The Rest-Frame Instant Form of Relativistic Perfect Fluids with

The Rest-Frame Instant Form of Relativistic Perfect Fluids with Equation of State ρ = ρ(n, s) and of Non-Dissipative Elastic Materials.

arXiv:hep-th/0003095v1 13 Mar 2000

Luca Lusanna Sezione INFN di Firenze L.go E.Fermi 2 (Arcetri) 50125 Firenze, Italy E-mail [email protected]

and Dobromila Nowak-Szczepaniak Institute of Theoretical Physics University of Wroclaw pl. M.Borna 9 50-204 Wroclaw, Poland E-mail [email protected]

Abstract

For perfect fluids with equation of state ρ = ρ(n, s), Brown [1] gave an action principle depending only on their Lagrange coordinates αi (x) without Clebsch potentials. After a reformulation on arbitrary spacelike hypersurfaces in Minkowski spacetime, the Wigner-covariant rest-frame instant form of these perfect fluids 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 fluids 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

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I. INTRODUCTION

Stability of stellar models for rotating stars, gravity-fluid 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 fluids by means of action functionals both in special and general relativity. Usually, besides the thermodynamical variables n (particle number density), ρ (energy density), p (pressure), T (temperature), s (entropy per particle), which are spacetime scalar fields whose values represent measurements made in the rest frame of the fluid (Eulerian observers), one characterizes the fluid motion by its unit timelike 4-velocity vector field U µ (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: “; µ” denotes a covariant derivative] i) particle number conservation, (nU µ );µ = 0; ii) absence of entropy exchange between neighbouring flow lines, (nsU µ );µ = 0; iii) the requirement that the fluid flow lines should be fixed 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 first 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 µ with a set of spacetime scalar fields α ˜ i (z), i = 1, 2, 3, interpreted as “Lagrangian (or comoving) coordinates for the fluid” labelling the fluid flow lines (physically determined by the average particle motions) passing through the points inside the boundary (on the boundary they are fixed: either the α ˜ i (z o , ~z)’s have a compact boundary Vα (z o ) or they have assigned boundary conditions at spatial infinity). This requires the choice of an arbitrary spacelike hypersurface on which the αi ’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 fluids 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 ǫ(+ − −−), ǫ = ±] whose points have locally coordinates z µ . Let 4 gµν (z) be its 4-metric with determinant 4 g = |det 4 gµν |. Given a perfect fluid with Lagrangian coordinates α ˜(z) = {α ˜ i (z)}, unit 4-velocity vector field U µ (z) and particle number density n(z), let us introduce the number flux vector

2

J µ (α ˜ i (z)) , n(z)U µ (z) = q 4 g(z)

(1.1)

√ and√ the densitized fluid number flux vector or material current [ǫ0123 = 1/ 4 g; ˜ i ) describes the orientation of the volume in thematerial space] ∂α ( 4 gǫµνρσ ) = 0; η123 (α q

J µ (α ˜ i (z)) = − ⇒ ⇒ ⇒

4 gǫµνρσ η αi (z))∂ν α ˜ 1 (z)∂ρ α ˜ 2 (z)∂σ α ˜ 3 (z), 123 (˜

q

ǫ4 gµν (z)J µ (α ˜ i(z))J ν (α ˜ i (z)) |J(α ˜ i(z))| q √4 = η123 (α ˜ i(z)) , n(z) = 4 g(z) g µ

i

∂µ J (α ˜ (z)) =

q

4g

[n(z)U µ (z)];µ = 0, q

J µ (α ˜ i (z))∂µ α ˜ i (z) = [

i 4 gnU µ ](z)∂ α µ ˜ (z)

= 0.

(1.2)

This shows that the fluid flow lines, whose tangent vector field is the fluid 4-velocity timelike vector field U µ , are identified by α ˜i = const. and that the particle number conservation is automatic. Moreover, if the entropy for particle is a function only of the fluid Lagrangian coordinates, s = s(˜ αi ), the assumed form of J µ also implies automatically the absence of entropy exchange between neighbouring flow lines, (nsU µ ),µ = 0. Since U µ ∂µ s(α ˜ i ) = 0, the perfect fluid is locally adiabatic; instead for an isentropic fluid we have ∂µ s = 0, namely s = const.. Even if in general the timelike vector field U µ (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 fluid flow line in that point (namely we split the tangent space Tz M 4 at z in the U µ (z) direction and in the orthogonal complement) and consider 3!1 [U µ ǫµνρσ dz ν ∧ dz ρ ∧ dz σ ](z) as the infinitesimal 3-volume on it at z. Then the 3-form 1 η[z] = [η123 (˜ α)dα ˜ 1 ∧ dα ˜ 2 ∧ dα ˜ 3 ](z) = n(z)[U µ ǫµνρσ dz ν ∧ dz ρ ∧ dz σ ](z), (1.3) 3! may be interpreted as the number of particles in this 3-volume. If V is a volume around z R R on the spacelike hypersurface, then V η is the number of particle in V and V sη is the total entropy contained in the flow lines included in the volume V. Note that locally η123 can be set to unity by an appropriate choice of coordinates. In Ref. [1] it is shown that the action functional S[4 gµν , α ˜] = −

Z

d4 z

q

4 g(z) ρ(

|J(α ˜ i(z))| q

4 g(z)

, s(˜ αi (z))),

(1.4)

has a variation with respect to the 4-metric, which gives rise to the correct stress tensor ∂ρ |s − ρ for a perfect fluid [see Appendix A]. T µν = (ρ + p)U µ U ν − ǫ p 4 g µν with p = n ∂n The Euler-Lagrange equations associated to the variation of the Lagrangian coordinates are [1] [Vµ = µUµ is the Taub current, see Appendix A] 1 δS 1 ∂s √4 = ǫµνρσ Vµ;ν ηijk ∂ρ α ˜ j ∂σ α ˜k − n T = 0, i ˜ 2 ∂α ˜i g δα 1 1 δS √4 ∂µ α ˜ i = ǫαβγδ Vα;β U ν ǫνµγδ − T ∂µ s = 2V[µ;ν] U ν − T ∂µ s = 0. i ˜ 2 g δα 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 satisfied and the Euler-Lagrange equations are equivalent to the Euler equations. In Minkowski spacetime R R 3 o ˜ i (z)), while the the conserved particle number is N = Vα (z o ) d z n(z) U (z) = Vα (z o ) d3 z J o (α R R ˜ i ). Moreconserved entropy per particle is Vα (z o ) d3 z s(z) n(z) U o (z) = Vα (z o ) d3 z s(z) J o (α µν over, the conservation laws T ,ν = 0 will generate the conserved 4-momentum and angular momentum of the fluid. However, in Ref. [1] there are only some comments on the Hamiltonian description implied by this particular action. This description of perfect fluids fits 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 find the Wigner covariant rest-frame instant form of the dynamics of a perfect fluid, on the special Wigner hyperplanes orthogonal to the total 4-momentum of the fluid. In this way we will get the description of the global rest frame of the fluid as a whole; instead, the 4-velocity vector field U µ defines the local rest frame in each point of the fluid by means of the projector 4 g µν − ǫU µ U ν . This approach will also produce automatically the coupling of the fluid to ADM metric and tetrad gravity with the extra property of allowing a well defined 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 fluid 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 defined by an embedding R × Σ → M 4 , (τ, ~σ ) 7→ z µ (τ, ~σ ) ∈ Στ and with Σ an abstract 3-surface diffeomorphic to R3 , with Στ its copy embedded in M 4 labelled by the value τ (the scalar mathematical “time” parameter τ labels the leaves of the foliation, ~σ are curvilinear coorˇ dinates on Στ and σ A = (σ τ = τ, σ rˇ) are Στ -adapted holonomic coordinates for M 4 ). See Appendix C for the notations on spacelike hypersurfaces. In this way one gets a parametrized field 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 configuration variables the embedding coordinates z µ (τ, ~σ) of the points of the spacelike hypersurface Στ [the only ones carrying Lorentz indices] and then to define the fields on Στ so that they know the hypersurface Στ of τ -simultaneity [for a Klein-Gordon field φ(x), this ˜ ~σ ) = φ(z(τ, ~σ )): it contains the non-local information about the embedding]. new field is φ(τ, Then one rewrites the Lagrangian of the given isolated system in the form required by the coupling to an external gravitational field, makes the previous 3+1 splitting of Minkowski spacetime and interpretes all the fields of the system as the new fields on Στ (they are Lorentz scalars, having only surface indices). Instead of considering the 4-metric as describing a gravitational field (and therefore as an independent field as it is done in metric gravity, where one adds the Hilbert action to the action for the matter fields), here one 4

replaces the 4-metric with the the induced metric gAˇBˇ [z] = zAµˇηµν zBνˇ on Στ [a functional ˇ of z µ ; zAµˇ = ∂z µ /∂σ A are flat inverse tetrad fields on Minkowski spacetime with the zrˇµ ’s tangent to Στ ] and considers the embedding coordinates z µ (τ, ~σ ) as independent fields [this ˇ is not possible in metric gravity, because in curved spacetimes zAµˇ 6= ∂z µ /∂σ A are not tetrad fields so that holonomic coordinates z µ (τ, ~σ ) do not exist]. From this Lagrangian, besides a Lorentz-scalar form of the constraints of the given system, we get four extra primary first class constraints ττ rˇτ Hµ (τ, ~σ ) = ρµ (τ, ~σ) − lµ (τ, ~σ)Tsys (τ, ~σ ) − zrˇµ (τ, ~σ )Tsys (τ, ~σ ) ≈ 0,

(1.6)

ττ rˇτ [ here Tsys (τ, ~σ ) = M(τ, ~σ), Tsys (τ, ~σ) = Mrˇ(τ, ~σ ), 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(µ) (τ, ~σ ), H(ν) (τ, ~σ )} = 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 µ rˇ µ is given by zτµ = N[z](f lat) lµ + N[z](f σ ) is the normal to Στ in z µ (τ, ~σ ) and lat) zrˇ , where l (τ, ~ rˇ N[z](f lat) (τ, ~σ ), N[z](f σ ) are the flat lapse and shift functions defined through the metric lat) (τ, ~ like in general relativity: however, now they are not independent variables but functionals of z µ (τ, ~σ ). R The Dirac Hamiltonian contains the piece d3 σλµ (τ, ~σ )Hµ (τ, ~σ ) with λµ (τ, ~σ ) Dirac multipliers. It is possible to rewrite the integrand in the form [γ rˇsˇ = −ǫ 3 g rˇsˇ is the inverse of the spatial metric grˇsˇ = 4 grˇsˇ = −ǫ 3 grˇsˇ, with 3 grˇsˇ of positive signature (+ + +)] def

λµ (τ, ~σ )Hµ (τ, ~σ ) = [(λµ lµ )(lν Hν ) − (λµ zrµ )(γ rˇsˇzsˇν Hν )](τ, ~σ ) = def

= N(f lat) (τ, ~σ )(lµ Hµ )(τ, ~σ ) − N(f lat)ˇr (τ, ~σ )(γ rˇsˇzsˇν Hν )(τ, ~σ ),

(1.7)

with the (non-holonomic form of the) constraints (lµ Hµ )(τ, ~σ) ≈ 0, (γ rˇsˇzsˇµ Hµ )(τ, ~σ) ≈ 0, satisfying the universal Dirac algebra of the ADM constraints. In this way we have defined new flat lapse and shift functions N(f lat) (τ, ~σ ) = λµ (τ, ~σ )lµ (τ, ~σ ), N(f lat)ˇr (τ, ~σ ) = λµ (τ, ~σ )zrˇµ (τ, ~σ ).

(1.8)

which have the same content of the arbitrary Dirac multipliers λµ (τ, ~σ ), namely they multiply primary first 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)ˇr , defined starting from the metric. Since the Hamilton equations imply zτµ (τ, ~σ ) = λµ (τ, ~σ ), it is only through the equations of motion that the two types of functions are identified. Instead in general relativity the lapse and shift functions defined starting from the 4-metric are the coefficients (in the canonical part Hc of the Hamiltonian) of secondary first class constraints satisfying the Dirac algebra independently from the equations of motion. For the relativistic perfect fluid with equation of state ρ = ρ(n, s) in Minkowski spacetime, we have only to replace the external 4-metric 4 gµν with gAˇBˇ (τ, ~σ) = 4 gAˇBˇ (τ, ~σ ) and the scalar fields for the Lagrangian coordinates with αi (τ, ~σ) = α ˜ i (z(τ, ~σ )); now either the αi (τ, ~σ )’s have a compact boundary Vα (τ ) ⊂ Στ or have boundary conditions at spatial infinity. For each value of τ , one could invert αi = αi (τ, ~σ ) to ~σ = ~σ (τ, αi ) and use the αi ’s as a special coordinate system on Στ inside the support Vα (τ ) ⊂ Στ : z µ (τ, ~σ (τ, αi)) = zˇµ (τ, αi). 5

By going to Στ -adapted coordinates such that η123 (α) = 1 we get [γ = |det grˇsˇ|; q √4 √ g = |det gAˇBˇ | = N γ]

√

g=

√ ˇ ˇ J A (αi (τ, ~σ )) = [N γnU A ](τ, ~σ ),

J τ (αi (τ, ~σ )) = [−ǫrˇuˇvˇ∂rˇα1 ∂uˇ α2 ∂vˇ α3 ](τ, ~σ ) = −det (∂rˇαi)(τ, ~σ ), J rˇ(αi (τ, ~σ )) = [

3 X

∂τ αi ǫrˇuˇvˇ∂uˇ αj ∂vˇ αk ](τ, ~σ ) =

i=1;i,j,k cyclic

1 = ǫrˇuˇvˇǫijk [∂τ αi ∂uˇ αj ∂vˇαk ](τ, ~σ ), 2 ⇒

|J| n(τ, ~σ ) = √ (τ, ~σ ) = N γ

q

ǫgAˇBˇ J AˇJ Bˇ (τ, ~σ), √ N γ

(1.9)

3 τ i σ )) giving with N = Vα (τ ) d σJ (α (τ, ~ R 3 τ the conserved particle number and Vα (τ ) d σ(s J )(τ, ~σ) giving the conserved entropy per particle. The action becomes

R

S=

Z

dτ d3 σL(z µ (τ, ~σ ), αi(τ, ~σ )) =

|J(αi(τ, ~σ ))| √ , s(αi (τ, ~σ))) = dτ d3 σ(N γ)(τ, ~σ )ρ( √ (N γ)(τ~σ ) Z √ = − dτ d3 σ(N γ)(τ, ~σ )

=−

Z

ρ( q

1 γ(τ, ~σ)

s h

(J τ )2 − 3 guˇvˇ

J uˇ + N uˇ J τ J vˇ + N vˇJ τ i (τ, ~σ ; αi(τ, ~σ )), s(αi (τ, ~σ ))), N N (1.10)

rˇ with N = N[z](f lat) , N rˇ = N[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 ρ = µn. In Section III we will define the “external” and “internal” centers of mass of the dust. In Section IV we will study Dixon’s multipoles of a perfect fluid 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 fluids and we will make some comments on non-isentropic fluids. In Section VI we will define 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 fluids 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 fluids.

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 definition of other types of Dixon’s multipoles.

7

II. DUST.

Let us consider first the simplest case of an isentropic perfect fluid, a dust with p = 0, s = const., and equation of state ρ = µn. In this case the chemical potential µ is the rest mass-energy of a fluid particle: µ = m (see Appendix A). Eq.(1.10) implies that the Lagrangian density is [we shall use the notation gAˇBˇ = 4 gAˇBˇ with signature ǫ(+ − −−), ǫ = ±1; by using the notation with lapse and shift functions given in Appendix C we get: gτ τ = ǫ(N 2 − 3 grˇsˇN rˇN sˇ), gτ rˇ = −ǫ 3 grˇsˇN sˇ, grˇsˇ = −ǫ 3 grˇsˇ with r ˇ r ˇ s ˇ 3 grˇsˇ of positive signature (+++), g τ τ = Nǫ2 , g τ rˇ = −ǫ N , g rˇsˇ = −ǫ(3 g rˇsˇ − NNN2 ); the inverse N2 of the spatial 4-metric 4 grˇsˇ is denoted γ rˇsˇ = 4 γ rˇsˇ = −ǫ 3 g rˇsˇ, where 3 g rˇsˇ is the inverse of the √ √ 3-metric 3 grˇsˇ and we use γ = 3 grˇsˇ] q √ √ L(αi , z µ ) = − gρ = −µ gn = −µ ǫgAˇBˇ J AˇJ Bˇ = q

= −µN (J τ )2 − 3 grˇsˇY rˇY sˇ = −µNX,

Y rˇ = X=

1 rˇ (J + N rˇJ τ ), N q

(J τ )2

−

3g

rˇsˇY

rˇY sˇ

=

√

g √ n = γn, N

(2.1)

with J τ , J rˇ given in Eqs.(1.9). The momentum conjugate to αi is Y tˇ 3 gtˇrˇ ǫrˇuˇvˇ ∂uˇ αj ∂vˇ αk ∂L = µ |i,j,k cyclic = ∂∂τ αi X Y rˇ Y tˇ 3 gtˇrˇ ǫrˇuˇvˇǫijk ∂uˇ αj ∂vˇαk = µ Trˇi , =µ 2X X

Πi =

,

def

Ttˇi =

1 gˇ ǫrˇuˇvˇ ǫijk ∂uˇ αj ∂vˇ αk = gtˇrˇ (ad Jiˇr ), 2 trˇ

(2.2)

−1 i where ad Jiˇr = (det J)Jiˇ r = ∂rˇα ) of the r is the adjoint matrix of the Jacobian J = (Jiˇ i transformation from the Lagrangian coordinates α (τ, ~σ ) to the Eulerian ones ~σ on Στ . The momentum conjugate to z µ is ri h (J τ )2 ∂L τ Y (τ, ~σ ). + µ zrµ J ρµ (τ, ~σ) = − µ (τ, ~σ ) = µ lµ ∂zτ X X

(2.3)

The following Poisson brackets are assumed ′

′

{z µ (τ, ~σ), ρν (τ, ~σ } = −ηνµ δ 3 (~σ − ~σ ), ′ ′ {αi (τ, ~σ ), Πj (τ, ~σ )} = δji δ 3 (~σ − ~σ ).

(2.4)

We can express Y rˇ/X in terms of Πi with the help of the inverse (T −1 )rˇi of the matrix Ttˇi 1 Y rˇ = (T −1 )rˇi Πi , X µ 8

(2.5)

where (T −1 )rˇi =

3 rˇsˇ

g ∂sˇαi . det (∂uˇ αk )

(2.6)

From the definition of X we find µJ τ . X=q µ2 + 3 guˇvˇ(T −1 )uˇi (T −1 )vˇj Πi Πj

(2.7)

Consequently, we can get the expression of the velocities of the Lagrangian coordinates in terms of the momenta ∂τ αi = −

J rˇ∂rˇαi (N rˇJ τ − NY rˇ) ∂rˇαi = , Jτ Jτ

(2.8)

namely h

∂τ αi = ∂rˇαi N rˇ − N(T −1 )rˇi Πi

q

i

µ2 + 3 guˇvˇ (T −1 )uˇi (T −1 )vˇj Πi Πj .

Now ρµ can be expressed as a function of the z’s, α’s and Π’s: q

µ2 + 3 guˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj + zrˇµ J τ (T −1 )rˇi Πi .

ρµ = lµ J τ

(2.9)

(2.10)

Since the Lagrangian is homogenous in the velocities, the Hamiltonian is only HD =

Z

d3 σλµ (τ, ~σ )Hµ (τ, ~σ),

(2.11)

where the Hµ are the primary constraints ˇ

Hµ = ρµ − lµ M + zrˇµ MR ≈ 0, M = T ττ = Jτ

q

µ2 + 3 guˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj ,

Mrˇ = T τ rˇ = J τ (T −1 )rˇi Πi .

