?)n r1 ...rnAB ? n 汛 3 (考 ? 灰(Ts )) 糸 = ?? |灰=0 . IT (Ts , 灰 ) A (u(ps ))?B (u(ps )) ?考 r1 ...?考 rn n=0 n!

(4.15)

For a non analytic f (x) we have < T ?糸 , f > = + dTs dTs 1 ?1 ...?n ?糸 ? n f (x) |x=xs (Ts ) + tT (Ts ) ?1 ?x ...?x?n n=0 n!
﹢ d4 k ? (?i)n ?1 ...?n ?糸 ?ik ﹞xs (Ts ) (Ts ), f ( k ) e k?1 ...k?n tT (2羽 )4 n ! n=N +1 N

(4.16)

?1 ...?n ? and, as shown in Ref. [20], from the knowledge of the moments tT (Ts ) for all n > N we ?糸 can get T (x) and, thus, all the moments with n ≒ N . In Appendix D other types of Dixon*s multipoles are analyzed. From this study it turns out that the multipolar expansion(4.15) may be rearranged with the help of the Hamilton ? equations implying ?? T ?糸 = 0, so that for analytic ?uid con?gurations from Eq.(D5) we get

T ?糸 (x)[汐] = ??u(? (ps )?A (u(ps )) +

?

糸)

1 ? 老(? dTs 汛 4 (x ? xs (Ts )) ST (Ts )[汐] u糸 ) (ps ) + 2 ?x老 ﹢ (?1)n ?n ?1 ..?n ?糸 + (Ts ), dTs 汛 4 (x ? xs (Ts )) IT ? ? n 1 n! ?x ...?x n=2
糸)

dTs 汛 4 (x ? xs (Ts )) P A +

T ?糸 (w + 汛z ) = ??u(? (ps ) ?A (u(ps ))P A 汛 3 (考 ? 灰 (Ts )) + 29

?汛 3 (考 ? 灰(ts )) 1 老(? ( u ( p )) + + ST (Ts , 灰 )[汐]u糸 ) (ps )?r s 老 2 ?考 r ﹢ (?)n n + 3 ? r1 ...rn 而 而 + u (ps )u糸 (ps )IT (Ts , 灰 ) + n ! n + 1 n=2 1 r1 ...rn r而 糸 ? + [u? (ps )?糸 (Ts , 灰) + r (u(ps )) + u (ps )?r (u(ps ))]IT n r1 ...rn s1 s2 糸 + ?? (Ts , 灰 ) ? s1 (u(ps ))?s2 (u(ps ))[IT n + 1 (r1 ...rns1 )s2 (r ...r s )s ? (IT (Ts , 灰 ) + IT 1 n 2 1 (Ts , 灰)) + n (r1 ...rn s1 s2 ) + IT (Ts , 灰 )] ,
(? ..? |?|? )糸

(4.17)

?1 ..?n ?糸 n?1) JT 1 n?1 n (Ts ), with the quantities (Ts ) = 4(n where for n ≡ 2 and 灰 = 0 IT +1 ?1 ..?n ?糸老考 JT (Ts ) being the Dixon 22+n -pole inertial moment tensors given in Eqs.(D7) [the r1 r2 而 而 quadrupole and related inertia tensor are proportional to IT (Ts )]. ? The equations ?? T ?糸 = 0 imply the Papapetrou-Dixon-Souriau equations for the &pole? ?糸 dipole* system PT (Ts ) and ST (Ts )[汐] [see Eqs.(D1) and (D4); here 灰 = 0] ? dPT (Ts ) ? = 0, dTs ?糸 (Ts )[汐] ? dST [? = 2PT (Ts ) u糸 ] (ps ) = ?2?P r ?[r? (u(ps )) u糸 ] (ps ) > 0. dTs

(4.18)

The Cartesian Dixon*s multipoles could be re-expressed in terms of either spherical or STF (symmetric tracefree) multipoles [21] [both kinds of tensors are associated with the irreducible representations of the rotation group: one such multipole of order l has exactly 2l + 1 independent components].

30

V. ISENTROPIC AND NON-ISENTROPIC FLUIDS.

Let us now consider isentropic (s = const.) perfect ?uids. For them we have from Eqs.(1.9), (1.10) and (2.1) [in general ? is not the chemical potential but only a parameter] n= |J | X ﹟ =﹟ , N 污 污

|J | X 老 = 老(n) = 老( ﹟ ) = ?f ( ﹟ ), N 污 污 X ﹟ L = ??N 污f ( ﹟ ). 污 Some possible equations of state for such ?uids are (see also Appendix A): 1) p = 0, dust: this implies X 老(n) = ?n = ? ﹟ , 污 X X f(﹟ ) = ﹟ , 污 污 ?X i.e. 1 =﹟ . 污

(5.1)

X ) ?f ( ﹟ 污

(5.2)

(n) ? 老(n) (k = ?1 because otherwise 老 = const., ? = 0, f = ?T s). 2) p = k老(n) = n ?老 ?n For k = 1 one has the photon gas. The previous di?erential equation for 老(n) implies 3

X 老(n) = (an)k+1 = ?nk+1 = ?( ﹟ )k+1 , 污 X X f ( ﹟ ) = ( ﹟ )k+1 , 污 污
X ) ?f ( ﹟ 污

(? = ak+1 ),

i.e.

?X

k+1 X = ﹟ ( ﹟ )k , 污 污

(5.3)

[for k ↙ 0 we recover 1)]. More in general one can have k = k (s): this is a non-isentropic perfect ?uid with 老 = 老(n, s). (n) ? 老(n) (污 = 1) [22]. It is an isentropic polytropic perfect ?uid 3) p = k老污 (n) = n ?老 ?n 1 (污 = 1 + n ). The di?erential equation for 老(n) implies [a is an integration constant; the ?老 chemical potential is ? = ?n |s ] 老(n) = an [1 ? k (an)污 ?1 ]
X
1 污 ?1

=

an [1 ? k (an) n ]n
X
1

,

i.e.

﹟ ﹟ X 污 污 f(﹟ ) = , = 1 1 X n X 污 ? 1 污 [1 ? k (a ﹟ ) ]n [1 ? k (a ﹟污 ) ] 污?1 污 X ) ?f ( ﹟ 污
污 1 X X 1 1 = ﹟ [1 ? k (a ﹟ )污 ?1 ]? 污?1 = ﹟ [1 ? k (a ﹟ ) n ]?(n+1) . 污 污 污 污

?X

(5.4)

31

Instead in Ref. [23,24] a polytropic perfect ?uid is de?ned by the equation of state [see the last of Eqs.(B4); m is a mass] 老(n, ns) = mn + k (s ) (mn)污 , 污?1 (5.5)

and has pressure p = k (s)(mn)污 = (污 ? 1)(老 ? mn) and chemical potential or speci?c ∩ 污 (mn)污 ?1 of Eq.(B3). enthalpy ? = mc2 + mk (s) 污 ? 1 4) p = p(老), barotropic perfect ?uid. In the isentropic case one gets 老 = 老(n) by solving (n) p(老(n)) = n ?老 ? 老(n). ?n 5)Relativistic ideal (Boltzmann) gas [2] (this is a non-isentropic case): p = nkB T
2

and 老 = mc2 n忙(汕 ) ? p,

?=

老+p = mc2 忙(汕 ), n

(5.6)

K3 (汕 ) mc with 汕 = k , 忙(汕 ) = K (Ki are modi?ed Bessel functions). One gets the equation of 2 (汕 ) BT state 老 = 老(n, s) by solving the di?erential equation

mc2 n ?老 |s = mc2 忙( ?老 ). ?n n ?n |s ? 老 5a) Ultrarelativistic case 汕 << 1 (mc2 << kB T ): since we have 忙(汕 ) >
2 4

(5.7)
4 +汕 汕 2 4/3

+ O (汕 3),

c n we get 老 = 3nkB T + m + O (kB T 汕 4), namely p > 1 老 and 老 > 老(n) = ?n . 2kB T 3 2 + O (汕 ?2), 5b) Non-relativistic case 汕 >> 1 (kB T << mc ): since we have 忙(汕 ) > 1 + 25 汕 ?老 we get 老 > mc2 n+ 3 p, so that we have to solve the di?erential equation 3n ?n |s ?5老+2mc2 n > 2 0. Its solution is 老(n, s) > mc2 n + k (s)n5/3 . To ?nd k (s) let us use the de?nition of (s) ?老 temperature: p = nkB T = kB ?老 | = kB n5/3 ?k = n ?n |s ? 老 = 2 n5/3 k (s). This leads to the ?s n ?s 3
2s 2(s?so )

2ds , whose solution is k (s) = he 3kB = e 3kB with h = e 3kB = const.. equation dln k (s) = 3 kB Therefore, in this case we get [it is a polytropic like in Eq.(5.5) with 污 = 5/3 and k (s) = 2 ?5/3 m e 3
2(s?so ) 3kB

?

2so

]
5 2(s?so ) 3kB

老(n, s) > mc2 n + n 3 e

and T =

s?so ) 2 5 2(3 1 ?老 |n > n 3 e kB . n ?s 3k B

(5.8)

The action for these ?uids is [s = s(汐i )] S= d而 d3 考L(汐i (而, 考), z ? (而, 考 )) = ? ﹟ d而 d3 考N 污老(n, s), (5.9)

and we have as in Section II
’u ’v ’ 1 2 3 J 而 = ??r ?r ’汐 ?u ’ 汐 ?v ’汐 , 3 ’ Jr = i=1 ’u ’v ’ j k ?而 汐i ?r ?u ’ 汐 ?v ’汐 ,

i, j, k, = cyclic, |J | =
r ’ r ’ 而 v ’ v ’ 而 N 2 (J 而 )2 ? 3 g r ’s ’[J + N J ][J + N J ] = NX,

32

X=
’ Yr =

1 r ’ 而 (J ’ + N r J ), N

r ’ s ’ (J 而 )2 ? 3 g r ’s ’Y Y ,

’ ?X Yr Tr ’i = , i ??而 汐 NX r ’u ’v ’ j k 3 ?u Tt ’r ’i = ? gt ’ 汐 ?v ’汐 , ’? 3 r ’ s ’ gr (J 而 )2 ? X 2 ?X ’s ’Y Y = = , ?N NX NX 3 s ’ ?X ’s ’Y 而 gu = ? J . ’ ?N u NX

(i, j, k

cyclic),

(5.10)

In the cases 1), 2) and 3) [in case 3) we rename ? the constant a] the canonical momenta can be written in the form 旭i (而, 考 ) = ? ?
’ ?L(而, 考 ) Yr Tr ?f (x) ’i (而, 考 ), = ? | X x = ﹟污 i ??而 汐 (而, 考 ) ?x X ’ Yr =? ’i (T ? 1 )r 旭i ?f (x) ’ X( | X )? 1 = Y r (X ), ? ?x x= ﹟污 ’ 而 ’ ’ i Nr J ? NY r Jr ?r ’汐 i = ?r ?而 汐i = ? ’汐 = J而 J而 X ?f (x) ’ i i ?1 r | X ]?1 , = Nr ?r ) ’j 旭j 而 [ ’汐 + N?r ’ 汐 (T ?J ?x x= ﹟污

老? (而, 考 ) = ?

?L ?L(而, 考 ) 3 s ’r ’ ?L ? ?zs (而, 考 ) = = ? l? ’? g ? ’ ?z而 (而, 考) ?N ?N r X X ) ) ?f ( ﹟ ?Nf ( ﹟ ﹟ 污 污 3 s ’r ’ = ? 污 l? ? ?zs g N (而, 考 ) = ’? ’ ?N ?N r ?f (x) X ? ﹟ | X l? + 污Xf ( ﹟ ) + [(J 而 )2 ? X 2 ] = X 污 ?x x= ﹟污 ? ?f (x) 而 r ’ X J Y zr |x= ﹟ + ’? (而, 考 ) = 污 X ?x ?f (x) ? ﹟ X 污Xf ( ﹟ ) + [(J 而 )2 ? X 2 ] | X l? (而, 考 ) ? = X 污 ?x x= ﹟污

?

’i ? ? J 而 (T ? 1 )r 旭i zr ’? (而, 考 ) = ﹟ ﹟ ’i = ? 污G(X, J 而 , 污 )l? + J 而 (T ?1 )r 旭i zr ’? (而, 考 ), with (J 而 )2 X 2 (J 而 )2 ? (﹟ ) ?f (x) ? n2 ?f (n) X X (J 而 )2 污 污 污 ) = f(﹟ ) + | X = f (n) + . G( ﹟ , X ﹟ 污 污 污 ?x x= ﹟污 n ?n 污

(5.11)

To get the Hamiltonian expression of the constraints H? (而, 考 ) > 0, we have to ?nd the 33

r ’ s ’ 而 2 r ’ solution X of the equation X 2 + 3 gr ’s ’Y (X )Y (X ) = (J ) with Y (X ) given by the second line of Eq.(5.11). This equation may be written in the following forms

X 2 ?2 + A2 ( X ( ﹟ )2 污 n2

?f (x) ?2 X ) = B 2 , or |x= ﹟ 污 ?x ?f (x) A2 B2 ?2 ( | X , 1+ 2 ) ? 1 = ? + A2 ?x x= ﹟污 污 (?2 + A2 ) A2 ?f (n) ?2 B2 1+ 2 ( ) ?1 = , ? + A2 ?n 污 (?2 + A2 )

or

?1 r ’j A2 = 3 gr ) ’i 旭i (T ?1 )s 旭j , ’s ’(T 2 2 而 2 B = ? (J ) ,

? ?

X=



污n =



污F (

B2 A2 (J 而 )2 A2 ﹟ ? , ) = , ), 污 F ( ?2 污 ?2 + A2 污 (?2 + A2 )

B2 A2 ’i , )l? + J 而 (T ?1 )r 旭i zr ’? = ?2 + A2 污 (?2 + A2 ) ’ = Ml? + Mr zr ’? . ﹟ ? 老? = ? 污 G (

(5.12)

Therefore, all the dependence on the metric and on the Lagrangian coordinates and ﹟ ?1 r ’j ) ’i 旭i (T ?1 )s 旭j /?2 , their momenta is concentrated in the 3 functions 污 , A2 /?2 = 3 gr ’s ’(T B 2 /?2污 = (J 而 )2 /污 . Let us consider various cases. 1) p = 0, dust. As in Section II the equation for X and the constraints are X2 [?2 + A2 ] = B 2 , B = ?2 + A2 ?|J 而 |

X=﹟

? 1 )r ’i 旭 (T ?1 )s ’j 旭 ?2 + 3 gr ’s ’(T i j

↙A2 ↙0 |J 而 |[1 ?
’ Yr =?

A2 + O (A4 )], 2? 2

’i | J 而 | (T ? 1 )r 旭i X ?1 r (T ) ’i 旭i = ? , ? 1 )r ’i 旭 (T ?1 )s ’j 旭 ? ?2 + 3 gr ’s ’(T i j

’i ? 1 )r ’i 旭 (T ?1 )s ’j 旭 l + J 而 (T ?1 )r 旭i zr 老? = |J 而 | ?2 + 3 gr ’s ’(T i j ? ’? .

