Multipolar Expansions for Closed and Open Systems of
Relativistic Particles.
arXiv:hep-th/0402181v1 23 Feb 2004
David Alba
Dipartimento di Fisica
Universita’ di Firenze
L.go E.Fermi 2 (Arcetri)
50125 Firenze, Italy
E-mail: [email protected]
Luca Lusanna
Sezione INFN di Firenze
L.go E.Fermi 2 (Arcetri)
50125 Firenze, Italy
E-mail: [email protected]
Massimo Pauri
Dipartimento di Fisica
Universita’ di Parma
Parco Area Scienze 7/A
43100 Parma, Italy
E-mail: [email protected]
1
Abstract
Dixon’s multipoles for a system of N relativistic positive-energy scalar particles are evaluated
in the rest-frame instant form of dynamics. The Wigner hyper-planes (intrinsic rest frame of the
isolated system) turn out to be the natural framework for describing multipole kinematics. Classical
concepts like the barycentric tensor of inertia turn out to be extensible to special relativity only by
means of the quadrupole moments of the isolated system. Two new applications of the multipole
technique are worked out for systems of interacting particles and fields. In the rest-frame of the
isolated system of either free or interacting positive energy particles it is possible to define a unique
world-line which embodies the properties of the most relevant centroids introduced in the literature
as candidates for the collective motion of the system. This is no longer true, however, in the case of
open subsystems of the isolated system. While effective mass, 3-momentum and angular momentum
in the rest frame can be calculated from the definition of the subsystem energy-momentum tensor,
the definitions of effective center of motion and effective intrinsic spin of the subsystem are not
unique. Actually, each of the previously considered centroids corresponds to a different worldline in the case of open systems. The pole-dipole description of open subsystems is compared to
their description as effective extended objects. Hopefully, the technique developed here could be
instrumental for the relativistic treatment of binary star systems in metric gravity.
December 28, 2013
2
I.
INTRODUCTION.
An important area of research is nowadays the construction of templates for gravitational
waves emitted by binary systems. Analytically, this can be done within the framework of
PN approximations by means of essentially non-relativistic multipole expansions for compact
bodies. On the other hand, since the main emission are supposed to take place in a region
where the PN approximation fails, it would be desirable to have at disposal a relativistic
treatment of multipolar expansions as a preliminary kinematical tool for dealing with open
gravitating systems. This paper focuses just on the construction of a suitable relativistic
kinematical background by an N-body system as a tool. A preliminary extension of the
results of this paper to relativistic fluids is given in [1].
Our starting point will be the result recently obtained[2] concerning a complete treatment of the kinematics of the relativistic N-body problem in the rest-frame instant form of
dynamics [3, 4, 5, 6]. This program required the re-formulation of the theory of isolated
relativistic systems on arbitrary space-like hyper-surfaces [7]. and has been based essentially upon Dirac’s reformulation[8] of classical field theory (suitably extended to particles)
on arbitrary space-like hyper-surfaces (equal time and Cauchy surfaces). For each isolated
system (containing any combination of particles, strings and fields) one obtains in this way a
re-formulation of the standard theory as a parametrized Minkowski theory[3]. This program
shows the extra bonus of being naturally prepared for the coupling to gravity in its ADM
formulation. The price to be paid is that the functions z µ (τ, ~σ ) describing the embedding of
the space-like hyper-surface in Minkowski space-time become configuration variables in the
action principle. Since the action is invariant under separate τ -reparametrizations and spacediffeomorphisms, first class constraints emerge ensuring the independence of the description
from the choice of the 3+1 splitting. The embedding configuration variables z µ (τ, ~σ ) are
thus the gauge variables associated with this particular kind of general covariance.
We summarize here the main results that are necessary to understand all the subsequent
technical developments. First of all, recall that since the intersection of a time-like world-line
with a space-like hyper-surface corresponding to a value τ of the time parameter is identified
by 3 numbers ~σ = ~η(τ ) instead of four, in parametrized Minkowski theories each particle
must have a well defined sign of the energy: therefore the two topologically disjoint branches
of the mass hyperboloid cannot be described simultaneously as in the standard manifestly
Lorentz-covariant theory, and there are no more mass-shell constraints. Each particle with
a definite sign of the energy is described by the canonical coordinates ~ηi (τ ), ~κi (τ ) while the
4-position of the particles are given by xµi (τ ) = z µ (τ, ~ηi (τ )). The following 4-momenta pµi (τ )
are ~κi -dependent solutions of p2i − m2i = 0 with the chosen sign of the energy.
In order to exploit the separate spatial and time reparametrization invariances of
parametrized Minkowski theories, we can first of all restrict the foliation to space-like hyperplanes as leaves. For each configuration of the isolated system with time-like 4-momentum,
we further restrict to the special leaves defined by hyper-planes orthogonal to the conserved
system 4-momentum (Wigner hyper-planes). This foliation is fully determined by the configuration of the isolated system. One gets in this way[3] the definition of the Wignercovariant rest-frame instant form of dynamics for any isolated system whose configurations
have well defined and finite Poincaré generators with time-like total 4-momentum (see Ref.[9]
for the traditional forms of dynamics). Finally, this formulation casts some light on the
3
long standing problem of defining a relativistic center of mass. As well known, no definition of this concept can enjoy all the properties of its non-relativistic counterpart. See
Refs.[10, 11, 13, 14, 15] for a partial bibliography of all the existing attempts.
As shown in Appendix A of Ref.[2] only four first class constraints survive in the restframe instant form on Wigner hyper-planes. The original configuration variables z µ (τ, ~σ),
~ηi (τ ) and their conjugate momenta ρµ (τ, ~σ ), ~κi (τ ) are reduced to:
µ
µ
i) a decoupled particle
p x̃s (τ ), ps (the only remnant of the space-like hyper-surface) with
a positive mass ǫs = p2s determined by the first class constraint ǫs − Msys ≈ 0, Msys being
the invariant mass of the isolated system. As a gauge fixing to the constraint, the rest-frame
Lorentz scalar time Ts = x̃sǫ·ps s is put equal to the mathematical time: Ts − τ ≈ 0. Here,
x̃µs (τ ) is a non-covariant canonical variable for the 4-center of mass. After the elimination
of Ts and ǫs by means of such pair of second class constraints, we are left with a decoupled
free point (point particle clock) of mass Msys and canonical 3-coordinates ~zs = ǫs (~x̃s − pp~os x̃o ),
s
~ks = p~s [16]. The non-covariant canonical x̃µ (τ ) must not be confused with the 4-vector
s
ǫs
xµs (τ ) = z µ (τ, ~σ = 0) identifying the origin of the 3-coordinates ~σ inside the Wigner hyperplanes. The world-line xµs (τ ) is arbitrary because it depends on xµs (0) while its 4-velocity
ẋµs (τ ) depends on the Dirac multipliers associated with the remaining 4 (or 3 after having
imposed Ts − τ ≈ 0) first class constraints (see Section II). Correspondingly, this worldline may be considered as an arbitrary centroid for the isolated system. The unit time-like
µ
4-vector uµ (ps ) = pµs /ǫs (= x̃˙ s (τ ) after having imposed Ts − τ ≈ 0) is orthogonal to the
Wigner hyper-planes and describes their orientation in the chosen inertial frame.
ii) the particle canonical variables ~ηi (τ ), ~κi (τ ) inside the Wigner hyper-planes. They are
Wigner spin-1 3-vectors, like the coordinates P
~σ . They are restricted by the three first class
constraints (the rest-frame conditions) ~κ+ = N
κi ≈ 0. Since the role of the relativistic
i=1 ~
µ
decoupled 4-center of mass is taken by x̃s (τ ) (or, after the gauge fixing Ts − τ ≈ 0, by an
external 3-center of mass ~zs , defined in terms of x̃µs and pµs [2]), the rest-frame conditions
imply that the internal canonical 3-center of mass ~q+ = ~σcom is a gauge variable that can
be eliminated through a gauge fixing [17]. This amounts in turn to a definite choice of the
world-line xµs (τ ) of the centroid.
All this leads to a doubling of viewpoints and concepts:
1) The external viewpoint, taken by an arbitrary inertial Lorentz observer, who describes
the Wigner hyper-planes determined by the time-like configurations of the isolated system.
A change of inertial observer by means of a Lorentz transformation rotates the Wigner
hyper-planes and induces a Wigner rotation of the 3-vectors inside each Wigner hyperplane.
Every such hyperplane inherits an induced internal Euclidean structure while an external
realization of the Poincaré group induces an internal Euclidean action.
Then, three external concepts of 4-center of mass can be defined by using the external
realization of the Poincaré algebra (each one corresponding to a different 3-location inside
the Wigner hyper-planes):
a) the external non-covariant canonical 4-center of mass (also named 4-center of spin) x̃µs
(with 3-location ~σ̃),
b) the external non-covariant non-canonical Møller 4-center of energy Rsµ (with 3-location
~σR ),
4
c) the external covariant non-canonical Fokker-Pryce 4-center of inertia Ysµ (with 3location ~σY ).
Only the canonical non-covariant center of mass x̃µs (τ ) is relevant to the Hamiltonian
treatment with Dirac constraints, while only the Fokker-Pryce Ysµ is a 4-vector by construction. See Ref.[2] for the construction of the 4-centers starting from the corresponding
3-centers (3-center of spin[14], 3-center of energy [12], 3-center of inertia[13, 14]), which are
group-theoretically defined in terms of generators of the external Poincare’ group.