(2.12)

satisfying ′

{Hµ (τ, ~σ ), Hν (τ, ~σ )} = 0.

(2.13)

One finds that {Hµ (τ, ~σ), HD } = 0. Therefore, there are only the four first class constraints Hµ (τ, ~σ) ≈ 0. They describe the arbitrariness of the foliation: physical results do not depend on its choice. The conserved Poincaré generators are (the suffix “s” denotes the hypersurface Στ ) pµs

=

Z

Jsµν =

d3 σρµ (τ, ~σ ), Z

d3 σ[z µ (τ, ~σ )ρν (τ, ~σ ) − z ν (τ, ~σ)ρµ (τ, ~σ )],

and one has 9

(2.14)

{z µ (τ, ~σ ), pνs } = −η µν , Z

d3 σHµ (τ, ~σ ) = pµs − +

Z

Z

h

h

d3 σ lµ J τ

q

(2.15) i

µ2 + 3 guˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj (τ, ~σ ) + i

d3 σ zrˇµ J τ (T −1 )rˇi Πi (τ, ~σ ) ≈ 0.

(2.16)

Let us now restrict ourselves to spacelike hyperplanes Στ by imposing the gauge-fixings ζ µ(τ, ~σ ) = z µ (τ, ~σ) − xµs (τ ) − bµrˇ (τ )σ rˇ ≈ 0, ′

′

{ζ µ (τ, ~σ), Hν (τ, ~σ )} = −ηνµ δ 3 (~σ − ~σ ),

(2.17)

where xµs (τ ) is an arbitrary point of Στ , chosen as origin of the coordinates σ rˇ, and bµrˇ (τ ), rˇ = 1, 2, 3, are three orthonormal vectors such that the constant (future pointing) normal to the hyperplane is lµ (τ, ~σ ) ≈ lµ = bµτ = ǫµ αβγ bαˇ1 (τ )bβˇ2 (τ )bγˇ3 (τ ).

(2.18)

Therefore, we get zrˇµ (τ, ~σ ) ≈ bµrˇ (τ ), zτµ (τ, ~σ ) ≈ x˙ µs (τ ) + b˙ rµˇ (τ )σ rˇ, grˇsˇ(τ, ~σ ) ≈ −ǫδrˇsˇ, γ rˇsˇ(τ, ~σ) ≈ −ǫδ rˇsˇ,

γ(τ, ~σ ) ≈ 1.

(2.19)

By introducing the Dirac brackets for the resulting second class constraints ∗

{A, B} = {A, B} −

Z

d3 σ[{A, ζ µ (τ, ~σ )}{Hµ (τ, ~σ ), B} − {A, Hµ (τ, ~σ )}{ζ µ(τ, ~σ ), B}], (2.20)

we find that, by using Eq.(2.15) and (2.16) [with xµs (τ ) = z µ (τ, ~σ) − bµrˇ (τ )σ rˇ − ζ µ(τ, ~σ ) and with the assumption {bµrˇ (τ ), pνs } = 0], we get {xµs (τ ), pνs (τ )}∗ = −η µν .

(2.21)

The ten degrees of freedom describing the hyperplane are xµs (τ ) with conjugate momentum pµs and six variables φλ (τ ), λ = 1, .., 6, which parametrize the orthonormal tetrad bµAˇ(τ ), with their conjugate momenta Tλ (τ ). The preservation of the gauge-fixings ζ µ(τ, ~σ ) ≈ 0 in time implies d µ ζ (τ, ~σ ) = {ζ µ(τ, ~σ ), HD } = −λµ (τ, ~σ) − x˙ µs (τ ) − b˙ µrˇ (τ )σ rˇ ≈ 0, dτ

(2.22)

so that one has [by using b˙ µτ = 0 and b˙ rµˇ (τ )bνrˇ (τ ) = −brµˇ (τ )b˙ νrˇ (τ )] ˜ µ (τ ) + λ ˜ µ ν (τ )bν (τ )σ rˇ, λµ (τ, ~σ ) ≈ λ rˇ µ µ ˜ λ (τ ) = −x˙ s (τ ), ˜ µν (τ ) = −λ ˜ νµ (τ ) = 1 [b˙ µrˇ (τ )bν (τ ) − brµˇ (τ )b˙ ν (τ )]. λ rˇ rˇ 2 10

(2.23)

Thus, the Dirac Hamiltonian becomes ˜ µ (τ )H ˜ µν (τ )H ˜ µ (τ ) − 1 λ ˜ µν (τ ), HD = λ 2

(2.24)

and this shows that the gauge fixings ζ µ(τ, ~σ ) ≈ 0 do not transform completely the constraints Hµ (τ, ~σ ) ≈ 0 in their second class partners; still the following ten first class constraints are left ˜ µ (τ ) = H −

Z

Z

d3 σHµ (τ, ~σ) = pµs − q

d3 σ J τ lµ µ2 + δuˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj + bµrˇ (T −1 )rˇl Πl

˜ µν (τ ) = bµrˇ (τ ) H

Z

d3 σσ rˇ Hν (τ, ~σ ) − bνrˇ (τ )

Z

Z

q

= Ssµν (τ ) −

−

[brµˇ (τ )bντ

−

brνˇ (τ )bµτ ]

d3 σσ rˇ Hµ (τ, ~σ ) =

Z

d3 σσ rˇ J τ (T −1 )sˇl Πl (τ, ~σ) ≈ 0.

Here Ssµν is the spin part of the Lorentz generators Jsµν = xµs pνs − xνs pµs + Ssµν , Z

(τ, ~σ) ≈ 0,

d3 σσ rˇ J τ µ2 + δuˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj (τ, ~σ) +

+ [brµˇ (τ )bνsˇ (τ ) − brνˇ (τ )bµsˇ (τ )]

Ssµν = bµrˇ (τ )

d3 σσ rˇρν (τ, ~σ ) − brνˇ (τ )

Z

d3 σσ rˇρµ (τ, ~σ ).

(2.25)

(2.26)

As shown in Ref. [10] instead of finding φλ (τ ), Tλ (τ ), one can use the redundant variables bµAˇ(τ ), Ssµν (τ ), with the following Dirac brackets assuring the validity of the orthonormality ˇˇ µναβ condition η µν − bµAˇ η Ab bνBˇ = 0 [Cγδ = ηγν ηδα η µβ + ηγµ ηδβ η να − ηγν ηδβ η µα − ηγµ ηδα η νβ are the structure constants of the Lorentz group] {Ssµν , bρAˇ}∗ = η ρν bµAˇ − η ρµ bνAˇ µναβ γδ {Ssµν , Ssαβ }∗ = Cγδ Ss ,

(2.27)

˜ µ (τ ) ≈ 0 has zero Dirac bracket with itself and with H ˜ µν (τ ) ≈ 0, these last so that, while H six constraints have the Dirac brackets ˜ µν (τ ), H ˜ αβ (τ )}∗ = C µναβ H ˜ γδ (τ ) ≈ 0. {H γδ

(2.28)

We have now only the variables: xµs , pµs , bµAˇ , Ssµν , αi , Πi with the following Dirac brackets: {xµs (τ ), pνs (τ )}∗ = −η µν , {Ssµν (τ ), bρAˇ (τ )}∗ = η ρν bµAˇ(τ ) − η ρµ bνAˇ(τ ),

µναβ γδ {Ssµν (τ ), Ssαβ (τ )}∗ = Cγδ Ss (τ ), ′

′

{αi (τ, ~σ ), Πj (τ, ~σ )}∗ = δji δ 3 (~σ − ~σ ).

(2.29)

After the restriction to spacelike hyperplanes we have zrˇµ (τ, ~σ) ≈ bµrˇ (τ ), so that zτµ (τ, ~σ) ≈ (µ) rˇ ˜ µ (τ ) − λ ˜ µν (τ )brˇν (τ )σ rˇ. N[z](f lat) (τ, ~σ )lµ (τ, ~σ) + N[z](f σ) brˇ (τ, ~σ ) ≈ x˙ µs (τ ) + b˙ rµˇ (τ )σ rˇ = −λ lat) (τ, ~ 11

As said in the Introduction only now we get the coincidence of the two definitions of flat lapse and shift functions (this point was missed in the older treatments of parametrized Minkowski theories): ˜ µ (τ )lµ − lµ λ ˜ µν (τ )bν (τ )σ sˇ = N(τ, ~σ ), N[z](f lat) (τ, ~σ) ≈ N(f lat) (τ, ~σ ) = −λ sˇ µ µ ˜ ˜ µν (τ )bν (τ )σ sˇ = Nrˇ(τ, ~σ ). N[z](f lat)ˇr (τ, ~σ) ≈ N(f lat)ˇr (τ, ~σ ) = −λµ (τ )brˇ (τ ) − brˇ (τ )λ sˇ

(2.30)

Let us now restrict ourselves to configurations with ǫp2s > 0 and let us use the Wigner ◦ boost Lµ ν (ps , ps ) to boost to rest the variables bµAˇ, Ssµν of the following non-Darboux basis xµs , pµs , bµAˇ , Ssµν , αi , Πi of the Dirac brackets {., .}∗ . The following new non-Darboux basis is obtained [˜ xµs is no o more a fourvector; we choose the sign η = sign ps positive] ∂ǫB 1 A ρ (u(ps )) νρ ǫν (u(ps ))ηAB Ss = 2 ∂psµ q psν pµs 1 q = xµs − q [psν Ssνµ + ǫp2s (Ssoµ − Ssoν )] = ǫp2s ǫp2s (pos + η p2s ) x˜µs = xµs +

q

pµs + 2 ǫp2s η µo

S¯sAr prs

1 q q )+ q S¯so¯r prs ], = xµs − q [ηAµ (S¯so¯A − 2 2 2 o 2 o ǫps ps + ǫps ǫps (ps + ǫps )

pµs = pµs ,

αi = αi , Πi = Πi , µ A bA rˇ = ǫµ (u(ps ))brˇ ,

1 ∂ǫB (u(ps )) µ ρσ ∂ǫB σ (u(ps )) ν S˜sµν = Ssµν − ǫA ps − σ ps )Ss = ρ (u(ps ))ηAB ( 2 ∂psµ ∂psν q 1 βµ ν βν µ µν q q [psβ (Ss ps − Ss ps ) + ǫp2s (Ssoµ pνs − Ssoν pµs )], = Ss + 2 o 2 ǫps (ps + ǫps )

Jsµν = x˜µs pνs − x˜νs pµs + S˜sµν .

(2.31)

We have {˜ xµs , pνs }∗ = 0, δ is (prs bsAˇ − pss brAˇ ) oi r ∗ ˜ q , {Ss , bAˇ} = pos + ǫp2s {S˜ij , brˇ}∗ = (δ ir δ js − δ is δ jr )bsˇ, s

A

A

µναβ ˜γδ {S˜sµν , S˜sαβ }∗ = Cγδ Ss ,

12

(2.32)

and we can define B µν A B B A S¯sAB = ǫA µ (u(ps ))ǫν (u(ps ))Ss ≈ [brˇ (τ )bτ − brˇ (τ )bτ ]

Z

−

q

d3 σ σ rˇ J τ µ2 + δuˇvˇ(T −1 )uˇi Πi (T −1 )vˇj Πj (τ, ~σ ) −

B [bA rˇ (τ )bsˇ (τ )

−

A bB rˇ (τ )bsˇ (τ )]

Z

d3 σ σ rˇ J τ (T −1 )sˇl Πl (τ, ~σ ).

(2.33)

Let us now add six more gauge-fixings by selecting the special family of spacelike hyperplanes Στ W orthogonal to pµs (this is possible for ǫp2s > 0), which can be called the ‘Wigner foliation’ of Minkowski spacetime. This can be done by requiring (only six conditions are independent) TAµˇ (τ ) = bµAˇ (τ ) − ǫµA=Aˇ(u(ps )) ≈ 0

µ A A bA ˇ (τ ) = ǫµ (u(ps ))bA ˇ. ˇ (τ ) ≈ ηA A

⇒

(2.34)

Now the inverse tetrad bµAˇ is equal to the polarization vectors ǫµA (u(ps )) [see Appendix C] and the indices ‘ˇ r’ are forced to coincide with the Wigner spin-1 indices ‘r’, while o¯ = τ is a Lorentz-scalar index. One has S¯sAB ≈ (ηrA ητB − ηrB ητA )S¯sτ r − − (ηrA ηsB − ηrB ηsA )S¯srs , S¯srs ≈

Z

d3 σ J τ [σ r (T −1 )sl Πl − σ s (T −1 )rl Πl ] (τ, ~σ ),

S¯sτ r ≈ −S¯srτ = −

Z

d3 σ J τ σ r

q

µ2 + δuv (T −1 )ui Πi (T −1 )vj Πj (τ, ~σ ).

(2.35)

The comparison of S¯sAB with S˜sµν yields S˜suv = δ ur δ vt S¯srt δ vr S¯srt ps t q S˜sov = − . p0s + ǫp2s

(2.36)

The time constancy of TAµˇ ≈ 0 with respect to the Dirac Hamiltonian of Eq.(2.24) gives d µ [b (τ ) − ǫµr (u(ps ))] = {bµrˇ (τ ) − ǫµr (u(ps )), HD }∗ = dτ rˇ 1 ˜ αβ ˜ µα (τ )brˇα (τ ) ≈ 0 (τ ){brµˇ (τ ), Ssαβ (τ )}∗ = λ = λ 2 ˜ µν (τ ) ≈ 0, ⇒λ

(2.37)

˜ µν (τ ) ≈ 0 so that the independent gauge-fixings contained in Eqs.(2.34) and the constraints H form six pairs of second class constraints. ˜ µ (τ )] Besides Eqs.(2.19), now we have [remember that x˙ µs (τ ) = −λ

13

lµ = bµτ = uµ (ps ), zτµ (τ ) = x˙ µs (τ ) = N(τ ) =

q

g(τ )uµ (ps ) − x˙ sν (τ )ǫµr (u(ps ))ǫνr (u(ps )),

q

g(τ ) = [x˙ sµ (τ )uµ (ps ))],

x˙ 2s ,

√

γ = 1,

3

gτ τ = grs = −ǫ grs = −ǫδrs , µ gτ r = −ǫx˙ sµ ǫr (u(ps )) = −ǫδrs N s , N r = δ ru x˙ sµ ǫµu (u(ps )), ǫ ǫ Nr 1 g τ r = − x˙ sµ δ ru ǫµr (u(ps )) = −ǫ 2 , gτ τ = = 2 , g N g N µ ν N rN s x˙ sµ ǫu (u(ps ))x˙ sν ǫv (u(ps )) rs ) = −ǫ(δ − ). g rs = −ǫ(δ rs − δ ru δ sv [x˙ s · u(ps )]2 N2

(2.38)

On the hyperplane Στ W all the degrees of freedom z µ (τ, ~σ ) are reduced to the four degrees ˜ µ (τ )H ˜ µ (τ ) with of freedom x˜µs (τ ), which replace xµs . The Dirac Hamiltonian is now HD = λ ˜ µ (τ ) = pµ − H s −

Z

h

q

d3 σ J τ uµ (ps ) µ2 + δuv (T −1 )ui Πi (T −1 )vj Πj − i

− ǫµr (u(ps )) µ(T −1)rl Πl (τ, ~σ ) ≈ 0.

(2.39)

To find 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) ˜ αβ , T σˇ }∗ = {H B   σβ α  [η ǫ (u(p )) − η σα ǫβB (u(ps ))]  = δ ˇ s BB B   C=  ˜ γδ }∗ = {TAρˇ, H {TAρˇ, TBσˇ }∗ = 0   γ ρδ ργ δ . = δAA ˇ [η ǫA (u(ps )) − η ǫA (u(ps ))] 

˜ αβ , H ˜ γδ }∗ ≈ 0 {H



(2.40)

Since the constraints are ! redundant, this matrix has ! the following left and right null aαβ = aβα 0 eigenvectors: [aαβ arbitrary], . Therefore, one has to find a left 0 ǫB σ (u(ps )) ¯ CC ¯ = C C¯ = D, such that C¯ and D have the same left and right and right quasi-inverse C, null eigenvectors. One finds C¯ = ¯ = C C¯ = D = CC

0γδµν 1 B [η ǫ (u(ps )) − ησµ ǫB ν (u(ps ))] 4 σν µ 1 (η α η β − ηνα ηµβ ) 2 µ ν 1 0ρAµν (η ρ η D 2 τ A

1 [η ǫD (u(ps )) − ηδτ ǫD γ (u(ps ))] 4 γτ δ BD 0στ

0αβD τ − ǫDρ (u(ps ))ǫAτ (u(ps ))

!

!

(2.41)

and the new Dirac brackets are 1 ˜ γδ }∗ [ηγτ ǫD (u(ps )) − ηδτ ǫD (u(ps ))]{T τ , B}∗ + {A, B}∗∗ = {A, B}∗ − [{A, H δ γ D 4 B ∗ ˜ µν + {A, TBσ }∗ [ησν ǫB µ (u(ps )) − ησµ ǫν (u(ps ))]{H , B} ].

(2.42)

˜ αβ , B}∗∗ = 0 is immediate, we must use the relation b ˇ T µ ǫDρ = −T ρˇ While the check of {H Aµ D A [at this level we have TAµˇ = TAµ ] to check {TAρ , B}∗∗ = 0. 14

Then, we find the following brackets for the remaining variables x˜µs , pµs , αi , Πi {˜ xµs , pνs }∗∗ = −η µν , ′ ′ {αi (τ, ~σ ), Πj (τ, ~σ )}∗∗ = δji δ 3 (~σ − ~σ ),

(2.43)

and the following form of the generators of the “external” Poincaré group pµs , Jsµν = x˜µs pνs − x˜νs pµs + S˜sµν , δ ir S¯srs pss q S˜soi = − , pos + ǫp2s S˜ij = δ ir δ js S¯rs . s

(2.44)

s

˜ µ (τ ) ≈ 0, {H ˜ µ, H ˜ ν }∗∗ = 0, of Let us come back to the four first class constraints H Eq.(2.25). They can be rewritten in the following form [from Eqs.(1.9), (2.6) we have J τ = −det (∂r αi ), (T −1)ri = δ rs ∂s αi /det (∂u αk )] ˜ µ (τ ) = ǫs − Msys ≈ 0, H(τ ) = uµ (ps )H Msys = =

Z

=−

3

d σ J Z

Z

τ

d3 σM(τ, ~σ) = q

h

Z

µ2 + δuv (T −1 )ui Πi (T −1 )vj Πj (τ, ~σ ) =

d3 σ det (∂r

~ p (τ ) def H = P~sys = =

Z

v u u k t 2 α ) µ

+ δ uv

i ∂u αi ∂v αj Π Π σ), i j (τ, ~ [det (∂r αk )]2

d3 σMr (τ, ~σ ) =

d3 σµ J τ (T −1 )rl Πl (τ, ~σ) = −

Z

h

i

d3 σ µ δ rs ∂s αi Πi (τ, ~σ) ≈ 0,

(2.45)

where Msys is the invariant mass of the fluid. The first 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 Στ W vanishes. q There is no more a restriction on pµs in this special gauge, because uµ (ps ) = pµs / ǫp2s 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) = −λ(τ ) = x˙ µs (τ )uµ (ps ), Nr = N[z](f lat)r = N(f lat)r = −λr (τ ) = −x˙ µs (τ )ǫrµ (u(ps )),

(2.46)

so that the velocity of the origin of the coordinates on the Wigner hyperplane is x˙ µs (τ ) = ǫ[−λ(τ )uµ (ps ) + λr (τ )ǫµr (u(ps )),

[u2 (ps ) = ǫ, ǫ2r (u(ps )) = −ǫ].