?

(5.13)

2) p = k老, k = ?1. The equation for X is X2 X ( ﹟ )2 污 [?2 + A2 2 X 2k ] = B , ) (k + 1)2 ( ﹟ 污 or

A2 1 B2 [1 + 2 ( , X 2k ? 1)] = ? + A2 (k + 1)2 ( ﹟ 污 (?2 + A2 ) ) 污 34

Y =?

r ’



X 1?k ’i ) (T ? 1 )r 旭i 污( ﹟ 污

?(k + 1)

, (5.14)

﹟ X (J 而 )2 X 2 ’i 老? = ? 污 ( ﹟ )k?1 [(k + 1) ? k () ﹟ ]l? + J 而 (T ?1 )r 旭i zr ’? . 污 污 污

Let us de?ne Z as the deviation of X from dust (for A2 ↙ 0 [?而 汐i = 0]: we have Z ↙ 1.
| X = ﹟ |B 2 ? +A2

Z . Then we get the following equation for Z Z2 ?2 A2 (?2 + A2 )k?1 污 k ?2k = 1, + Z ?2 + A2 (k + 1)2 B 2k 2 ?2k Z 2 [汐2 + 汕k Z ] = 1, ?2 ↙A2 ↙0 1, 汐 = 2 ? + A2 A2 (?2 + A2 )k?1 污 k 2 汕k = ↙A2 ↙0 0. (k + 1)2 B 2k
2

or

(5.15)

We may consider the following subcases: 2a) k = m = ?1, with the equation Z1 = Z 2 , Z1 X=﹟ |B | Z1 , ?2 + A2 or
2 1?m 汕m Z1 + 汐 2 Z 1 ? 1 = 0.

2 ?m [汐2 + 汕m Z1 ] = 1,

(5.16)

i) p = 老 (k = m = 1), with the equation
2 ? A2 ) 污 (?2 + A2 )(4 B 1 ? 汕1 污 Z1 = = , 汐2 4? 2 B 2
2

X=

4B ? A2 污 2?

2

=

1 (J 而 )2 3 ? 1 )r ’i 旭 (T ?1 )s ’j 旭 , + gr 4? 2 ’s ’(T i j 2? 污

f or

A2 < 4

B2 . 污

(5.17)

ii) p = 2老 (k = m = 2), with the equation
2 2 汐 2 Z1 ? Z1 + 汕2 = 0,

Z1 =

1 [1 ㊣ 2汐 2

2 1 ? 4汐2 汕2 ],

(5.18)

iii) p = ?2老 (k = m = ?2), with the equation
2 3 2 汕? 2 Z 1 + 汐 Z 1 ? 1 = 0,

(5.19)

2b) k =

1 , m

with the equation

35

Z2 = Z m ,

m |B | 2 , Z 2 ?2 + A2 m 2 m?1 汐 2 Z2 + 汕1 ? 1 = 0, /m Z2 2

X=﹟

(5.20)

1 i) p = 2 老 (k = 1 , m = 2), with the equation 2 2 2 汐 2 Z2 + 汕1 /2 Z2 ? 1 = 0, 1 2 4 2 Z2 = 2 [?汕1 汕1 /2 ㊣ /2 + 4汐 ]. 2汐

(5.21)

老 (k = ? 1 , m = ?2), with the equation ii) p = ? 1 2 2
2 ?1 3 2 ?1 2 汕? 1/2 (Z2 ) + 汐 (Z2 ) ? 1 = 0. 1 2 iii) p = 3 老, photon gas (k = 1 , m = 3), 汕1 /3 = 3 3 2 2 汐 2 Z2 + 汕1 /3 Z2 ? 1 = 0, 9A2 污 1/3 , 16B 2/3 (?2 +A2 )2/3

(5.22) with the equation

or

3 2 Z2 + pZ2 + r = 0,

with p = Z2 X = =

2 汕1 /3 > 0, 汐2

r=?

1 < 0, 汐2

2 汕1 1 /3 ↙ A2 ↙ 0 1 , Y2 ? p = Y2 ? 3 3汐 2 2 汕1 |B | |B | 3A2 (?2 + A2 )1/3 污 1/3 3/2 /3 ﹟ 2 ﹟ ) = ) , ( Y ? ( Y ? 2 2 3汐 2 16B 2/3 ? + A2 ?2 + A2

a b

= =

Y23 + aY2 + b = 0, 4 汕1 1 27A4 (?2 + A2 )2/3 污 2/3 /3 ? p2 = ? 4 = ? < 0, 3 3汐 28 B 4/3 6 汕1 1 1 ?2 + A2 27A6 污 /3 (2p3 + 27r ) = (2 ? 27) = ?1 , 27 27汐2 汐4 ?2 211 ?4 B 2 6 汕1 1 1 (?2 + A2 )2 27A6 污 b2 a3 /3 2 ↙0 + = (27 ? 4 ) = 1 ? ↙ > 0, A 4 27 4 ﹞ 27汐4 汐4 4? 4 210 ?4 B 2 4 ? ? b + 2 b2 a3 + 4 27
1/3

Y2

=

?

b + 2

b2 a3 + 4 27

1/3

↙A2 ↙0 1,

X

= ?

|J 而 | ? 1 21/3

27A6 污 1 3A2 污 1/3 1 ? + + 16(J 而 )2/3 21/3 211 ?6 (J 而 )2 27A6 污 + 211 ?6 (J 而 )2 1? 27A6 污 210 ?6 (J 而 )2

1?

27A6 污 210 ?6 (J 而 )2 ↙ A2 ↙ 0

1/3

?

?1+

1/3 3/2

↙A2 ↙0 |J 而 |[1 ?

9A2 污 1/3 + O (A4 )], 32(J 而 )2/3 36

老?

= =

? 1 ﹟ X ?2/3 (J 而 )2 X ’i ? 污( ﹟ ) [4 ? ( ﹟ )2 ]l? + J 而 (T ?1 )r 旭i zr ’? = 3 污 污 污 Ml? + Mr zr? .

(5.23)

In the case of the photon gas we get a closed analytical form for the constraints. 1 3) p = k老污 , 污 = 1 + n , 污 = 1 (n = 0), with the equation X2 X2 X ( ﹟ )2 污
2污 X ?2 + A2 1 ? k (? ﹟ )污 ?1 污?1 = B 2 , or 污 X 1 2(n+1) = B 2 , or ?2 + A2 1 ? k (? ﹟ ) n 污 X 1 2(n+1) B2 A2 n [1 ? k ( ? , ] ? 1 = 1+ 2 ) ﹟ ? + A2 污 污 (?2 + A2 )

’ Yr =?

X X 1 ’i 旭i , [1 ? k ( ﹟ ) n ]n+1 (T ?1 )r ? 污
(J 而 )2 污 X ﹟ [1 污 X 2+ n ? k? n ( ﹟ ) 污
1 1

﹟ 老? = ? 污

?

l? 1 X n n+1 ] k (? ﹟ ) 污

’i + J 而 (T ? 1 )r 旭i zr ’? .

(5.24)

Let us de?ne Z as the deviation of X from dust (for A2 ↙ 0 [?而 汐i = 0]: we have Z ↙ 1, | X = ﹟ |B Z ). Then we get the following equation for Z 2 2
? +A

Z2

1 A2 (?2 + A2 )k?1 污 k ?|B |Z ?2 + 1 ? k( ﹟ ﹟ 2 )n 2 2 2 2 k 2 ? +A (k + 1) B 污 ? +A

2(n+1)

= 1.

(5.25)

In conclusion, only in the cases of the dust and of the photon gas we get the closed analytic form of the constraints ( i.e. of the density of invariant mass M(而, 考), because for ’ ’i the momentum density we have Mr = J 而 (T ? 1 )r 旭i independently from the type of perfect ?uid). In all the other cases we have only an implicit form for them depending on the solution X of Eq.(5.12) and numerical methods should be used.

37

VI. COUPLING TO ADM METRIC AND TETRAD GRAVITY.

Let us now assume to have a globally hyperbolic, asymptotically ?at at spatial in?nity spacetime M 4 with the spacelike leaves 曳而 of the foliations associated with its 3+1 splittings, di?eomorphic to R3 [25每27,11]. In 考而 -adapted coordinates 考 A = (考 而 = 而 ; 考 ) corresponding to a holonomic basis [d考 A , ?A = ?/?考 A ] for tensor ?elds we have [25] [N and N r are the lapse and shift functions; 3 grs is the 3-metric of 曳而 ; lA (而, 考 ) is the unit normal vector ?elds to 曳而 ]
4

gAB = gAB = {4 g而 而 = ?(N 2 ? 3 grs N r N s ); 4 g而 r = ?? 3 grs N s ; 4 grs = ?? 3 grs } = = ?lA lB + →AB , (6.1)

→AB = 4 gAB ? ?lA lB .

3 (a) 3 A set of 曳而 -adapted tetrad and cotetrad ?elds is (a) = (1), (2), (3); 3 er (a) and er = e(a)r are triad and cotriad ?elds on 曳而 ] 4 A (曳) E(o) 4 A (曳) E(a) (o) 4 (曳) EA (a) 4 (曳) EA

= ?lA = (

1 Nr ; ? ), N N 3 r = (0; e(a) ), = lA = (N ; 0),
(a) 3 (a) = (N (a) = N r 3 er ; er ).

(6.2)

In these coordinates the energy-momentum tensor of the perfect ?uid is T AB = ??(老 + p)U A U B + p 4 g AB . If we use the notation 忙 = ?lA U A = ?NU 而 , we get TAB = 4 gAC 4 gBC T CD = [?lA lC + →AC ][?lB lD + →BD ]T CD = = ElA lB + jA lB + lA jB + SAB , E = T AB lA lB = ??[(老 + p)忙2 ? p], jA = ?→AC T CB lB = ??(老 + p)忙→AB U B , SAB = →AC →BD T CD = ??[(老 + p)→AC U C →BD U D ? p→AB ].
?

(6.3)

(6.4)

The same decomposition can be referred to a non-holonomic basis [牟A = {牟l = Nd而 ; 牟r = ? ?1 ?A ? = (l; r )] in which we have [忙 ? = ?? = (?而 ? N r ?r ); ?r }; A lA d考 r + N r d而 }, XA ?U ? = {Xl = N l r ? = 1 ? 3 grs U ? s , so that U ? may be interpreted as the generalized boost velocity of ? rU ?U ? ? ? A with respect to ? lA [28,29]] U
4 4 ?ll = ?; 4 g ?lr = 0; 4 g ?rs = 4 grs = ?? 3 grs } = g ?A ?B ? = { g ?A lB = ?? lA ?B ?, ? +→ ??

38

? ? ? lA lA = (?; 0), ? = (1; 0), ? ll = → ? lr = 0, ? rs = ?? 3 grs , → → 4 ?B ?A ?A ?A T ?B ? = ? +p g ?U ?B ? = ??(老 + p)U ? ? ? ? ?A ? ? ? = E lA ?B ?, ? +S ? jB ? + lA ? lB ? + jA ? lB

? ?A ?B = (0; ?? 3 grs ), → ?B ?U

?B ?? ? ? 2 ? p], ?=T ?A lA E ? = ??[(老 + p)忙 ? lB ?B ? ?A ? 3 grs U ? s ), ?C ? jl = 0; ? jr = ??(老 + p)忙 lB jA ? = (? ?C ?T ? = ?→ ?D ? ?B ?A ?C ?A = S ?D ?T ?C ?→ ?B ? = → ?ll = S ?lr = 0; S ?rs = ??[?p 3 grs + (老 + p)3 gru U ? u 3 gsv U ? v )]. = (S

(6.5)

? and ? E jr are the energy and momentum densities determined by the Eulerian observers ?rs is called the ※spatial stress tensor§. on 曳而 , while S The non-holonomic basis is used to get the 3+1 decomposition (projection normal and 3 ? T ?糸 [when one does not has parallel to 曳而 ) of Einstein*s equations with matter 4 G?糸 = 8?c 羽G an action principle for matter, one cannot use the Hamiltonian ADM formalism]: in this ? ?c3 ? ll ? ?c3 ? lr ? ll = ? lr = way one gets four restrictions on the Cauchy data [ 4 G T and 4 G T ; they 8羽G 8羽G become the secondary ?rst class superhamiltonian and supermomentum constraints in the ADM theory; k = c3 /8羽G]
? 2 ? R + (3 K )2 ? 3 Krs 3 K rs (而, 考 ) = E (而, 考 ), k ? 1 ?r (而, 考 ), (3 K rs ? 3 g rs 3 K )|s (而, 考) = J k 3

(6.6)

? ?c3 ? rs ? rs = T . By introducing the extrinsic curvature and the spatial Einstein*s equations 4 G 8羽G 3 Krs , this last equations are written in a ?rst order form [it corresponds to the Hamilton ? rs = ?c3 ﹟污 (3 K rs ? 3 g rs 3 K ); ※|§ denotes the equations of the ADM theory for 3 grs and 3 旭 8羽G covariant 3-derivative]

?而 3 Krs (而, 考 ) = N [3 Rrs + 3 K 3 Krs ? 2 3 Kru 3 K u s ] ?

?而 3 grs (而, 考 ) = Nr|s + Ns|r ? 2N 3 Krs (而, 考 ),
?

? N|s|r + N u |s 3 Kur + N u |r 3 Kus + N u 3 Krs|u (而, 考 ) ? 13 1 ? ??S ?u u ) (而, 考). grs (E ? (S rs + k 2
?

(6.7)

The matter equations T ?糸 ;糸 = 0 become a generalized continuity equation [entropy con? ?B ? A servation when the particle number conservation law is added to the system] ? lA ? =0 ?T ;B ? ?C ? B ?A and generalized Euler equations → ? = 0 [LX is the Lie derivative with respect to the ?B ?T ;C vector ?eld X] ? + N? ?rs 3 Krs + E ? 3 K ) ? 2? ? (而, 考), ?而 E j r | r = N (S j r N|r + LN E
?

? r ?rs |s (而, 考 ) = ?rs N|s ? EN ? |r + L ? ?而 ? jr + N S N (2 3 K rs ? js + ? j r 3K ) ? S N j (而, 考 ).