2) The internal viewpoint, taken by an observer inside the Wigner hyper-planes. This
viewpoint is associated to a unfaithful internal realization of the Poincaré algebra: the
total internal 3-momentum of the isolated system vanishes due to the rest-frame conditions.
The internal energy and angular momentum are the invariant mass Msys and the spin (the
angular momentum with respect to x̃µs (τ )) of the isolated system, respectively. By means
of the internal realization of the Poincaré algebra we can define three internal 3-centers of
mass: the internal canonical 3-center of mass (or 3-center of spin) ~q+ , the internal Møller
~ + and the internal Fokker-Pryce 3-center of inertia ~y+ . However, due to
3-center of energy R
~ + ≈ ~y+ . As a natural gauge fixing
the rest-frame condition ~κ+ ≈ 0, they all coincide: ~q+ ≈ R
~
to the rest-frame conditions, we can add the vanishing of the internal Lorentz boosts K
~ + /Msys ). This is equivalent to locate the internal canonical
(recall that they are equal to −R
3-center of mass ~q+ in ~σ = 0, i.e. in the origin xµs (τ ) = z µ (τ, ~0). Upon such gauge fixings
and with Ts − τ ≈ 0, the world-line xµs (τ ) becomes uniquely determined except for the
arbitrariness in the choice of xµs (0) [ uµ (ps ) = pµs /ǫs ]
xµs (τ ) = xµs (0) + uµ (ps )Ts ,
(1.1)
and, modulo xµs (0), it coincides with the external covariant non-canonical Fokker-Pryce 4center of inertia, xµs (τ ) = xµs (0) + Ysµ [2].
The doubling of concepts, the external non-covariant canonical 4-center of mass x̃µs (τ )
(or the external 3-center of mass ~zs when Ts − τ ≈ 0) and the internal canonical 3-center
of mass ~q+ ≈ 0 replaces the separation of the non-relativistic 3-center of mass due to the
Abelian translation symmetry. Correspondingly, the non-relativistic conserved 3-momentum
is replaced by the external ~ps = ǫs~ks , while, as already said, the internal 3-momentum
vanishes, ~κ+ ≈ 0, as a definition of rest frame.
In the gauge where ǫs ≡ Msys , Ts ≡ τ , the canonical basis ~zs , ~ks , ~ηi , ~κi is restricted by
P
the three pairs of second class constraints ~κ+ = N
κi ≈ 0, ~q+ ≈ 0, so that 6N canonical
i=1 ~
variables describe the N particles like in the non-relativistic case. We still need a canonical
transformation ~ηi , ~κi 7→ ~q+ [≈ 0], ~κ+ [≈ 0], ρ~a , ~πa [a = 1, .., N −1] identifying a set of relative
canonical variables. The final 6N-dimensional canonical basis is ~zs , ~ks , ρ~a , ~πa . To get this
result we need a highly non-linear canonical transformation[2], which can be obtained by
exploiting the Gartenhaus-Schwartz singular transformation [18].
At the end we obtain the Hamiltonian for the relative motions as a sum of N square roots,
each one containing a squared mass and a quadratic form in the relative momenta, which
goes into the non-relativistic Hamiltonian for relative motions in the limit c → ∞. This
fact has the following implications:
5
a) if one tries to make the inverse Legendre transformation to find the associated Lagrangian, it turns out that, due to the presence of square roots, the Lagrangian is a hyperelliptic function of ρ~˙aqalready in the free case. A closed form exists only for N=2,
m1 = m2 = m: L = −m 4 − ρ~˙2 . This exceptional case already shows that the existence
of the limiting velocity c forbids a linear relation between the spin (center-of-mass angular
momentum) and the angular velocity.
b) the N quadratic forms in the relative momenta appearing in the relative Hamiltonian
cannot be simultaneously diagonalized. In any case the Hamiltonian is a sum of square roots,
so that concepts like reduced masses, Jacobi normal relative coordinates and tensor of inertia
cannot be extended to special relativity. As a consequence, for example, a relativistic static
orientation-shape SO(3) principal bundle approach [19, 20] can be implemented only by
using non-Jacobi relative coordinates.
c) the best way of studying rotational kinematics, viz the non-Abelian rotational symmetry associated with the conserved internal spin, is based on the canonical spin bases with
the associated concepts of spin frames and dynamical body frames introduced in Ref.[2]. It
is remarkable that they can be build as in the non-relativistic case [21] starting from the
canonical basis ρ~a , ~πa .
Let us clarify this important point. In the non-relativistic N-body problem it is easy
to make the separation of the absolute translational motion of the center of mass from the
relative motions, due to the Abelian nature of the translation symmetry group. This implies
that the associated Noether constants of motion (the conserved total 3-momentum) are in
involution, so that the center-of-mass degrees of freedom decouple. Moreover, the fact that
the non-relativistic kinetic energy of the relative motions is a quadratic form in the relative
velocities allows the introduction of special sets of relative coordinates, the Jacobi normal
relative coordinates that diagonalize the quadratic form and correspond to different patterns
of clustering of the centers of mass of the particles. Each set of Jacobi normal relative
coordinates organizes the N particles into a hierarchy of clusters, in which each cluster of
two or more particles has a mass given by an eigenvalue (reduced masses) of the quadratic
form; Jacobi normal coordinates join the centers of mass of pairs of clusters.
However, the non-Abelian nature of the rotation symmetry group whose associated
Noether constants of motion (the conserved total angular momentum) are not in involution, prevents the possibility of a global separation of absolute rotations from the relative
motions, so that there is no global definition of absolute vibrations. Consequently, an isolated deformable body can undergo rotations by changing its own shape (see the examples
of the falling cat and of the diver). It was just to deal with these problems that the theory of
the orientation-shape SO(3) principal bundle approach[19] has been developed. Its essential
content is that any static (i.e. velocity-independent) definition of body frame for a deformable
body must be interpreted as a gauge fixing in the context of a SO(3) gauge theory. Both
the laboratory and the body frame angular velocities, as well as the orientational variables
of the static body frame, become thereby unobservable gauge variables. This approach is
associated with a set of point canonical transformations, which allow to define the body
frame components of relative motions in a velocity-independent way.
Since in many physical applications (e.g. nuclear physics, rotating stars,...) angular
velocities are viewed as measurable quantities, one would like to have an alternative formulation complying with this requirement and possibly generalizable to special relativity.
6
This program has been first accomplished in a previous paper [21] and then relativistically
extended (Ref.[2]). Let us summarize, therefore, the main points of our formulation.
First of all, for N ≥ 3, we have constructed (see Ref.[21]) a class of non-point canonical transformations which allow to build the already quoted canonical spin bases: they are
connected to the patterns of the possible clusterings of the spins associated with relative
motions. The definition of these spin bases is independent of Jacobi normal relative coordinates, just as the patterns of spin clustering are independent of the patterns of center-of-mass
Jacobi clustering. We have found two basic frames associated to each spin basis: the spin
frame and the dynamical body frame. Their construction is guaranteed by the fact that in
the relative phase space, besides the natural existence of a Hamiltonian symmetry left action
of SO(3) [22, 23], it is possible to define as many Hamiltonian non-symmetry right actions of
SO(3) [24] as the possible patterns of spin clustering. While for N=3 the unique canonical
spin basis coincides with a special class of global cross sections of the trivial orientationshape SO(3) principal bundle, for N ≥ 4 the existing spin bases and dynamical body frames
turn out to be unrelated to the local cross sections of the static non-trivial orientation-shape
SO(3) principal bundle, and evolve in a dynamical way dictated by the equations of motion.
In this new formulation both the orientation variables and the angular velocities become, by
construction, measurable quantities in each canonical spin basis.
For each N, every allowed spin basis provides a physically well-defined separation between
rotational and vibrational degrees of freedom. The non-Abelian nature of the rotational symmetry implies that there is no unique separation of absolute rotations and relative motions.
The unique body frame of rigid bodies is replaced here by a discrete number of evolving
dynamical body frames and of spin canonical bases, both of which are grounded in patterns
of spin couplings, direct analog of the coupling of quantum angular momenta.
As anticipated at the outset, we want to complete our study of relativistic kinematics for
the N-body system by first evaluating the rest-frame Dixon multipoles [25, 26] and then by
analyzing the role of Dixon’s multipoles for open subsystems. The basic technical tool will be
the standard definition of the energy momentum tensor of the N positive-energy free particles
on the Wigner hyperplane. It will be seen, however, that in order to get a sensible extension
of this definition to open subsystems, a physically significant convention is required. On the
whole, it turns out that the Wigner hyperplane is the natural framework for reorganizing a
lot of kinematics connected with multipoles. Only in this way, moreover, a concept like the
barycentric tensor of inertia can be introduced in special relativity, specifically by means of
the quadrupole moments.
A first application of the formalism is done for an isolated system of N positive-energy
particles with mutual action-at-a-distance interactions. Then the formalism is applied to
the case of an open n < N particle subsystem of an isolated system consisting of N charged
positive-energy particles (with Grassmann-valued electric charges to regularize the Coulomb
self-energies) plus the electro-magnetic field [5]. In the rest frame of the isolated system a
suitable definition of the energy-momentum tensor of the open subsystem allows to define
its effective mass, 3-momentum and angular momentum.