(2.47)

The Dirac Hamiltonian is now ~ p (τ ), HD = λ(τ )H(τ ) − ~λ(τ ) · H 15

(2.48)

µ

xµs , HD }∗∗ = −λ(τ )uµ (ps ). Therefore, while the old xµs had a velocity x˙ µs and we have x˜˙ s = {˜ not parallel to the normal lµ = uµ (ps ) to the hyperplane as shown by Eqs.(2.47), the new µ x˜µs has x˜˙ s klµ and no classical zitterbewegung. Moreover, we have that Ts = l · x˜s = l · xs is the Lorentz-invariant rest frame time. q The canonical variables x˜µs , pµs , may be replaced by the canonical pairs ǫs = p2s , Ts = ps · x˜s /ǫs [to be gauge fixed with Ts − τ ≈ 0]; ~ks = p~s /ǫs = ~u(ps ), ~zs = ǫs (~x˜s − pp~os xõs ) ≡ ǫs ~qs . 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 ~2 P + Hrel ]. The role of the “external” center of mass mass from the relative motion [H = 2M is taken by the Wigner hyperplane, identified by the point x˜µs (τ ) and by its normal pµs . 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-fixing Ts − τ ≈ 0 [identifying the time parameter τ , 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 λ(τ ) = −ǫ]. ~ p (τ ). After this gauge fixing the Dirac Hamiltonian would be pure gauge: HD = −~λ(τ ) · H However, if we wish to reintroduce the evolution in the time τ ≡ 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 Poincaré group as Hamiltonian] ~ p (τ ). HD = Msys − ~λ(τ ) · H

(2.49)

The Hamilton equations for αi(τ, ~σ ) in the Wigner covariant rest-frame instant form are equivalent to the hydrodynamical Euler equations: ∂τ αi (τ, ~σ ) = {αi(τ, ~σ ), HD } = δ uv ∂u αi ∂v αj Πj q =− (τ, ~σ ) + ∂u αm ∂v αn det (∂r αk ) µ2 + δ uv [det Π Π (∂r αk )]2 m n + λr (τ )∂r αi (τ ),

∂τ Πi (τ, ~σ ) = {Πi (τ, ~σ ), Hd} = (ijk cyclic) s ∂u αm ∂v αn ∂ h suv j k 2 + δ uv ǫ ∂ α ∂ α Πm Πn + µ = u v ∂σ s [det (∂r αk )]2 n δuv ∂v αm slt ǫ ǫipq ∂l αp ∂t αq ) det∂u(∂αr αl ) Πm Πn i (δim δsu − det (∂r αl ) q + (τ, ~σ) + ∂u αm ∂v αn Π Π µ2 + δ uv [det (∂r αk )]2 m n ∂r αm ∂ h suv j k ǫ ∂ α ∂ α Πm + u v ∂σ s det (∂t αl ) i ∂r αm slt p q + (δim δrs − ǫ ǫ ∂ α ∂ α )Π (τ, ~σ ). ipq l t m det (∂t αl ) + λr (τ )

◦

(2.50)

In this special gauge we have bµA ≡ Lµ A (ps , ps ) (the standard Wigner boost for timelike R 3 τ −1 vs µν µν r ruv u Poincaré orbits), Ss ≡ Ssys [Ssys = ǫ d σ σ J (T ) Πs (τ, ~σ )], 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 defined inside the Wigner hyperplane and conjugate to P~sys ] become gauge variables [the natural gauge fixing ~ sys ≈ 0, implying λr (τ ) = 0, so that it coincides to the rest-frame condition P~sys ≈ 0 is X with the origin xµs (τ ) = z µ (τ, ~σ = 0) of the Wigner hyperplane]. The variable x˜µs 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 fields 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 definition of the relativistic center of mass of a perfect fluid configuration, 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 finds after the canonical reduction to Wigner hyperplane that there is an origin xµs (τ ) for these hyperplanes (a covariant non-canonical centroid) and a noncovariant canonical coordinate x˜µs (τ ) 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 Poincaré group. Instead, all the degrees of freedom of the isolated system (here the perfect fluid configuration) are described by canonical variables on the Wigner hyperplane restricted by the rest-frame condition P~sys ≈ 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 τ = 0 there is another realization of the Poincaré group, the “internal” Poincaré group. Its generators are built by using the invariant mass Msys and the 3-momentum P~sys , determined by the constraints (2.25), as the generators of the translations and by using the spin tensor S¯sAB as the generator of the Lorentz subalgebra τ

P = Msys =

Z

P r = P~sys = − r

τr

r

r Ssys

K =J J =

h

q

i

d3 σ J τ µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj (τ, ~σ ), Z

h

i

d3 σ J τ (T −1 )ri Πi (τ, ~σ ) ≈ 0,

= S¯sτ r ≡

Z

h

q

i

d3 σσ r J τ µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj (τ, ~σ),

Z h i 1 ruv ¯uv ruv = ǫ Ss ≡ ǫ d3 σσ u J τ (T −1 )vi Πi (τ, ~σ ). 2

(3.1)

By using the methods of Ref. [13] (where there is a complete discussion of many definitions 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 field 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 field Ref. [17]]. The non-canonical “internal” Møller 3-center of energy and the associated spin 3-vector are ~rsys = −

Z i h q ~ K 1 3 τ 2 + δ (T −1 )ui (T −1 )vj Π Π (τ, ~ µ σ ), = − d σ ~ σ J uv i j Pτ 2P τ

18

~ sys = J~ − ~rsys × P~ , Ω r {rsys , P s } = δ rs ,

r {rsys , Pτ} =

Pr , Pτ

1 rsu u ǫ Ωsys , (P τ )2 1 ~ u ~ {Ωrsys , Ωssys } = ǫrsu (Ωusys − τ 2 (Ω sys · P ) P ), (P ) r s {rsys , rsys }=−

{Ωrsys , P τ } = 0.

(3.2)

r s r The canonical “internal” 3-center of mass ~qsys [{qsys , qsys } = 0, {qsys , P s } = δ rs , s rsu u {J , qsys } = ǫ qsys ] is r

~ sys J~ × Ω q = ~qsys = ~rsys − q (P τ )2 − P~ 2(P τ + (P τ )2 − P~ 2 ) ~ K J~ × P~ q = −q + +q (P τ )2 − P~ 2 (P τ )2 − P~ 2 (P τ + (P τ )2 − P~ 2 ) ~ · P~ ) P~ (K q q + , P τ (P τ )2 − P~ 2 P τ + (P τ )2 − P~ 2 ≈ ~rsys

P~ ≈ 0;

f or

{~qsys , P τ } =

P~ ≈ 0, Pτ

~q sys = J~ − ~qsys × P~ = S ~ × P~ (J~ · P~ ) P~ K P τ J~ q +q −q =q ≈ (P τ )2 − P~ 2 (P τ )2 − P~ 2 (P τ )2 − P~ 2 P τ + (P τ )2 − P~ 2 ~sys , ≈S

P~ ≈ 0,

f or

r Ssys = ǫruv

~q sys , P~ } = {S ~q sys , ~qsys } = 0, {S

Z

d3 σ σ u J τ (T −1 )vs Πs (τ, ~σ ),

{Sqr sys , Sqs sys } = ǫrsu Squsys .

(3.3)

The “internal” non-canonical Fokker-Pryce 3-center of inertia’ ~ysys is ~sys × P~ ~sys × P~ S S q q ~ysys = ~qsys + q = ~rsys + , (P τ )2 − P~ 2 (P τ + (P τ )2 − P~ 2 ) P τ (P τ )2 − P~ 2 ~qsys = ~rsys +

P τ (P τ

r s {ysys , ysys }=

~sys × P~ S

+

q

Pτ

(P τ )2

Pτ h

+

q

(P τ )2 − P~ 2~ysys

q

(P τ )2

, 2 ~ −P ~sys · P~ ) P u (S

i

u q +q , ǫrsu Ssys τ 2 2 τ 2 2 τ ~ ~ (P ) − P (P ) − P (P + (P τ )2 − P~ 2 )

q

P~ ≈ 0 ⇒ ~qsys ≈ ~rsys ≈ ~ysys .

1

− P~ 2 )

=

P τ ~rsys +

(3.4)

The Wigner 3-vector ~qsys is therefore the canonical 3-center of mass of the perfect fluid r configuration [since ~qsys ≈ ~rsys , it also describe that point z µ (τ, ~qsys ) = xµs (τ ) + qsys ǫµr (u(ps )) where the energy of the configuration is concentrated]. 19

There should exist a canonical transformation from the canonical basis αi (τ, ~σ ), Πi (τ, ~σ), i i to a new basis ~qsys , P~ = P~φ , αrel (τ, ~σ ), Πrel i (τ, ~σ ) containing relative variables αrel (τ, ~σ), Πrel i (τ, ~σ ) with respect to the true center of mass of the perfect fluid configuration. To identify this final canonical basis one shall need the methods of Ref. [16]. The gauge fixing ~qsys ≈ 0 [it implies ~λ(τ ) = 0] forces all three internal center-ofmass variables to coincide with the origin xµs of the Wigner hyperplane. We shall denote qsys )µ x(~ (τ ) = xµs (0) + τ uµ(ps ) the origin in this gauge (it is a special centroid among the many s possible ones; xµs (0) is arbitrary). ~ sys = ~qsys ≈ 0 one As we shall see in the next Section, by adding the gauge fixings X µ can show that the origin xs (τ ) 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 )µ 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 Poincaré algebra of Eq. (2.44) with the generators pµs , Jsij = x˜is pjs − x˜js pis + δ ir δ js Sφrs , Ksi = δir S rs ps

~ ×~ (S p )r

s φ Jsoi = xõs pis − x˜is pos − po +ǫ = xõs pis − x˜is pos + δ ir pφo +ǫss [for xõs = 0 this is the Newton-Wigner s s s decomposition of Jsµν ] we can build three “external” collective 3-positions (all located on ~ s connected with the the Wigner hyperplane): i) the “external canonical 3-center of mass Q µ “external” canonical non-covariant center of mass x˜s ; ii) the “external” Møller 3-center of ~ s connected with the “external” non-canonical and non-covariant Møller center of energy R energy Rsµ ; iii) the “external” Fokker-Pryce 3-center of inertia connected with the “external” covariant non-canonical Fokker-Price center of inertia Ysµ (when there are the gauge fixings ~σsys ≈ 0 it coincides with the origin xµs ). 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 defining the canonical relative variables with respect to the center of mass. ~ s , the Møller R ~ s and the Fokker-Pryce The three “external” 3-variables, the canonical Q Y~s built by using the rest-frame “external” realization of the Poincaré algebra are

~ ~ s = (~x˜s − p~s xõ ) − Ssys × ~ps , ~s = − 1 K R pos pos s pos (pos + ǫs ) o~ ~ ~ ~ s + Ssys × p~s = ps Rs + ǫs Ys , ~ s = ~x˜s − ~ps xõ = ~zs = R Q s pos ǫs pos (pos + ǫs ) pos + ǫs ~ ~ ~ s + Ssys × p~s = R ~ s + Ssys × ~ps , Y~s = Q ǫs (pos + ǫs ) pos ǫs 1 ~ s = J~s − R ~ s × p~s , {Rsr , Rss } = − o 2 ǫrsu Ωus , Ω (ps ) ~sys · ~ps ) pu i (S 1 rsu h u s S + ǫ , {Ysr , Yss } = sys ǫs pos ǫs (pos + ǫs ) ~ s = p~s · R ~ s = p~s · Y~s = ~ks · ~zs , p~s · Q

20

~ s = Y~s = R ~ s, p~s = 0 ⇒ Q

(3.5) ◦µ

with the same velocity and coinciding in the Lorentz rest frame where ps = ǫs (1; ~0) In Ref. [13] in a one-time framework without constraints and at a fixed time, it is shown ~ s and R ~ s ] satisfies the condition {K r , Y s } = Y r {Y s , po } for that the 3-vector Y~s [but not Q s s s s s being the space component of a 4-vector Ysµ . In the enlarged canonical treatment including ~ s, R ~ s , Y~s , to time variables, it is not clear which are the time components to be added to Q rebuild 4-dimesnional quantities x˜µs , Rsµ , Ysµ , in an arbitrary Lorentz frame Γ, in which the origin of the Wigner hyperplane is the 4-vector xµs = (xos ; ~xs ). We have x˜µs (τ ) = (˜ xos (τ ); ~x˜s (τ )) = xµs −

µ i h 1 νµ oµ oν psν ps ), p S + ǫ (S − S sν s s s s ǫs (pos + ǫs ) ǫ2s

pµs ,

q q ~ks · ~zs 0 o 2 2 ~ ~ ~ = 1 + ks (Ts + ) = 1 + ks (Ts + ks · ~qs ) 6= xs , ps = ǫs 1 + ~ks2 , ǫs ~ ~x˜s = ~zs + (Ts + ks · ~zs )~ks = ~qs + (Ts + ~ks · ~qs )~ks , ~ps = ǫs~ks . (3.6) ǫs ǫs q

xõs

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 xµs (τ ) = ˜ ) in some ~σ ˜ and the worldline of z µ (τ, ~0) in ~σ = 0, the pseudo worldline of x˜µs (τ ) = z µ (τ, ~σ µ µ the Fokker-Pryce 4-vector Ys (τ ) = z (τ, ~σY ) in some ~σY [on this worldline one can put the “internal center of mass” with the gauge fixing ~qφ ≈ 0 (~qφ ≈ ~rφ ≈ ~yφ due to P~φ ≈ 0)]; one also has Rsµ = z µ (τ, ~σR ). Since we have Ts = u(ps ) · xs = u(ps ) · x˜s ≡ τ on the Wigner hyperplane labelled by τ , we require that also Ysµ , Rsµ have time components such that they too satisfy u(ps ) · Ys = u(ps ) · Rs = Ts ≡ τ . Therefore, it is reasonable to assume that x˜µs , Ysµ and Rsµ satisfy the following equations consistently with Eqs.(3.2), (3.3) when Ts ≡ τ and ~qsys ≈ 0 ~ s + ~ps xõ ) = x˜µs = (˜ xos ; ~x ˜s ) = (˜ xos ; Q pos s ~ks · ~zs ~zs qsys )µ = (˜ xos ; + (Ts + )~ks ) = x(~ + ǫµu (u(ps ))˜ σu, s ǫs ǫs ~s ) = Ysµ = (˜ xos ; Y ~ks · ~zs ~sys × p~s S 1 = (˜ xos ; [~zs + ] + (T + )~ks ) = s ǫs ǫs [1 + uo (ps )] ǫs ~sys × p~s )r (S = x˜µs + ηrµ = ǫs [1 + uo(ps )] qsys )µ = x(~ + ǫµu (u(ps ))σYu , s ~ s) = Rµ = (˜ xo ; R s

s

~ks · ~zs ~sys × p~s 1 S [~zs − )~ks ) = ] + (T + s ǫs ǫs uo (ps )[1 + uo(ps )] ǫs ~sys × p~s )r (S µ µ = x˜s − ηr = ǫs uo (ps )[1 + uo(ps )] = (˜ xos ;

21

qsys )µ = x(~ + ǫµu (u(ps ))σRu , s

Ts = u(ps ) · xs(~qsys ) = u(ps ) · x˜s = u(ps ) · Ys = u(ps ) · Rs , r

σ ˜ = = ≈ σYr = = σRr = = ≈

ǫrµ (u(ps ))[uν (ps )Ssνµ + Ssoµ ] = − = [1 + uo(ps )] rs s rs s Ssys ps Ssys u (ps ) τr r −Ssys + = ǫ r + ≈ s φ o ǫs [1 + u (ps )] 1 + uo (ps ) rs s rs s Ssys u (ps ) Ssys u (ps ) r ǫs qsys + ≈ , o 1 + u (ps ) 1 + uo (ps ) ~sys × p~s )u (S qsys )µ µ r ǫrµ (u(ps ))[x(~ − Y ] = σ ˜ − ǫ (u(p )) = ru s s s ǫs [1 + uo (ps )] rs s Ssys u (ps ) r r σ ˜r + = ǫs rsys ≈ ǫs qsys ≈ 0, 1 + uo (ps ) ~sys × p~s )u (S qsys )µ ǫrµ (u(ps ))[x(~ − Rsµ ] = σ ˜ r + ǫru (u(ps )) o = s ǫs u (ps )[1 + uo(ps )] rs s S rs us (ps ) [1 − uo (ps )]Ssys u (ps ) r σ ˜ r − o sys = ǫ r + ≈ s sys o o o u (ps )[1 + u (ps )] u (ps )[1 + u (ps )] rs s [1 − uo(ps )]Ssys u (ps ) , o o u (ps )[1 + u (ps )] qsys )µ ǫrµ (u(ps ))[x(~ s

qsys )µ ⇒ x(~ (τ ) = Ysµ , s

x˜µs ]

f or ~qsys ≈ 0,

(3.7)

namely in the gauge ~qsys ≈ 0 the external Fokker-Pryce non-canonical center of inertia qsys )µ coincides with the origin x(~ (τ ) carrying the “internal” center of mass (coinciding with s the “internal” Möller 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 find the center-of-mass canonical basis, then, in the gauge ~qsys ≈ 0 and Ts ≈ τ , the perfect fluid configurations would have the four-momentum density peaked qsys )µ i on the worldline x(~ (Ts ); the canonical variables αrel (τ, ~σ), Πrel i (τ, ~σ ) would characterize s the relative motions with respect to the “monopole” configuration describing the center of mass of the fluid configuration. The “monopole” configurations would be identified 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 Fokker~sys |/P τ . Pryce 4-vector extends to a worldtube with radius (the Moller radius) |S

22

IV. DIXON’S MULTIPOLES IN MINKOWSKI SPACETIME.

Let us now look at other properties of a perfect fluid configuration 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 fluid configuration 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 √ ˇˇ ◦ σ ) = ηµν ∂Aˇ[ gT AB [α]zBνˇ ](τ, ~σ ) = 0, µ (τ, ~ µ ∂z ∂zAˇ ∂L ∂L ◦ (τ, ~σ ) = 0, − ∂Aˇ ∂φ ∂∂Aˇφ ∂L

− ∂Aˇ

(4.1)

where we introduced the energy-momentum tensor [with a different sign with respect to the standard convention to conform with Ref. [1]] h 2 δS i ˇˇ (τ, ~σ ) = T AB (τ, ~σ )[α] = √ g δgAˇBˇ

= =

h

h

ˇˇ

− ρg AB + n ˇ

ˇ

ˇ

ˇˇ

ˇ

ˇ

J AJ B i ∂ρ ˇˇ |s (g AB − ) (τ, ~σ) = ∂n gCˇ Dˇ J Cˇ J Dˇ ˇ

i

− ǫρU A U B + p(g AB − ǫU A U B ) (τ, ~σ ) ˇ

ˇ

J AJ B q 2 3 = −ǫµnU U = −ǫ 2 √ τ µ + guˇvˇ(T −1 )uˇi (T −1 )vˇj Πi Πj . N γJ Aˇ

dust

ˇ B

(4.2)

√ When ∂Aˇ [ gzBµˇ ] = 0, as it happens on the Wigner hyperplanes in the gauge Ts − τ ≈ 0, ◦ ~λ(τ ) = 0, we get the conservation of the energy-momentum tensor T AˇBˇ , i.e. ∂ ˇT AˇBˇ = 0. A Otherwise, there is compensation coming from the dynamics of the surface. As shown in Eq.(A9) the conserved, manifestly Lorentz covariant energy-momentum ∂ρ |s − ρ)] is tensor of the perfect fluid with equation of state ρ = ρ(n, s) [so that p = n ∂n T µν (x)[˜ α] = = = dust

J µJ ν i ∂ρ 4 µν |s ( g − 4 ) (x) = ∂n gαβ J α J β

h

− ǫρ 4 g µν + n

h

− ǫ(ρ + p)U µ U ν + p 4 g µν (x)

h

i

− ǫρU µ U ν + p(4 g µν − ǫU µ U ν ) (x) = h

i

= −ǫµ n U µ U ν (x),

i

ˇ

nU µ = J µ = −ǫµνρσ ∂ν α ˜ 1 ∂ρ α ˜ 2 ∂σ α ˜ 3 = nzAµˇ U A = ˇ

= zAµˇ J A = zτµ J τ + zrˇµ J rˇ.