(6.8)

39

?rs would follow from an equation of state or dynamical equation of The equation for S the sources [for perfect ?uids it is the particle number conservation]. This formulation is the starting point of many approaches to the post-Newtonian approximation (see for instance Refs. [23,30]) and to numerical gravity (see for instance Refs. [31,32]). Instead with the action principle for the perfect ?uid described with Lagrangian coordinates (containing the information on the equation of state and on the particle number and entropy conservations) coupled to the action for tetrad gravity of Ref. [25] (it is the ADM action of metric gravity re-expressed in terms of a new parametrization of tetrad ?elds) we get S = ??k
3 3 s 3 d而 d3 考 {N 3 e ?(a)(b)(c) 3 er (a) e(b) ?rs(c) +

e 3 ?1 3 3 s 3 ( Go )(a)(b)(c)(d) 3 er (b) (N(a)|r ? ?而 e(a)r ) e(d) (N(c)|s ? ?而 e(c) s )}(而, 考 ) + 2N |J | ﹟ ? d而 d3 考 {N 污老( ﹟ , s)}(而, 考). N 污 +

(6.9)

The superhamiltonian and supermomentum constraints of Ref. [25] are modi?ed in the following way by the presence of the perfect ?uid H = Ho + M > 0, Hr > 成r = 成o r + Mr > 0,

(6.10)

with Mr = J 而 (T ?1 )ri 旭i = ??r 汐i 旭i and with M given by Eq.(2.12) for the dust and by Eq.(5.23) for the photon gas. In the case of dust the explicit Hamiltonian form of the energy and momentum densities is M(而, 考) = J 而 (而, 考) ?2 + [3 grs (T ?1 (汐))ri(T ?1 (汐))sj 旭i 旭j ](而, 考) = = ?det (?r 汐i (而, 考 )) ?2 + 3 g uv ?u 汐m ?v 汐n 旭m 旭n (而, 考 ), [det (?r 汐k )]2

Mr (而, 考) = 3 grs J 而 (而, 考 )[(T ?1 (汐))si 旭i ](而, 考 ) = ??r 汐i (而, 考)旭i (而, 考).

(6.11)

In any case, all the dependence of M and Mr on the metric and on the La﹟ grangian coordinates and their momenta is concentrated in the 3 functions 污 , A2 /?2 = 3 uv i j 1 3 ?1 r ’j u 汐 ?v 汐 旭i 旭?J , B 2 /?2 污 = (J 而 )2 /污 = 污 [det (?r 汐i )]2 . gr ) ’i 旭i (T ?1 )s 旭j /?2 = g ?? 2 [det (? 汐k )]2 ’s ’(T u The study of the canonical reduction to the 3-orthogonal gauges [26,27] will be done in a future paper.

40

VII. NON-DISSIPATIVE ELASTIC MATERIALS.

With the same formalism we may describe relativistic continuum mechanics [any relativistic material (non-homogeneous, pre-stressed,...) in the non-dissipative regime] and in particular a relativistic elastic continuum [6] [see also Refs. [33,34] and their bibliography] in the rest-frame instant form of dynamics. Now the scalar ?elds 汐 ? i (z (而, 考 )) = 汐i (而, 考 ) describe the idealized ※molecules§ of the ’ material in an abstract 3-dimensional manifold called the ※material space§, while J A (而, 考 ) = ﹟ ’ [N 污nU A ](而, 考) is the matter number current with future-oriented timelike 4-velocity vector ’ ?eld U A (而, 考 ); n is a scalar ?eld describing the local rest-frame matter number density. The ? i quantity ?A ? i (z ) is called the ※relativistic deformation gradient§ in ’ 汐 (而, 考 ) = zA ’ (而, 考 )?? 汐 曳而 -adapted coordinates. The material space inherits a Riemannian (symmetric and positive de?nite) 3-metric from the spacetime M 4
j i Gij = 4 g ?糸 ?? 汐 ? i ?糸 汐 ? j = 4 g AB ?A ’汐 . ’ 汐 ?B ’’

(7.1)

Its inverse Gij carries the information about the actual distances of adjacent molecules in the local rest frame. For an ideal ?uid the 3-form 灰 of Eq(1.3) gives the volume element in the material space, which is su?cient to describe the mechanical properties of an ideal ?uid. Since | ﹟ is a scalar, we can evaluate n in the local rest frame at z where we have that n = N|J 污 1 ? ﹟ ; 0) and ?o 汐 ? i |z = 0 [in the local adapted non-holonomic basis we have U (z ) = (
? 4g ?o ?o ? 4

﹟4 ﹟4 ? 1 ﹟3 ? 1 ﹟ 4 1 0 1 0 ﹟ ﹟ o ?? ?, , 4g ?AB = ? g ?A ??o ?o g ? = 污/ ? 4 g g ? = g ? / ? g ?B ? = ? ?o ? = 3 rs , 3 0 ?g 0 ? grs ﹟4 ﹟4 ? 1 ﹟ 4 ? g ?o g ?= g ? ]. We get at z ?o ?/ ﹟ 4 ﹟ ? g Jo ?o ?o ? i n = o ﹟4 = 灰123 det ?k 汐 ?rs | = 灰123 det Gij . (7.2) ? ﹟4 = 灰123 det ?k 汐 ? i det |4 g U g ? g ? ﹟ Therefore, we have n = 灰123 det Gij . Moreover, the material space of elastic materials, which have not only volume rigidity but also shape rigidity, is equipped with a Riemannian (symmetric and de?nite positive) 3-metric (M ) 污ij (汐i ), the ※material metric§, which is frozen in the material and it is not a dynamical object of the theory. It describes the ※would be§ local rest-frame space distance between neighbouring ※molecules§, measured in the locally relaxed state of the material. To measure (M ) the components 污ij (汐i ) we have to relax the material at di?erent points 汐i (而, 考 ) separately, since global relaxation of the material may not be possible [the material space may not be isometric with any 3-dimensional subspace of M 4 , as in classical non-linear elastomechanics, when the material exhibits internal stresses frozen in it]. The components (M ) (M ) 污ij = 污ij (汐i (而, 考 )) are given functions, which describe axiomatically the properties of the material (the theory is fully invariant with respect to reparametrizations of the material space). (M ) For ideal ?uids 污ij = 汛ij ; this also holds for non-pre-stressed materials without ※internal§ or ※frozen§ stresses. 41

Now the material space volume element 灰 has 灰123 (汐i ) = det 污ij (汐i),
(M )

﹟ so that n = 灰123 det Gij =

det 污ij

(M )



det Gij ,

(7.3)

and it cannot be put = 1 like for perfect ?uids. (M ) The pull-back of the material metric 污ij to M 4 is
j i 污A ’汐 ’ 汐 ?B ’B ’ = 污ij ?A (M ) (M )

satisf ying

B 污A ’B ’ U = 0.

(M )



(7.4)

The next step is to de?ne a measure of the di?erence between the induced 3-metric (M ) Gij (?汐i ) and the constitutive metric 污ij (汐i ), to be taken as a measure of the deformation of the material and as a de?nition of a ※relativistic strain tensor§, locally vanishing when there is a local relaxation of the material. Some existing proposal for such a tensor in M 4 are [33,34]: i) 1 4 (M ) (1) SA ’B ’ ? ?UA ’ UB ’ ? 污A ’B ’ ) ’B ’ = ( gA 2
(1) ’

(7.5)

B = 0 [but it must satisfy the involved matrix which vanishes at relax and satis?es SA ’B ’U (1) 4 inequality 2 det S ≡ det ( g ? ?UU )].

ii)
’ ’’ (M )

1 A ’ ’ ’ A S (2)A B ’ = (K B ’ ? 汛B ’ ), 2
’ ’ ’

B S (2)A B ’ U = 0,





(7.6)

B A 4 AC = U A [the 4-velocity ?eld is an eigenvector of (污 C with K A B ’U ’ ), K B ’ UB ’ = g ’B ’ ? ?UC the K -matrix].

iii)

1 S (3) = ? ln K, 2

(7.7)

with the same K -matrix as in ii). However a simpler proposal [6] is to de?ne a ※relativistic strain tensor§ in the material space Si j = 污ik Gkj ,
(M )

(7.8)

j with locally Si j = 汛i when there is local relaxation of the material (in this case physical spacelike distances between material points near a point z ? (而, 考 ) agree with their material ﹟ ﹟ ﹟ (M ) distances). Since n = 灰123 det Gij = det 污ij det Gij , we have n = det Si j for the local rest-frame matter number density. We can now de?ne the local rest-frame energy per unit volume of the material n(而, 考 )e(而, 考 ), where e denotes the molar local rest-frame energy (moles = number of particles)

e(而, 考 ) = m + uI (而, 考).

(7.9)

Here m is the molar local rest mass, uI is the amount of internal energy (per mole of the material) of the elastic deformations, accumulated in an in?nitesimal portion during the deformation from the locally relaxed state to the actual state of strain. 42

For isotropic media uI may depend on the deformation only via the invariants of the strain tensor. Let us notice that for an anisotropic material (like a crystal) the energy uI may depend upon the orientation of the deformation with respect to a speci?c axis, re?ecting the microscopic composition of the material: this information may be encoded in a vector ?eld 1 i j E i (而, 考 ) in the material space and one may assume uI = uI (G? ij E E ). (M ) The function e = e[汐i , Gij , 污ij , ...] describes the dependence of the energy of the material upon its state of strain and plays the role of an ※equation of state§ or ※constitutive equation§ of the material. In the weak strain approximation of an isotropic elastic continuum (Hooke approximation) the function uI depends only on the linear (h = Si i ) and quadratic (q = Si j S j i ) invariants of the strain tensor and coincides with the standard formula of linear elasticity 1 [V = n = ﹟ 1 j is the speci?c volume],
det Si

uI = 竹(V )h2 + 2?(V )q + O (cubic invariants), where 竹 and ? are the Lam? e coe?cients. The action principle for this description of relativistic materials is S [4 g, 汐i, ?汐i ] = d而 d3 考L(而, 考 ) = ? ﹟ d而 d3 考 (N 污 )(而, 考) n(而, 考 ) e(而, 考).

(7.10)

(7.11)


A It is shown in Refs. [6,33] that the canonical stress-energy- momentum tensor TB ’ = ’ ’ ’ ?L A A i A pi ?B ’ 汐 ? 汛B ’ L, where pi = ? ?? ’ 汐i is the relativistic Piola-Kirchho? momentum density, A ?L and coincides with the symmetric energy-momentum tensor TA ’B ’ = ?2 4 A ’ , which satis?es ? g ’B

T AB ; B ’ = 0 due to the Euler-Lagrange equations. This energy-momentum tensor may be written in the following form ﹟ TA ’B ’ ], ’B ’ + ZA ’B ’ = N 污 n [eUA

’’

(7.12)

i j 4 where ZA ’B ’ = Zij ?A ’ 汐 ?B ’ 汐 is the pull-back from the material space to M of the ※response tensor§ of the material

Zij = 2

?e , ?Gij


1 so that de(G) = Zij dGij . 2

(7.13)

B = 0, may be called the relativistic ※stress or Cauchy§ The part 而A ’B ’U ’B ’ , 而A ’B ’ = nZA tensor and contains the ※stress-strain relation§ through the dependence of Zij on Si j implied by the consitutive equation e = e[汐, 污 (M ) , G, ..] of the material. For an isotropic elastic material we get 1 Zij = V pG? ij + B污ij ?e ,B= where p = ? ?V 2 ?e , V ?h (M )

+ CGij ,

(7.14)

C=

2 ?e V ?q

and we get (7.15)

1 1 de(V, h, q ) = ?pdV + V Bdh + V Cdq. 2 2 43

The response parameters describe the reaction of the material to the strain: p is the ※isotropic stress§, while B and C give the anisotropic response as in non-relativistic elesticity 1 [perfect ?uids have e = e(V ), B = C = 0, Zij = V pG? ij and de(V ) = ?pdV is the Pascal law]. See Ref. [35] for a di?erent description of relativistic Hooke law in linear elasticity: there is a 4-dimensional deformation tensor S?糸 = 1 (4 ?? 缶糸 + 4 ?糸 缶? ) and the constitutive equations 2 of the material are given in the form T ?糸 = C (?糸 )(汐汕 ) S汐汕 . In Ref. [6] the theory is also extended to the thermodynamics of isentropic ?ows (no heat conductivity). The function e is considered also as a function of entropy S = S (汐) and de = 1 Z dGij is generalized to 2 ij 1 de = Zij dGij ? SdT. 2 Then e is replaced with the Helmholtz free energy f = e ? T S so to obtain 1 df = Zij dGij ? SdT, 2 (7.17) (7.16)

[for perfect ?uids we get de(V, S ) = ?pdV + T dS , df (V, T ) = ?pdV ? SdT ]. This suggests to consider the temperature T as a strain and the entropy S as the corresponding stress and ’ 而 而 to introduce an extra scalar ?eld 汐而 (而, 考) so that T = const.U A ?A ’ 汐 . The potential 汐 (而, 考 ) has the microscopic interpretation as the retardation of the proper time of the molecules with respect to the physical time calculated over averaged spacetime trajectories of the idealized continuum material. ﹟ In this case the action principle becomes S = ? d而 d3 考 N 污nf (G, T ) (而, 考 ) and one ﹟ gets the conserved energy-momentum tensor TA ’B ’ = N 污n[(f + T S )UA ’B ’ ]. ’ + ZA ’ UB In all these cases one can develop the rest-frame instant form just in the same way as it was done in Section II and III for perfect ?uids, even if it is not possible to obtain a closed form of the invariant mass.

44

VIII. CONCLUSIONS.

In this paper we have studied the Hamiltonian description in Minkowski spacetime associated with an action principle for perfect ?uids with an equation of state of the form 老 = 老(n, s) given in Ref. [1], in which the ?uid is descrbed only in terms of Lagrangian coordinates. This action principle can be reformulated on arbitrary spacelike hypersurfaces embedded in Minkowski spacetime (covariant 3+1 splitting of Minkowski spacetime) along the lines of Refs. [10,11]. At the Hamiltonian leve the canonical Hamiltonian vanishes and the theory is governed by four ?rst class constraints H? (而, 考 ) > 0 implying the independence of the description from the choice of the 3+1 splitting of Minkowski spacetime. These constraints can be obtained in closed form only for the &dust* and for the &photon gas*. For other types of perfect ?uids one needs numerical calculations. After the inclusion of the coupling to the gravitational ?eld one could begin to think to formulate Hamiltonian numerical gravity with only physical degrees of freedom and hyperbolic Hamilton equations for them [like the form (2.50) of the relativistic Euler equations for the dust]. After the canonical reduction to 3+1 splittings whose leaves are spacelike hyperplanes, we consider all the con?gurations of the perfect ?uid whose conserved 4-momentum is timelike. For each of these con?gurations we can select the special foliation of Minkowski spacetime with spacelike hyperplanes orthogonal to the 4-momentum of the con?guration, This gives rise to the ※Wigner-covariant rest-frame instant form of dynamics§ [10,11] for the perfect ?uids. After a discussion of the ※external§ and ※internal§ centers of mass and realizations of the Poincar? e algebra, rest-frame Dixon*s Cartesian multipoles [20] of the perfect ?uid are studied. It is also shown that the formulation of non-dissipative elastic materials of Ref. [6], based on the use of Lagrangian coordinates, allows to get the rest-frame instant form for these materials too. Finally it is shown how to make the coupling to the gravitational ?eld by giving the ADM action for the perfect ?uid in tetrad gravity. Now it becomes possible to study the canonical reduction of tetrad gravity with the perfect ?uids as matter along the lines of Refs. [25每27].

45

APPENDIX A: RELATIVISTIC PERFECT FLUIDS.