Then we evaluate the rest-frame Dixon multipoles of the energy-momentum tensor of the
open subsystem with respect to various centroids describing possible collective centers of
motion. Unlike the case of isolated systems, each centroid generates a different world-line
and there are many candidates for the effective center of motion and the effective intrinsic
spin. Two centroids (namely the center of energy and Tulczyjew centroid) appear to be
7
preferable due to their properties. The case n = 2 is studied explicitly. It is also shown that
the pole-dipole description of the 2-particle cluster can be replaced by a description of the
cluster as an extended system (its effective spin frame can be evaluated). This can be done,
however, at the price of introducing an explicit dependence on the action of the external
electro-magnetic field upon the cluster. By comparing the effective parameters of an open
cluster of n1 + n2 particles to the effective parameters of the two clusters with n1 and n2
particles, it turns out in particular that only the effective center of energy appears to be a
viable center of motion for studying the interactions of open subsystems.
A review of the rest-frame instant form of dynamics for N scalar free positive-energy
particles and some new original results on the canonical transformation to the internal
center of mass and relative variables are given in Section II.
In Section III we evaluate the energy momentum tensor in the Wigner hyper-planes.
Dixon’s multipoles are introduced in Section IV. A special study of monopole, dipole and
quadrupole moments is given and the multipolar expansion is defined.
After the extension of the previous results to isolated systems with mutual action-at-adistance interactions, we study in Section V the behavior of open subsystems of isolated
systems, the centroids which are good candidates for the description the collective center of
motion, and discuss the determination of the effective parameters (mass, spin, momentum,
variables relative to the center of motion) for the open subsystem.
Some comments about standing problems are given in the Conclusions.
The non-relativistic N-particle multipolar expansion is given in Appendix A, while in
Appendix B contains is a review of symmetric trace-free (STF) tensors. The GartenhausSchwartz transformation is summarized in Appendix C. Finally, other properties of Dixon’s
multipoles are reported in Appendix D.
8
II.
REVIEW OF THE REST-FRAME INSTANT FORM.
We briefly review the treatment of N free scalar positive-energy particles in the framework
of parametrized Minkowski theory (see Appendices A and B and Section II of Ref.[2]). Each
particle is described by a configuration 3-vector ~ηi (τ ). The particle world-line is xµi (τ ) =
z µ (τ, ~ηi (τ )), where z µ (τ, ~σ) are the embedding configuration variables describing the spacelike hyper-surface Στ .
The foliation is defined by an embedding R × Σ → M 4 , (τ, ~σ ) 7→ z µ (τ, ~σ ), with Σ an
abstract 3-surface diffeomorphic to R3 . Στ is the Cauchy surface of equal time. The metric
induced on Στ is gAB [z] = zAµ ηµν zBν , a functional of z µ , and the embedding coordinates
z µ (τ, ~σ ) are considered as independent fields. We use the notation σ A = (τ, σ ř ) of Refs.[3, 5].
The zAµ (σ) = ∂z µ (σ)/∂σ A are flat cotetrad fields on Minkowski space-time with the zřµ ’s
A
tangent to Στ . The dual tetrad fields are zµA (σ) = ∂σ∂z µ(z) , zµA (τ, ~σ ) zBµ (τ, ~σ) = δBA . While in
Ref.[2] we used the metric convention ηµν = ǫ(+ − −−) with ǫ = ±, in this paper we shall
use ǫ = 1 like in Ref.[3].
p
p
p
p
g(τ,
~
σ
)
=
−det
g
(τ,
~
σ
)
and
γ(τ,
~
σ
)
=
If we put
AB
−det gřš (τ, ~σ), we have
q
g µ
ř ǔ
ř
µ
řš µ
µ
l + gτ ř γ zš (τ, ~σ ) (γ gǔš = δš ), l (τ, ~σ ) = √1γ ǫµ αβγ z1̌α z2̌β z3̌γ (τ, ~σ)
zτ (τ, ~σ ) =
γ
(normal to Στ ; l2p
(τ, ~σ ) = 1; lµp
(τ, ~σ ) zřµ (τ, ~σ ) = 0) and d4 z = zτµ (τ, ~σ ) dτ d3 Σµ =
dτ [zτµ (τ, ~σ ) lµ (τ, ~σ )] γ(τ, ~σ ) d3 σ = g(τ, ~σ) dτ d3 σ.
The system is described by the action[3, 4, 5]
S =
Z
3
dτ d σ L(τ, ~σ) =
L(τ, ~σ) = −
L(τ ) = −
N
X
i=1
N
X
i=1
Z
dτ L(τ ),
δ 3 (~σ − ~ηi (τ ))mi
q
gτ τ (τ, ~σ ) + 2gτ ř (τ, ~σ)η̇iř (τ ) + gřš (τ, ~σ )η̇iř (τ )η̇iš (τ ),
q
mi gτ τ (τ, ~ηi (τ )) + 2gτ ř (τ, ~ηi (τ ))η̇iř (τ ) + gřš (τ, ~ηi (τ ))η̇iř (τ )η̇iš (τ ).
The action is invariant under separate τ - and ~σ -reparametrizations.
The canonical momenta are
9
(2.1)
N
X
∂L(τ, ~σ )
=
δ 3 (~σ − ~ηi (τ ))mi
ρµ (τ, ~σ ) = − µ
∂zτ (τ, ~σ )
i=1
zτ µ (τ, ~σ ) + zřµ (τ, ~σ)η̇iř (τ )
p
gτ τ (τ, ~σ ) + 2gτ ř (τ, ~σ )η̇iř (τ ) + gřš (τ, ~σ )η̇iř (τ )η̇iš (τ )
= [(ρν lν )lµ + (ρν zřν )γ řš zšµ ](τ, ~σ ),
κiř (τ ) = −
=
∂L(τ )
=
∂ η̇iř (τ )
= mi p
gτ ř (τ, ~ηi (τ )) + gřš (τ, ~ηi (τ ))η̇iš (τ )
gτ τ (τ, ~ηi (τ )) + 2gτ ř (τ, ~ηi (τ ))η̇iř (τ ) + gřš (τ, ~ηi (τ ))η̇iř (τ )η̇iš (τ )
′
,
′
{z µ (τ, ~σ), ρν (τ, ~σ } = −ηνµ δ 3 (~σ − ~σ ),
{ηiř (τ ), κjš (τ )} = −δij δšř .
(2.2)
The canonical Hamiltonian Hc is zero, but there are the primary first class constraints
Hµ (τ, ~σ ) = ρµ (τ, ~σ) − lµ (τ, ~σ)
− zřµ (τ, ~σ )γ řš (τ, ~σ)
N
X
i=1
N
X
i=1
3
δ (~σ − ~ηi (τ ))
q
m2i − γ řš (τ, ~σ )κiř (τ )κiš (τ ) −
δ 3 (~σ − ~ηi (τ ))κiš ≈ 0,
so that the Dirac Hamiltonian is HD =
multipliers.
R
(2.3)
d3 σλµ (τ, ~σ )Hµ (τ, ~σ ), where λµ (τ, ~σ ) are Dirac
The conserved Poincaré generators are (the suffix “s” denotes the hypersurface Στ )
pµs
=
Jsµν
Z
=
Z
d3 σρµ (τ, ~σ ),
d3 σ[z µ (τ, ~σ )ρν (τ, ~σ ) − z ν (τ, ~σ )ρµ (τ, ~σ )].
(2.4)
After the restriction to space-like hyper-planes, z µ (τ, ~σ ) = xµs (τ ) + bµš (τ ) σ š , the Dirac
Hamiltonian is reduced to HD = λµ (τ )H̃µ (τ ) + λµν (τ )H̃µν (τ ) (only ten Dirac multipliers
survive) while the remaining ten constraints are given by
10
µ
H̃ (τ ) =
Z
3
µ
d σH (τ, ~σ ) =
H̃µν (τ ) = bµř (τ )
pµs
N q
N
X
X
µ
2
2
−l
mi + ~κi (τ ) + bř (τ )
κiř (τ ) ≈ 0,
µ
i=1
Z
i=1
d3 σσ ř Hν (τ, ~σ ) − bνř (τ )
= Ssµν (τ ) − [bµř (τ )bντ − bνř (τ )bµτ ]
− [břµ (τ )bνš (τ ) − bνř (τ )bµš (τ )]
N
X
i=1
Z
N
X
d3 σσ ř Hµ (τ, ~σ ) =
ηiř (τ )
i=1
q
m2i + ~κ2i (τ ) −
ηiř (τ )κši (τ ) ≈ 0.
(2.5)
Here Ssµν is the spin part of the Lorentz generators
Jsµν = xµs pνs − xνs pµs + Ssµν ,
Z
Z
µ
µν
3
ř ν
ν
Ss = bř (τ ) d σσ ρ (τ, ~σ ) − bř (τ ) d3 σσ ř ρµ (τ, ~σ ).
(2.6)
The condition l˙µ = 0 that the unit normal lµ = ǫµ αβγ bα1̌ (τ ) bβ2̌ (τ ) bγ3̌ (τ ), l2 = 1, to the
hyper-planes be τ -indepedent, implies three gauge fixing constraints lµ = const. for the
first class constraints H̃µν (τ ) ≈ 0. These imply in turn that the břµ (τ )’s depend only on
three Euler angles describing a rotating spatial frame. Therefore foliations with parallel
hyper-planes have only 7 independent first class constraints.