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

23

(4.3)

ˇˇ

ˇ

ˇ

ˇ

ˇ

α] = T AB (τ, ~σ )[α] = zµA (τ, ~σ )zνB (τ, ~σ )T µν (x = z(τ, ~σ ))[˜ ˜ ◦ z], = zµA (τ, ~σ )zνB (τ, ~σ )T µν (τ, ~σ )[α = α

(4.4)

On Wigner hyperplanes, where Eqs.(2.38) hold and where we have z µ (τ, ~σ ) = xµs (τ ) + ǫµu (u(ps ))σ u , 4 µν

η

h

= ǫ uµ (ps )uν (ps ) −

N = x˙ s · u(ps ), Yr =

3 X

i

ǫµr (u(ps ))ǫνr (u(ps )) ,

r=1

N r = δ ru x˙ sµ ǫµu (u(ps )),

Jr + N rJτ J r + δ ru x˙ sµ ǫµu (u(ps ))J τ = , N x˙ s · u(ps )

q

ǫgAB J A J B q = (J τ )2 − δrs Y r Y s N µJ τ dust , = q µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj

n=

we get [Aˇ = A]

T µν [xβs (τ ) + ǫβu (u(ps ))σ u ][α] = = [δAτ x˙ µs (τ ) + δAr ǫµr (u(ps ))][δBτ x˙ νs (τ ) + δBs ǫνs (u(ps ))]T AB (τ, ~σ) = = x˙ µs (τ )x˙ νs (τ )T τ τ (τ, ~σ ) + ǫµr (u(ps ))ǫνs (u(ps ))T rs (τ, ~σ) + + [x˙ µs (τ )ǫνr (u(ps )) + x˙ νs (τ )ǫµr (u(ps ))]T rτ (τ, ~σ), ǫρ + [x˙ s · u(ps )]2 i ǫ (J τ )2 ∂ρ − ) (τ, ~σ ) + n |s ( ∂n [x˙ s · u(ps )]2 [x˙ s · u(ps )]2 [(J τ )2 − δuv Y u Y v ] q i h Jτ dust 2 + δ (T −1 )ui (T −1 )vj Π Π (τ, ~ σ ), = −ǫ µ uv i j [x˙ s · u(ps )]2 h δ ru x˙ sµ ǫµu (u(ps )) T rτ (τ, ~σ ) = T τ r (τ, ~σ ) = − ǫρ − [x˙ s · u(ps )]2 ∂ρ δ ru x˙ sµ ǫµu (u(ps )) − n |s (ǫ + ∂n [x˙ s · u(ps )]2 i J rJ τ ) (τ, ~σ) + [x˙ s · u(ps )]2 [(J τ )2 − δuv Y u Y v ] q i h Jr dust 2 + δ (T −1 )ui (T −1 )vj Π Π (τ, ~ µ σ ), = −ǫ uv i j [x˙ s · u(ps )]2 h ˙ sµ ǫµu (u(ps ))x˙ sν ǫνv (u(ps )) rs rs ru sv x T (τ, ~σ ) = ǫρ(δ − δ δ )− [x˙ s · u(ps )]2 h

T τ τ (τ~σ ) = −

24

(4.5)

x˙ sµ ǫµu (u(ps ))x˙ sν ǫνv (u(ps )) ∂ρ rs |s ǫ(δ − δ ru δ sv )+ ∂n [x˙ s · u(ps )]2 i JrJs + (τ, ~σ ) [x˙ s · u(ps )]2 [x˙ s · u(ps )]2 [(J τ )2 − δuv Y u Y v ] q i h J rJ s dust 2 + δ (T −1 )ui (T −1 )vj Π Π (τ, ~ µ σ ). = −ǫ uv i j [x˙ s · u(ps )]2 J τ −n

(4.6)

Since we have x˙ µs (τ ) = −λµ (τ ) = ǫ[uµ (ps )uν (ps ) − ǫµr (u(ps ))ǫνr (u(ps ))]x˙ sν (τ ) = h

i

= ǫ − uµ (ps )λ(τ ) + ǫµr (u(ps ))λr (τ ) ,

x˙ 2s (τ ) = λ2 (τ ) − ~λ(τ ) > 0, −λ(τ )uµ (ps ) + λr (τ )ǫµr (u(ps )) x˙ µ (τ ) q =ǫ , Usµ (τ ) = q s x˙ 2s (τ ) λ2 (τ ) − ~λ2 (τ )

(4.7)

the timelike worldline described by the origin of the Wigner hyperplane is arbitrary (i.e. gauge dependent): xµs (τ ) may be any covariant non-canonical centroid. As already said the hreal “external” center of imass is the canonical non-covariant x˜µs (Ts ) = µ xµs (Ts )− ǫs (po1+ǫs) psν Ssνµ +ǫs (Ssoµ +Ssoν psνǫ2ps ) : it describes a decoupled point particle observer. s s ~ sys = ~qsys ≈ 0, implying λ(τ ) = −1, ~λ(τ ) = 0 [gτ τ = ǫ, N = 1, In the gauge Ts − τ ≈ 0, X r µ N = gτ r = 0], we get x˙ s (Ts ) = uµ (ps ). Therefore, in this gauge, we have the centroid qsys )µ xµs (Ts ) = x(~ (Ts ) = xµs (0) + Ts uµ (ps ), s

(4.8)

~ sys = ~qsys ≈ 0. which carries the fluid “internal” collective variable X In this gauge we get the following form of the energy-momentum tensor [Y r = J r ] T µν [xs(~qsys )β (Ts ) + ǫβu (u(ps ))σ u ][α] = uµ (ps )uν (ps )T τ τ (Ts , ~σ ) + + [uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]T rτ (Ts , ~σ ) + + ǫµr (u(ps ))ǫνs (u(ps ))T rs (Ts , ~σ ), h

T τ τ (Ts , ~σ ) = − ǫρ + +n dust

i ∂ρ (J τ )2 |s (ǫ − τ 2 ) (Ts , ~σ) ∂n (J ) − δuv Y u Y v h

q

i

q

i

= −ǫ J τ µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj (Ts , ~σ ), h

T rτ (Ts , ~σ ) = − n dust

h

i ∂ρ J rJ τ |s τ 2 (Ts , ~σ ) ∂n (J ) − δuv Y u Y v

= −ǫ J r µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj (Ts , ~σ ), h

T rs (Ts , ~σ ) = ǫρδ rs − n

∂ρ rs |s ǫδ + ∂n

i JrJs + τ 2 (Ts , ~σ ) (J ) − δuv Y u Y v

25

h J rJ s q

dust

= −ǫ

Jτ

i

µ2 + δuv (T −1 )ui (T −1 )vj Πi Πj (Ts , ~σ ),

with total 4 − momentum PTµ [α]

=

Z

d3 σT µν [xµs (Ts ) + ǫµu (u(ps ))σ u ][α]uν (ps ) =

= −P τ uµ (ps ) − P r ǫµr (u(ps )) ≈ −P τ uµ (ps ) = = −Msys uµ (ps ) ≈ −pµs , and total mass M[α] = PTµ [α]uµ (ps ) = −P τ = −Msys .

(4.9)

The stress tensor of the perfect fluid configuration on the Wigner hyperplanes is T rs (Ts , ~σ). 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 fluid as seen from a local observer at rest (see the Eckart decomposition in Appendix B): h

T µν [˜ α] = ρ[α, Π] uµ (ps )uν (ps ) + + P[α, Π] [η µν − uµ (ps )uν (ps )] + + uµ (ps )q ν [α, Π] + uν (ps )q µ [α, Π] + i

rs + Tan [α, Π] ǫµr (u(ps ))ǫνs (u(ps )) (Ts , ~σ ),

ρ[α, Π] = T τ τ , 1 X uu P[α, Π] = T , 3 u q µ [α, Π] = ǫµr (u(ps ))T rτ , 1 X uu rs Tan [α, Π] = T rs − δ rs T , 3 u

uv δuv Tan [α, Π] = 0,

(4.10)

where i) the constant normal uµ (ps ) to the Wigner hyperplanes replaces the hydrodynamic velocity field of the fluid; ii) ρ[α, Π](Ts , ~σ ) is the energy density; iii) P[α, Π](Ts , ~σ) is the analogue of the pressure (sum of the thermodynamical pressure and of the non-equilibrium bulk stress or viscous pressure); iv) q µ [α, Π](Ts , ~σ ) is the analogue of the heat flow; rs v) Tan [α, Π](Ts , ~σ ) is the shear (or anisotropic) stress tensor. We can now study the manifestly Lorentz covariant Dixon multipoles [20] for the perfect fluid configurationon the Wigner hyperplanes in the gauge λ(τ ) = −1, ~λ(τ ) = 0 [so qsys )µ that x˙ µs (Ts ) = uµ (ps ), x¨µs (Ts ) = 0, xµs (Ts ) = x(~ (Ts ) = xµs (0) + uµ (ps )Ts ] with res µ qsys )µ spect to the origin an arbitrary timelike worldline w (Ts ) = z µ (Ts , ~η(Ts )) = x(~ (Ts ) + s 26

qsys )µ ǫµr (u(ps ))η r (Ts ). Since we have z µ (Ts , ~σ ) = x(~ (Ts )+ǫµr (u(ps ))σ r = w µ (Ts )+ǫµr (u(ps ))[σ r − s def η r (Ts )] = w µ (Ts ) + δz µ (Ts , ~σ ) [for ~η (Ts ) = 0 we get the multipoles with respect to the origin of coordinates], we obtain [ (µ1 ..µn ) means symmetrization, while [µ1 ..µn ] means antisymmetrization; tµT1 ..µn µν (Ts , ~η = 0) = tµT1 ..µn µν (Ts )] (µ ...µn )(µν)

tTµ1 ...µn µν (Ts , ~η) = tT 1 =

Z

(Ts , ~η) =

qsys )β d3 σ δz µ1 (Ts , ~σ )...δz µn (Ts , ~σ ) T µν [x(~ (Ts ) + ǫβu (u(ps ))σ u ][α] = s

= ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ǫµA (u(ps ))ǫνB (u(ps )) ITr1..rn AB (Ts , ~η ) = h

= ǫµr11 (u(ps ))...ǫµrnn (u(ps )) uµ (ps )uν (ps )ITr1 ...rnτ τ (Ts , ~η) + + ǫµr (u(ps ))ǫνs (u(ps ))ITr1 ...rnrs (Ts , ~η ) + i

+ [uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]ITr1 ...rnrτ (Ts , ~η) , ITr1 ..rn AB (Ts , ~η)

=

Z

d3 σ [σ r1 − η r1 (Ts )]...[σ rn − η rn (Ts )]T AB (Ts , ~σ)[α],

tTµ1 ...µn µν (Ts , ~η ) = 0,

uµ1 (ps )

n = 0 (monopole) ITτ τ (Ts ) = P τ ,

F or~η = 0 tTµ1 ...µnµ µ (Ts ) =

Z

ITrτ (Ts ) = P r ,

d3 σδxµs 1 (~σ )...δxµs n (~σ )T µ µ [xs(~qsys )β (Ts ) + ǫβu (u(ps ))σ u ][α] =

def µ1 = ǫr1 (u(ps ))...ǫµrnn (u(ps ))ITr1 ...rnA A (Ts )

1 1 ITr1 r2 A A (Ts ) = IˇTr1 r2 A A (Ts ) − δ r1 r2 δuv ITuvA A (Ts ) = iTr1 r2 (Ts ) − δ r1 r2 δuv iuv T (Ts ), 3 2 1 δuv IˇTuvA A (Ts ) = 0, IˇTr1 r2 A A (Ts ) = iTr1 r2 (Ts ) − δ r1 r2 δuv iuv T (Ts ), 3 iTr1 r2 (Ts ) = ITr1 r2 A A (Ts ) − δ r1 r2 δuv ITuvA A (Ts ), t˜Tµ1 ...µn (Ts ) = tTµ1 ...µn µν (Ts )uµ (ps )uν (ps ) = = ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ITr1 ..rn τ τ (Ts ), r1 t˜µT1 (Ts ) = ǫµr11 (u(ps ))ITr1 τ τ (Ts ) = −P τ ǫµr11 (u(ps ))rsys ,

t˜Tµ1 µ2 (Ts ) = ǫµr11 (u(ps ))ǫµr22 (u(ps ))ITr1 r2 τ τ (Ts ), 1 1 ITr1 r2 τ τ (Ts ) = IˆTr1 r2 τ τ (Ts ) − δ r1 r2 δuv ITuvτ τ (Ts ) = ˜iTr1 r2 (Ts ) − δ r1 r2 δuv˜iuv T (Ts ), 3 2 1 δuv IˆTuvτ τ (Ts ) = 0, IˆTr1 r2 τ τ (Ts ) = ˜iTr1 r2 (Ts ) − δ r1 r2 δuv˜iuv T (Ts ), 3 ˜iTr1 r2 (Ts ) = ITr1 r2 τ τ (Ts ) − δ r1 r2 δuv ITuvτ τ (Ts ).

(4.11)

The Wigner covariant multipoles ITr1 ..rn τ τ (Ts ), ITr1 ..rn rs (Ts ), ITr1 ..rn rτ (Ts ) are the mass, 27

stress and momentum multipoles respectively. The quantities IˇTr1 r2 A A (Ts ) and iTr1 r2 (Ts ) are the traceless quadrupole moment and the inertia tensor defined by Thorne in Ref. [21]. The quantities ITr1 r2 τ τ (Ts ) and ˜iTr1 r2 (Ts ) are Dixon’s definitions of quadrupole moment and of tensor of inertia respectively. Moreover, Dixon’s definition of “center of mass” of an extended object is t˜µT1 (Ts ) = 0 r or ITrτ τ (Ts ) = −P τ rsys = 0: therefore the quantity ~rsys defined in the previous equation is r s rs a non-canonical [{rsys , rsys } = Ssys ] candidate for the “internal” center of mass of the field qsys )µ configuration: its vanishing is a gauge fixing for P~ ≈ 0 and implies xµs (Ts ) = x(~ (Ts ) = s xµs (0) + uµ (ps )Ts . As we have seen in the previous Section ~rsys is the “internal” Møller 3-center of energy and we have ~rsys ≈ ~qsys ≈ ~ysys . dr r dI rτ τ (T ) ◦ When ITrτ τ (Ts ) = 0, the equations 0 = TdTs s = −P τ dTsys = = − P r implies the correct s ~

◦

rsys momentum-velocity relation PPτ = d~dT ≈ 0. s Then there are the related Dixon multipoles

(µ ...µ )µ

pTµ1 ...µn µ (Ts ) = tTµ1 ...µn µν (Ts )uν (ps ) = pT 1 n (Ts ) = = ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ǫµA (u(ps ))ITr1 ..rn Aτ (Ts ), uµ1 (ps ) n=0

pTµ1 ...µn µ (Ts ) = 0, ⇒ pµT (Ts ) = PTµ [α] = −ǫǫµA (u(ps ))P A ≈ −ǫpµs ,

pµT1 ..µn µ (Ts )uµ (ps ) = t˜Tµ1 ...µn (Ts ) = ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ITr1 ..rn τ τ (Ts ).

(4.12)

The spin dipole is defined as [µν]

ν]

rAτ STµν (Ts )[α] = 2pT (Ts ) = 2ǫ[µ r (u(ps ))ǫA (u(ps ))IT (Ts ) = = Ssµν = rs τr = ǫµr (u(ps ))ǫνs (u(ps ))Ssys + [ǫµr (u(ps ))uν (ps ) − ǫνr (u(ps ))uµ (ps )]Ssys , τr r uµ (ps )STµν (Ts )[α] = −ǫνr (u(ps ))Ssys = −t˜νT (Ts ) = P τ ǫνr (u(ps ))rsys ,

(4.13)

with uµ (ps )STµν (Ts )[α] = 0 when t˜µT1 (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]τ ν] rsτ barycentric spin dipole is STµν (Ts )[α] = 2ǫ[µ (Ts ) = r (u(ps ))ǫs (u(ps ))IT (Ts ), so that IT µν r s ǫµ (u(ps ))ǫν (u(ps ))ST (Ts )[α]. As shown in Ref. [20], if the fluid configuration has a compact support W on the Wigner hyperplanes ΣW τ and if f (x) is a C ∞ complex-valued scalar function on Minkowski spacetime R with compact support [so that its Fourier transform f˜(k) = d4 xf (x)eik·x is a slowly increasing entire analytic function on Minkowski spacetime (|(xo + iy o)qo ...(x3 + iy 3)q3 f (xµ + iy µ)| < o 3 Cqo ...q3 eao |y |+...+a3|y | , aµ > 0, qµ positive integers for every µ and Cqo ...q3 > 0), whose inverse R d4 k −ik·x ˜ ], we have [we consider ~η = 0 with δz µ = δxµs ] is f (x) = (2π) 4 f (k)e 28

< T µν , f > = =

Z

d4 xT µν (x)f (x) =

Z

dTs

Z

d3 σf (xs + δxs )T µν [xs (Ts ) + δxs (~σ )][α] =

dTs

Z

=

Z

dTs

Z

d4 k ˜ f(k)e−ik·xs (Ts ) (2π)4

Z

∞ X d4 k ˜ (−i)n −ik·xs (Ts ) f(k)e kµ1 ...kµn tTµ1 ...µn µν (Ts ), 4 (2π) n! n=0

∞ X

3

dσ

Z

d4 k ˜ f (k)e−ik·[xs(Ts )+δxs (~σ )] T µν [xs (Ts ) + δxs (~σ )][α] = (2π)4

=

Z

(−i)n [kµ ǫµu (u(ps ))σ u ]n = n! n=0 =

Z

dTs

Z

d3 σT µν [xs (Ts ) + δxs (~σ )][α]

(4.14)

and, but only for f (x) analytic in W [20], we get =

Z

∞ X

1 µ1 ...µnµν ∂ n f (x) dTs |x=xs(Ts ) , tT (Ts ) µ1 ∂x ...∂xµn n=0 n!

⇓

∞ X

∂n (−1)n T (x)[α] = n! ∂xµ1 ...∂xµn n=0 µν

Z

dTs δ 4 (x − xs (Ts ))tTµ1 ...µn µν (Ts ) =

= ǫµA (u(ps ))ǫνB (u(ps ))T AB (Ts , ~σ )[α] = ∞ X (−)n r1 ...rnAB ∂ n δ 3 (~σ − ~η(Ts )) µ ν = ǫA (u(ps ))ǫB (u(ps )) |η~=0 . IT (Ts , ~η ) ∂σ r1 ...∂σ rn n=0 n!

(4.15)

For a non analytic f (x) we have N X

< T ,f > =

Z

dTs

1 µ1 ...µn µν ∂ n f (x) |x=xs (Ts ) + tT (Ts ) µ1 ∂x ...∂xµn n=0 n!