As in Ref. [1] let us consider a perfect ?uid in a curved spacetime M 4 with unit 4velocity vector ?eld U ? (z ), Lagrangian coordinates 汐 ? i (z ), particle number density n(z ), energy density 老(z ), entropy per particle s(z ), pressure p(z ), temperature T (z ). Let J ? (z ) = 4 g (z )n(z )U ? (z ) the densitized particle number ?ux vector ?eld, so that we have n = ﹟ ? 4 g?糸 J ? J 糸 / 4 g . Other local thermodynamical variables are the chemical potential or speci?c enthalpy (the energy per particle required to inject a small amount of ?uid into a ?uid sample, keeping the sample volume and the entropy per particle s constant) ?= 1 (老 + p ), n (A1)

the physical free energy (the injection energy at a constant number density n and constant total entropy) a= 老 ? T s, n (A2)

and the chemical free energy (the injection energy at constant volume and constant total entropy) f= 1 (老 + p) ? T s = ? ? T s. n (A3)

Since the local expression of the ?rst law of thermodynamics is d老 = ?dn + nT ds, or dp = nd? ? nT ds, or d(na) = f dn ? nsdT, (A4)

an equation of state for a perfect ?uid may be given in one of the following forms 老 = 老(n, s), or p = p(?, s), or a = a(n, T ). (A5)

By de?nition, the stress-energy-momentum tensor for a perfect ?uid is T ?糸 = ?? 老U ? U 糸 + p(4 g ?糸 ? ?U ? U 糸 ) = ??(老 + p)U ? U 糸 + p 4 g ?糸 , and its equations of motion are T ?糸 ;糸 = 0, 1 (nU ? );? = ﹟4 ?? J ? = 0. g (A7) (A6)

As shown in Ref. [1] an action functional for a perfect ?uid depending upon J ? (z ), 4 g?糸 (z ), s(z ) and 汐 ? i (z ) requires the introduction of the following Lagrange multipliers to implement all the required properties: i) 牟(z ): it is a scalar ?eld named &thermasy*; it is interpreted as a potential for the ?uid 1 ?老 | . In the Lagrangian it is interpreted as a Lagrange multiplier for temperature T = n ?s n implementing the ※entropy exchange constraint§ (sJ ? ),? = 0.

46

ii) ?(z ): it is a scalar ?eld; it is interpreted as a potential for the chemical free energy f . In the Lagrangian it is interpreted as a Lagrange multipliers for the ※particle number conservation constraint§ J ? ,? = 0. iii) 汕i (z ): they are three scalar ?elds; in the Lagrangian they are interpreted as Lagrange multipliers for the ※constraint§ 汐 ? i ,? J ? = 0 that restricts the ?uid 4-velocity vector to be directed along the ?ow lines 汐 ? i = const. Given an arbitrary equation of state of the type 老 = 老(n, s), the action functional is S [4 g?糸 , J ? , s, 汐 ?, ?, 牟, 汕i] = |J | , s) + 4g + J ? [?? ? + s?? 牟 + 汕i ?? 汐 ? i ]}. d4 z {?
4 g老( ﹟

(A8)

By varying the 4-metric we get the standard stress-energy-momentum tensor 2 T ?糸 = ﹟4 汛S = ??老U ? U 糸 + p(4 g ?糸 ? ?U ? U 糸 ) = ??(老 + p)U ? U 糸 + p 4 g ?糸 , g 汛 4 g?糸 (A9)

where the pressure is given by p=n ?老 |s ? 老. ?n (A10)

The Euler-Lagrange equations for the ?uid motion are 汛S 汛J ? 汛S 汛? 汛S 汛牟 汛S 汛s 汛S 汛汐 ?i 汛S 汛汕i = ?U? + ?? ? + s?? 牟 + 汕i ?? 汐 ? i = 0, = ??? J ? = 0, = ??? (sJ ? ) = 0, =?
4g

?老 + J ? ?? 牟 = 0, ?s

= ??? (汕i J ? ) = 0, = J ? ?? 汐 ? i = 0. (A11)

The second equation is the particle number conservation, the third one the entropy exchange constraint and the last one restricts the ?uid 4-velocity vector to be directed along the ?ow lines 汐 ? i = const.. The ?rst equation gives the Clebsch or velocity-potential representation of the 4-velocity U? (the scalar ?elds in this representation are called Clebsch or velocity potentials). The ?fth equations imply the constancy of the 汕i *s along the ?uid ?ow lines, so that these Lagrange multipliers can be expressed as a function of the Lagrangian coordinates. The fourth equation, after a comparison with the ?rst law of thermodynamics, 1 ?老 | for the ?uid temperature. leads to the identi?cation T = U ? ?? 牟 = n ?s n Moreover, one can show that the Euler-Lagrange equations imply the conservation of the stress-energy-momentum tensor T ?糸 ;糸 = 0. This equations can be split in the projection 47

along the ?uid ?ow lines and in the one orthogonal to them: i) The projection along the ?uid ?ow lines plus the particle number conservation give U? T ?糸 ;糸 = ? ?老 U ? ?? s = 0, which is veri?ed due to the entropy exchange constraint. There?s fore, the ?uid ?ow is locally adiabatic, that is the entropy per particle along the ?uid ?ow lines is conserved. ii) The projection orthogonal to the ?uid ?ow lines gives the Euler equations, relating the ?uid acceleration to the gradient of pressure
糸 (4 g?糸 ? ?U? U糸 )T 糸汐 ;汐 = ??(老 + p)U?;糸 U 糸 ? (汛? ? ?U? U 糸 )?糸 p.

(A12)

?老 By using p = n ?n |s ? 老, it is shown in Ref. [1] that these equations can be rewritten as 糸 2(?U[? );糸 ] U 糸 = ??(汛? ? U? U 糸 )

1 ?老 |n ?糸 s. n ?s

(A13)

The use of the entropy exchange constraint allows the rewrite the equations in the form 2V[?;糸 ] U 糸 = T ?? s, (A14)

where V? = ?U? is the Taub current (important for the description of circulation and vorticity), which can be identi?ed with the 4-momentum per particle of a small amount of ?uid to be injected in a larger sample of ?uid without changing the total ?uid volume or the entropy per particle. Now from the Euler-Lagrange we get 2V[?;糸 ] U 糸 = ?2(?[? ? + s?[? 牟 + 汕i ?[? 汐 ? i);糸 ] U 糸 = (s?[? 牟);糸 ] U 糸 = T ?? s, (A15)

and this result implies the validity of the Euler equations. In the non-relativistic limit (nU ? );? = 0, T ?糸 ;糸 = 0 become the particle number (or mass) conservation law, the entropy conservation law and the Euler-Newton equations. See Refs. [23,30] for the post-Newtonian approximation. We refer to Ref. [1] for the complete discussion. The previous action has the advantage on other actions that the canonical momenta conjugate to ? and 牟 are the particle number density and entropy density seen by Eulerian observers at rest in space. The action evaluated on the solutions of the equations of motion is d4 z 4 g (z )p(z ). In Ref. [1] there is a study of a special class of global Noether symmetries of this action associated with arbitrary functions F (汐 ?, 汕i , s). It is shown that for each F there is a con﹟ servation equation ?? (F J ? ) = 0 and a Noether charge Q[F ] = 曳 d3 考 污n (?l? U ? )F (汐 ?, 汕i , s) [曳 is a spacelike hypersurface with future pointing unit normal l? and with a 3-metric with ﹟ determinant 污 ]. For F = 1 inside a volume V in 曳 we get the conservation of particle number within a ?ow tube de?ned by the bundle of ?ow lines contained in the volume V. The factor ?l? U ? is the relativistic &gamma factor* characterizing a boost from the Lagrangian observers with 4-velocity U ? to the Eulerian observers with 4-velocity l? ; thus n(?l? U ? ) is the particle number density as seen from the Eulerian observers. These symmetries describe the changes of Lagrangian coordinates 汐 ? i and the fact that both the Lagrange multipliers ? and 牟 are constant along each ?ow line (so that it is possible to transform any solution 48

to the ?uid equations of motion into a solution with ? = 牟 = 0 on any given spacelike hypersurface). However, the Hamiltonian formulation associated with this action is not trivial, because the many redundant variables present in it give rise to many ?rst and second class constraints. In particular we get: 1) second class constraints: A) 羽J 而 > 0, J 而 ? 羽? > 0; B) 羽s > 0, sJ 而 ? 羽牟 > 0; C) 羽汕s > 0, 汕s J 而 ? 羽汐s > 0. 2) ?rst class constraints: 羽J r > 0, so that the J r *s are gauge variables. Therefore the physical variables are the ?ve pairs: ?, 羽? ; 牟, 羽牟 ; 汐 ? i, 羽汐r and one could study the associated canonical reduction. In Ref. [1] [see its rich bibliography for the references] there is a systematic study of the action principles associated to the three types of equations of state present in the literature, ?rst by using the Clebsch potentials and the associated Lagrange multipliers, then only in terms of the Lagrangian coordinates by inserting the solution of some of the Euler-Lagrange equations in the original action and eventually by adding surface terms. 1) Equation of state 老 = 老(n, s). One has the action S [n, U ? , ?, 牟, s, 汐 ? r , 汕r ; 4 g?糸 ] = ? d4 x
4g

老(n, s) ? nU ? (?? ? ? 牟?? s + 汕r ?? 汐 ?r )

(A16)

﹟ ? 1 ?老 汐 ? 2 ?考 汐 ? 3 灰123 (汐 ? r ), one can If one knows s = s(汐 ? r ) and J ? = J ? (汐 ? r ) = ? 4 g??糸老考 ?糸 汐 ? = S ? d4 x?? [(? + s牟)J ? ], and one can show that it has the form de?ne S ?=S ?[? S 汐r ] = ? d4 x
4 g老( ﹟

|J | , s ). 4g

(A17)

2) Equation of state: p = p(?, s) [V ? = ?U ? Taub vector] S(p) = s(p) [V ? , ?, 牟, s, 汐 ? r , 汕r ; 4 g?糸 ] = ?p V? = d4 x 4 g p(?, s) ? |V | ? (?? ? + s?? 牟 + 汕r ?? 汐 ?r ) ?? |V |
?

(A18)

or by using one of its EL equations V? = ? (?? ? + s?? 牟 + 汕r ?? 汐 ? r ) to eliminate V ? one gets Schutz*s action [? determined by ?2 = ?V ? V? ] ?(p) [?, 牟, s, 汐 S ? r , 汕r ; 4 g?糸 ] = d4 x
4 gp(?, s)

(A19)

3) Equation of state a = a(n, T ). The action is S(a) [J ? , ?, 牟, 汐 ? r , 汕r ; 4 g?糸 ] = |J | d4 x |J |a( ﹟4 , ?? 牟J ? ) ? J ? (?? ? + 汕r ?? 汐 ?r ) g
g

(A20)

At the end of Ref. [1] there is the action for ※isentropic§ ?uids and for their particular case of a ※dust§ (used in Ref. [9] as a reference ?uid in canonical gravity). 49

|J | J ? ?(a) [?, 牟, s, 汐 , |J | ?? 牟) . or S ? r , 汕r ] = S(a) ? d4 x |J |a( ﹟ 4

n) ? sT with s = const. (constant The isentropic ?uids have equation of state a(n, T ) = 老(n ∩ value of the entropy per particle). By introducing ? = ? + s牟, the action can be written in the form

S(isentrpic)[J ? , ? , 汐 ? r , 汕r , 4 g ?糸 ] = or



d4 x ?

4 g老( ﹟

|J | ∩ ? ) + J ( ? ? + 汕r ?? 汐 ?r ) ? 4g

(A21)

?(isentropic) [? S 汐r ; 4 g?糸 ] = ?

d4 x

4 g老( ﹟

|J | ) 4g

(A22)

The dust has equation of state 老(n) = ?n, namely a(n, T ) = ? ? sT so that we get zero ∩ ?老 pressure p = n ?n ? 老 = 0. Again with ? = ? + s牟 the action becomes S(dust) J ? , ? , 汐 ? r , 汕r , 4 g ?糸 ] =
∩ ∩

d4 x ? ?|J | + J ? (?? ? + 汕r ?? 汐 ?r )



(A23)

1 (?? ? + 汕r ?? 汐 ? r ) [In Ref. [9]: M = ?n rest mass (energy) density and or with U? = ? ? ∩ T = ? /?, Wr = ?汕r , Z r = 汐 ? r ; U? = ??? T + Wr ?? Z r ]

S(dust) [T, Z r , M, Wr ; 4 g?糸 ] = ? or



1 2

d4 x

4 g (?n)

U? 4 g ?糸 U糸 ? ? ,

(A24)

?(dust) [? S 汐r ; 4 g?糸 ] = ?

d4 x?|J |

(A25)

In Ref. [9] there is a study of the action (A24) since the dust is used as a reference ?uid in general relativity. At the Hamiltonian level one gets: r i) 3 pairs of second class constraints [羽W (而, 考 ) > 0, 羽Z r (而, 考) ? Wr (而, 考)羽T (而, 考) > 0], which r allow the elimination of Wr (而, 考) and 羽W (而, 考 ); ii) a pair of second class constraints [ 羽M (而, 考 ) > 0 plus the secondary M (而, 考 ) ? 2 羽T (而, 考 ) > 0], which allow the elimination of M, 羽M . ﹟ ﹟ 2 3 rs u v
污 羽T + g (羽T ?r T +羽Z u ?r Z )(羽T ?s T +羽Z v ?s Z )

50

APPENDIX B: COVARIANT RELATIVISTIC THERMODYNAMICS OF EQUILIBRIUM AND NON-EQUILIBRIUM.