When p2s > 0, the restriction to Wigner hyper-planes is done by imposing bµǍ = (bµτ =
◦
◦
lµ ; břµ ) ≈ bµA = Lµ ν=A (ps , =s ), where Lµ ν (ps , =s ) is the standard Wigner boost for time-like
◦
Poincare’ orbits.
The indices ř = r are Wigner spin 1 indices and we have bµτ = Lµ o (ps , =s ) =
p
◦
uµ (ps ) = pµs / p2s , bµr = ǫµr (u(ps )) = Lµ r (ps , =s ).
On the Wigner hyperplane [28], z µ (τ, ~σ ) = xµs (τ ) + ǫµr (u(ps )) σ r , we have the following
constraints and Dirac Hamiltonian[3, 5]
11
µ
H̃ (τ ) =
pµs
N q
N
X
X
µ
2
2
− u (ps )
mi + ~κi + ǫr (u(ps ))
κir =
µ
i=1
i=1
N
N q
i
h
X
X
µ
µ
2
2
mi + ~κi + ǫr (u(ps ))
κir ≈ 0,
= u (ps ) ǫs −
i=1
i=1
or
ǫs − Msys ≈ 0,
p~sys = ~κ+ =
N
X
i=1
HD
Msys =
N q
X
m2i + ~κ2i ,
i=1
~κi ≈ 0,
= λ (τ )H̃µ (τ ) = λ(τ )[ǫs − Msys ] − ~λ(τ )
µ
λ(τ ) ≈ −ẋsµ (τ )uµ (ps ),
N
X
~κi ,
i=1
λr (τ ) ≈ −ẋsµ (τ )ǫµr (u(ps )),
µ
x̃˙ s (τ ) = −λ(τ )uµ (ps ),
◦
ẋµs (τ ) = {xµs (τ ), HD } = λν (τ ){xµs (τ ), Hµ (τ )} ≈
≈ −λµ (τ ) = −λ(τ )uµ (ps ) + ǫµr (u(ps ))λr (τ ),
ẋ2s (τ ) = λ2 (τ ) − ~λ2 (τ ) > 0,
ẋs · u(ps ) = −λ(τ ),
ẋµ (τ )
−λ(τ )uµ (ps ) + λr (τ )ǫµr (u(ps ))
q
Usµ (τ ) = ps
=
,
ẋ2s (τ )
λ2 (τ ) − ~λ2 (τ )
Z τ
Z
µ
µ
µ
µ
⇒
xs (τ ) = xs (0) − u (ps )
dτ1 λ(τ1 ) + ǫr (u(ps ))
0
τ
dτ1 λr (τ1 ).
(2.7)
0
While the Dirac multiplier λ(τ ) is determined by the gauge fixing Ts −τ ≈ 0, the 3 Dirac’s
multipliers ~λ(τ ) describe the classical zitterbewegung of the centroid xµs (τ ) = z µ (τ, ~0) which
is the origin of the 3-coordinates on the Wigner hyperplane. Each gauge-fixing χ
~ (τ ) ≈ 0
to the 3 first class constraints ~κ+ ≈ 0 (defining the internal rest-frame 3-center of mass
~σcm ) gives a different determination of the multipliers ~λ(τ ) [29]. Therefore each gauge-fixing
(~
χ)µ
identifies a different world-line for the covariant non-canonical centroid xs (τ ). Of course,
inside the Wigner hyperplane, three degrees of freedom of the isolated system [30] become
gauge variables. The natural gauge fixing for eliminating the first class constraints ~κ+ ≈ 0
is χ
~ (τ ) = ~σcom = ~q+ ≈ 0 [vanishing of the location of the internal canonical 3-center of mass,
see after Eq.(2.16)]. We have that ~q+ ≈ 0 implies λř (τ ) = 0: in this way the internal 3-center
12
(~
q )µ
(~
q )µ
of mass is located in a unique centroid xs + (τ ) = z µ (τ, ~σ = 0) [ẋs +
µ
= x̃˙ s = uµ (ps )].
Note that the constant xµs (0) [and, therefore, also x̃µs (0)] is arbitrary, reflecting the arbitrariness in the absolute location of the origin of the internal coordinates on each hyperplane
in Minkowski space-time. The centroid xµs (τ ) corresponds to the unique special relativistic
center-of-mass-type world-line for isolated systems of Refs. [31, 32], which unifies previous
proposals of Synge, Møller and Pryce.
The only remaining canonical variables describing the Wigner hyperplane in the final
Dirac brackets are a non-covariant canonical coordinate x̃µs (τ ) [35] and pµs . The point with
coordinates x̃µs (τ ) is the decoupled canonical external 4-center of mass of the isolated system,
which can be interpreted as a decoupled observer with his parametrized clock (point particle
µ
clock). Its velocity x̃˙ s (τ ) is parallel to pµs , so that it has no classical zitterbewegung.
The relation between xµs (τ ) and x̃µs (τ ) (~σ̃ is its 3-location on the Wigner hyperplane) is
[2, 3]
x̃µs (τ ) = (x̃os (τ ); ~x̃s (τ )) = z µ (τ, ~σ̃) = xµs (τ ) −
h
µ i
1
νµ
oµ
oν psν ps
p
S
+
ǫ
(S
−
S
) , (2.8)
sν
s
s
s
s
ǫs (pos + ǫs )
ǫ2s
After the separation of the relativistic canonical non-covariant external 4-center of mass
the N particles are described on the Wigner hyperplane by the 6N Wigner spin-1
P
κi ≈ 0.
3-vectors ~ηi (τ ), ~κi (τ ) restricted by the rest-frame condition ~κ+ = N
i=1 ~
x̃µs (τ ),
The various spin tensors and vectors are [3]
Jsµν = xµs pνs − xνs pµs + Ssµν = x̃µs pνs − x̃νs pµs + S̃sµν ,
Ssµν = [uµ (ps )ǫν (u(ps )) − uν (ps )ǫµ (u(ps ))]S̄sτ r + ǫµ (u(ps ))ǫν (u(ps ))S̄srs ≡
N
q
h
iX
≡ ǫµr (u(ps ))uν (ps ) − ǫν (u(ps ))uµ (ps )
ηir m2i c2 + ~κ2i +
i=1
N
h
iX
µ
ν
ν
µ
+ ǫr (u(ps ))ǫs (u(ps )) − ǫr (u(ps ))ǫr (u(ps ))
ηir κsi ,
i=1
B
µν
S̄sAB = ǫA
µ (u(ps ))ǫν (u(ps ))Ss ,
S̄srs
≡
N
X
(ηir κsi
i=1
S̃sµν = Ssµν + p
−
1
p
+ ǫp2s )
ǫp2s (pos
S̃sij = δ ir δ js S̄srs ,
~ ≡ S̄
~=
S̄
N
X
i+1
ηis κri ),
~ηi × ~κi ≈
S̄sτ r
i=1
ηir
i=1
q
m2i c2 + ~κ2i ,
i
h
p
psβ (Ssβµ pνs − Ssβν pµs ) + p2s (Ssoµ pνs − Ssoν pµs ) ,
S̃soi = −
N
X
≡−
N
X
δ ir S̄srs pss
p ,
pos + ǫp2s
~ηi × ~κi − ~η+ × ~κ+ =
13
N
−1
X
a=1
ρ~a × ~πa .
(2.9)
µ ν
ν µ
µν
Note that while Lµν
s = xs ps − xs ps and Ss are not constants of the motion due to the
µ ν
ν µ
µν
classical zitterbewung (when ~λ(τ ) 6= 0), both L̃µν
s = x̃s ps − x̃s ps and S̃s are conserved.
The canonical variables x̃µs , pµs for the external 4-center of mass, can be replaced by the
canonical pairs[36]
ps · xs
ps · x̃s
=
,
ǫs
ǫs
p~s
~zs = ǫs (~x̃s − o x̃os ),
ps
Ts =
p
ǫs = ± ǫp2s ,
~ks = p~s ,
ǫs
(2.10)
which make explicit the interpretation of it as a point particle clock. The inverse transformation is
~ks · ~zs
= 1 + ~ks2 (Ts +
),
ǫs
p~s = ǫs~ks .
x̃os
q
~
~x̃s = ~zs + (Ts + ks · ~zs )~ks ,
ǫs
ǫs
pos
q
= ǫs 1 + ~ks2 ,
(2.11)
This non-point canonical transformation can be summarized as [ǫs − Msys ≈ 0, ~κ+ =
PN
κi ≈ 0]
i=1 ~
x̃µs ~ηi
pµs ~κi
−→
ǫs ~zs ~ηi
Ts ~ks ~κi
.
(2.12)
After the addition of the gauge-fixing Ts − τ ≈ 0 [37], the invariant mass Msys of the
system, which is also the internal energy of the isolated system, replaces the non-relativistic
Hamiltonian Hrel for the relative degrees of freedom: this reminds of the frozen HamiltonJacobi theory, in which the time evolution can be re-introduced by using the energy generator
of the Poincaré group as Hamiltonian [38].
After the gauge fixings Ts − τ ≈ 0 [implying λ(τ ) = −1], the final Hamiltonian, the
embedding of the Wigner hyperplane into Minkowski space-time and the Hamilton equations
become
14
HD = Msys − ~λ(τ ) · ~κ+ ,
z µ (τ, ~σ ) = xµs (τ ) + ǫµr (u(ps ))σ r =
=
xµs (0)
µ
+ u (ps )τ +
ǫµr (u(ps ))
Z
r
σ +
o
τ
dτ1 λr (τ1 ) ,
with
ẋµs (τ ) = uµ (ps ) + ǫµr (u(ps ))λr (τ ),
~κi (τ )
◦
~η˙ i (τ ) = p 2
− ~λ(τ ),
2
(τ
)
mi + ~κl
⇒
◦
~κ˙ i (τ ) = 0.