+

Z

dTs

Z

µν

∞ X d4 k ˜ (−i)n −ik·xs (Ts ) f (k)e kµ1 ...kµn tTµ1 ...µn µν (Ts ), (2π)4 n! n=N +1

(4.16)

and, as shown in Ref. [20], from the knowledge of the moments tTµ1 ...µn µ (Ts ) for all n > N we can get T µν (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 ∂µ T µν = 0, so that for analytic fluid configurations from Eq.(D5) we get ◦

ν)

T µν (x)[α] = −ǫu(µ (ps )ǫA (u(ps ))

Z

dTs δ 4 (x − xs (Ts )) P A +

1 ∂ ρ(µ dTs δ 4 (x − xs (Ts )) ST (Ts )[α] uν) (ps ) + 2 ∂xρ Z ∞ X (−1)n ∂n + dTs δ 4 (x − xs (Ts )) ITµ1 ..µn µν (Ts ), µ µ n 1 n! ∂x ...∂x n=2 +

Z

ν)

T µν (w + δz) = −ǫu(µ (ps ) ǫA (u(ps ))P A δ 3 (~σ − ~η (Ts )) + 29

∂δ 3 (~σ − ~η(ts )) 1 ρ(µ + + ST (Ts , ~η )[α]uν) (ps )ǫrρ (u(ps )) 2 ∂σ r ∞ X (−)n h n + 3 µ + u (ps )uν (ps )ITr1 ...rnτ τ (Ts , ~η ) + n! n + 1 n=2 1 + [uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]ITr1 ...rnrτ (Ts , ~η) + n + ǫµs1 (u(ps ))ǫνs2 (u(ps ))[ITr1 ...rns1 s2 (Ts , ~η ) − n + 1 (r1 ...rns1 )s2 (r ...r s )s − (IT (Ts , ~η ) + IT 1 n 2 1 (Ts , ~η)) + n i (r1 ...rn s1 s2 ) + IT (Ts , ~η )] , (µ ..µ

(4.17)

|µ|µ )ν

JT 1 n−1 n (Ts ), with the quantities where for n ≥ 2 and ~η = 0 ITµ1 ..µn µν (Ts ) = 4(n−1) n+1 JTµ1 ..µn µνρσ (Ts ) being the Dixon 22+n -pole inertial moment tensors given in Eqs.(D7) [the quadrupole and related inertia tensor are proportional to ITr1 r2 τ τ (Ts )]. ◦ The equations ∂µ T µν = 0 imply the Papapetrou-Dixon-Souriau equations for the ‘poledipole’ system PTµ (Ts ) and STµν (Ts )[α] [see Eqs.(D1) and (D4); here ~η = 0] dPTµ (Ts ) ◦ = 0, dTs dSTµν (Ts )[α] ◦ [µ ν] = 2PT (Ts ) uν] (ps ) = −2ǫP r ǫ[µ r (u(ps )) u (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 fluids. For them we have from Eqs.(1.9), (1.10) and (2.1) [in general µ is not the chemical potential but only a parameter] n=

|J| X √ =√ , N γ γ

|J| X ρ = ρ(n) = ρ( √ ) = µf ( √ ), N γ γ X √ L = −µN γf ( √ ). γ

(5.1)

Some possible equations of state for such fluids are (see also Appendix A): 1) p = 0, dust: this implies X ρ(n) = µn = µ √ , γ

i.e.

∂f ( √Xγ )

X X f(√ ) = √ , γ γ

∂X

1 =√ . γ

(5.2)

− ρ(n) (k 6= −1 because otherwise ρ = const., µ = 0, f = −T s). 2) p = kρ(n) = n ∂ρ(n) ∂n For k = 13 one has the photon gas. The previous differential equation for ρ(n) implies X ρ(n) = (an)k+1 = µnk+1 = µ( √ )k+1 , γ ∂f ( √Xγ )

X X f ( √ ) = ( √ )k+1 , γ γ

∂X

(µ = ak+1 ),

i.e.

k+1 X = √ ( √ )k , γ γ

(5.3)

[for k → 0 we recover 1)]. More in general one can have k = k(s): this is a non-isentropic perfect fluid with ρ = ρ(n, s). − ρ(n) (γ 6= 1) [22]. It is an isentropic polytropic perfect fluid 3) p = kργ (n) = n ∂ρ(n) ∂n (γ = 1 + n1 ). The differential equation for ρ(n) implies [a is an integration constant; the ∂ρ chemical potential is µ = ∂n |s ] ρ(n) =

an [1 − k(an)γ−1 ] X

1 γ−1

=

an 1

[1 − k(an) n ]n

,

i.e.

X

√ √ X γ γ f(√ ) = , = 1 1 X γ−1 γ [1 − k(a √Xγ ) n ]n [1 − k(a √γ ) ] γ−1

∂f ( √Xγ ) ∂X

γ 1 X X 1 1 = √ [1 − k(a √ )γ−1 ]− γ−1 = √ [1 − k(a √ ) n ]−(n+1) . γ γ γ γ

31

(5.4)

Instead in Ref. [23,24] a polytropic perfect fluid is defined by the equation of state [see the last of Eqs.(B4); m is a mass] ρ(n, ns) = mn +

k(s) (mn)γ , γ−1

(5.5)

and has pressure p = k(s)(mn)γ = (γ − 1)(ρ − mn) and chemical potential or specific ′ γ (mn)γ−1 of Eq.(B3). enthalpy µ = mc2 + mk(s) γ−1 4) p = p(ρ), barotropic perfect fluid. In the isentropic case one gets ρ = ρ(n) by solving p(ρ(n)) = n ∂ρ(n) − ρ(n). ∂n 5)Relativistic ideal (Boltzmann) gas [2] (this is a non-isentropic case): p = nkB T

and ρ = mc2 nΓ(β) − p,

µ=

ρ+p = mc2 Γ(β), n

(5.6)

2

K3 (β) with β = kmc , Γ(β) = K (Ki are modified Bessel functions). One gets the equation of 2 (β) BT state ρ = ρ(n, s) by solving the differential equation

mc2 n ∂ρ |s = mc2 Γ( ∂ρ ). ∂n n ∂n |s − ρ

(5.7) 4 + β2 β 4/3

5a) Ultrarelativistic case β 1 (kB T 0, A 4 27 4 · 27α4 α4 4µ4 210 µ4 B 2 4

b − + 2

⇓ X

2 β1/3 > 0, α2

(5.22)

h

|J τ | − 1 21/3

s

b2 a3 1/3 b + + − 4 27 2 2 1/3

s

b2 a3 1/3 + →A2 →0 1, 4 27 6

27A γ 1 3A γ + 1/3 1 − 11 6 τ 2 τ 2/3 16(J ) 2 2 µ (J )

6

−1+

→A2 →0 |J τ |[1 −

27A γ 11 2 µ6 (J τ )2

v u u + t1 −

v u u + t1 −

27A6 γ 1/3 − 210 µ6 (J τ )2

27A6 γ 1/3 i3/2 →A2 →0 210 µ6 (J τ )2

9A2 γ 1/3 + O(A4 )], 32(J τ )2/3

36

ρµ

⇓ 1 √ X −2/3 (J τ )2 X µ γ( √ ) [4 − ( √ )2 ]lµ + J τ (T −1 )rˇi Πi zrˇµ = 3 γ γ γ r Mlµ + M zrµ .

= =

(5.23)

In the case of the photon gas we get a closed analytical form for the constraints. 3) p = kργ , γ = 1 + n1 , γ 6= 1 (n 6= 0), with the equation X2 X2 X ( √ )2 γ

2γ i X µ2 + A2 1 − k(µ √ )γ−1 γ−1 = B 2 , or γ h X 1 2(n+1) i = B 2 , or µ2 + A2 1 − k(µ √ ) n γ i h X 1 2(n+1) B2 A2 n [1 − k(µ , ] − 1 = 1+ 2 ) √ µ + A2 γ γ(µ2 + A2 ) h

Y rˇ = −

X X 1 [1 − k( √ ) n ]n+1 (T −1 )rˇi Πi , µ γ

√ ρµ = µ γ

(J τ )2 γ X √ [1 γ

1

1

− kµ n ( √Xγ )2+ n −

lµ 1 k(µ √Xγ ) n ]n+1

+ J τ (T −1 )rˇi Πi zrˇµ .

(5.24)

Let us define Z as the deviation of X from dust (for A2 → 0 [∂τ αi = 0]: we have Z → 1, X = √ |B| Z). Then we get the following equation for Z 2 2 µ +A

Z2

h

i 1 2(n+1) A2 (µ2 + A2 )k−1 γ k µ|B|Z µ2 n √ + 1 − k( = 1. ) √ µ2 + A2 (k + 1)2 B 2k γ µ2 + A2

(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(τ, ~σ), because for the momentum density we have Mrˇ = J τ (T −1 )rˇi Πi independently from the type of perfect fluid). 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 flat at spatial infinity spacetime M 4 with the spacelike leaves Στ of the foliations associated with its 3+1 splittings, diffeomorphic to R3 [25–27,11]. In στ -adapted coordinates σ A = (σ τ = τ ; ~σ ) corresponding to a holonomic basis [dσ A , ∂A = ∂/∂σ A ] for tensor fields we have [25] [N and N r are the lapse and shift functions; 3 grs is the 3-metric of Στ ; lA (τ, ~σ ) is the unit normal vector fields to Στ ] 4

gAB = gAB = {4 gτ τ = ǫ(N 2 − 3 grs N r N s ); 4 gτ r = −ǫ 3 grs N s ; 4 grs = −ǫ 3 grs } = = ǫlA lB + △AB ,

△AB = 4 gAB − ǫlA lB .

(6.1)

A set of Στ -adapted tetrad and cotetrad fields is (a) = (1), (2), (3); 3 er(a) and 3 er(a) = 3 e(a)r are triad and cotriad fields on Στ ]

4 A (Σ) E(a)

1 Nr ; − ), N N 3 r = (0; e(a) ),

(o) 4 (Σ) EA

= lA = (N; ~0),

(a) 4 (Σ) EA

= (N (a) = N r 3 er(a) ; 3 er(a) ).

4 A (Σ) E(o)

= ǫlA = (

(6.2)

In these coordinates the energy-momentum tensor of the perfect fluid is T AB = −ǫ(ρ + p)U A U B + p 4 g AB .

(6.3)

If we use the notation Γ = ǫlA U A = ǫNU τ , we get TAB = 4 gAC 4 gBC T CD = [ǫlA lC + △AC ][ǫlB lD + △BD ]T CD = = ElA lB + jA lB + lA jB + SAB , E = T AB lA lB = −ǫ[(ρ + p)Γ2 − p], jA = ǫ△AC T CB lB = −ǫ(ρ + p)Γ△AB U B , SAB = △AC △BD T CD = −ǫ[(ρ + p)△AC U C △BD U D − p△AB ]. ¯

(6.4)

The same decomposition can be referred to a non-holonomic basis [θA = {θl = Ndτ ; θr = ¯ = ǫ¯lA¯U¯ A¯ = dσ r + Nqr dτ }, XA¯ = {Xl = N −1 (∂τ − N r ∂r ); ∂r }; A¯ = (l; r)] in which we have [Γ ¯ s , so that U¯ r may be interpreted as the generalized boost velocity of ǫU¯ l = 1 − 3 grs U¯ r U ¯ ¯ U¯ A with respect to ¯lA [28,29]] 4

g¯A¯B¯ = {4 g¯ll = ǫ; 4 g¯lr = 0; 4 g¯rs = 4 grs = −ǫ 3 grs } = ¯ A¯B¯ , = ǫ¯lA¯¯lB¯ + △ 38

¯lA¯ = (1; ~0), ¯lA¯ = (ǫ; ~0), ¯ ll = △ ¯ lr = 0, ¯ rs = −ǫ 3 grs , △ △

¯ A¯B¯ U¯ B¯ = (0; −ǫ 3 grs ), △

¯A¯ U¯B¯ + p 4 g¯A¯B¯ = T¯A¯B¯ = −ǫ(ρ + p)U = E¯ ¯lA¯¯lB¯ + ¯jA¯¯lB¯ + ¯lA¯¯jB¯ + S¯A¯B¯ , ¯¯ ¯ 2 − p], E¯ = T¯AB ¯lA¯¯lB¯ = −ǫ[(ρ + p)Γ ¯ A¯C¯ T¯ C¯ B¯ lB¯ = (¯jl = 0; ¯jr = −ǫ(ρ + p)Γ ¯ 3 grs U ¯ s ), ¯jA¯ = ǫ△ ¯ B¯ D¯ T¯ C¯ D¯ = ¯ A¯C¯ △ S¯A¯B¯ = △ = (S¯ll = S¯lr = 0; S¯rs = −ǫ[−p 3 grs + (ρ + p)3 gru U¯ u 3 gsv U¯ v )].

(6.5)

E¯ and ¯jr are the energy and momentum densities determined by the Eulerian observers on Στ , while S¯rs is called the “spatial stress tensor”. The non-holonomic basis is used to get the 3+1 decomposition (projection normal and ◦ ǫc3 T µν [when one does not has parallel to Στ ) of Einstein’s equations with matter 4 Gµν = 8πG an action principle for matter, one cannot use the Hamiltonian ADM formalism]: in this ◦ ǫc3 ¯ ll ◦ ǫc3 ¯ lr ¯ ll = ¯ lr = way one gets four restrictions on the Cauchy data [ 4 G T and 4 G T ; they 8πG 8πG become the secondary first class superhamiltonian and supermomentum constraints in the ADM theory; k = c3 /8πG] i ◦ 2 ¯ R + (3 K)2 − 3 Krs 3 K rs (τ, ~σ ) = E(τ, ~σ ), k ◦ 1 (3 K rs − 3 g rs 3 K)|s (τ, ~σ) = J¯r (τ, ~σ ), k h

3

(6.6)

◦ ǫc3 ¯ rs ¯ rs = T . By introducing the extrinsic curvature and the spatial Einstein’s equations 4 G 8πG 3 Krs , this last equations are written in a first order form [it corresponds to the Hamilton ˜ rs = ǫc3 √γ(3 K rs − 3 g rs 3 K); “|” denotes the equations of the ADM theory for 3 grs and 3 Π 8πG covariant 3-derivative]

h

i

∂τ 3 grs (τ, ~σ ) = Nr|s + Ns|r − 2N 3 Krs (τ, ~σ ), ◦

∂τ 3 Krs (τ, ~σ ) = N[3 Rrs + 3 K 3 Krs − 2 3 Kru 3 K u s ] −

− N|s|r + N u |s 3 Kur + N u |r 3 Kus + N u 3 Krs|u (τ, ~σ ) − i 1 1h − (S¯rs + 3 grs (E¯ − S¯u u ) (τ, ~σ). k 2

(6.7)

◦

The matter equations T µν ;ν = 0 become a generalized continuity equation [entropy con◦ ¯¯ servation when the particle number conservation law is added to the system] ¯lA¯T AB ;B¯ = 0 ◦ ¯ A¯B¯ T B¯ C¯ ;C¯ = 0 [LX is the Lie derivative with respect to the and generalized Euler equations △ vector field X] h

◦

h

i

¯ + N ¯j r |r = N(S¯rs 3 Krs + E¯ 3 K) − 2¯j r N|r + L ~ E¯ (τ, ~σ), ∂τ E N

◦ ¯ |r + L ~ ¯j r (τ, ~σ ). ∂τ ¯jr + N S¯rs |s (τ, ~σ ) = N(2 3 K rs ¯js + ¯j r 3 K) − S¯rs N|s − EN N

h

i

h

i

39

(6.8)

The equation for S¯rs would follow from an equation of state or dynamical equation of the sources [for perfect fluids 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 fluid 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 fields) we get S = −ǫk 3

Z

dτ d3 σ {N 3 e ǫ(a)(b)(c) 3 er(a) 3 es(b) 3 Ωrs(c) +

e 3 −1 ( Go )(a)(b)(c)(d) 3 er(b) (N(a)|r − ∂τ 3 e(a)r ) 3 es(d) (N(c)|s − ∂τ 3 e(c) s )}(τ, ~σ ) + 2N Z |J| √ − dτ d3 σ{N γρ( √ , s)}(τ, ~σ). N γ +

(6.9)

The superhamiltonian and supermomentum constraints of Ref. [25] are modified in the following way by the presence of the perfect fluid H = Ho + M ≈ 0, Hr ≈ Θr = Θo r + Mr ≈ 0,

(6.10)

with Mr = J τ (T −1 )ri Πi = −∂r αi Πi 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 q

M(τ, ~σ) = J τ (τ, ~σ) µ2 + [3 grs (T −1 (α))ri(T −1 (α))sj Πi Πj ](τ, ~σ) = i

s

= −det (∂r α (τ, ~σ )) µ2 + 3 g uv

i ∂u αm ∂v αn Π Π (τ, ~σ ), m n [det (∂r αk )]2

Mr (τ, ~σ) = 3 grs J τ (τ, ~σ )[(T −1 (α))si Πi ](τ, ~σ ) = −∂r αi (τ, ~σ)Πi (τ, ~σ).

(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 γ, A2 /µ2 = 3 uv i j 3 u α ∂v α Πi Π−J , B 2 /µ2 γ = (J τ )2 /γ = γ1 [det (∂r αi )]2 . grˇsˇ(T −1 )rˇi Πi (T −1 )sˇj Πj /µ2 = g µ∂2 [det (∂u αk )]2 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 fields α ˜ i (z(τ, ~σ )) = αi (τ, ~σ ) describe the idealized “molecules” of the ˇ material in an abstract 3-dimensional manifold called the “material space”, while J A (τ, ~σ ) = √ ˇ [N γnU A ](τ, ~σ) is the matter number current with future-oriented timelike 4-velocity vector ˇ field U A (τ, ~σ ); n is a scalar field describing the local rest-frame matter number density. The quantity ∂Aˇαi (τ, ~σ ) = zAµˇ(τ, ~σ )∂µ α ˜ i (z) is called the “relativistic deformation gradient” in Στ -adapted coordinates. The material space inherits a Riemannian (symmetric and positive definite) 3-metric from the spacetime M 4 ˇˇ

Gij = 4 g µν ∂µ α ˜ i ∂ν α ˜ j = 4 g AB ∂Aˇ αi ∂Bˇ αj .

(7.1)

Its inverse Gij carries the information about the actual distances of adjacent molecules in the local rest frame. For an ideal fluid the 3-form η of Eq(1.3) gives the volume element in the material space, which is sufficient to describe the mechanical properties of an ideal fluid. Since √ is a scalar, we can evaluate n in the local rest frame at z where we have that n = N|J| γ 1 µ ~ √ ; 0) and ∂o α ˜ i |z = 0 [in the local adapted non-holonomic basis we have U (z) = ( ǫ 4 g¯o¯o¯

√4 √ √ √ 1 0 1 0 √ √ ¯¯ 4 , 4 g¯AB = ǫ g¯A¯B¯ = ǫ g¯ = γ/ ǫ 4 g¯o¯o¯, 4 g¯−1 = 3 g¯−1 / ǫ 4 g¯o¯o¯ = 3 rs , 3 0 −g 0 − grs √4 √4 −1 √ 4 ǫ g¯o¯o¯/ g¯ = g¯ ]. We get at z √ 4 q √ ǫ g¯o¯o¯ Jo i n = o √4 = η123 det ∂k α (7.2) ˜ √4 = η123 det ∂k α ˜ i det |4 g¯rs | = η123 det Gij . U g¯ g¯ √ Therefore, we have n = η123 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 definite positive) 3-metric (M ) γij (αi ), 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 γij (αi ) we have to relax the material at different points αi (τ, ~σ ) 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 ) γij = γij (αi (τ, ~σ )) 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 fluids γij = δij ; this also holds for non-pre-stressed materials without “internal” or “frozen” stresses. 41

Now the material space volume element η has i

η123 (α ) =

r

(M ) det γij (αi),

so that n = η123

√

det Gij

=

r

(M )

det γij

√

det Gij ,

(7.3)

and it cannot be put = 1 like for perfect fluids. (M ) The pull-back of the material metric γij to M 4 is (M )

(M )

γAˇBˇ = γij ∂Aˇαi ∂Bˇ αj

satisf ying

ˇ

(M )

γAˇBˇ U B = 0.