In this Appendix we shall collect some results on relativistic ?uids which are well known but scattered in the specialized literature. We shall use essentially Ref. [2], which has to be consulted for the relevant bibliography. See also Ref. [36]. Firstly we remind some notions of covariant thermodynamics of equilibrium. Let us remember that given the stress-energy-momentum tensor of a continuous medium ?糸 T , the densities of energy and momentum are T oo and c?1 T ro respectively [so that dP ? = c?1 灰T ?糸 d曳糸 is the 4-momentum that crosses the 3-area element d曳糸 in the sense of its normal (灰 = ?1 if the normal is spacelike, 灰 = +1 if it is timelike)]; instead, cT or is the energy ?ux in the positive r direction, while T rs is the r component of the stress in the plane perpendicular to the s direction (a pressure, if it is positive). A local observer with timelike 4-velocity u? (u2 = ?c2 ) will measure energy density c?2 T ?糸 u? u糸 and energy ?ux ?T ?糸 u? n糸 along the direction of a unit vector n? in his rest frame. For a ?uid at thermal equilibrium with T ?糸 = 老U ? U 糸 ? ? cp2 (4 g ?糸 ? ?U ? U 糸 ) [U ? is the hydrodynamical 4-velocity of the ?uid] with particle number density n, speci?c volume S 1 V =n and entropy per particle s = kB (kB is Boltzmann*s constant) in its rest frame, the n energy density is 老c2 = n(mc2 + e), (B1)

where e is the mean internal (thermal plus chemical) energy per particle and m is particle*s rest mass. From a non-relativistic point of view, by writing the equation of state in the form s = s(e, V ) the temperature and the pressure emerge as partial derivatives from the ?rst law of thermodynamics in the form (Gibbs equation) ds(e, V ) = 1 (de + pdV ). T (B2)

If ?clas = e + pV ? T s is the non-relativistic chemical potential per particle, its relativistic version is ? = mc2 + ?class = ? ? T s, [? = get
老c2 +p n


(B3)

is the speci?c enthalpy, also called chemical potential as in Appendix A] and we ? n = 老c2 + p ? nT s = 老c2 + p ? kB T S, ∩ kB T dS = d(老c2 ) ? ? dn = d(老c2 ) ? (? ? T s)dn, ∩ d(老c2 ) = ? dn + T d(ns) = ?dn + nT ds.


or (B4)

By introducing the ※thermal potential§ 汐 = 2 汕 = kc , these two equations take the form BT

? kB T



=

??T s kB T

and the inverse temperature

51

p ns = 汕 (老 + 2 ) ? 汐n, kB c dS = 汕d老 ? 汐dn. S=

(B5)

Let us remark that in Refs. [23,29,37] one uses di?erent notations, some of which are ∩ given in the following equation (in Ref. [23] 老 is denoted e and 老 is denoted r ) p p p ∩ ∩ ∩ ∩ ∩ = n(mc2 + e) + 2 = 老 h = 老 (c2 + e + 2 ∩ ) = 老 (c2 + h ), (B6) 2 c c c老 ﹟ ∩ ∩ where 老 = nm is the rest-mass density [r? = 4 g老 is called the coordinate rest-mass ∩ ∩ density] and e = e/m is the speci?c internal energy [so that 老 h is the ※e?ective inertial mass of the ?uid; in the post-Newtonian approximation of Ref. [23] it is shown that 考 = ﹟ o c?2 (T oo + s T ss ) + O (c?4) = c?2 4 g (?To + Tss ) + O (c?4) has the interpretation of equality of the ※passive§ and the ※active§ gravitational mass]. For the speci?c enthalpy or chemical ∩ potential we get [?/m = h = c2 + h is called enthalpy] 老+ ?= p m p 1 ∩ (老 + 2 ) = ∩ (老 + 2 ) = mh = m(c2 + h ). n c 老 c (B7)

See Ref. [29] for a richer table of conversion of notations. Relativistically, we must consider, besides the stress-energy-momentum tensor T ?糸 and the associated 4-momentum P ? = V d3 曳糸 T ?糸 , a particle ?ux density n? (one n? a for each constituent a of the system) and the entropy ?ux density s? . At thermal equilibrium all these a priori unrelated 4-vectors must all be parallel to the hydrodynamical 4-velocity n? = nU ? , s? = sU ? , P ? = P U ?, (B8)
2

c Analogously, we have V ? = V U ? (V = 1/n is the speci?c volume), 汕 ? = 汕U ? = kB U? T ∩ [a related 4-vector is the equilibrium parameter 4-vector i? = ? 汕 ? ]. Since ?U? T ?糸 = 老U 糸 , we get the ?nal manifestly covariant form of the previous two equations (now the hydrodynamical 4-velocity is considered as an extra thermodynamical variable)

p ns ? U = 2 汕 ? ? 汐n? ? ?汕糸 T 糸? , kB c dS ? = ?汐dn? ? ?汕糸 dT 糸? . S ? = SU ? = Global thermal equilibrium imposes ?? 汐 = ?? 汕糸 + ?糸 汕? = 0. As a consequence we get d( p ? 汕 ) = n? d汐 + ?T 糸? d汕糸 , c2

(B9)

(B10)

(B11)

namely the basic variables n? , T 糸? and S ? can all be generated from partial derivatives of the ※fugacity§ 4-vector (or ※thermodynamical potential§) 52

耳? (汐, 汕竹 ) =

p ? 汕 , c2 ?耳? n? = , ?汐

T(糸? mat) =

?耳? , ?汕糸

S ? = 耳? ? 汐n? ? 汕糸 T(糸? mat) ,

(B12)

once the equation of state is known. Here T(糸? mat) is the canonical or material (in general non symmetric) stress tensor, ensuring that reversible ?ows of ?eld energy are not accompanied by an entropy ?ux. This ?nal form remains valid (at least to ?rst order in deviations) for ※states that deviate from equilibrium§, when the 4-vectors S ? , n? ,... are no more parallel; the extra information in this equation is precisely the standard linear relation between entropy ?ux and heat ?ux. The second law of thermodynamics for relativistic systems is ?? S ? ≡ 0, which becomes a strict equality in equilibrium. The fugacity 4-vector 耳? is evaluated by using the covariant relativistic statistical theory for thermal equilibrium [2] starting from a grand canonical ensemble with density matrix 老 ? by maximizing the entropy S = ?T r (? 老ln 老 ?) subject to the constraints T r 老 ? = 1, T r (? 老n ? ) = n, ? 竹) = P 竹: this gives (in the large volume limit) T r (? 老P
? +汕 ? P , 老 ? = Z ?1 e汐n with ?耳? d曳? , ln Z = →曳 ??

n=

→曳

?n? d曳? ,

P? =

→曳

?T(?糸 mat) d曳糸 ,

(B13)

[it is assumed that the members of the ensemble are small (macroscopic) subregions of one extended body in thermal equilibrium, whose worldtubes intersect an arbitrary spacelike hypersurface in small 3-areas →曳]. Therefore, one has to ?nd the grand canonical partition function Z (V? , 汕? , i? ) =
n

ei? n Qn (V? , 汕? ),

?

where Qn (V? , 汕? ) =

V?

d考n (q, p)e?汕?P ,

?

(B14)

is the canonical partition function for ?xed volume V? and d考n is the invariant microcanonical density of states. For an ideal Boltzmann gas of N free particles of mass m [see Section III for its equation of state] it is d考n (p, m) = 1 N!
N N

汛 4 (P ?

pi )
i=1 i=1

o 2 2 4 2 V ? p? i 牟 (pi )汛 (pi ? ?m )d pi .

(B15)

Following Ref. [38] [using a certain type of gauge ?xings to the ?rst class constraints ? ?m2 > 0] in Ref. [10] Qn was evaluated in the rest-frame instant form on the Wigner hyperplane (this method can be extended to a gas of molecules, which are N-body bound states): p2 i Qn = 1 V m2 K2 (m汕 ) N ! 2羽 2 汕
N

.

(B16)

The same results may be obtained by starting from the covariant relativistic kinetic theory of gas [see Ref. [39,40]; in Ref. [2] there is a short review] whose particles interact 53

only by collisions by using Synge*s invariant distribution function N (q, p) [41] [the number of particle worldlines with momenta in the range (p? , d肋 ) that cross a target 3-area d曳? in M 4 in the direction of its normal is given by dN = N (q, p)d肋灰v ?d曳? (= Nd3 qd3 p for the 3-space ﹟ q o = const.); d肋 = d3 p/v o 4 g is the invariant element of 3-area on the mass-shell]. One = ?? (Nv ? ) [v ? is the particle velocity obtained from arrives at a transport equation for N , dN d而 the Hamilton equation implied by the one-particle Hamiltonian H = ? 4 g ?糸 (q )p? p糸 = m (it is the energy after the gauge ?xing q o > 而 to the ?rst class contraint 4 g ?糸 (q )p? p糸 ? ?m2 > 0); ?? is the covariant gradient holding the 4-vector p? (not its components) ?xed] with a ※collision term§ C [N ] describing the collisions; for a dilute simple gas dominated by binary collisions one arrives at the Boltzmann equation [for C [N ] = 0 one solution is the relativistic version of the Maxwell-Boltzmann distribution function, i.e. the classical J“ uttner?汕 ? P ? 2 Synge one N = const.e /4羽m K2 (m汕 ) for the Boltzmann gas [41]]. The H-theorem ? ? [?? S ≡ 0, where S (q ) = ? [Nln(Nh3 ) ? N ]v ? d肋 is the entropy ?ux] and the results at thermal equilibrium emerge [from the balance law ?? ( Nf v ? d肋 ) = f C [N ]d肋 (f is an arbitrary tensorial function) one can deduce the conservation laws ?? n? = ?? T 糸? = 0, where n? = Nv ? d肋 , T糸 ? = Np糸 v ? d肋 ; the vanishing of entropy production at local 糸 thermal equilibrium gives Neq (q, p) = h?3 e汐(q)+汕糸 (q)P in the case of Boltzmann statistic 糸? and one gets (U ? = 汕 ? /汕 ) n? Neq v ? d肋 = nU ? , Teq = 老U 糸 U ? ? ?p(4 g 糸? ? ?U 糸 U ? ), eq = ? ? ? ?糸 Seq = p汕 ? 汐neq ? 汕糸 Teq ; one obtains the equations for the Boltzmann ideal gas given in Section III]. One can study the small deviations from thermal equilibrium [N = Neq (1+ f ), where Neq is an arbitrary local equilibrium distribution] with the linearized Boltzmann equation and then by using either the Chapman-Enskog ansatz of quasi-stationarity of small deviations (this ignores the gradients of f and gives the standard Landau-Lifshitz and Eckart phenomenological laws; one gets Fourier equation for heat conduction and the Navier-Stokes equation for the bulk and shear stresses; however one has parabolic and not hyperbolic equations implying non-causal propagation) or with the Grad method in the 14-moment approximation. This method retains the gradients of f [there are 5 extra thermodynamical variables, which can be explicitly determined from 14 moments among the in?nite set of moments Np? p糸 p老 ...d4 p of kinetic theory; no extra auxiliary state variables are introduced to specify a non-equilibrium state besides T ?糸 , n? , S ? ] and gives phenomenological laws which are the kinetic equivalent of M“ uller extended thermodynamics and its various developments; now the equations are hyperbolic, there is no causality problem but there are problems with shock waves. See Ref. [2] for the bibliography and for a review of the non-equilibrium phenomenological laws (see also Ref. [42]) of Eckart, Landau-Lifshitz, of the various formulations of extended thermodynamics, of non-local thermodynamics. While in Ref. [43] it is said that the di?erence between causal hyperbolic theories and acausual parabolic one is unobservable, in Ref. [44][see also Ref. [24]] there is a discussion of the cases in which hyperbolic theories are relevant. See also the numerical codes of Refs. [45,31,32]. In phenomenological theories the starting point are the equations ?? T ?糸 = ?? n? = 0, ?? S ? ≡ 0. There is the problem of how to de?ne a 4-velocity and a rest-frame for a given non-equilibrium state. Another problem is how to specify a non-equilibrium state completely at the macroscopic level: a priori one could need an in?nite number of auxiliary quantities (vanishing at equilibrium) and an equation of state depending on them. The basic postulate 54

of extended thermodynamics is the absence of such variables. Regarding the rest frame problem there are two main solutions in the literature connected with the relativistic description of ※heat ?ow§: i) Eckart theory. One considers a local observer in a simple ?uid who is at rest with respect to the average motion of the particles: its 4-velocity U(? eck ) is parallel by de?nition ? to the particle ?ux n , namely n? = n(eck) U(? eck ) . This local observer sees ※heat ?ow§ as a ?ux of energy in his rest frame:
? ?U(eck)? T ?糸 = 老(eck) U(? eck ) + q(eck ) , so that we get ?糸 ? ? 糸 糸 糸 ?糸 T = 老(eck) U(? eck ) U(eck ) + q(eck ) U(eck ) + U(eck ) q(eck ) + P(eck ) , 糸? ?糸 4 ?糸 糸 P(?糸 ? ?U(? eck ) = P(eck ) = ?(p + 羽(eck ) )( g eck ) U(eck ) + 羽(eck ) , ? P(?糸 eck ) U(eck )糸 = q(eck )? U(eck ) = 0, 糸 羽(eck)?糸 (4 g ?糸 ? ?U(? eck ) U(eck ) ) = 0,

(B17)

(B18)

?糸 where p is the thermodynamic pressure, 羽(eck) the bulk viscosity and 羽( eck ) the shear stress. This description has the particle conservation law ?? n? = 0. ii) Landau-Lifshitz theory. One considers a di?erent observer (drifting slowly in the direction of heat ?ow with a 3-velocity vD = q/nmc2 ) whose 4-velocity U(? ll) is by de?nition such to give a vanishing ※heat ?ow§, i.e. there is no net energy ?ux in his rest frame: ? ? 糸 U(? ll) T? n糸 = 0 for all vectors n? orthogonal to U(ll) . This implies that U(ll) is the timelike ? eigenvector of T ?糸 , T ?糸 U(ll)糸 = ?老(ll) U(ll) , which is unique if T ?糸 satis?es a positive energy condition. Now we get ?糸 糸 T ?糸 = 老(ll) U(? ll) U(ll) + P(ll) , 糸? ?糸 4 ?糸 糸 P(?糸 ? ?U(? ll) = P(ll) = ?(p + 羽(ll) )( g ll) U(ll) + 羽(ll) ,

P(?糸 ll) U(ll)糸 = 0,
? n? = n(ll) U(? ll) + j(ll) ,

糸 羽(ll)?糸 (4 g ?糸 ? ?U(? ll) U(ll) ) = 0,

j(ll)? U(? ll) = 0

(j = ?nvD = ?q/mc2 ).

(B19)

This observer in his rest frame does not see a heat ?ow but a particle drift. This description has the simplest form of the energy-momentum tensor. One has n(eck) = n(ll) ch ?, 老(eck) = 老(ll) ch2 ? + p(ll) sh2 ? = 羽 ?糸 j? j糸 /n2 (eck ) , with ch ? =
2 U(? 1 ? vD /c2 , so that there are insigni?cant ll) U(eck )? [the di?erence is a Lorentz factor di?erences for many practical purposes if deviations from equilibrium are small]. The angle ? > j/n > vD /c > q/nmc2 is a dimensionless measure of the deviation from equilibrium [n(ll) ? n(eck) and 老(ll) ? 老(eck) are of order ?2 ].