◦
~κi (τ )=mi q
~η˙ i (τ ) + ~λ(τ )
2
1 − [~η˙ i (τ ) + ~λ(τ )]2
,
(2.13)
The particles’ world-lines in Minkowski space-time and the associated momenta are
xµi (τ ) = z µ (τ, ~ηi (τ )) = xµs (τ ) + ǫµr (u(ps ))ηir (τ ),
q
µ
pi (τ ) = m2i + ~κ2i (τ )uµ (ps ) + ǫµr (u(ps ))κir (τ ) ⇒ p2i = m2i .
(2.14)
The external rest-frame instant form realization of the Poincaré generators [40] with non~ 2 ≈ −M 2 S̄
~ 2 , is obtained from Eq.(2.9):
fixed invariants p2 = ǫ2 ≈ M 2 , −p2 S̄
s
s
sys
s
s
sys
pµs ,
Jsµν = x̃µs pνs − x̃νs pµs + S̃sµν ,
pos
=
p
ǫ2s
+
~p2s
q
q
q
2
2
2
~
= ǫs 1 + ks ≈ Msys + p~s = Msys 1 + ~ks2 ,
p~s = ǫs~ks ≈ Msys~ks ,
Jsij
Ksi
=
x̃is pjs
−
x̃js pis
ir js
+δ δ
N
X
i=1
(ηir κsi − ηis κri ) = zsi ksj − zsj ksi + δ ir δ js ǫrsu S̄su ,
N
X
1
p
+ −
(ηir κsi − ηis κri ) =
δ ir pss
=
=
−
ǫs + ǫ2s + p~2s
i=1
q
q
ir s rsu u
δ ks ǫ S̄s
δ ir pss ǫrsu S̄su
2 +~
q
p 2
≈ x̃os pis − x̃is Msys
(2.15)
.
p2s −
= − 1 + ~ks2 zsi −
2
M
+
~
p
M
sys +
2
~
sys
s
1 + 1 + ks
Jsoi
x̃os pis
x̃is
p
ǫ2s
p~2s
15
On the other hand the internal realization of the Poincaré algebra is built inside the
Wigner hyperplane by using the expression of S̄sAB given by Eq.(2.9) [41]
Msys = HM =
N q
X
m2i + ~κ2i ,
i=1
~κ+ =
J~ =
N
X
~κi (≈ 0),
i=1
N
X
i=1
~ =−
K
~ηi × ~κi ,
1
J r = S̄ r = ǫruv S̄ uv ≡ S̄sr ,
2
N q
X
~ +,
m2i + ~κ2i ~ηi = −Msys R
K r = J or = S̄sτ r ,
i=1
2
2
Π = Msys
− ~κ2+ ≈ Msys
> 0,
2
~ 2 ≈ −ǫM 2 S̄
~2
W 2 = −ǫ(Msys
− ~κ2+ )S̄
s
sys s .
(2.16)
The constraints ǫs − Msys ≈ 0, ~κ+ ≈ 0 have the following meaning:
i) the constraint ǫs − Msys ≈ 0 is the bridge connecting the external and internal realizations [42];
~ ≈ 0 (i.e. R
~ + ≈ ~q+ ≈ ~y+ ≈ 0) [43], imply
ii)the constraints ~κ+ ≈ 0, together with K
an unfaithful internal realization, in which the only non-zero generators are the conserved
energy and spin of an isolated system.
The determination of ~q+ for the N particle system has been carried out by the group
theoretical methods of Ref.[44] in Section III of Ref.[2]. Given a realization of the ten
Poincaré generators on the phase space, one can build three 3-position variables in terms
of them only. For N free scalar relativistic particles on the Wigner hyperplane with
p~sys = ~κ+ ≈ 0 within the internal realization (2.16) they are:
i) a canonical internal 3-center of mass (or 3-center of spin) ~q+ ;
~ +;
ii) a non-canonical internal Møller 3-center of energy R
iii) a non-canonical internal Fokker-Pryce 3-center of inertia ~y+ .
~ + ≈ ~y+ .
It can be shown[2] that, due to ~κ+ ≈ 0, they all coincide: ~q+ ≈ R
~ + ≈ ~y+ ≈ 0 imply ~λ(τ ) ≈ 0 and force the
Therefore the gauge fixings χ
~ (τ ) = ~q+ ≈ R
three internal collective variables to coincide with the origin of the coordinates, which now
becomes
q+ )µ
x(~
(Ts ) = xµs (0) + uµ (ps )Ts .
s
(2.17)
~ + ≈ ~y+ ≈ 0
As shown in Section IV, the addition of the gauge fixings χ
~ (τ ) = ~q+ ≈ R
also implies that the Dixon center of mass of an extended object[27] and the Pirani[45] and
(~
q (µ
Tulczyjew[34, 46, 47] centroids [48] all simultaneously coincide with the centroid xs + (τ ).
16
~ s,
The external realization (2.15) allows to build the analogous external 3-variables ~qs , R
µ
Y~s . Eq.(4.4) of Ref.[2] shows the construction of the associated external 4-variables x̃s , Ysµ ,
Rsµ and their locations ~σ̃, ~σY , ~σR on the Wigner hyperplane. It appears that the external
Fokker-Pryce non-canonical covariant 4-center of inertia Ysµ coincides with the centroid
(2.17).
In Ref.[2] there is the definition of the following sequence of canonical transformations
~ηi
~κi
−→
~η+ ρ~a
~κ+ ~πa
−→
~q+ ρ~qa
~κ+ ~πqa
, a = 1, .., N − 1,
(2.18)
leading to the canonical separation of the internal 3-center of mass (~q+ , ~κ+ ) from the internal
relative variables ρ~qa , ~πqa . Since the rest-frame condition ~κ+ ≈ 0 implies [2] ρ~qa ≈ ρ~a ,
~πqa ≈ ~πa , in the gauge ~q+ ≈ 0 and in terms of the associated Dirac brackets we get an
internal reduced phase space whose canonical basis is ρ~qa ≡ ρ~a , ~πqa ≡ ~πa , a = 1, .., N − 1.
The intermediate linear point canonical transformation in (2.18) is [actually P
this is a
family
of
canonical
transformations,
since
the
γ
’s
are
any
set
of
numbers
satisfying
ai
i γai =
P
P
1
0, a γai γaj = δij − N , i γai γbi = δab ]
1 X
γai ρ~a ,
~ηi = ~η+ + √
N a
√ X
1
~κi =
~κ+ + N
γai ~πa ,
N
a
1 X
~ηi ,
N i
X
=
~κi ,
~η+ =
~κ+
i
ρ~a =
√
N
X
γai ~ηi ,
i
1 X
~πa = √
γai ~κi .
N i
(2.19)
The second canonical transformation has been defined in Section V of Ref.[2] by using
a singular Gartenhaus-Schwartz transformation (see Appendix C), but it was not written
explicitly for ~κ+ 6= 0. By using Eqs. (C14) and (C15) we get the following results:
1) For N = 2 (γ1 1 = −γ1 2 =
√1 )
2
we have
17
1
ρ~,
2
1
~η2 = ~η+ − ρ~,
2
~η1 = ~η+ +
~η+ =
1
~κ+ + ~π ,
2
1
~κ2 = ~κ+ − ~π ,
2
~κ1 =
1
(~η1 + ~η2 ),
2
ρ~ = ~η1 − ~η2 ,
~κ+ = ~κ1 + ~κ2 ,
~π =
1
(~κ1 − ~κ2 ),
2
~ = ~q+ × ~κ+ + S
~q ,
J~ = ~η1 × ~κ1 + ~η2 × ~κ2 = ~η+ × ~κ+ + S
~+ =
R
p
~ = ρ~ × ~π ,
S
~q = ρ~q × ~πq ,
S
p
p
p
m21 + ~κ21 ~η1 + m22 + ~κ22 ~η2
1 m21 + ~κ21 − m22 + ~κ22
p
p
p
= ~η+ + p 2
ρ~,
2 m1 + ~κ21 + m22 + ~κ22
m21 + ~κ21 + m22 + ~κ22
~+ +
~q+ = R
~q × ~κ+
S
qp
+ p
.