(7.4)

The next step is to define a measure of the difference between the induced 3-metric (M ) Gij (∂αi ) and the constitutive metric γij (αi ), to be taken as a measure of the deformation of the material and as a definition 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 (M ) (1) SAˇBˇ = (4 gAˇBˇ − ǫUAˇUBˇ − γAˇBˇ ) 2

(7.5)

ˇ

(1)

which vanishes at relax and satisfies SAˇBˇ U B = 0 [but it must satisfy the involved matrix inequality 2 det S (1) ≥ det (4 g − ǫUU)]. ii) ˇ

ˇˇ

1 ˇ ˇ ˇ S (2)A Bˇ = (K A Bˇ − δBAˇ ), 2 ˇ

(M )

ˇ

ˇ

S (2)A Bˇ U B = 0,

(7.6)

ˇ

ˇ

with K A Bˇ = 4 g AC (γCˇ Bˇ − ǫUCˇ UBˇ ), K A Bˇ U B = U A [the 4-velocity field is an eigenvector of 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 define a “relativistic strain tensor” in the material space (M )

Si j = γik Gkj ,

(7.8)

with locally Si j = δij when there is local relaxation of the material (in this case physical spacelike distances between material q points near a point z µ (τ, ~σ ) agree with their material √ √ √ (M ) distances). Since n = η123 det Gij = det γij det Gij , we have n = det Si j for the local rest-frame matter number density. We can now define the local rest-frame energy per unit volume of the material n(τ, ~σ )e(τ, ~σ ), where e denotes the molar local rest-frame energy (moles = number of particles) e(τ, ~σ ) = m + uI (τ, ~σ).

(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 infinitesimal 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 specific axis, reflecting the microscopic composition of the material: this information may be encoded in a vector field i j E i (τ, ~σ ) in the material space and one may assume uI = uI (G−1 ij E E ). (M ) The function e = e[αi , Gij , γij , ...] 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 [V = n1 = √ 1 j is the specific volume], det Si

uI = λ(V )h2 + 2µ(V )q + O(cubic invariants),

(7.10)

where λ and µ are the Lamé coefficients. The action principle for this description of relativistic materials is 4

i

i

S[ g, α , ∂α ] =

Z

3

dτ d σL(τ, ~σ ) = −

Z

√ dτ d3 σ(N γ)(τ, ~σ) n(τ, ~σ ) e(τ, ~σ).

(7.11) ˇ

It is shown in Refs. [6,33] that the canonical stress-energy- momentum tensor TBA ˇ = ˇ ˇ ˇ ∂L A A i A pi ∂Bˇ α − δBˇ L, where pi = − ∂∂ ˇ αi is the relativistic Piola-Kirchhoff momentum density, A , which satisfies and coincides with the symmetric energy-momentum tensor TAˇBˇ = −2 ∂ 4∂L g AˇBˇ ˇˇ

T AB ;Bˇ = 0 due to the Euler-Lagrange equations. This energy-momentum tensor may be written in the following form √ TAˇBˇ = N γ n [eUAˇBˇ + ZAˇBˇ ],

(7.12)

where ZAˇBˇ = Zij ∂Aˇ αi ∂Bˇ αj is the pull-back from the material space to M 4 of the “response tensor” of the material Zij = 2

∂e , ∂Gij

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

(7.13)

ˇ

The part τAˇBˇ = nZAˇBˇ , τAˇBˇ U B = 0, may be called the relativistic “stress or Cauchy” tensor and contains the “stress-strain relation” through the dependence of Zij on Si j implied by the consitutive equation e = e[α, γ (M ) , G, ..] of the material. For an isotropic elastic material we get h

(M )

Zij = V pG−1 ij + Bγij ∂e ,B= where p = − ∂V

2 ∂e , V ∂h

C=

2 ∂e V ∂q

i

+ CGij ,

(7.14)

and we get

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

(7.15)

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 [perfect fluids have e = e(V ), B = C = 0, Zij = V pG−1 ij and de(V ) = −pdV is the Pascal law]. See Ref. [35] for a different description of relativistic Hooke law in linear elasticity: there is a 4-dimensional deformation tensor Sµν = 12 (4 ∇µ ξν + 4 ∇ν ξµ ) and the constitutive equations of the material are given in the form T µν = C (µν)(αβ) Sαβ . In Ref. [6] the theory is also extended to the thermodynamics of isentropic flows (no heat conductivity). The function e is considered also as a function of entropy S = S(α) and de = 12 Zij dGij is generalized to 1 de = Zij dGij − SdT. 2

(7.16)

Then e is replaced with the Helmholtz free energy f = e − T S so to obtain 1 df = Zij dGij − SdT, 2

(7.17)

[for perfect fluids 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 ˇ to introduce an extra scalar field ατ (τ, ~σ) so that T = const.U A ∂Aˇ ατ . The potential ατ (τ, ~σ) 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. h √ i R In this case the action principle becomes S = − dτ d3 σ N γnf (G, T ) (τ, ~σ ) and one √ gets the conserved energy-momentum tensor TAˇBˇ = N γn[(f + T S)UAˇUBˇ + ZAˇBˇ ]. 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 fluids, 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 fluids with an equation of state of the form ρ = ρ(n, s) given in Ref. [1], in which the fluid 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 first class constraints Hµ (τ, ~σ ) ≈ 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 fluids one needs numerical calculations. After the inclusion of the coupling to the gravitational field 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 configurations of the perfect fluid whose conserved 4-momentum is timelike. For each of these configurations we can select the special foliation of Minkowski spacetime with spacelike hyperplanes orthogonal to the 4-momentum of the configuration, This gives rise to the “Wigner-covariant rest-frame instant form of dynamics” [10,11] for the perfect fluids. After a discussion of the “external” and “internal” centers of mass and realizations of the Poincaré algebra, rest-frame Dixon’s Cartesian multipoles [20] of the perfect fluid 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 field by giving the ADM action for the perfect fluid in tetrad gravity. Now it becomes possible to study the canonical reduction of tetrad gravity with the perfect fluids as matter along the lines of Refs. [25–27].

45

APPENDIX A: RELATIVISTIC PERFECT FLUIDS.

As in Ref. [1] let us consider a perfect fluid in a curved spacetime M 4 with unit 4velocity vector field U µ (z), Lagrangian coordinates α ˜ i (z), particle number density n(z), energy density ρ(z), entropy per particle s(z), pressure p(z), temperature T (z). Let J µ (z) = q 4 g(z)n(z)U µ (z) the densitized particle number flux vector field, so that we have n = q √ ǫ 4 gµν J µ J ν / 4 g. Other local thermodynamical variables are the chemical potential or specific enthalpy (the energy per particle required to inject a small amount of fluid into a fluid sample, keeping the sample volume and the entropy per particle s constant) µ=

1 (ρ + p), n

(A1)

the physical free energy (the injection energy at a constant number density n and constant total entropy) a=

ρ − T s, n

(A2)

and the chemical free energy (the injection energy at constant volume and constant total entropy) f=

1 (ρ + p) − T s = µ − T s. n

(A3)

Since the local expression of the first law of thermodynamics is dρ = µdn + nT ds,

or

dp = ndµ − nT ds,

or

d(na) = f dn − nsdT,

(A4)

an equation of state for a perfect fluid may be given in one of the following forms ρ = ρ(n, s),

or

p = p(µ, s),

or

a = a(n, T ).

(A5)

By definition, the stress-energy-momentum tensor for a perfect fluid is T µν = −ǫ ρU µ U ν + p(4 g µν − ǫU µ U ν ) = −ǫ(ρ + p)U µ U ν + p 4 g µν ,

(A6)

and its equations of motion are T µν ;ν = 0,

1 (nU µ );µ = √4 ∂µ J µ = 0. g

(A7)

As shown in Ref. [1] an action functional for a perfect fluid depending upon J µ (z), 4 gµν (z), s(z) and α ˜ i (z) requires the introduction of the following Lagrange multipliers to implement all the required properties: i) θ(z): it is a scalar field named ‘thermasy’; it is interpreted as a potential for the fluid | . In the Lagrangian it is interpreted as a Lagrange multiplier for temperature T = n1 ∂ρ ∂s n implementing the “entropy exchange constraint” (sJ µ ),µ = 0.

46

ii) ϕ(z): it is a scalar field; 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 µ ,µ = 0. iii) βi (z): they are three scalar fields; in the Lagrangian they are interpreted as Lagrange multipliers for the “constraint” α ˜ i ,µ J µ = 0 that restricts the fluid 4-velocity vector to be directed along the flow lines α ˜ i = const. Given an arbitrary equation of state of the type ρ = ρ(n, s), the action functional is |J| , s) + 4g + J µ [∂µ ϕ + s∂µ θ + βi ∂µ α ˜ i ]}.

S[4 gµν , J µ , s, α ˜, ϕ, θ, βi] =

Z

q

d4 z{−

4 gρ( √

(A8)

By varying the 4-metric we get the standard stress-energy-momentum tensor 2 T µν = √4

δS = −ǫρU µ U ν + p(4 g µν − ǫU µ U ν ) = −ǫ(ρ + p)U µ U ν + p 4 g µν , g δ 4 gµν

(A9)

where the pressure is given by p=n

∂ρ |s − ρ. ∂n

(A10)

The Euler-Lagrange equations for the fluid motion are δS δJ µ δS δϕ δS δθ δS δs δS δα ˜i δS δβi

= µUµ + ∂µ ϕ + s∂µ θ + βi ∂µ α ˜ i = 0, = −∂µ J µ = 0, = −∂µ (sJ µ ) = 0, q

=−

4g

∂ρ + J µ ∂µ θ = 0, ∂s

= −∂µ (βi J µ ) = 0, = J µ ∂µ α ˜ i = 0.

(A11)

The second equation is the particle number conservation, the third one the entropy exchange constraint and the last one restricts the fluid 4-velocity vector to be directed along the flow lines α ˜ i = const.. The first equation gives the Clebsch or velocity-potential representation of the 4-velocity Uµ (the scalar fields in this representation are called Clebsch or velocity potentials). The fifth equations imply the constancy of the βi ’s along the fluid flow lines, so that these Lagrange multipliers can be expressed as a function of the Lagrangian coordinates. The fourth equation, after a comparison with the first law of thermodynamics, | for the fluid temperature. leads to the identification T = U µ ∂µ θ = n1 ∂ρ ∂s n Moreover, one can show that the Euler-Lagrange equations imply the conservation of the stress-energy-momentum tensor T µν ;ν = 0. This equations can be split in the projection 47

along the fluid flow lines and in the one orthogonal to them: i) The projection along the fluid flow lines plus the particle number conservation give Uµ T µν ;ν = − ∂ρ U µ ∂µ s = 0, which is verified due to the entropy exchange constraint. There∂s fore, the fluid flow is locally adiabatic, that is the entropy per particle along the fluid flow lines is conserved. ii) The projection orthogonal to the fluid flow lines gives the Euler equations, relating the fluid acceleration to the gradient of pressure (4 gµν − ǫUµ Uν )T να ;α = −ǫ(ρ + p)Uµ;ν U ν − (δµν − ǫUµ U ν )∂ν p.

(A12)

∂ρ By using p = n ∂n |s − ρ, it is shown in Ref. [1] that these equations can be rewritten as

2(µU[µ );ν] U ν = −ǫ(δµν − Uµ U ν )

1 ∂ρ |n ∂ν s. n ∂s

(A13)

The use of the entropy exchange constraint allows the rewrite the equations in the form 2V[µ;ν] U ν = T ∂µ s,

(A14)

where Vµ = µUµ is the Taub current (important for the description of circulation and vorticity), which can be identified with the 4-momentum per particle of a small amount of fluid to be injected in a larger sample of fluid without changing the total fluid volume or the entropy per particle. Now from the Euler-Lagrange we get 2V[µ;ν] U ν = −2(∂[µ ϕ + s∂[µ θ + βi ∂[µ α ˜ i);ν] U ν = (s∂[µ θ);ν] U ν = T ∂µ s,

(A15)

and this result implies the validity of the Euler equations. In the non-relativistic limit (nU µ );µ = 0, T µν ;ν = 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 ϕ and θ are the particle number density and entropy density seen by Eulerian observers at rest in space. The action evaluated R 4 q 4 on the solutions of the equations of motion is d z 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 (˜ α, βi , s). It is shown that for each F there is a conR √ servation equation ∂µ (F J µ ) = 0 and a Noether charge Q[F ] = Σ d3 σ γn (ǫlµ U µ )F (˜ α, βi , s) [Σ is a spacelike hypersurface with future pointing unit normal lµ and with a 3-metric with √ determinant γ]. For F = 1 inside a volume V in Σ we get the conservation of particle number within a flow tube defined by the bundle of flow lines contained in the volume V. The factor ǫlµ U µ is the relativistic ‘gamma factor’ characterizing a boost from the Lagrangian observers with 4-velocity U µ to the Eulerian observers with 4-velocity lµ ; thus n(ǫlµ U µ ) is the particle number density as seen from the Eulerian observers. These symmetries describe the changes of Lagrangian coordinates α ˜ i and the fact that both the Lagrange multipliers ϕ and θ are constant along each flow line (so that it is possible to transform any solution 48

to the fluid equations of motion into a solution with ϕ = θ = 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 first and second class constraints. In particular we get: 1) second class constraints: A) πJ τ ≈ 0, J τ − πϕ ≈ 0; B) πs ≈ 0, sJ τ − πθ ≈ 0; C) πβs ≈ 0, βs J τ − παs ≈ 0. 2) first class constraints: πJ r ≈ 0, so that the J r ’s are gauge variables. Therefore the physical variables are the five pairs: ϕ, πϕ ; θ, πθ ; α ˜ i, παr 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, first 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 ρ = ρ(n, s). One has the action µ

r

4

S[n, U , ϕ, θ, s, α ˜ , βr ; gµν ] = −

Z

q

d4 x

4g

h

ρ(n, s) − nU µ (∂µ ϕ − θ∂µ s + βr ∂µ α ˜r )

i

(A16)

√ ˜ 1 ∂ρ α ˜ 2 ∂σ α ˜ 3 η123 (α ˜ r ), one can If one knowsR s = s(α ˜ r ) and J µ = J µ (α ˜ r ) = − 4 gǫµνρσ ∂ν α define S˜ = S − d4 x∂µ [(ϕ + sθ)J µ ], and one can show that it has the form ˜ αr ] = − S˜ = S[˜

Z

|J| , s). 4g

q

d4 x

4 gρ( √

(A17)

2) Equation of state: p = p(µ, s) [V µ = µU µ Taub vector] S(p) = s(p) [V µ , ϕ, θ, s, α ˜ r , βr ; 4 gµν ] = Z q h i ∂p Vµ = d4 x 4 g p(µ, s) − |V | − (∂µ ϕ + s∂µ θ + βr ∂µ α ˜r ) ∂µ |V |

(A18)

◦

or by using one of its EL equations Vµ = − (∂µ ϕ + s∂µ θ + βr ∂µ α ˜ r ) to eliminate V µ one gets Schutz’s action [µ determined by µ2 = −V µ Vµ ] S˜(p) [ϕ, θ, s, α ˜ r , βr ; 4 gµν ] =

Z

q

d4 x

4 gp(µ, s)

(A19)

3) Equation of state a = a(n, T ). The action is µ

r

4

S(a) [J , ϕ, θ, α ˜ , βr ; gµν ] =

Z

h i |J| d4 x |J|a( √4 , ∂µ θJ µ ) − J µ (∂µ ϕ + βr ∂µ α ˜r ) g h

i

R µ or S˜(a) [ϕ, θ, s, α ˜ r , βr ] = S(a) − d4 x |J|a( √|J|4 , J|J| ∂µ θ) . g

(A20)

At the end of Ref. [1] there is the action for “isentropic” fluids and for their particular case of a “dust” (used in Ref. [9] as a reference fluid in canonical gravity). 49

− sT with s = const. (constant The isentropic fluids have equation of state a(n, T ) = ρ(n) n ′ value of the entropy per particle). By introducing ϕ = ϕ + sθ, the action can be written in the form ′

S(isentrpic)[J µ , ϕ , α ˜ r , βr , 4 g µν ] =

Z

h

d4 x −

i |J| ′ µ r ) + J (∂ ϕ + β ∂ α ˜ ) µ r µ 4g

q

4 gρ( √

(A21)

or S˜(isentropic) [˜ αr ; 4 gµν ] = −

Z

|J| ) 4g

q

d4 x

4 gρ( √

(A22)

The dust has equation of state ρ(n) = µn, namely a(n, T ) = µ − sT so that we get zero ′ ∂ρ pressure p = n ∂n − ρ = 0. Again with ϕ = ϕ + sθ the action becomes µ

′

r

4 µν

S(dust) J , ϕ , α ˜ , βr , g ] =

Z

h

′

d4 x − µ|J| + J µ (∂µ ϕ + βr ∂µ α ˜r )

i

(A23)

˜ r ) [In Ref. [9]: M = µn rest mass (energy) density and or with Uµ = − µ1 (∂µ ϕ + βr ∂µ α ′ T = ϕ /µ, Wr = −βr , Z r = α ˜ r ; Uµ = −∂µ T + Wr ∂µ Z r ] ′

1 S(dust) [T, Z , M, Wr ; gµν ] = − 2 ′

r

4

Z

q

d4 x

4 g(µn)

Z

d4 xµ|J|

or S˜(dust) [˜ αr ; 4 gµν ] = −

Uµ 4 g µν Uν − ǫ ,

(A24)

(A25)

In Ref. [9] there is a study of the action (A24) since the dust is used as a reference fluid in general relativity. At the Hamiltonian level one gets: r i) 3 pairs of second class constraints [πW σ ) ≈ 0, πZ~ r (τ, ~σ) − Wr (τ, ~σ)πT (τ, ~σ) ≈ 0], which ~ (τ, ~ r allow the elimination of Wr (τ, ~σ) and πW (τ, ~σ ); ~ ii) a pair of second class constraints [ πM (τ, ~σ ) ≈ 0 plus the secondary M(τ, ~σ ) − 2 πT (τ, ~σ ) ≈ 0], which allow the elimination of M, πM . √ √ 2 3 rs u v γ

πT + g (πT ∂r T +πZ ~ u ∂r Z )(πT ∂s T +πZ ~ v ∂s Z )

50

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

In this Appendix we shall collect some results on relativistic fluids 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 µν T , the densities of energy and momentum are T oo and c−1 T ro respectively [so that dP µ = c−1 ηT µν dΣν is the 4-momentum that crosses the 3-area element dΣν in the sense of its normal (η = −1 if the normal is spacelike, η = +1 if it is timelike)]; instead, cT or is the energy flux 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 uµ (u2 = ǫc2 ) will measure energy density c−2 T µν uµ uν and energy flux ǫT µν uµ nν along the direction of a unit vector nµ in his rest frame. For a fluid at thermal equilibrium with T µν = ρU µ U ν − ǫ cp2 (4 g µν − ǫU µ U ν ) [U µ is the hydrodynamical 4-velocity of the fluid] with particle number density n, specific volume V = n1 and entropy per particle s = kBnS (kB is Boltzmann’s constant) in its rest frame, the energy density is ρc2 = 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 first law of thermodynamics in the form (Gibbs equation) ds(e, V ) =

1 (de + pdV ). T

(B2)

If µclas = e+pV −T s is the non-relativistic chemical potential per particle, its relativistic version is ′

µ = mc2 + µclass = µ − T s, [µ = get

ρc2 +p n

(B3)

is the specific enthalpy, also called chemical potential as in Appendix A] and we ′

µ n = ρc2 + p − nT s = ρc2 + p − kB T S, ′ kB T dS = d(ρc2 ) − µ dn = d(ρc2 ) − (µ − T s)dn, ′ d(ρc2 ) = µ dn + T d(ns) = µdn + nT ds. By introducing the “thermal potential” α = 2 β = kcB T , these two equations take the form

51

′

µ kB T

=

µ−T s kB T

or (B4)

and the inverse temperature

p ns = β(ρ + 2 ) − αn, kB c dS = βdρ − αdn. S=

(B5)

Let us remark that in Refs. [23,29,37] one uses different notations, some of which are ′ given in the following equation (in Ref. [23] ρ is denoted e and ρ is denoted r) p p p ′ ′ ′ ′ ′ = n(mc2 + e) + 2 = ρ h = ρ (c2 + e + 2 ′ ) = ρ (c2 + h ), (B6) 2 c c cρ √ ′ ′ where ρ = nm is the rest-mass density [r∗ = 4 gρ is called the coordinate rest-mass ′ ′ density] and e = e/m is the specific internal energy [so that ρ h is the “effective inertial mass of the fluid; in the post-Newtonian approximation of Ref. [23] it is shown that σ = √ P c−2 (T oo + s T ss ) + O(c−4) = c−2 4 g(−Too + Tss ) + O(c−4) has the interpretation of equality of the “passive” and the “active” gravitational mass]. For the specific enthalpy or chemical ′ potential we get [µ/m = h = c2 + h is called enthalpy] ρ+

µ=

p m p 1 ′ (ρ + 2 ) = ′ (ρ + 2 ) = mh = m(c2 + h ). n c ρ c

(B7)

See Ref. [29] for a richer table of conversion of notations. Relativistically, we must consider, besides the stress-energy-momentum tensor T µν and R the associated 4-momentum P µ = V d3 Σν T µν , a particle flux density nµ (one nµa for each constituent a of the system) and the entropy flux density sµ . At thermal equilibrium all these a priori unrelated 4-vectors must all be parallel to the hydrodynamical 4-velocity nµ = nU µ

,

sµ = sU µ ,

P µ = P U µ,

(B8) 2

Analogously, we have V µ = V U µ (V = 1/n is the specific volume), β µ = βU µ = kBc T U µ ′ [a related 4-vector is the equilibrium parameter 4-vector iµ = µ β µ ]. Since ǫUµ T µν = ρU ν , we get the final manifestly covariant form of the previous two equations (now the hydrodynamical 4-velocity is considered as an extra thermodynamical variable) p ns µ U = 2 β µ − αnµ − ǫβν T νµ , kB c dS µ = −αdnµ − ǫβν dT νµ . S µ = SU µ =

(B9)

Global thermal equilibrium imposes ∂µ α = ∂µ βν + ∂ν βµ = 0.