55

One can decompose T ?糸 , n? , in terms of any 4-velocity U ? that falls within a cone of ? ? ? angle > ? containing U(? eck ) and U(ll) ; each choice U gives a particle density n(U ) = ?u? n ?糸 ? and energy density 老(U ) = U? U糸 T which are independent of U if one neglects terms of order ?2 . Therefore, one has: i) if Seq (老(U ), n(U )) is the equilibrium entropy density, then S (U ) = ?U? S ? = Seq + O (?2 );
?老/n ii) if p(U ) = ? ? | is the (reversible) thermodynamical pressure de?ned as work done 1/n S/n in an isentropic expansion (o? equilibrium this de?nition allows to sepate it from the bulk stress 羽 (U ) in the stress-energy-momentum tensor) and peq is the pressure at equilibrium, then p(U ) = Peq (老(U ), n(U )) + O (?2). By postulating that the covariant Gibbs relation remains valid for arbitrary in?nitesimal displacements (汛n? , 汛T ?糸 , ..) from an equilibrium state, one gets a covariant o?-equilibrium thermodynamics based on the equation

S ? = p(汐, 汕 )汕 ? ? 汐n? ? 汕糸 T ?糸 ? Q? (汛n糸 , 汛T 糸老, ..), ?? S ? = ?汛n? ?? 汐 ? 汛T ?糸 ?糸 汕? ? ?? Q? ≡ 0,

(B20)

with Q? of second order in the displacements and 汐, 汕? arbitrary. At equilibrium one recovers ? ?糸 ? ? ? Seq = p汕 ? ? 汐n? eq ? 汕糸 Teq , ?? Seq = 0 [with U = 汕 /汕 and (for viscous heat-conducting ?uids, but not for super?uids) ?? 汐 = ?? 汕糸 + ?糸 汕? = 0]. If we choose 汕 ? = U ? /kB T parallel to n? of the given o?-equilibrium state, we are in the ※Eckart frame§, U ? = U(? eck ) , and we get S = ?U(eck)? S ? = Seq + ?U(eck)? Q? ,
? ? 4 ?糸 糸 4 ?糸 糸 考( ? ?U(? ? ?U(? eck ) = ( g eck ) U(eck ) )S糸 = 汕q(eck ) ? ( g eck ) U(eck ) )Q糸 , ? 老 糸 4 ?糸 ? ?U(? q( eck ) U(eck ) )T糸 U(eck )老 , eck ) = ?( g

(B21)

so that to linear order we get the standard relation between entropy ?ux 考(eck) and heat ?ux q(eck) 考(eck) = q(eck) + (possible 2nd order term). kB T (B22)

?糸 ? 4 糸 If we choose U ? = U(? ll) , the timelike eigenvector of T , so that U(ll)? T糸 ( g老 ? ?U(糸ll) U(ll)老 ) = 0, we are in the ※Landau-Lifshitz frame§ and we get ? ? 糸 4 ?糸 糸 4 ?糸 ? ?U(? ? ?U(? 考( ll) U(ll) )Q糸 , ll) U(ll) )S糸 = ?汐j(ll) ? ( g ll) = ( g ? 4 ?糸 糸 j( ? ?U(? ll) = ( g ll) U(ll) )n糸 ,

(B23)

so that at linear order we get the standard relation between entropy ?ux 考(ll) and di?usive ?ux j(ll) 56

考(ll) = ?

? j(ll) + (possible 2nd order term). kB T

(B24)

In the Landau-Lifschitz frame heat ?ow and di?usion are englobed in the di?usive ?ux j(ll) relative to the mean mass-energy ?ow.

The entropy inequality becomes (each term is of second order in the deviations from local equilibrium) 0 ≒ ?? S ? = ?汛n? ?? 汐 ? 汛T ?糸 ?糸 汕? ? ?? Q? , (B25)

with the ?tting conditions 汛n? U? = 汛T ?糸 U? U糸 = 0, which contain all information about the viscous stresses, heat ?ow and di?usion in the o?-equilibrium state (they are dependent on the arbitrary choice of the 4-velocity U ? ). Once a detailed form of Q? is speci?ed, linear relations between irreversible ?uxes 汛T ?糸 , 汛n? and gradients ?(? 汕糸 ) , ?? 汐 follow. A) Q? = 0 (like in the non-relativistic case). The spatial entropy ?ux 考 is only a strictly linear function of heat ?ux q and di?usion ?ux j . In this case the o?-equilibrium entropy density S = ?U? S ? is given by the equilibrium equation of state S = Seq (老, n). We have 0 ≒ ?? S ? = ?汛n? ?? 汐 ? 汛T ?糸 ?糸 汕? with ?tting conditions 汛n? U? = 汛T ?糸 U? U糸 = 0 and with U ? still arbitrary at ?rst order.
?糸 A1) Landau-Lifschitz frame and theory. U ? = U(? ll) is the timelike eigenvector of T . ?糸 This and the ?tting conditions imply 汛T ?糸 U(ll)糸 = 0. The shear and bulk stresses 羽( ll) , 羽(ll) are identi?ed by the decomposition ?糸 糸 4 ?糸 ? ?U(? 汛T ?糸 = 羽( ll) U(ll) ), ll) + 羽(ll) ( g ? ?糸 羽( ll) U(ll)糸 = 羽(ll) ? = 0.

(B26)

The inequality ?? S ? ≡ 0 becomes
? ?糸 ? ?j( ll) ?? 汐 ? 汕羽(ll) < ?糸 汕? > ?汕羽(ll) ?? U(ll) ≡ 0, 汕 4 汕 汐 < X?糸 > = [(4 g? ? ?U(汐 ll) U(ll)? )( g糸 ? ?U(ll) U(ll)糸 ) ? 1 汕 ? (4 g?糸 ? ?U(ll)? U(ll)糸 )(4 g 汐汕 ? ?U(汐 ll) U(ll) )]X汐汕 , 3 ? ? j( ll) = 汛n ,

(B27)

[the < .. > operation extracts the purely spatial, trace-free part of any tensor].
? ?糸 If the equilibrium state is isotropic (Curie*s principle) and if we assume that (j( ll) , 羽(ll) , 羽(ll) ) are ※linear and purely local§ functions of the gradiants, ?? S ? ≡ 0 implies ? 4 ?糸 糸 j( ? ?U(? ll) = ?百( g ll) U(ll) )?? 汐,

百 > 0,

(B28)

[it is a mixture of Fourier*s law of heat conduction and of Fick*s law of di?usion, stemming from the relativistic mass-energy equivalence], 57

and the standard Navier-Stokes equations (汎S , 汎V are shear and bulk viscosities) 羽(ll)?糸 = ?2汎S < ?糸 汕? >, 1 羽(ll) = 汎V ?? U(? ll) . 3 (B29)

? A2) Eckart frame and theory. U(? eck ) parellel to n . Now we have the ?tting condition 汛n? = 0. The heat ?ux appears in the decomposition of 汛T ?糸 [a(eck)? = U(糸eck) ?糸 U(eck)? is the 4-acceleration] ?糸 ? ? 糸 4 ?糸 糸 糸 ? ?U(? 汛T ?糸 = q( eck ) U(eck ) ). eck ) U(eck ) + U(eck ) q(eck ) + 羽(eck ) + 羽(eck ) ( g

(B30)

The inequality ?? S ? ≡ 0 becomes
? ?糸 ? q( eck ) (?? 汐 ? 汕a(eck )? ) ? 汕 (羽(eck ) ?糸 U(eck )? + 羽(eck ) ?? U(eck ) ) ≡ 0.

(B31)

With the simplest assumption of linearity and locality, we obtain Fourier*s law of heat conduction [it is not strictly equivalent to the Landau-Lifshitz one, because they di?er by spatial gradients of the viscous stresses and the time-derivative of the heat ?ux]
? 糸 4 ?糸 ? ?U(? q( eck ) U(eck ) )(?糸 T + T ?而 U(eck )糸 ), eck ) = ?百( g

(B32)

[the term depending on the acceleration is sometimes referred to as an e?ect of the ※inertia ?糸 of heat§], and the same form of the Navier-Stokes equations for 羽( eck ) , 羽(eck ) (they are not strictly equivalent to the Landau-Lifshitz ones, because they di?er by gradients of the drift vD = q/nmc2 ). For a simple ?uid Fourier*s law and Navier-Stokes equations (9 equations) and the conservation laws ?? T ?糸 = ?? n? = 0 (5 equations) determine the 14 variables T ?糸 , n? from suitable initial data. However, these equations are of mixed parabolic-hyperbolic- elliptic type and, as said, one gets acausality and instability. Kinetic theory gives Q? = ? 1 2 Neq f 2 p? d肋 = 0, (B33)

for a gas up to second order in the deviation (N ? Neq ) = Neq f [Q? = 0 requires small gradients and quasi-stationary processes]. Two alternative classes of phenomenological theories are B) Linear non-local thermodynamics (NLT). This theory gives a rheomorphic rather than causal description of the phenomenological laws: transport coe?cients at an event x are taken to depend, not on the entire causal past of x, but only on the past history of a ※comoving local ?uid element§. It is a linear theory restricted to small deviations from equilibrium, which can be derived from the linearized Boltzmann equation by projector-operator techniques (and probably inherits its causality properties). Instead of writing (汛n? (x), 汛T?糸 (x)) = 考 (U, T )(??? 汐(x), ??(? 汕糸 ) (x)), 58

this local phenomenological law is generalized to (汛n? (x, xo ), 汛T?糸 (x, xo )) = ∩ ∩ ∩ xo )(??? 汐(xxo ), ??(? 汕糸 ) (xxo )). C) Local non-linear extended thermodynamics (ET).

﹢ ?﹢

dxo 考 (xo ?



It is more relevant for relativistic astrophysics, where correlation and memory e?ects are not of primary interest and, instead, one needs a tractable and consistent transport theory coextensive at the macroscopic level with Boltzmann*s equation. It is assumed that the second order term Q? (汛n糸 , 汛T 糸老 , ..) does not depend on auxiliary variables vanishing at equilibrium: this ansatz is the phenomenological equivalent of Grad*s 14-moment approximation in kinetic theory. These theories are called ※second-order theories§ and many of them are analyzed in Ref. [46]; when the dissipative ?uxes are subject to a conservation equation, these theories are called of causal ※divergence type§ like the ones of Refs. [47每49]. Another type of theory (extended irreversible thermodynamics; in general these theories are not of divergence type) was developed in Refs. [50每53]: in it there are transport equations for the dissipative ?uxes rather than conservation laws. For small deviations one retains only the quadratic terms in the Taylor expansion of Q? (leading to ※linear§ phenomenological laws): this implies 5 new undetermined coe?cients 1 Q? = U ? [汕o 羽 2 + 汕1 q ? q? + 汕2 羽 ?糸 羽?糸 ] ? 汐o 羽q ? ? 汐1 羽 ?糸 q糸 , 2 (B34)

with 汕i > 0 from ?U? Q? > 0 (the 汕i *s are &relaxation times*). A ?rst-order change of rest frame produces a second-order change in Q? [going from the Landau-Lifshitz frame to the Eckart one, one gets 汐(eck)i ? 汐(ll)i = 汕(ll)1 ? 汕(eck)1 = [(老 + p)T ]?1 , 汕(ll)0 = 汕(eck)o , 汕(ll)2 = 汕(eck)2 , and the phenomenological laws are now invariant to ?rst order]. In the ※Eckart frame§ the phenomenological laws take the form
? 4 ?糸 糸 ?1 ? ?U(? ?糸 T + a(eck)糸 + 汕(eck)1 ?而 q(eck)糸 ? q( eck ) = ?百T ( g eck ) U(eck ) )[T 老 ? 汐(eck)o ?糸 羽(eck) ? 汐(eck)1 ?老 羽( eck )糸 ],

羽(eck)?糸 = ?2汎S [< ?糸 U(eck)? > +汕(eck)2 ?而 羽(eck)?糸 ? 汐(eck)1 < ?糸 q(eck)? >], 1 ? 羽(eck) = ? 汎V [?? U(? eck ) + 汕(eck )o ?而 羽(eck ) ? 汐(eck )o ?? q(eck ) ], 3 (B35)

which reduce to the equation of the standard Eckart theory if the 5 relaxation (汕i ) and coupling (汐i ) coe?cients are put equal to zero. See for instance Ref. [54] for a complete treatment and also Ref. [55]. For appropriate values of these coe?cients these equations are hyperbolic and, therefore, causal and stable. The transport equations can be understood [44] as evolution equations for the dissipative variables as they describe how these ?uxes evolve from an initial arbitrary state to a ?nal steady one [the time parameter 而 is usually interpreted as the relaxation time of the dissipative processes]. In the case of a gas the new coe?cients can be found explicitly [52] [see also Ref. [56] for a recent approach to relativistic 59

interacting gases starting from the Boltzmann equation], and they are purely thermodynamical functions. Wave front speeds are ?nite and comparable with the speed of sound. A problem with these theories is that they do not admit a regular shock structure (like the Navier-Stokes equations) once the speed of the shock front exceeds the highest characteristic velocity (a ※subshock§ will form within a shock layer for speeds exceeding the wave-front velocities of thermo-viscous e?ects). The situation slowly ameliorates if more moments are taken into account [54]. In the approach reviewed in Ref. [54] the extra indeterminacy associated to the new 5 coe?cients is eliminated (at the price of high non-linearity) by annexing to the usual conservation and entropy laws a new phenomenological assumption (in this way one obtains a causal divergence type theory): ?老 A?糸老 = I ?糸 , in which A老?糸 and I ?糸 are symmetric tensors with the following traces A?糸 糸 = ?n? , I ? ? = 0. (B37) (B36)

These conditions are modelled on kinetic theory, in which A老?糸 represents the third moment of the distribution function in momentum space, and I ?糸 the second moment of the collision term in Boltzmann*s equation. The previous equations are central in the determination of the distribution function in Grad*s 14-moment approximation. The phenomenological theory is completed by the postulate that the state variables S ? , A老?糸 , I ?糸 are invariant functions of T ?糸 , n? only. The theory is an almost exact phenomenological counterpart of the Grad approximation. See Ref. [37] for the beginning (only non viscous heta conducting materials are treated) of a derivation of extended thermodynamics from a variational principle. Everything may be rephrased in terms of the Lagrangian coordinates of the ?uid used in this paper. What is lacking in the non-dissipative case of heat conduction is the functional form of the o?-equilibrium equation of state reducing to 老 = 老(n, s) at thermal equilibrium. In the dissipative case the system is open and T ?糸 , P ? , n? are not conserved. See Ref. [57] for attempts to de?ne a classical theory of dissipation in the Hamiltonian framework and Ref. [58] about Hamiltonian molecular dynamics for the addition of an extra degree of freedom to an N-body system to transform it into an open system (with the choice of a suitable potential for the extra variable the equilibrium distribution function of the N-body subsystem is exactly the canonical ensemble). However, the most constructive procedure is to get (starting from an action principle) the Hamiltonian form of the energy-momentum of a closed system, like it has been done in Ref. [17] for a system of N charged scalar particles, in which the mutual action-at-a-distance interaction is the complete Darwin potential extracted from the Lienard-Wiechert solution in the radiation gauge (the interactions are momentum- and, therefore, velocity-dependent). In this case one can de?ne an open (in general dissipative) subsystem by considering a cluster of n < N particles and assigning to it a non-conserved energy-momentum tensor built with all the terms of the original energy-momentum tensor which depend on the canonical variables of the n particles (the other N ? n particles are considered as external ?elds). 60

APPENDIX C: NOTATIONS ON SPACELIKE HYPERSURFACES.