p
p
p
p
2
2
2
2
2
2
2
2
2
2
2
2 2
2
( m1 + ~κ1 + m2 + ~κ2 ) ( m1 + ~κ1 + m2 + ~κ2 + ( m1 + ~κ1 + m2 + ~κ2 ) − ~κ+ )
(2.20)
Then after some straightforward algebra we get (note that in Ref.[2] we used the notation
M(f ree) = Msys and M2(f ree) = Π)
18
q
2
2
~πq · ~κ+
1q 2
i+1 m1 − m2
2
2
2
,
+ (−)i+1
mi + ~κi =
M(f ree) + ~κ+ 1 + (−)
2
2
M(f ree)
M(f ree)
q
q
q
q
q
def
2
2
M(f ree) = m1 + ~κ1 + m22 + ~κ22 = M2(f ree) + ~κ2+ ≈ M(f ree) = m21 + ~πq2 + m22 + ~πq2 ,
q
q
2 ~πq · ~κ+ m21 − m22 q 2
def
2
2
2
2
+
m1 + ~κ1 − m2 + ~κ2 =
M(f ree) + ~κ2+ = E,
2
M(f ree)
M(f ree)
h
q
M(f ree) ~κ
q +
~πq · ~κ+ 1 − ( M2(f ree) + ~κ2+ − M(f ree) )
+
2
2
2
~
κ
+
M(f ree) M(f ree) + ~κ+
i
q
def
+(m21 − m22 ) M2(f ree) + ~κ2+ = ~πq + F ~κ+ ,
~π = ~πq +
A
~κ+ · ρ~q ~πq
def
q
= ρ~q + C ~πq ,
2
B M(f ree) M2
+
~
κ
+
(f ree)
p
p
m2 + ~κ21
m2 + ~κ22
A ~πq · ~κ+
q
+q 2
, B =1+
A= q 1
,
2
2
2
2
2
2
M
M
+
~
κ
m2 + ~πq
m1 + ~πq
(f ree)
+
(f ree)
q
1 q
~κ+
2
2
~πq = ~π − q
( m1 + ~κ1 − m22 + ~κ22 ) −
2
2
2
M(f ree) − ~κ+
q
~κ+ · ~π
def
2
2
−
~
κ
]
= ~π − D ~κ+ ,
− 2 [M(f ree) − M(f
+
ree)
~κ+
A ~κ+ · ρ~
q
~πq ,
ρ~q = ρ~ +
2
2
−
~
κ
M(f ree) M(f
+
ree)
ρ~ = ρ~q −
~q = S
~ − D ρ~ × ~κ+ ,
S
~+ + G S
~ q × ~κ+ ,
~q+ = R
1
1
q
q
=q
,
2
2
2
2
2
2
M(f ree) (M(f ree) + M(f
−
~
κ
)
M
+
~
κ
(
M
+
~
κ
+
M
)
(f
ree)
+
+
+
ree)
(f ree)
(f ree)
h
i
E
~q × ~κ+ ,
~η+ = ~q+ − q
ρ~q + C ~πq − G S
2
2
2 M(f ree) + ~κ+
G=
1
E
~q × ~κ+ ,
ρ~q + C ~πq − G S
(−)i+1 − q
2
M2(f ree) + ~κ2+
1
i+1
+ (−) F ~κ+ + (−)i+1 ~πq .
~κi =
2
~ηi = ~q+ +
19
(2.21)
2) For N > 2 the results concerning the coordinates (and also ~q+ ) are much more involved
due to the complexity of Eqs. (C15) so that we give only the following results for the
momenta
M(f ree)
q
Xq
Xs
X
2
2
2
2
=
mi + ~κi = M(f ree) + ~κ+ ≈ M(f ree) =
m2i + N
γai γbi ~πqa · ~πqb ,
i
q
m2i + ~κ2i =
i
1
√
N
ab
X
γai ~κ+ · ~πqa +
M(f ree)
a
s
q
X
γai γbi ~πqa · ~πqb M2(f ree) + ~κ2+ ,
+ m2i + N
ab
h
M(f ree)
~κ+ √ X
~κ+ · ~πqa
γai ~πqa + (
− 1)
+ N
~κ+ +
N
M(f ree)
~κ2+
a
s
i
X
X
~κ+
√
γai m2i + N
+
γai γbi ~πqa · ~πqb .
M(f ree) N i
ab
~κi =
20
(2.22)
III.
THE ENERGY-MOMENTUM TENSOR ON THE WIGNER HYPERPLANE.
A. The Euler-Lagrange Equations and the Energy-Momentum Tensor of
Parametrized Minkowski Theories.
◦
The Euler-Lagrange equations associated with the Lagrangian (2.1) are (the symbol ’=’
means evaluated on the solutions of the equations of motion)
∂L
∂L √
◦
σ ) = ηµν ∂A [ gT AB zBν ](τ, ~σ ) = 0,
µ (τ, ~
µ
∂z
∂zA
∂L
∂L
− ∂τ
=
∂~ηi
∂ ~η˙
− ∂A
i
1T
∂gAB
gτ r + grs η̇is
◦
|~σ=~ηi = 0,
− ∂τ p
√ ]|~σ=~ηi
u
u
v
2 g
∂~ηi
gτ τ + 2gτ u η̇i + guv η̇i η̇i
AB
−[
(3.1)
where we have introduced the energy-momentum tensor [here η̇iA (τ ) = (1; ~η˙ i (τ ))]
N
T
AB
X
2 δS
mi η̇iA (τ )η̇iB (τ )
(τ, ~σ). (3.2)
(τ, ~σ ) = −[ √
](τ, ~σ) = −
δ 3 (~σ − ~ηi (τ )) p
g δgAB
gτ τ + 2gτ u η̇iu + guv η̇iu η̇iv
i=1
Because of the delta functions, the Euler-Lagrange equations for the fields z µ (τ, ~σ ) are
◦
trivial (0 = 0) everywhere except at the positions of the particles. They may be rewritten in
a form valid for every isolated system
1
√
◦
∂A T AB zBµ = − √ ∂A [ gzBµ ]T AB .
g
(3.3)
√
When ∂A [ gzBµ ] = 0, as it happens on the Wigner hyper-planes in the gauge ~q+ ≈ 0 and
◦
Ts − τ ≈ 0, we get the conservation of the energy-momentum tensor T AB , i.e. ∂A T AB = 0.
Otherwise, there is a compensation coming from the dynamics of the surface.
On the Wigner hyperplane, where we have
xµi (τ ) = z µ (τ, ~ηi (τ )) = xµs (τ ) + ǫµr (u(ps ))ηir (τ ),
ẋµi (τ ) = zτµ (τ, ~ηi (τ )) + zrµ (τ, ~ηi (τ ))η̇ir (τ ) = ẋµs (τ ) + ǫµr (u(ps ))η̇ir (τ ),
ẋ2i (τ ) = gτ τ (τ, ~ηi (τ )) + 2gτ r (τ, ~ηi (τ ))η̇ir (τ ) + grs (τ, ~ηi (τ ))η̇ir (τ )η̇is (τ ) =
2
ẋ2 (τ ) + 2ẋsµ (τ )ǫµ (u(ps ))η̇ r (τ ) − ~η˙ (τ ),
s
r
i
i
q
= m2i − γ rs (τ, ~ηi (τ ))κir (τ )κis (τ )lµ (τ, ~ηi (τ )) − κir (τ )γ rs (τ, ~ηi (τ ))zsµ (τ, ~ηi (τ )) =
q
= m2i + ~κ2i (τ )uµ (ps ) + ǫµr (u(ps ))κri (τ ) ⇒ p2i = m2i ,
Z
N
X
µ
pµi (τ ),
ps =
d3 σρµ (τ, ~σ ) ≈
(3.4)
pµi (τ )
i=1
21
the energy-momentum tensor T AB (τ, ~σ ) takes the form
ττ
T (τ, ~σ) = −
i=1
T (τ, ~σ) = −
N
X
T rs (τ, ~σ) = −
N
X
τr
B.
N
X
i=1
i=1
mi
δ 3 (~σ − ~ηi (τ )) q
ẋ2s (τ )
δ 3 (~σ − ~ηi (τ )) q
ẋ2s (τ )
δ 3 (~σ − ~ηi (τ )) q
ẋ2s (τ )
+
2ẋsµ (τ )ǫµr (u(ps ))
2
− ~η˙ i (τ )
mi η̇ir (τ )
+
2ẋsµ (τ )ǫµr (u(ps ))
2
− ~η˙ i (τ )
mi η̇ir (τ )η̇is (τ )
+
2ẋsµ (τ )ǫµr (u(ps ))
2
− ~η˙ i (τ )
,
,
.
(3.5)
The Energy-Momentum Tensor in the Standard Lorentz-Covariant Theory.
With the position xµ = z µ (τ, ~σ ), the same form is obtained from the energy momentum
tensor of the
manifestly Lorentz covariant theory with Lagrangian SS =
R
R p
P standard
dτ LS (τ ) = − N
m
dτ
ẋ2i (τ ), restricted to positive energies[51]
i
i=1
2 δS S
|x=z(τ,~σ) =
T µν (z(τ, ~σ )) = − √
g δgµν
Z
N
X
ẋµ (τ1 )ẋν (τ1 ) 4 δ xi (τ1 ) − z(τ, ~σ ) =
=
mi dτ1 ip 2 i
ẋi (τ1 )
i=1
= ǫµA (u(ps ))ǫνB (u(ps ))T AB (τ, ~σ ).