(B10)

p µ β ) = nµ dα + ǫT νµ dβν , c2

(B11)

As a consequence we get d(

namely the basic variables nµ , T νµ and S µ can all be generated from partial derivatives of the “fugacity” 4-vector (or “thermodynamical potential”) 52

p µ β , c2 ∂φµ nµ = , ∂α

φµ (α, βλ ) =

νµ T(mat) =

∂φµ , ∂βν

νµ S µ = φµ − αnµ − βν T(mat) ,

(B12)

νµ once the equation of state is known. Here T(mat) is the canonical or material (in general non symmetric) stress tensor, ensuring that reversible flows of field energy are not accompanied by an entropy flux. This final form remains valid (at least to first order in deviations) for “states that deviate from equilibrium”, when the 4-vectors S µ , nµ ,... are no more parallel; the extra information in this equation is precisely the standard linear relation between entropy flux and heat flux. The second law of thermodynamics for relativistic systems is ∂µ S µ ≥ 0, which becomes a strict equality in equilibrium. The fugacity 4-vector φµ is evaluated by using the covariant relativistic statistical theory for thermal equilibrium [2] starting from a grand canonical ensemble with density matrix ρˆ by maximizing the entropy S = −T r(ˆ ρln ρˆ) subject to the constraints T r ρˆ = 1, T r(ˆ ρn ˆ ) = n, λ λ ˆ T r(ˆ ρP ) = P : this gives (in the large volume limit) ˆµ

ρˆ = Z −1 eαˆn+βµ P , with Z ǫφµ dΣµ , ln Z = △Σ

n=

Z

△Σ

µ

ǫn dΣµ ,

µ

P =

Z

△Σ

µν dΣν , ǫT(mat)

(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 △Σ]. Therefore, one has to find the grand canonical partition function Z(Vµ , βµ , iµ ) =

X

µ

eiµ n Qn (Vµ , βµ ),

where Qn (Vµ , βµ ) =

n

Z

Vµ

µ

dσn (q, p)e−βµP ,

(B14)

is the canonical partition function for fixed volume Vµ and dσn 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 dσn (p, m) =

N N Y X 1 Z 4 pi ) 2Vµ pµi θ(poi )δ(p2i − ǫm2 )d4 pi . δ (P − N! i=1 i=1

(B15)

Following Ref. [38] [using a certain type of gauge fixings to the first class constraints − ǫm2 ≈ 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): p2i

Qn =

iN 1 h V m2 K (mβ) . 2 N! 2π 2 β

(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 (pµ , dω) that cross a target 3-area dΣµ in M 4 in the direction of its normal is given by dN = N(q, p)dωηv µdΣµ (= Nd3 qd3 p for the 3-space √ q o = const.); dω = d3 p/v o 4 g is the invariant element of 3-area on the mass-shell]. One = ∇µ (Nv µ ) [v µ is the particle velocity obtained from arrives at a transport equation for N, dN dτ q the Hamilton equation implied by the one-particle Hamiltonian H = ǫ 4 g µν (q)pµ pν = m (it is the energy after the gauge fixing q o ≈ τ to the first class contraint 4 g µν (q)pµ pν − ǫm2 ≈ 0); ∇µ is the covariant gradient holding the 4-vector pµ (not its components) fixed] 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−βµ P µ 2 Synge one N = const.e /4πm K2 (mβ) for the Boltzmann gas [41]]. The H-theorem R µ µ [∇µ S ≥ 0, where S (q) = − [Nln(Nh3 ) − N]v µ dω is the entropy flux]R and the results R at thermal equilibrium emerge [from the balance law ∇µ ( Nf v µ dω) = f C[N]dω (f is an arbitrary tensorial function) one can deduce the conservation laws ∇µ nµ = ∇µ T νµ = 0, R R where nµ = Nv µ dω, Tν µ = Npν v µ dω; the vanishing of entropy production at local ν thermal equilibrium gives Neq (q, p)R = h−3 eα(q)+βν (q)P in the case of Boltzmann statistic νµ and one gets (U µ = β µ /β) nµeq = Neq v µ dω = nU µ , Teq = ρU ν U µ − ǫp(4 g νµ − ǫU ν U µ ), µ µ µ µν Seq = pβ − αneq − βν 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,R which can be explicitly determined from 14 moments among the infinite set of moments Npµ pν pρ ...d4 p of kinetic theory; no extra auxiliary state variables are introduced to specify a non-equilibrium state besides T µν , nµ , S µ ] 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 difference 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 ∂µ T µν = ∂µ nµ = 0, ∂µ S µ ≥ 0. There is the problem of how to define 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 infinite 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 flow”: i) Eckart theory. One considers a local observer in a simple fluid who is at rest with µ is parallel by definition respect to the average motion of the particles: its 4-velocity U(eck) µ to the particle flux n , namely µ nµ = n(eck) U(eck) .

(B17)

This local observer sees “heat flow” as a flux of energy in his rest frame: µ µ ǫU(eck)µ T µν = ρ(eck) U(eck) + q(eck) , so that we get µν µ µ µ ν ν ν µν , q(eck) + P(eck) U(eck) + U(eck) U(eck) + q(eck) T = ρ(eck) U(eck) µν νµ µ µν ν P(eck) = P(eck) = ǫ(p + π(eck) )(4 g µν − ǫU(eck) U(eck) + π(eck) , µ ν U(eck) ) = 0, π(eck)µν (4 g µν − ǫU(eck)

µ µν = 0, U(eck)ν = q(eck)µ U(eck) P(eck)

(B18)

µν where p is the thermodynamic pressure, π(eck) the bulk viscosity and π(eck) the shear stress. This description has the particle conservation law ∂µ nµ = 0. ii) Landau-Lifshitz theory. One considers a different observer (drifting slowly in the µ direction of heat flow with a 3-velocity ~vD = ~q/nmc2 ) whose 4-velocity U(ll) is by definition such to give a vanishing “heat flow”, i.e. there is no net energy flux in his rest frame: µ µ µ U(ll) Tµν nν = 0 for all vectors nµ orthogonal to U(ll) . This implies that U(ll) is the timelike µ µν µν µν eigenvector of T , T U(ll)ν = ǫρ(ll) U(ll) , which is unique if T satisfies a positive energy condition. Now we get µν µ ν , U(ll) + P(ll) T µν = ρ(ll) U(ll) µν νµ µ µν ν P(ll) = P(ll) = ǫ(p + π(ll) )(4 g µν − ǫU(ll) U(ll) + π(ll) , µν P(ll) U(ll)ν = 0, µ µ nµ = n(ll) U(ll) + j(ll) ,

µ ν π(ll)µν (4 g µν − ǫU(ll) U(ll) ) = 0, µ j(ll)µ U(ll) =0

(~j = −n~vD = −~q/mc2 ).

(B19)

This observer in his rest frame does not see a heat flow but a particle drift. This description has the simplest form of the energy-momentum tensor. One has n(eck) = n(ll) ch ϕ, ρ(eck) = ρ(ll) ch2 ϕ + p(ll) sh2 ϕ = π µν jµ jν /n2(eck) , with ch ϕ = q

µ 2 U(ll) U(eck)µ [the difference is a Lorentz factor 1 − ~vD /c2 , so that there are insignificant differences for many practical purposes if deviations from equilibrium are small]. The angle ϕ ≈ j/n ≈ vD /c ≈ q/nmc2 is a dimensionless measure of the deviation from equilibrium [n(ll) − n(eck) and ρ(ll) − ρ(eck) are of order ϕ2 ].

55

One can decompose T µν , nµ , in terms of any 4-velocity U µ that falls within a cone of µ µ angle ≈ ϕ containing U(eck) and U(ll) ; each choice U µ gives a particle density n(U) = ǫuµ nµ µν and energy density ρ(U) = Uµ Uν T which are independent of U µ if one neglects terms of order ϕ2 . Therefore, one has: i) if Seq (ρ(U), n(U)) is the equilibrium entropy density, then S(U) = ǫUµ S µ = Seq + O(ϕ2 ); ∂ρ/n ii) if p(U) = − ∂1/n |S/n is the (reversible) thermodynamical pressure defined as work done in an isentropic expansion (off equilibrium this definition allows to sepate it from the bulk stress π(U) in the stress-energy-momentum tensor) and peq is the pressure at equilibrium, then p(U) = Peq (ρ(U), n(U)) + O(ϕ2). By postulating that the covariant Gibbs relation remains valid for arbitrary infinitesimal displacements (δnµ , δT µν , ..) from an equilibrium state, one gets a covariant off-equilibrium thermodynamics based on the equation

S µ = p(α, β)β µ − αnµ − βν T µν − Qµ (δnν , δT νρ, ..), ∇µ S µ = −δnµ ∂µ α − δT µν ∇ν βµ − ∇µ Qµ ≥ 0,

(B20)

with Qµ of second order in the displacements and α, βµ arbitrary. At equilibrium one recovers µ µν µ Seq = pβ µ − αnµeq − βν Teq , ∇µ Seq = 0 [with U µ = β µ /β and (for viscous heat-conducting fluids, but not for superfluids) ∂µ α = ∇µ βν + ∇ν βµ = 0]. If we choose β µ = U µ /kB T parallel to nµ of the given off-equilibrium state, we are in the µ “Eckart frame”, U µ = U(eck) , and we get S = ǫU(eck)µ S µ = Seq + ǫU(eck)µ Qµ , µ µ µ µ ν ν σ(eck) = (4 g µν − ǫU(eck) U(eck) )Sν = βq(eck) − (4 g µν − ǫU(eck) U(eck) )Qν , µ µ ν )TνρU(eck)ρ , U(eck) = −(4 g µν − ǫU(eck) q(eck)

(B21)

so that to linear order we get the standard relation between entropy flux ~σ(eck) and heat flux ~q(eck) ~σ(eck) =

~q(eck) + (possible 2nd order term). kB T

(B22)

µ If we choose U µ = U(ll) , the timelike eigenvector of T µν , so that U(ll)µ Tνµ (4 gρν − ν ǫU(ll) U(ll)ρ ) = 0, we are in the “Landau-Lifshitz frame” and we get µ µ µ µ ν ν U(ll) )Qν , − (4 g µν − ǫU(ll) U(ll) )Sν = −αj(ll) = (4 g µν − ǫU(ll) σ(ll) µ µ ν j(ll) = (4 g µν − ǫU(ll) U(ll) )nν ,

(B23)

so that at linear order we get the standard relation between entropy flux ~σ(ll) and diffusive flux ~j(ll) 56

~σ(ll) = −

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

(B24)

In the Landau-Lifschitz frame heat flow and diffusion are englobed in the diffusive flux ~j(ll) relative to the mean mass-energy flow.

The entropy inequality becomes (each term is of second order in the deviations from local equilibrium) 0 ≤ ∇µ S µ = −δnµ ∂µ α − δT µν ∇ν βµ − ∇µ Qµ ,

(B25)

with the fitting conditions δnµ Uµ = δT µν Uµ Uν = 0, which contain all information about the viscous stresses, heat flow and diffusion in the off-equilibrium state (they are dependent on the arbitrary choice of the 4-velocity U µ ). Once a detailed form of Qµ is specified, linear relations between irreversible fluxes δT µν , δnµ and gradients ∇(µ βν) , ∂µ α follow. A) Qµ = 0 (like in the non-relativistic case). The spatial entropy flux ~σ is only a strictly linear function of heat flux ~q and diffusion flux ~j. In this case the off-equilibrium entropy density S = ǫUµ S µ is given by the equilibrium equation of state S = Seq (ρ, n). We have 0 ≤ ∇µ S µ = −δnµ ∂µ α − δT µν ∇ν βµ with fitting conditions δnµ Uµ = δT µν Uµ Uν = 0 and with U µ still arbitrary at first order. µ A1) Landau-Lifschitz frame and theory. U µ = U(ll) is the timelike eigenvector of T µν . µν , π(ll) This and the fitting conditions imply δT µν U(ll)ν = 0. The shear and bulk stresses π(ll) are identified by the decomposition µ µν ν ), U(ll) + π(ll) (4 g µν − ǫU(ll) δT µν = π(ll)

µ µν U(ll)ν = π(ll) π(ll) µ = 0.

(B26)

The inequality ∇µ S µ ≥ 0 becomes µ µν µ −j(ll) ∂µ α − βπ(ll) < ∇ν βµ > −βπ(ll) ∇µ U(ll) ≥ 0,

µ j(ll) = δnµ ,

β α U(ll)ν ) − U(ll)µ )(4 gνβ − ǫU(ll) < Xµν > = [(4 gµα − ǫU(ll) 1 β α − (4 gµν − ǫU(ll)µ U(ll)ν )(4 g αβ − ǫU(ll) )]Xαβ , U(ll) 3

(B27)

[the < .. > operation extracts the purely spatial, trace-free part of any tensor]. µ µν If the equilibrium state is isotropic (Curie’s principle) and if we assume that (j(ll) , π(ll) , π(ll) ) µ are “linear and purely local” functions of the gradiants, ∇µ S ≥ 0 implies µ µ ν j(ll) = −κ(4 g µν − ǫU(ll) U(ll) )∂µ α,

κ > 0,

(B28)

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

and the standard Navier-Stokes equations (ζS , ζV are shear and bulk viscosities) 1 µ π(ll) = ζV ∇µ U(ll) . 3

π(ll)µν = −2ζS < ∇ν βµ >,

(B29)

µ parellel to nµ . Now we have the fitting condition A2) Eckart frame and theory. U(eck) ν δnµ = 0. The heat flux appears in the decomposition of δT µν [a(eck)µ = U(eck) ∇ν U(eck)µ is the 4-acceleration] µ µν µ µ ν ν ν U(eck) ). + π(eck) (4 g µν − ǫU(eck) q(eck) + π(eck) U(eck) + U(eck) δT µν = q(eck)

(B30)

The inequality ∇µ S µ ≥ 0 becomes µ µν µ q(eck) (∂µ α − βa(eck)µ ) − β(π(eck) ∇ν U(eck)µ + π(eck) ∇µ 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 differ by spatial gradients of the viscous stresses and the time-derivative of the heat flux] µ µ ν U(eck) )(∂ν T + T ∂τ U(eck)ν ), = −κ(4 g µν − ǫU(eck) q(eck)

(B32)

[the term depending on the acceleration is sometimes referred to as an effect of the “inertia µν of heat”], and the same form of the Navier-Stokes equations for π(eck) , π(eck) (they are not strictly equivalent to the Landau-Lifshitz ones, because they differ by gradients of the drift ~vD = ~q/nmc2 ). For a simple fluid Fourier’s law and Navier-Stokes equations (9 equations) and the conservation laws ∇µ T µν = ∇µ nµ = 0 (5 equations) determine the 14 variables T µν , nµ 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 Qµ = −

1Z Neq f 2 pµ dω 6= 0, 2

(B33)

for a gas up to second order in the deviation (N − Neq ) = Neq f [Qµ = 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 coefficients 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 fluid 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 (δnµ (x), δTµν (x)) = σ(U, T )(−∂µ α(x), −∇(µ βν) (x)), 58

this local phenomenological law is generalized to (δnµ (~x, xo ), δTµν (~x, xo )) = ′ ′ ′ xo )(−∂µ α(~xxo ), −∇(µ βν) (~xxo )).

R∞

−∞

′

dxo σ(xo −

C) Local non-linear extended thermodynamics (ET). It is more relevant for relativistic astrophysics, where correlation and memory effects 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 Qµ (δnν , δT νρ , ..) 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 fluxes are subject to a conservation equation, these theories are called of causal “divergence type” like the ones of Refs. [47–49]. Another type of theory (extended irreversible thermodynamics; in general these theories are not of divergence type) was developed in Refs. [50–53]: in it there are transport equations for the dissipative fluxes rather than conservation laws. For small deviations one retains only the quadratic terms in the Taylor expansion of Qµ (leading to “linear” phenomenological laws): this implies 5 new undetermined coefficients 1 Qµ = U µ [βo π 2 + β1 q µ qµ + β2 π µν πµν ] − αo πq µ − α1 π µν qν , 2

(B34)

with βi > 0 from ǫUµ Qµ > 0 (the βi ’s are ‘relaxation times’). A first-order change of rest frame produces a second-order change in Qµ [going from the Landau-Lifshitz frame to the Eckart one, one gets α(eck)i − α(ll)i = β(ll)1 − β(eck)1 = [(ρ + p)T ]−1 , β(ll)0 = β(eck)o , β(ll)2 = β(eck)2 , and the phenomenological laws are now invariant to first order]. In the “Eckart frame” the phenomenological laws take the form µ µ ν = −κT (4 g µν − ǫU(eck) U(eck) )[T −1 ∂ν T + a(eck)ν + β(eck)1 ∂τ q(eck)ν − q(eck) ρ − α(eck)o ∂ν π(eck) − α(eck)1 ∇ρ π(eck)ν ],

π(eck)µν = −2ζS [< ∇ν U(eck)µ > +β(eck)2 ∂τ π(eck)µν − α(eck)1 < ∇ν q(eck)µ >], 1 µ µ ], + β(eck)o ∂τ π(eck) − α(eck)o ∇µ q(eck) π(eck) = − ζV [∇µ U(eck) 3

(B35)

which reduce to the equation of the standard Eckart theory if the 5 relaxation (βi ) and coupling (αi ) coefficients are put equal to zero. See for instance Ref. [54] for a complete treatment and also Ref. [55]. For appropriate values of these coefficients 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 fluxes evolve from an initial arbitrary state to a final steady one [the time parameter τ is usually interpreted as the relaxation time of the dissipative processes]. In the case of a gas the new coefficients 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 finite 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 effects). 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 coefficients 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): ∇ρ Aµνρ = I µν ,

(B36)

in which Aρµν and I µν are symmetric tensors with the following traces Aµν ν = −nµ ,

I µ µ = 0.