Let us ?rst review some preliminary results from Refs. [10] needed in the description of physical systems on spacelike hypersurfaces. Let {曳而 } be a one-parameter family of spacelike hypersurfaces foliating Minkowski spacetime M 4 with 4-metric 灰?糸 = ?(+ ? ??), ? = ㊣ [? = +1 is the particle physics convention; ? = ?1 the general relativity one] and giving a 3+1 decomposition of it. At ?xed 而 , let z ? (而, 考 ) be the coordinates of the points on 曳而 in M 4 , {考 } a system of coordinates on 曳而 . ’ ’ ’ = (而, r }) [the notation A ’) with r ’ = 1, 2, 3 will be used; note that If 考 A = (考 而 = 而 ; 考 = {考 r ’ A ’ = 而 and A ’=r A ’ = 1, 2, 3 are Lorentz-scalar indices] and ?A ’ = ?/?考 , one can de?ne the vierbeins
? ? zA ’ z (而, 考 ), ’ (而, 考 ) = ?A ? ? ?B ’ zB ’ zA ’ = 0, ’ ? ?A

(C1)

so that the metric on 曳而 is
? 糸 gA ’B ’ (而, 考 ) = zA ’ (而, 考 ), ’ (而, 考 )灰?糸 zB

?g而 而 (而, 考 ) > 0,
2

3 污 (而, 考 ) = ?det || gr ’s ’(而, 考 ) || = det || gr ’s ’(而, 考 )||,

? g (而, 考) = ?det || gA ’B ’ (而, 考 ) || = (det || zA ’ (而, 考 ) ||) ,

(C2)

3 3 where gr ’s ’ = ?? gr ’s ’ with gr ’s ’ having positive signature (+ + +). r ’s ’ 3 r ’s ’ r ’u ’ r ’ If 污 (而, 考 ) = ?? g is the inverse of the 3-metric gr ’s ’(而, 考 ) [污 (而, 考 )gu ’s ’(而, 考 ) = 汛s ’ ], the ’ ’C ’ ’B ’ A A A (而, 考 )gc inverse g (而, 考 ) of gA ’B ’ (而, 考 ) [g ’ ] is given by ’’ b (而, 考 ) = 汛B

污 (而, 考) , g (而, 考) 污 污 u ’r ’ 3 u ’r ’ ’ g而 r (而, 考) = ?[ g而 u ’ 污 ](而, 考 ) = ?[ g而 u ’ g ](而, 考 ), g g 污 ’s ’ ’s ’ u ’r ’ v ’s ’ gr (而, 考 ) = 污 r (而, 考) + [ g而 u ’ g而 v ’ 污 污 ](而, 考 ) = g 污 ’s ’ 3 ’s ’ = ?? 3 g r (而, 考 ) + [ g而 u ’r ’ 3gv ](而, 考 ), ’ g而 v ’ gu g g 而 而 (而, 考 ) = so that 1 = g 而 C (而, 考)gC而 ’ (而, 考 ) is equivalent to g (而, 考) ’s ’ = g而 而 (而, 考 ) ? 污 r (而, 考 )g而 r ’(而, 考 )g而 s ’(而, 考 ). 污 (而, 考) We have
? z而 (而, 考 ) = ( ’

(C3)

(C4)

g ? r ’s ’ ? l + g而 r ’污 zs ’ )(而, 考 ), 污

(C5)

and
? AB 糸 (而, 考 )zB 灰 ?糸 = zA ’ (而, 考 ) = ’ (而, 考 )g ? r ? 糸 ’s ’ 糸 = (l l + zr ’ 污 zs ’ )(而, 考 ), ’’

(C6)

61

where 1 汐 汕 污 z3 )(而, 考), l? (而, 考) = ( ﹟ ?? 汐汕污 z’ 1 z’ 2 ’ 污 l2 (而, 考 ) = 1,
? l? (而, 考 )zr ’ (而, 考 ) = 0,

(C7)

is the unit (future pointing) normal to 曳而 at z ? (而, 考 ). For the volume element in Minkowski spacetime we have
? ? d4 z = z而 (而, 考 )d而 d3 曳? = d而 [z而 (而, 考 )l? (而, 考 )] 污 (而, 考 )d3 考 =

=

g (而, 考)d而 d3 考.

(C8)

Let us remark that according to the geometrical approach of Ref. [12],one can use Eq.(C5) in the form
? ? ’ z而 (而, 考 ) = N (而, 考 )l? (而, 考 ) + N r (而, 考)zr ’ (而, 考 ),

where N =

g/污 =



’s ’g g g而 而 ? 污 r 而r ’ 而s ’ =



r ’ s ’r ’ 3 s ’r ’ ’s ’g g g而 而 + ?3 g r = ??g而 s 而r ’ 而s ’ and N = g而 s ’污 ’ g

’ are the standard lapse and shift functions N[z ](f lat) , N[r z ](f lat) of the Introduction, so that

r ’ s ’ 2 3 r ’ s ’ g而 而 = ?N 2 + gr ’s ’N N = ?[N ? gr ’s ’N N ], s ’ 3 s ’ g而 r ’ = gr ’s ’N = ?? gr ’s ’N , 而而 ?2 g = ?N , ’ ’ g而 r = ??N r /N 2 , r ’ s ’ r ’ s ’ ’s ’ ’s ’ ’s ’ gr = 污r + ? NNN = ??[3 g r ? NNN 2 2 ], ? s ’r ’ ? + zs = l? ?N ’? 污 ?N r ’ = l? ﹟ 3 d z = N 污d而 d 考 . ? ? ?z而 4 ? ?N 3 s ’r ’ ? ? ?zs ’? g ?N r ’,

﹟ ﹟ ? ? The rest frame form of a timelike fourvector p? is p ? = 灰 ?p2 (1; 0) = 灰 ?o 灰 ?p2 , p 2 = p2 , ? where 灰 = sign po. The standard Wigner boost transforming p ? into p? is L? 糸 (p, p) = ?? 糸 (u(p)) =
? ? ??

p? p糸 (p? + p )(p糸 + p糸 ) ? = 灰糸 +2 2 ? = ? ?p p﹞ p +?p2 = 糸=0 糸=r
? 灰糸 ? ? ?

?

(u? (p) + u? (p))(u糸 (p) + u糸 (p)) , + 2 u (p )u 糸 (p ) ? 1 + u o (p )

?

? ? 2 ?? o (u(p)) = u (p) = p /灰 ?p , i ?? r (u(p)) = (?ur (p); 汛r ? ? ?

u i (p )u r (p ) ). 1 + u o (p )

(C9)

The inverse of L? 糸 (p, p) is L? 糸 (p, p), the standard boost to the rest frame, de?ned by 62

L? 糸 (p, p) = L糸 ? (p, p) = L? 糸 (p, p)|p↙?p .

?

?

?

(C10)

Therefore, we can de?ne the following vierbeins [the ?? r (u(p))*s are also called polarization vectors; the indices r, s will be used for A=1,2,3 and o ? for A = o]
? p ?? A (u(p)) = L A (p, ), A p AB ?A 灰?糸 ?糸 ? (u(p)) = L ? ( , p) = 灰 B (u(p)), ? 糸 ?o ? (u(p)) = 灰?糸 ?o (u(p)) = u? (p), rs 糸 rs r rs ?r ? (u(p)) = ?汛 灰?糸 ?r (u(p)) = (汛 us (p); 汛j ? 汛 汛jh ? ?

u h (p )u s (p ) ), 1 + u o (p ) (C11)

?A o (u(p)) = uA (p), which satisfy
糸 ? ?A ? (u(p))?A (u(p)) = 灰糸 , ? A ?A ? (u(p))?B (u(p)) = 灰B , AB 糸 灰 ?糸 = ?? ?B (u(p)) = u? (p)u糸 (p) ? A (u(p))灰 糸 灰AB = ?? A (u(p))灰?糸 ?B (u(p)), ? A ? ? ?A (u(p)) = p汐 ? (u(p)) = 0. p汐 ?p汐 ?p汐 ? 3 糸 ?? r (u(p))?r (u(p)), r =1

(C12)

The Wigner rotation corresponding to the Lorentz transformation 托 is R? 糸 (托, p) = [L(p, p)托?1 L(托p, p)] Ri j (托, p) = (托?1 ) j ?
i ? ? ? 糸

=

1 0 0 Ri j (托, p)

,

(托?1)i o p汕 (托?1 )汕 j ﹟ ? p老 (托?1 )老 o + 灰 ?p2 pi ((托?1 )o o ? 1)p汕 (托?1 )汕 j ﹟ 2 [(托?1 )o j ? ﹟ ? o ]. p + 灰 ?p p老 (托?1 )老 o + 灰 ?p2

(C13)

The polarization vectors transform under the Poincar? e transformations (a, 托) in the following way
?1 s ? 糸 ?? r (u(托p)) = (R )r 托 糸 ?s (u(p)).

(C14)

63

APPENDIX D: MORE ON DIXON*S MULTIPOLES.

Let us add other forms of the Dixon multipoles. In the case of the ?uid con?gurations treated in Section II and IV, the Hamilton equations generated by the Dirac Hamiltonian (2.49) in the gauge qsys > 0 [竹(而 ) = 0] imply [in Ref. ? [20] this is a consequence of ?? T ?糸 = 0] dp? T (Ts ) ? = 0, f or n = 0, dTs ?1 ...?n ? dpT (Ts ) ? ? ...? )? (? ...? )? = ?nu(?1 (ps )pT2 n (Ts ) + ntT 1 n (Ts ), dTs Let us de?ne for n ≡ 1
?1 ...?n ? bT (Ts ) = pT 1

n ≡ 1.

(D1)

=
?1 ...?n ? rn 1 ?r (Ts ) = ?1 (u(ps ))....??n (u(ps ))bT

(Ts ) = ?) r1 ..rn A而 (?1 n ?r1 (u(ps ))....?? (Ts ), rn (u(ps ))?A (u(ps ))IT 1 (r1 ...rn r )而 r1 ...rn 而 而 u? (ps )IT (Ts ) + ?? (Ts ), r (u(ps ))IT n+1
(? ...? )? (? ...?n ?)

(? ...?n ?)

?1 ...?n ? ?1 ...?n ? cT (Ts ) = cT 1 n (Ts ) = pT (Ts ) ? pT 1 ?n ? 1 = [?? r1 (u(ps ))...?rn ?A (u(ps )) ? ?)

(Ts ) =

r1 ..rn A而 (?1 n ? ?r (u(ps ))...?? (Ts ), rn (u(ps ))?A (u(ps ))]IT 1 (? ...? ?)

cT 1 n (Ts ) = 0, n ?1 ...?n ? r1 ...rn 而 而 rn 1 ?r (Ts ) = u? (ps )IT (Ts ) + ?1 (u(ps ))....??n (u(ps ))cT n+1 (r ...r r )而 r1 ...rn r而 + ?? (Ts ) ? IT 1 n (Ts )], r (u(ps ))[IT and then for n ≡ 2
?1 ...?n ?糸 ?1 ...?n ?糸 dT (Ts ) = dT 1 n (Ts ) = tT (Ts ) ? n + 1 (?1 ...?n ?)糸 (? ...? 糸 )? [tT (Ts ) + tT 1 n (Ts )] + ? n n + 2 (?1 ...?n ?糸 ) t (Ts ) = + n T n + 1 (?1 ?n ?) 糸 ?n ? 糸 1 = ?? ?r1 ...?rn ?A ?B + r1 ...?rn ?A ?B ? n n + 2 (?1 ?n ? 糸 ) (?1 n 糸) ? ? ..? ? ? (u(ps )) + ?r ...?? rn ?B ?A + 1 n r1 rn A B r1 ..rn AB IT (Ts ), (? ...?n ?)糸 (? ...? )(?糸 )

(D2)

dT 1

(Ts ) = 0,

64

?1 ...?n ?糸 rn 1 ?r (Ts ) = ?1 (u(ps ))....??n (u(ps ))dT

n?1 ? r1 ...rn 而 而 u (ps )u糸 (ps )IT (Ts ) + n+1 1 糸 ? + [u? (ps )?糸 r (u(ps )) + u (ps )?r (u(ps ))] n (r ...r r )而 r1 ...rn r而 [(n ? 1)IT (Ts ) + IT 1 n (Ts )] + r1 ...rn s1 s2 糸 + ?? (Ts ) ? s1 (u(ps ))?s2 (u(ps ))[IT n + 1 (r1 ...rns1 )s2 (r ...r s )s ? (IT (Ts ) + IT 1 n 2 1 (Ts )) + n (r ...r s s ) + IT 1 n 1 2 (Ts )]. (D3)

Then Eqs.(D1) may be rewritten in the form 1) n = 1 t?糸 T (Ts ) = tT
? (?糸 ) 糸 (Ts ) = p? T (Ts )u (ps ) + ?

1 d ?糸 ( ST (Ts )[汐] + 2b?糸 T (Ts )), 2 dTs

糸) t?糸 T (Ts ) = pT (Ts )u (ps ) +

d ?糸 ) b (Ts ) = P 而 u? (ps )u糸 (ps ) + P r u(? (ps )?糸 r (u(ps )) + dTs T ? 糸) r而 而 + ?( r (u(ps ))u (ps )IT (Ts ) + ? 糸) rs而 + ?( r (u(ps ))?s (u(ps ))IT (Ts ),
(?

?

d ?糸 ? [? r [? ST (Ts )[汐] = 2pT (Ts )u糸 ] (ps ) = 2P耳 ?r (u(ps ))u糸 ] (ps ) > 0, dTs 2) n = 2 [identity t老?糸 = tT T 2tT
(老?)糸 ? ?)糸 ?)糸 (老?)糸

+ tT

(老糸 )?

+ tT

(?糸 )老

]

(Ts ) = 2u(老 (ps )bT (Ts ) + u(老 (ps )ST (Ts )[汐] +
?

d 老?糸 (bT (Ts ) + c老?糸 T (Ts )), dTs

?糸 老 糸) t老?糸 T (Ts ) = u (ps )bT (Ts ) + ST (Ts )[汐]u (ps ) +

?