1) On arbitrary space-like hyper-surfaces we have
22
(3.6)
N
X
ẋµi (τ1 )ẋνi (τ1 ) 4 δ xi (τ1 ) − z(τ, ~σ ) =
T (z(τ, ~σ )) =
mi dτ1 p 2
ẋi (τ1 )
i=1
Z
N
X
dτ1
4
=
mi p 2
δ z(τ1 , ~ηi (τ1 )) − z(τ, ~σ )
ẋi (τ1 )
i=1
µν
Z
[zτµ (τ1 , ~ηi (τ1 )) + zrµ (τ1 , ~ηi (τ1 ))η̇ir (τ1 )]
[zτν (τ1 , ~ηi (τ1 )) + zrν (τ1 , ~ηi (τ1 ))η̇ir (τ1 )] =
Z
N
X
dτ1
δ 4 z(τ1 , ~ηi (τ1 )) − z(τ, ~σ )
=
mi p 2
ẋi (τ1 )
hi=1
zτµ (τ1 , ~ηi (τ1 )) zτν (τ1 , ~ηi (τ1 )) +
µ
ν
ν
µ
+ zτ (τ1 , ~ηi (τ1 ))zr (τ1 , ~ηi (τ1 )) + zτ (τ1 , ~ηi (τ1 ))zr (τ1 , ~ηi (τ1 )) η̇ir (τ1 ) +
i
+ zrµ (τ1 , ~ηi (τ1 ))zsν (τ1 , ~ηi (τ1 ))η̇ir (τ1 )η̇is (τ1 ) =
=
N
X
mi
p
δ 3 ~σ − ~ηi (τ ) (det |zAµ (τ, ~σ)|)−1
ẋ2i (τ )
hi=1
µ
ν
zτ (τ, ~ηi (τ1 )) zτ (τ, ~ηi (τ )) + zτµ (τ, ~ηi (τ ))zrν (τ, ~ηi (τ )) +
i
+ zτν (τ, ~ηi (τ ))zrµ (τ, ~ηi (τ )) η̇ir (τ ) + zrµ (τ, ~ηi (τ ))zsν (τ, ~ηi (τ ))η̇ir (τ )η̇is (τ ) =
since det |zAµ | =
N
X
mi
3
p
~
σ
−
~
η
(τ
)
δ
i
ẋ2i (τ ) g(τ, ~σ)
i=1
h
µ
ν
zτ (τ, ~ηi (τ1 )) zτ (τ, ~ηi (τ )) + zτµ (τ, ~ηi (τ ))zrν (τ, ~ηi (τ )) +
i
+ zτν (τ, ~ηi (τ ))zrµ (τ, ~ηi (τ )) η̇ir (τ ) + zrµ (τ, ~ηi (τ ))zsν (τ, ~ηi (τ ))η̇ir (τ )η̇is (τ ) , (3.7)
=
√
g=
p
p
γ(gτ τ − γ rs gτ r gτ s ), γ = |det grs |.
2) On arbitrary space-like hyper-planes, where it holds[3]
23
z µ (τ, ~σ ) = xµs (τ ) + bµu (τ )σ u ,
xµi (τ ) = xµs (τ ) + bµu (τ )ηiu (τ ),
√
zτµ (τ, ~σ ) = ẋµs (τ ) + ḃµu (τ )σ u = lµ / g τ τ − gτ r zrµ ,
zrµ (τ, ~σ ) = bµr (τ ),
gτ τ = [ẋµs + ḃµr σ r ]2 ,
gτ r = brµ [ẋµs + ḃµs σ s ],
grs = −δrs ,
γ rs = −δ rs ,
γ = 1,
X
2
g = gτ τ +
gτ r ,
g
ττ
=
r
µ
1/[lµ (ẋs
+ ḃµu σ u )]2 ,
g τ r = g τ τ gτ r = brµ (ẋµs + ḃµu σ u )/[lµ (ẋµs + ḃµu σ u )]2 ,
g rs = −δ rs + g τ τ gτ r gτ s = −δ rs + brµ (ẋµs + ḃµu σ u )brν (ẋνs + ḃνv σ v )/[lµ (ẋµs + ḃµu σ u )]2(3.8)
,
we have
T µν [xβs (τ ) + bβr (τ )σ r ] =
N
X
i=1
mi δ 3 ~σ − ~ηi (τ )
q
p
2
g(τ, ~σ) gτ τ (τ, ~σ) + 2gτ r (τ, ~σ )η̇i (τ ) − ~η˙ i (τ )
h
(ẋµs (τ ) + ḃµr (τ )ηir (τ ))(ẋνs (τ ) + ḃνs (τ )ηis (τ )) +
+ (ẋµs (τ ) + ḃµs (τ )ηis (τ ))bνr (τ ) + (ẋνs (τ ) + ḃνs (τ )ηis (τ ))bµr (τ ) η̇ir (τ ) +
i
+ bµr (τ )bνs (τ )η̇ir (τ )η̇is (τ ) .
(3.9)
3) Finally, on Wigner hyper-planes, where it holds[3]
z µ (τ, ~σ ) = xµs (τ ) + ǫµu (u(ps ))σ u ,
zrµ
g
ττ
g
g rs
=
=
=
=
xµi (τ ) = xµs (τ ) + ǫµu (u(ps ))ηiu (τ ),
ǫµr (u(ps ),
lµ = uµ (ps ),
zτµ = ẋµs (τ ),
[ẋs (τ ) · u(ps )]2 ,
gτ τ = ẋ2s ,
gτ r = ẋsµ ǫµr (u(ps )),
1/[ẋsµ uµ (ps )]2 ,
g τ r = ẋsµ ǫµr (u(ps ))/[ẋsµ uµ (ps )]2 ,
−δ rs + ẋsµ ǫµr (u(ps ))ẋsν ǫνs (u(ps ))/[ẋsµ uµ (ps )]2 ,
ds2 = ẋ2s (τ )dτ 2 + 2ẋs (τ ) · ǫr (u(ps ))dτ dσ r − d~σ 2 ,
we have
24
grs = −δrs ,
(3.10)
N
T
µν
[xβs (τ )
+
ǫβr (u(ps ))σ r ]
X
mi
1
q
=
2
ẋs (τ ) · u(ps ) i=1
ẋ2s (τ ) + 2ẋsβ (τ )ǫβr (u(ps ))η̇ir (τ ) − ~η˙ i (τ )
h
δ 3 ~σ − ~ηi (τ ) ẋµs (τ )ẋνs (τ ) +
µ
ν
ν
µ
+ ẋs (τ )ǫr (u(ps )) + ẋs (τ )ǫr (u(ps )) η̇ir (τ ) +
i
+ ǫµr (u(ps ))ǫνs (u(ps ))η̇ir (τ )η̇is (τ ) =
N
X
1
mi
q
2
ẋs (τ ) · u(ps ) i=1
ẋ2s (τ ) + 2ẋsβ (τ )ǫβr (u(ps ))η̇ir (τ ) − ~η˙ i (τ )
h
ẋµs (τ )ẋνs (τ )δ 3 ~σ − ~ηi (τ ) +
+ ẋµs (τ )ǫνr (u(ps )) + ẋνs (τ )ǫµr (u(ps )) δ 3 ~σ − ~ηi (τ ) η̇ir (τ ) +
i
µ
ν
3
r
s
+ ǫr (u(ps ))ǫs (u(ps ))δ ~σ − ~ηi (τ ) η̇i (τ )η̇i (τ ) .
(3.11)
=
Since the volume element on the Wigner hyperplane is uµ (ps )d3 σ, we obtain the following
total 4-momentum and total mass of the N free particle system (Eqs.(2.7) are used)
PTµ
=
=
Z
d3 σT µν [xβs (τ ) + ǫβu (u(ps ))σ u ]uν (ps ) =
N
X
i=1
◦
h
mi
q
λ2 (τ ) − [~η˙ i (τ ) + ~λ(τ )]2
− λ(τ )uµ (ps ) + [η̇ir (τ ) + λr (τ )]ǫµr (u(ps ))
= |λ(τ )=−1
N hq
X
i=1
N
i X
m2i c2 + ~κ2i (τ )uµ (ps ) + κri (τ )ǫµr (u(ps )) =
pµi (τ ) = pµs ,
Msys = PTµ uµ (ps ) = −λ(τ )
◦
= |λ(τ )=−1
N q
X
i
i=1
N
X
i=1
mi
q
λ2 (τ ) − [~η˙ i (τ ) + ~λ(τ )]2
m2i + ~κ2i (τ ),
(3.12)
i=1
which turn out to be in the correct form only if λ(τ ) = −1. This shows that the agreement
with parametrized Minkowski theories on arbitrary space-like hyper-surfaces is obtained only
on Wigner hyper-planes in the gauge Ts − τ ≈ 0, which indeed implies λ(τ ) = −1.
25
C.
The Phase-Space Version of the Standard Energy-Momentum Tensor.
The same result may be obtained by first rewriting the standard energy-momentum
tensor
in phase space and then imposing the restriction pµi (τ ) =
p
µ
2
rs
−
κir (τ )γ rs (τ, ~ηi (τ ))zsµ (τ, ~ηi (τ ))
→
pmi − γ (τ, ~ηi (τ ))κir (τ )κis (τ )l (τ, ~ηi (τ ))
µ
r µ
2
2
mi + ~κi (τ )u (ps ) + κi ǫr (u(ps )) [52] [the last equality refers to the Wigner hyperplane, see Eq.(2.14)]. We have:
T
µν
Z
N
q
X
1
dτ1 ẋ2i (τ1 )pµi (τ1 )pνi (τ1 )δ 4 (xi (τ1 ) − z(τ, ~σ )) =
(z(τ, ~σ )) =
mi
i=1
p
N
X
ẋ2i (τ ) µ
p
pi (τ )pνi (τ )δ 3 (~σ − ~ηi (τ )),
=
g(τ, ~σ)
i=1 mi
on W igner ′s hyper − planes
⇓
T
µν
[xβs (τ )
+
ǫβu (u(ps ))σ u ]
=
=
+
+
=
+
+
q
2
ẋ2s + 2ẋsβ ǫβu (u(ps ))η̇iu − ~η˙ i
p
(τ )pµi (τ )pνi (τ )δ 3 (~σ − ~ηi (τ )) =
mi ẋs · u(ps )
i=1
q
N
2 + 2ẋ ǫβ (u(p ))η̇ u − ~
˙2
X
ẋ
η
u
sβ
s
s
i
i
p
δ 3 (~σ − ~ηi (τ ))
(τ )
mi ẋs · u(ps )
i=1
h
(m2i + ~κ2i (τ ))uµ (ps )uν (ps ) +
q
r
ki (τ ) m2i + ~κ2i (τ ) uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps )) +
i
κri (τ )κsi (τ )ǫµr (u(ps ))ǫνs (u(ps )) =
q
N
X
λ2 (τ ) − [~η˙ i (τ ) + ~λ(τ )]2
3
p
δ (~σ − ~ηi (τ ))
−λ(τ )
i=1
p
q
m2i + ~κ2i (τ ) h
m2i + ~κ2i (τ )uµ (ps )uν (ps ) +
m
i
r
ki (τ ) uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps )) +
i
κri (τ )κsi (τ ) µ
ν
p
ǫr (u(ps ))ǫs (u(ps )) .