(B37)

These conditions are modelled on kinetic theory, in which Aρµν represents the third moment of the distribution function in momentum space, and I µν 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 µ , Aρµν , I µν are invariant functions of T µν , nµ 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 fluid used in this paper. What is lacking in the non-dissipative case of heat conduction is the functional form of the off-equilibrium equation of state reducing to ρ = ρ(n, s) at thermal equilibrium. In the dissipative case the system is open and T µν , P µ , nµ are not conserved. See Ref. [57] for attempts to define 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 define 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 fields). 60

APPENDIX C: NOTATIONS ON SPACELIKE HYPERSURFACES.

Let us first review some preliminary results from Refs. [10] needed in the description of physical systems on spacelike hypersurfaces. Let {Στ } be a one-parameter family of spacelike hypersurfaces foliating Minkowski spacetime M 4 with 4-metric ηµν = ǫ(+ − −−), ǫ = ± [ǫ = +1 is the particle physics convention; ǫ = −1 the general relativity one] and giving a 3+1 decomposition of it. At fixed τ , let z µ (τ, ~σ ) be the coordinates of the points on Στ in M 4 , {~σ } a system of coordinates on Στ . ˇ If σ A = (σ τ = τ ; ~σ = {σ rˇ}) [the notation Aˇ = (τ, rˇ) with rˇ = 1, 2, 3 will be used; note that ˇ Aˇ = τ and Aˇ = rˇ = 1, 2, 3 are Lorentz-scalar indices] and ∂Aˇ = ∂/∂σ A , one can define the vierbeins ∂Bˇ zAµˇ − ∂AˇzBµˇ = 0,

zAµˇ(τ, ~σ ) = ∂Aˇz µ (τ, ~σ),

(C1)

so that the metric on Στ is gAˇBˇ (τ, ~σ) = zAµˇ (τ, ~σ )ηµν zBνˇ (τ, ~σ ),

ǫgτ τ (τ, ~σ ) > 0, 2

g(τ, ~σ) = −det || gAˇBˇ (τ, ~σ ) || = (det || zAµˇ (τ, ~σ) ||) ,

γ(τ, ~σ ) = −det || grˇsˇ(τ, ~σ) || = det ||3grˇsˇ(τ, ~σ )||,

(C2)

where grˇsˇ = −ǫ 3 grˇsˇ with 3 grˇsˇ having positive signature (+ + +). If γ rˇsˇ(τ, ~σ ) = −ǫ 3 g rˇsˇ is the inverse of the 3-metric grˇsˇ(τ, ~σ) [γ rˇuˇ (τ, ~σ )guˇsˇ(τ, ~σ ) = δsˇrˇ], the ˇ ˇˇ ˇˇ inverse g AB (τ, ~σ ) of gAˇBˇ (τ, ~σ ) [g AC (τ, ~σ )gcˇˇb (τ, ~σ ) = δBAˇ ] is given by γ(τ, ~σ) , g(τ, ~σ) γ γ g τ rˇ(τ, ~σ) = −[ gτ uˇ γ uˇrˇ](τ, ~σ) = ǫ[ gτ uˇ 3 g uˇrˇ](τ, ~σ ), g g γ g rˇsˇ(τ, ~σ ) = γ rˇsˇ(τ, ~σ) + [ gτ uˇ gτ vˇ γ uˇrˇγ vˇsˇ](τ, ~σ ) = g γ = −ǫ 3 g rˇsˇ(τ, ~σ ) + [ gτ uˇ gτ vˇ 3 g uˇrˇ 3 g vˇsˇ](τ, ~σ ), g g τ τ (τ, ~σ ) =

(C3)

ˇ

σ ) is equivalent to so that 1 = g τ C (τ, ~σ)gCτ ˇ (τ, ~ g(τ, ~σ) = gτ τ (τ, ~σ ) − γ rˇsˇ(τ, ~σ )gτ rˇ(τ, ~σ)gτ sˇ(τ, ~σ ). γ(τ, ~σ)

(C4)

We have zτµ (τ, ~σ )

s

=(

g µ l + gτ rˇγ rˇsˇzsˇµ )(τ, ~σ ), γ

(C5)

and ˇˇ

η µν = zAµˇ (τ, ~σ)g AB (τ, ~σ )zBνˇ (τ, ~σ) = = (lµ lν + zrˇµ γ rˇsˇzsˇν )(τ, ~σ), 61

(C6)

where 1 lµ (τ, ~σ) = ( √ ǫµ αβγ zˇ1α zˇ2β zˇ3γ )(τ, ~σ), γ lµ (τ, ~σ )zrˇµ (τ, ~σ ) = 0,

l2 (τ, ~σ ) = 1,

(C7)

is the unit (future pointing) normal to Στ at z µ (τ, ~σ ). For the volume element in Minkowski spacetime we have q

d4 z = zτµ (τ, ~σ )dτ d3 Σµ = dτ [zτµ (τ, ~σ )lµ (τ, ~σ )] γ(τ, ~σ )d3 σ = q

g(τ, ~σ)dτ d3 σ.

=

(C8)

Let us remark that according to the geometrical approach of Ref. [12],one can use Eq.(C5) in the form zτµ (τ, ~σ ) = N(τ, ~σ )lµ (τ, ~σ ) + N rˇ(τ, ~σ)zrˇµ (τ, ~σ), where N =

q

g/γ =

√

gτ τ − γ rˇsˇgτ rˇgτ sˇ =

√

gτ τ + ǫ3 g rˇsˇgτ rˇgτ sˇ and N rˇ = gτ sˇγ sˇrˇ = −ǫgτ sˇ 3 g sˇrˇ

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

gτ τ = ǫN 2 + grˇsˇN rˇN sˇ = ǫ[N 2 − 3 grˇsˇN rˇN sˇ], gτ rˇ = grˇsˇN sˇ = −ǫ 3 grˇsˇN sˇ, g τ τ = ǫN −2 , g τ rˇ = −ǫN rˇ/N 2 , r ˇ s ˇ r ˇ s ˇ g rˇsˇ = γ rˇsˇ + ǫ NNN2 = −ǫ[3 g rˇsˇ − NNN2 ], ∂ ∂zτµ 4

∂ + zsˇµ γ sˇrˇ ∂N∂ rˇ = lµ = lµ ∂N √ d z = N γdτ d3 σ.

∂ ∂N

− ǫzsˇµ 3 g sˇrˇ ∂N∂ rˇ ,

√ √ ◦ ◦ The rest frame form of a timelike fourvector pµ is p µ = η ǫp2 (1; ~0) = η µo η ǫp2 , p 2 = p2 , ◦ where η = sign po. The standard Wigner boost transforming p µ into pµ is ◦

Lµ ν (p, p) = ǫµν (u(p)) =

◦µ

◦

◦

pµ pν (pµ + p )(pν + pν ) = ηνµ + 2 2 − = ◦ ǫp p· p +ǫp2 =

ηνµ

◦

µ

µ

q

ν=0

ǫµo (u(p))

ν=r

ǫµr (u(p)) = (−ur (p); δri − ◦

◦

(uµ (p) + uµ (p))(uν (p) + uν (p)) , + 2u (p)uν (p) − 1 + uo (p) ◦

µ

= u (p) = p /η ǫp2 ,

◦

ui (p)ur (p) ). 1 + uo (p)

(C9)

The inverse of Lµ ν (p, p) is Lµ ν (p, p), the standard boost to the rest frame, defined by 62

◦

◦

◦

Lµ ν (p, p) = Lν µ (p, p) = Lµ ν (p, p)|p~→−~p.

(C10)

Therefore, we can define the following vierbeins [the ǫµr (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] ◦

ǫµA (u(p)) = Lµ A (p, p), ◦

A p AB ǫA ηµν ǫνB (u(p)), µ (u(p)) = L µ ( , p) = η

ǫoµ¯ (u(p)) = ηµν ǫνo (u(p)) = uµ (p), ǫrµ (u(p)) = −δ rs ηµν ǫνr (u(p)) = (δ rs us (p); δjr − δ rs δjh

uh (p)us (p) ), 1 + uo(p)

ǫA o (u(p)) = uA (p),

(C11)

which satisfy ν µ ǫA µ (u(p))ǫA (u(p)) = ην , µ A ǫA µ (u(p))ǫB (u(p)) = ηB ,

η µν = ǫµA (u(p))η AB ǫνB (u(p)) = uµ (p)uν (p) − ηAB = ǫµA (u(p))ηµν ǫνB (u(p)), ∂ A ∂ µ ǫA (u(p)) = pα ǫ (u(p)) = 0. pα ∂pα ∂pα µ

3 X

ǫµr (u(p))ǫνr (u(p)),

r=1

(C12)

The Wigner rotation corresponding to the Lorentz transformation Λ is µ

◦

−1

◦

µ

R ν (Λ, p) = [L(p, p)Λ L(Λp, p)]

ν

=

1 0 0 Ri j (Λ, p)

!

,

(Λ−1)i o pβ (Λ−1 )β j √ − pρ (Λ−1 )ρ o + η ǫp2 pi ((Λ−1 )o o − 1)pβ (Λ−1 )β j √ 2 [(Λ−1 )o j − √ − o ]. p + η ǫp pρ (Λ−1 )ρ o + η ǫp2 i

Ri j (Λ, p) = (Λ−1 ) j −

(C13)

The polarization vectors transform under the Poincaré transformations (a, Λ) in the following way ǫµr (u(Λp)) = (R−1 )r s Λµ ν ǫνs (u(p)).

63

(C14)

APPENDIX D: MORE ON DIXON’S MULTIPOLES.

Let us add other forms of the Dixon multipoles. In the case of the fluid configurations treated in Section II and IV, the Hamilton equations generated by the Dirac Hamiltonian (2.49) in the gauge ~qsys ≈ 0 [~λ(τ ) = 0] imply [in Ref. ◦ [20] this is a consequence of ∂µ T µν = 0] dpµT (Ts ) ◦ = 0, f or n = 0, dTs dpTµ1 ...µn µ (Ts ) ◦ µ ...µ )µ (µ ...µ )µ = −nu(µ1 (ps )pT2 n (Ts ) + ntT 1 n (Ts ), dTs

n ≥ 1.

(D1)

Let us define for n ≥ 1 (µ ...µn µ)

bTµ1 ...µn µ (Ts ) = pT 1 = ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))bTµ1 ...µn µ (Ts ) =

(Ts ) = µ) (µ1 ǫr1 (u(ps ))....ǫµrnn (u(ps ))ǫA (u(ps ))ITr1 ..rn Aτ (Ts ), 1 (r ...r r)τ uµ (ps )ITr1 ...rnτ τ (Ts ) + ǫµr (u(ps ))IT 1 n (Ts ), n+1 (µ ...µ )µ

(µ ...µn µ)

cTµ1 ...µn µ (Ts ) = cT 1 n (Ts ) = pTµ1 ...µn µ (Ts ) − pT 1 = [ǫµr11 (u(ps ))...ǫµrnn ǫµA (u(ps )) −

(Ts ) =

µ)

− ǫr(µ1 1 (u(ps ))...ǫµrnn (u(ps ))ǫA (u(ps ))]ITr1 ..rn Aτ (Ts ), (µ ...µ µ)

cT 1 n (Ts ) = 0, n ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))cTµ1 ...µn µ (Ts ) = uµ (ps )ITr1 ...rnτ τ (Ts ) + n+1 (r ...r r)τ + ǫµr (u(ps ))[ITr1 ...rnrτ (Ts ) − IT 1 n (Ts )], and then for n ≥ 2 (µ ...µ )(µν)

dTµ1 ...µn µν (Ts ) = dT 1 n (Ts ) = tTµ1 ...µnµν (Ts ) − n + 1 (µ1 ...µn µ)ν (µ ...µ ν)µ [tT (Ts ) + tT 1 n (Ts )] + − n n + 2 (µ1 ...µn µν) t (Ts ) = + n T h n + 1 (µ1 µn µ) ν = ǫµr11 ...ǫµrnn ǫµA ǫνB − ǫr1 ...ǫrn ǫA ǫB + n n + 2 (µ1 µn µ ν) i ν) ǫ ..ǫ ǫ ǫ (u(ps )) + ǫr(µ1 1 ...ǫµrnn ǫB ǫµA + n r1 rn A B ITr1 ..rnAB (Ts ), (µ ...µn µ)ν

dT 1

64

(Ts ) = 0,

(D2)

n−1 µ u (ps )uν (ps )ITr1 ...rn τ τ (Ts ) + n+1 1 + [uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))] n (r ...r r)τ [(n − 1)ITr1 ...rn rτ (Ts ) + IT 1 n (Ts )] + + ǫµs1 (u(ps ))ǫνs2 (u(ps ))[ITr1 ...rns1 s2 (Ts ) − 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)

ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))dTµ1 ...µn µν (Ts ) =

Then Eqs.(D1) may be rewritten in the form 1) n = 1 (µν)

tµν T (Ts ) = tT

◦

(Ts ) = pµT (Ts )uν (ps ) +

1 d (STµν (Ts )[α] + 2bµν T (Ts )), 2 dTs

⇓

d µν b (Ts ) = P τ uµ (ps )uν (ps ) + P r u(µ (ps )ǫν) r (u(ps )) + dTs T ν) rτ τ + ǫ(µ r (u(ps ))u (ps )IT (Ts ) + ν) rsτ + ǫ(µ r (u(ps ))ǫs (u(ps ))IT (Ts ), ◦

(µ

ν) tµν T (Ts ) = pT (Ts )u (ps ) +

d µν ◦ [µ ν] ST (Ts )[α] = 2pT (Ts )uν] (ps ) = 2Pφr ǫ[µ r (u(ps ))u (ps ) ≈ 0, dTs (ρµ)ν

2) n = 2 [identity tρµν = tT T (ρµ)ν

2tT

◦

µ)ν

(ρν)µ

+ tT

µ)ν

(Ts ) = 2u(ρ (ps )bT (Ts ) + u(ρ (ps )ST (Ts )[α] + ◦

⇓

ρ(µ

µν ρ ν) tρµν T (Ts ) = u (ps )bT (Ts ) + ST (Ts )[α]u (ps ) +

(µν)ρ

+ tT

]

d ρµν (bT (Ts ) + cρµν T (Ts )), dTs

d 1 ρµν ( bT (Ts ) − cρµν T (Ts )), dTs 2

3) n ≥ 3 ◦

µ ...µ )µν

µ ...µ )(µν)

tTµ1 ...µnµν (Ts ) = dTµ1 ...µnµν (Ts ) + u(µ1 (ps )bT2 n (Ts ) + 2u(µ1 (ps )cT2 n (Ts ) + 2 µ ...µ (µ 2 µ ...µ (µν) d 1 µ1 ...µn µν = cT1 n (Ts )uν) (ps ) + (Ts ) + cT1 n (Ts )], [ bT n dTs n + 1 n

(D4)

This allows [20] to rewrite < T µν , f > in the following form µν

< T ,f > = +

Z

dTs

∞ X

Z

h d4 k ˜ ν) ρ(µ −ik·xs (Ts ) (µ u (ps )pT (Ts ) − ikρ ST (Ts )[α]uν) (ps ) + f(k)e 4 (2π)

i (−i)n kρ1 ...kρn ITρ1 ...ρn µν (Ts ) , n! n=2

65

(D5)

with (µ1 ...µn )(µν)

ITµ1 ...µn µν (Ts ) = IT

(Ts ) = dTµ1 ...µn µν (Ts ) − 2 2 µ ...µ (µ µ ...µ )(µν) − u(µ1 (ps )cT2 n (Ts ) + cT1 n (Ts )uν) (ps ) = n−1 n h n + 1 µ) = ǫµr11 ...ǫµrnn ǫµA ǫνB − ǫr(µ1 1 ...ǫµrnn ǫA ǫνB + n n + 2 (µ1 µn µ ν) i ν) ǫ ...ǫrn ǫA ǫB (u(ps )0ITr1 ..rn AB (Ts ) − + ǫr(µ1 1 ...ǫµrnn ǫB ǫµA + n r1 h 2 ν) (µ2 µn ) (µ ν)) n) − − u(µ1 (ps ) ǫµr12 ...ǫµrn−1 ǫ(µ ǫ − ǫ ...ǫ ǫ ǫ rn A r1 rn−1 rn A n−1 2 µ1 µn (µ (µ) ǫ ...ǫ ǫ − ǫr(µ1 1 ...ǫµrnn ǫA uν) (ps )](u(ps )0ITr1 ..rnAτ (Ts ), − n r1 rn A (µ1 ...µn µ)ν

IT

(Ts ) = 0,

n+3 µ u (ps )uν (ps )ITr1 ...rnτ τ (Ts ) + n+1 1 + [uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]ITr1 ...rnrτ (Ts ) + n + ǫµs1 (u(ps ))ǫνs2 (u(ps ))[ITr1 ...rn s1 s2 (Ts ) − n + 1 (r1 ...rns1 )s2 (r ...r s )s − (IT (Ts ) + IT 1 n 2 1 (Ts )) + n (r1 ...rn s1 s2 ) + IT (Ts )].

ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))ITµ1 ...µn µν (Ts ) =

Finally, a set of multipoles equivalent to the ITµ1 ...µn µν is n≥0 (µ1 ...µn )[µν][ρσ]

µ ...µ [µ[ρν]σ]

JTµ1 ...µnµνρσ (Ts ) = JT

(Ts ) = IT 1 n (Ts ) = h 1 ν]µ ...µ [ρσ] µ ...µ [µ[ρν]σ] u[µ(ps )pT 1 n (Ts ) + = tT1 n (Ts ) − n+ 1 i σ]µ1 ...µn [µν] [ρ + u (ps )pT (Ts ) = ν] σ]

h

i

r1 ..rn AB [ρ = ǫµr11 ..ǫµrnn ǫ[µ (Ts ) − r ǫs ǫA ǫB (u(ps ))IT h 1 σ] [ρ − u[µ (ps )ǫν] r (u(ps ))ǫs (u(ps ))ǫA (u(ps )) + n+1 i ν] [µ + u[ρ(ps )ǫσ] r (u(ps ))ǫs (u(ps ))ǫA (u(ps ))

ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ITrr1 ..rn sAτ (Ts ),

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

(D6)

uµ1 (ps )

µ ...µn−1 (µn µν)ρσ

JTµ1 ...µn µνρσ (Ts ) = JT 1

(Ts ) = 0,

n≥2 ITµ1 ...µnµν (Ts ) =

4(n − 1) (µ1 ...µn−1 |µ|µn )ν J (Ts ), n+1 T h

ν] σ]

i

r1 ..rn AB [ρ (Ts ) − ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))JTµ1 ...µnµνρσ (Ts ) = ǫ[µ r ǫs ǫA ǫB (u(ps ))IT h 1 σ] [ρ u[µ (ps )ǫν] − r (u(ps ))ǫs (u(ps ))ǫA (u(ps )) + n+1 i ν] [µ + u[ρ(ps )ǫσ] (u(p ))ǫ (u(p ))ǫ (u(p )) ITrr1 ..rnsAτ (Ts ). s s s A r s

(D7)

The JTµ1 ...µn µνρσ are the Dixon “2n+2 -pole inertial moment tensors” of the extended system: they [or equivalently the ITµ1 ...µn µν ’s] determine its energy-momentum tensor together ◦ with the monopole pµT and the spin dipole STµν . The equations ∂µ T µν = 0 are satisfied due to the equations of motion (D4) for PTµ and STµν [the so called Papapetrou-Dixon-Souriau equations given in Eqs.(4.18)] without the need of the equations of motion for the JTµ1 ...µn µνρσ . When all the multipoles JTµ1 ...µn µνρσ are zero [or negligible] one speaks of a pole-dipole field configuration of the perfect fluid.

67

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