老(?

d 1 老?糸 ( bT (Ts ) ? c老?糸 T (Ts )), dTs 2

3) n ≡ 3
?1 ...?n ?糸 ?1 ...?n ?糸 tT (Ts ) = dT (Ts ) + u(?1 (ps )bT2 n (Ts ) + 2u(?1 (ps )cT2 n (Ts ) + 2 ? ...? (? 2 ? ...? (?糸 ) d 1 ?1 ...?n ?糸 = cT1 n (Ts )u糸 ) (ps ) + (Ts ) + cT1 n (Ts )], [ bT n dTs n + 1 n ? ? ...? )?糸 ? ...? )(?糸 )

(D4)

This allows [20] to rewrite < T ?糸 , f > in the following form < T ,f > = +
?糸

dTs


d4 k ? 糸) 老(? f (k )e?ik﹞xs (Ts ) u(? (ps )pT (Ts ) ? ik老 ST (Ts )[汐]u糸 ) (ps ) + 4 (2羽 ) (D5)

(?i)n 老1 ...老n ?糸 (Ts ) , k老1 ...k老n IT n! n=2 65

with
?1 ...?n ?糸 IT (Ts ) = IT ?1 ...?n ?糸 (Ts ) = dT (Ts ) ? 2 2 ? ...? (? ? ...? )(?糸 ) ? u(?1 (ps )cT2 n (Ts ) + cT1 n (Ts )u糸 ) (ps ) = n?1 n n + 1 ?n ? 糸 (?1 n ?) 糸 1 = ?? ?r ...?? r1 ...?rn ?A ?B ? rn ?A ?B + 1 n n + 2 (?1 ?n ? 糸 ) r1 ..rn AB (?1 n 糸) ? (Ts ) ? ? ...?rn ?A ?B (u(ps )0IT + ?r ...?? rn ?B ?A + 1 n r1 2 ?n ) (? 糸 ) (?2 ?n ) (? 糸 )) 2 ? ? u(?1 (ps ) ?? r1 ...?rn?1 ?rn ?A ? ?r1 ...?rn?1 ?rn ?A n?1 2 ?1 ?n (? r1 ..rn A而 (?1 n (?) 糸 ) ? ...? ? ? ?r ...?? (Ts ), ? rn ?A u (ps )](u(ps )0IT 1 n r1 rn A (?1 ...?n )(?糸 )

IT
?1 ...?n ?糸 rn 1 ?r (Ts ) = ?1 (u(ps ))....??n (u(ps ))IT

(?1 ...?n ?)糸

(Ts ) = 0,

n+3 ? r1 ...rn 而 而 u (ps )u糸 (ps )IT (Ts ) + n+1 1 r1 ...rn r而 糸 ? + [u? (ps )?糸 (Ts ) + r (u(ps )) + u (ps )?r (u(ps ))]IT n r1 ...rn s1 s2 糸 + ?? (Ts ) ? s1 (u(ps ))?s2 (u(ps ))[IT n + 1 (r1 ...rns1 )s2 (r ...r s )s ? (IT (Ts ) + IT 1 n 2 1 (Ts )) + n (r1 ...rn s1 s2 ) + IT (Ts )].

(D6)

?1 ...?n ?糸 Finally, a set of multipoles equivalent to the IT is

n≡0
?1 ...?n ?糸老考 JT (Ts ) = JT

(Ts ) = IT 1 n (Ts ) = 1 糸 ]? ...? [老考] ? ...? [?[老糸 ]考] u[?(ps )pT 1 n (Ts ) + = tT1 n (Ts ) ? n+1 考]?1 ...?n [?糸 ] [老 + u (p s )p T (Ts ) =
糸 ] 考]

(?1 ...?n )[?糸 ][老考]

? ...? [?[老糸 ]考]

r1 ..rn AB ?n [? [老 1 = ?? (Ts ) ? r1 ..?rn ?r ?s ?A ?B (u(ps ))IT 1 考] ] [老 ? u[? (ps )?糸 r (u(ps ))?s (u(ps ))?A (u(ps )) + n+1 糸] ] [? + u[老(ps )?考 r (u(ps ))?s (u(ps ))?A (u(ps )) rr1 ..rn sA而 ?n 1 ?? (Ts ), r1 (u(ps ))...?rn (u(ps ))IT

[(n + 4)(3n + 5) linearly independent components], n≡1 66

u?1 (ps )

?1 ...?n ?糸老考 JT (Ts ) = JT 1

? ...?n?1 (?n ?糸 )老考

(Ts ) = 0,

n≡2
?1 ...?n ?糸 IT (Ts ) =

4(n ? 1) (?1 ...?n?1 |?|?n )糸 J (Ts ), n+1 T
糸 ] 考]

?1 ...?n ?糸老考 r1 ..rn AB rn 1 (Ts ) ? ?r (Ts ) = ?[r? ?[s老 ?A ?B (u(ps ))IT ?1 (u(ps ))....??n (u(ps ))JT 1 考] ] [老 u[? (ps )?糸 ? r (u(ps ))?s (u(ps ))?A (u(ps )) + n+1 糸] rr1 ..rn sA而 ] [? + u[老(ps )?考 (Ts ). r (u(ps ))?s (u(ps ))?A (u(ps )) IT

(D7)
?1 ...?n ?糸老考 The JT are the Dixon ※2n+2 -pole inertial moment tensors§ of the extended sys?1 ...?n ?糸 *s] determine its energy-momentum tensor together tem: they [or equivalently the IT ? ?糸 ? . The equations ?? T ?糸 = 0 are satis?ed due to with the monopole pT and the spin dipole ST ?糸 ? the equations of motion (D4) for PT and ST [the so called Papapetrou-Dixon-Souriau equa?1 ...?n ?糸老考 tions given in Eqs.(4.18)] without the need of the equations of motion for the JT . ?1 ...?n ?糸老考 are zero [or negligible] one speaks of a pole-dipole ?eld When all the multipoles JT con?guration of the perfect ?uid.

67

REFERENCES
[1] J.D.Brown, Class.Quantum Grav. 10, 1579 (1993). [2] W.Israel, ※Covariant Fluid Mechanics and Thermodynamics: An Introduction§, in ※Relativistic Fluid Dynamics§, eds. A.Anile and Y.Choquet-Bruhat, Lecture Notes in Math. n. 1385 (Springer, Berlin, 1989). [3] C.C.Lin, &Hydrodynamics of Helium II*, in ※Liquid Helium§, ed. G.Careri (Academic Press, New York, 1963). J.Serrin, &Mathematical Principles of Classical Fluid Mechanics*, in ※Handbuch der Physik§ vol. 8, eds. S.Fl“ ugge and C.Truesdell (Springer, Berlin, 1959). [4] J.Kijowski and W.M.Tulczyjew, ※A Symplectic Framework for Field Theories§, Lecture Notes in Physics vol. 107 (Springer, Berlin, 1979); &Relativistic Hydrodynamics of Isentropic Flows§, Mem.Acad.Sci.Torino V 6, 3 (1982). [5] H.P.K“ unzle and J.M.Nester, J.Math.Phys. 25, 1009 (1984). [6] J.Kijowski and G.Magli, Class. Quantum Grav. 15, 3891 (1998). [7] S.Adler and T.Buchert, Astron.Astrophys. 343, 317 (1999) astro-ph/9806320. [8] D.Bao, J.Marsden and R.Walton, Commun.Math.Phys. 99, 319 (1985). D.D.Holm, ※Hamiltonian Techniques for Relativistic Fluid Dynamics and Stability Theory§, in ※Relativistic Fluid Dynamica§, eds. A.Anile and Y.Choquet-Bruhat (Springer, Berlin, 1989). [9] J.D.Brown and K.Kuchar, Phys.Rev. D51, 5600 (1995). [10] L.Lusanna, Int.J.Mod.Phys. 12A, 645 (1997). [11] L.Lusanna, ※Towards a Uni?ed Description of the Four Interactions in Terms of DiracBergmann Observables§, invited contribution to the book of the Indian National Science Academy for the International Mathematics Year 2000 AD (hep-th/9907081). ※Tetrad Gravity and Dirac*s Observables§, talk given at the Conf. ※Constraint Dynamics and Quantum Gravity 99§, Villasimius 1999 (gr-qc/9912091).※The Rest-Frame Instant Form of Dynamics and Dirac*s Observables§, talk given at the Int.Workshop※Physical Variables in Gauge Theories§, Dubna 1999. [12] K.Kuchar, J.Math.Phys. 17, 777, 792, 801 (1976). [13] M.Pauri and M.Prosperi, J.Math.Phys. 16, 1503 (1975). [14] L.Lusanna and M.Materassi, ※The Canonical Decomposition in Collective and Relative Variables of a Klein-Gordon Field in the Rest-Frame Wigner-Covariant Instant Form§, to appear in Int.J.Mod.Phys. A (hep-th/9904202). [15] G.Longhi and M.Materassi, Int.J.Mod.Phys. A14, 3397 (1999)(hep-th/9809024); J.Math.Phys. 40, 480 (1999) (hep-th/9803128). [16] D.Alba, L.Lusanna and M.Pauri, ※Center of Mass, Rotational Kinematics and Multipolar Expansions for the Relativistic and Non-Relativistic N-Body Problems in the Rest-Frame Instant Form§, in preparation. [17] H.Crater and L.Lusanna, ※The Rest-Frame Darwin Potential from the Lienard-Wiechert Solution in the Radiation Gauge§, Firenze univ. preprint 2000 (hep-th/0001046). [18] D.Bini, G.Gemelli and R.Ru?ni, ※Spining Test Particles in General Relativity: Nongeodesic Motion in the Reissner-Nordstr“ om Spacetime§ talk at the III W.Fairbank Meeting and I ICRA Network Workshop ※The Lense-Thirring E?ect§, Roma-Pescara 1998. [19] W.Beiglb“ ock, Commun.Math.Phys. 5, 106 (1967). J.Ehlers and E.Rudolph, Gen.Rel.Grav. 8, 197 (1977). R.Schattner, Gen.Rel.Grav. 10, 377 and 395 (1978). 68

[20] W.G.Dixon, J.Math.Phys. 8, 1591 (1967). ※Extended Objects in General Relativity: their Description and Motion§, in ※Isolated Gravitating Systems in General Relativity§, ed. J.Ehlers (North-Holland, Amsterdam, 1979). [21] K.S.Thorne, ※The Theory of Gravitational Radiation: an Introductory Review§ in &Gravitational Radiation* eds. N.Deruelle and T.Piran, 1982 NATO-ASI School at Les Houches (North Holland, Amsterdam, 1983); Rev.Mod.Phys. 52, 299 (1980). [22] M.P.Ryan Jr. and L.C.Shepley, ※Homogeneous Relativistic Cosmologies§ (Princeton Univ. Press, Princeton, 1975). [23] L.Blanchet, T.Damour and G.Sch“ afer, Mon.Not.R.astr.Soc. 242, 289 (1990). [24] A.M.Anile, ※Relativistic Fluids and Magneto?uids§ (Cambridge Univ.Press, Cambridge, 1989). [25] L.Lusanna and S.Russo, ※Tetrad Gravity I): A New Formulation§, Firenze Univ. preprint 1998 (gr-qc/9807073). [26] L.Lusanna and S.Russo, ※Tetrad Gravity II): Dirac*s Observables§, Firenze Univ. preprint 1998 (gr-qc/9807074). [27] R.DePietri and L.Lusanna, ※Tetrad Gravity III): Asymptotic Poincar? e Charges, the Physical Hamiltonian and Void Spacetimes§, (gr-qc/9909025). [28] L.Smarr and J.W.York jr., Phys.Rev. D17, 2529 (1978). J.W.York jr., ※Kinematics and Dynamics of General Relativity§, in ※Sources of Gravitational Radiation§, ed. L.L.Smarr (Cambridge Univ.Press, Cambridge, 1979). [29] L.Smarr, C.Taubes and J.R.Wilson, ※General Relativistic Hydrodynamics: the Comoving, Eulerian and Velocity Potential Formalisms§, in ※Essays in General Relativity: a Festschrift for Abraham Taub§, ed. F.J.Tipler (Academic Press, New York, 1980). [30] H.Asada, M.Shibata and T.Futamase, ※Post-Newtonian Hydrodynamic Equations Using the (3+1) Formalism in General Relativity§, Osaka Univ. preprint (gr-qc/9606041). [31] J.A.Font, N.Stergioulas and K.D.Kokkotas, ※Nonlinear Hydrodynamical Evolution of Rotating Relativistic Stars: Numerical Methods and Code Tests§ (gr-qc/9908010). [32] T.W.Baumgarte, S.A.Hughes, L.Rezzolla, S.L.Shapiro and M.Shibata, ※Implementing Fully relativistic Hydrodynamics in Three Dimensions§ (gr-qc/9907098). [33] J.Kijowski and G.Magli, Rep.Math.Phys. 39, 99 (1997). [34] J.Kijowski and G.Magli,J.Geom.Phys. 9, 207 (1992). [35] S.Antoci and L.Mihic, ※A Four-Dimensional Hooke*s Law can Encompass Linear Elasticity and Inertia§ (gr-qc/9906094). [36] R.Maartens, ※Causal Thermodynamics in Relativity§, Lectures at the H.Rund Workshop on &Relativity and Thermodynamics* 1996 (astro-ph/9609119). [37] B.Carter, ※Covariant Theory of Conductivity in Ideal Fluid or Solid Media§, in ※Relativistic Fluid Dynamics§, eds. A.Anile and Y.Choquet-Bruhat, Lecture Notes in Math. n. 1385 (Springer, Berlin, 1989). [38] F.Karsch and D.E.Miller, Phys.Rev. D24, 2564 (1981). [39] J.M.Stewart, ※Non-Equilibrium Relativistic Kinetic Theory§, Lecture Notes Phys. n.10 (Springer, Berlin, 1971). [40] R.Hakim, J.Math.Phys. 8, 1315, 1379 (1967). [41] J.L.Synge, ※The Relativistic Gas§ (North-Holland, Amsterdam, 1957). [42] Ch.G.van Weert, ※Some Problems in Relativistic Hydrodynamics§, in ※Relativistic Fluid Dynamics§, eds. A.Anile and Y.Choquet-Bruhat, Lecture Notes in Math. n. 1385 69

[43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58]

(Springer, Berlin, 1989). L.Lindblom, Ann.Phys.(N.Y.) 247, 1 (1996). A.M.Anile, D.Pav? on and V.Romano, ※The Case for Hyperbolic Theories of Dissipation in Relativistic Fluids§, (gr-qc/9810014). J.M.Mart? i and E.M“ uller, ※Numerical Hydrodynamics in Special Relativity§, (astroph/9906333). R.Geroch and L.Lindblom, Ann.Phys.(N.Y.) 207, 394 (1991). I.Liu, I.M“ uller and T.Ruggeri, Ann.Phys.(N.Y.) 169, 191 (1986). R.Geroch and L.Lindblom, Phys.Rev. D41, 1855 (1990). E.Calzetta, Class.Quantum Grav. 15, 653 (1998), (gr-qc/9708048). I.M“ uller, Z.Phys. 198, 329 (1967). W.Israel, Ann.Phys.(N.Y.) 100, 310 (1976). W.Israel and J.M.Stewart, Ann.Phys.(N.Y.) 118, 341 (1979). D.Jou, G.Leblon and J.Casas Vasquez, ※Extended Thermodynamics§ 2nd edition (Springer, Heidelberg, 1996). I.M“ uller and T.Ruggeri, ※Rational Extended Thermodynamics§, 2nd edition (Springer, Berlin, 1998). J.Peitz and S.Appl, ※3+1 Formulation of Non-Ideal Hydrodynamics§, (gr-qc/9710107). Class.Quantum Grav. 16, 979 (1999). J.P.Uzan, Class.Quantum Grav. 15, 1063 and 3737 (1998). J.M.Stewart, Class.Quantum Grav. 15, 3731 (1998). P.J.Morrison, Physica 18D, 410 (1986). S.Nose*, J.Chem.Phys. 81, 511 (1984).

70


婝翑妀蟈諉
芢熱眈壽:
厙桴忑珜 | 厙桴華芞
All rights reserved Powered by 蹄扂訧蹋厙 koorio.com
copyright ©right 2014-2019﹝
恅紫訧蹋踱囀暌棚奜讕蝤畏觬陎硊裔蹅肢翕芛﹝zhit325@126.com