(3.13)
m2i + ~κ2i (τ )
N
X
The total 4-momentum and total mass are then
26
PTµ
Msys
Z
d3 σT µν [xβs (τ ) + ǫβu (u(ps ))σ u ]uν (ps ) =
p
2
N
X
m2i + ~κ2i (τ )
ẋ2s + 2ẋsβ ǫβu (u(ps ))η̇iu − ~η˙ i
(τ )
=
ẋs · u(ps )
mi
i=1
q
[ m2i + ~κ2i (τ )uµ (ps ) + κri (τ )ǫµr (u(ps ))] =
q
p
N
X
λ2 (τ ) − [~η˙ i (τ ) + ~λ(τ )]2 m2i + ~κ2i (τ ) µ
p
=
pi (τ ),
m
−λ(τ
)
i
i=1
q
p
N
X λ2 (τ ) − [~η˙ i (τ ) + ~λ(τ )]2 m2 + ~κ2 (τ ) q
µ
i
i
= PT uµ (ps ) =
m2i + ~κ2i (τ ). (3.14)
−λ(τ
)
m
i
i=1
=
q
◦ p
Since in this case we have mi / 1 − [~η˙ i (τ ) + ~λ(τ )]2 = m2i c2 + ~κ2i (τ ), the equations above
show that the total 4-momentum evaluated from the energy-momentum tensor of the standard theory restricted to positive energy particles is consistent [53] with the description on
the Wigner hyper-planes with its gauge freedom λ(τ ), ~λ(τ ), provided one works with Dirac
brackets of the gauge Ts ≡ τ , where one has λ(τ ) = −1 and
ẋµs (τ ) = uµ (ps ) + ǫµr (u(ps ))λr (τ ),
xµs (τ )
=
xµs (0)
µ
+ τ u (ps ) +
Therefore, for every ~λ(τ ), we have
27
ǫµr (u(ps ))
Z
0
τ
dτ1 λr (τ1 ),
(3.15)
T µν [xβs (Ts ) + ǫβu (u(ps ))σ u ] = ǫµA (u(ps ))ǫνB (u(ps ))T AB (Ts , ~σ) =
N
hq
X
3
=
δ (~σ − ~ηi (Ts )) m2i + ~κ2i (Ts )uµ (ps )uν (ps ) +
i=1
+ kir (Ts ) uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps )) +
i
κr (Ts )κs (Ts ) µ
+ pi 2 i 2
ǫr (u(ps ))ǫνs (u(ps )) ,
mi + ~κi (Ts )
N
q
X
ττ
3
T (Ts , ~σ ) =
δ (~σ − ~ηi (Ts )) m2i + ~κ2i (Ts )),
T τ r (Ts , ~σ ) =
T rs (Ts , ~σ ) =
i=1
N
X
i=1
N
X
i=1
δ 3 (~σ − ~ηi (Ts ))κri (Ts ),
κr (Ts )κs (Ts )
,
δ 3 (~σ − ~ηi (Ts )) pi 2 i 2
mi + ~κi (Ts )
PTµ = pµs = Muµ (ps ) + ǫµr (u(ps ))κr+ ≈ Muµ (ps ),
N q
X
m2i c2 + ~κ2i (Ts ),
M =
i=1
T µν [xβs (Ts ) + ǫβu (u(ps ))σ u ]
uν (ps ) = ǫµA (u(ps ))T Aτ (Ts , ~σ) =
N
X
=
δ 3 (~σ − ~ηi (Ts ))
i=1
q
[ m2i c2 + ~κ2i (Ts )uµ (ps ) + κri (Ts )ǫµr (u(ps ))],
T µ µ [xβs (Ts ) + ǫβu (u(ps ))σ u ] = T A A (Ts , ~σ ) =
N
X
m2
.
=
δ 3 (~σ − ~ηi (Ts )) p 2 i 2
mi + ~κi (Ts )
i=1
28
(3.16)
IV. DIXON’S MULTIPOLES FOR FREE PARTICLES ON THE WIGNER HYPERPLANE.
In this Section we shall define the special relativistic Dixon multipoles on the Wigner
hyperplane with Ts − τ ≡ 0 for the N-body problem [54] [see Eqs.(2.13) with xµs (τ ) =
Rτ
Rτ
(~
q )µ
xµo + uµ (ps ) Ts + ǫµr (u(ps )) o dτ1 λr (τ1 ) = xs + (τ ) + o dτ1 λr (τ1 )]. By comparison, a list of
the non-relativistic multipoles for N free particles is given in Appendix A .
Consider an arbitrary time-like world-line w µ (τ ) = z µ (τ, ~η(τ )) = xµs (τ )+ǫµr (u(ps )) η r (τ ) =
Rτ
(~
q )µ
xs + (τ ) + ǫµr (u(ps )) η̃ r (τ ) [η̃ r (τ ) = η r (τ ) + o dτ1 λr (τ1 )] and evaluate the Dixon multipoles
[25] [61] on the Wigner hyper-planes in the natural gauge with respect to the given world-line.
A generic point will be parametrized by
z µ (τ, ~σ) = xµs (τ ) + ǫµr (u(ps )) σ r =
Z
h
r
(~
q+ )µ
µ
= xs
(τ ) + ǫr (u(ps )) σ +
µ
= w (τ ) +
ǫµr (u(ps ))[σ r
r
τ
o
i
dτ1 λr (τ1 ) =
def
− η (τ )] = w µ (τ ) + δz µ (τ, ~σ ),
(4.1)
so that δzµ (τ, ~σ )uµ (ps ) = 0.
Rτ
While for ~η̃(τ ) = 0 [~η (τ ) = o dτ1 λr (τ1 )] we get the multipoles relative to the centroid
(~
q )µ
~ + ≈ ~q+ ≈
xµs (τ ), for ~η (τ ) = 0 we get those relative to the centroid xs + (τ ). In the gauge R
~
~y+ ≈ 0, where λ(τ ) = 0, it follows that ~η(τ ) = ~η̃(τ ) = 0 identifies the barycentric multipoles
(~
q )µ
with respect to the centroid xs + (τ ), which now carries the internal 3-center of mass.
A.
Dixon’s Multipoles.
Lorentz covariant Dixon’s multipoles and their Wigner covariant counterparts on the
Wigner hyper-planes are then defined as
29
(µ ...µn )(µν)
tTµ1 ...µnµν (Ts , ~η) = tT 1
(Ts , ~η ) =
= ǫµr11 (u(ps ))...ǫµrnn (u(ps )) ǫµA (u(ps ))ǫνB (u(ps ))qTr1 ..rnAB (Ts , ~η) =
Z
q+ )β
d3 σδz µ1 (Ts , ~σ )...δz µn (Ts , ~σ )T µν [x(~
(Ts ) + ǫβu (u(ps ))σ u ] =
s
Z
µ
ν
= ǫA (u(ps ))ǫB (u(ps )) d3 σδz µ1 (Ts , ~σ )....δz µn (Ts , ~σ )T AB (Ts , ~σ ) =
=
= ǫµr11 (u(ps ))...ǫµrnn (u(ps ))
N
q
h
X
r1
rn
rn
µ
ν
r1
u (ps )u (ps )
[ηi (Ts ) − η (Ts )]...[ηi (Ts ) − η (Ts )] m2i + ~κ2i (Ts ) +
+
+
qTr1 ...rnAB (Ts , ~η)
i=1
µ
ν
ǫr (u(ps ))ǫs (u(ps ))
N
X
κr (Ts )κs (Ts )
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )] pi 2 i 2
mi + ~κi (Ts )
i=1
[uµ (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]
N
i
X
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )]κri (Ts ) ,
i=1
=
Z
=
δτA δτB
+
d3 σ [σ r1 − η r1 (Ts )]...[σ rn − η rn ] T AB (Ts , ~σ ) =
N
X
[ηir1 (Ts )
i=1
+ δuA δvB
N
X
−η
r1
(Ts )]...[ηirn (Ts )
q
− η (Ts )] m2i + ~κ2i (Ts ) +
rn
κu (Ts )κv (Ts )
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )] pi 2 i 2
+
m
+
~
κ
(T
)
s
i
i
i=1
+ (δτA δuB + δuA δτB )
N
X
i=1
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )]κri (Ts ),
tTµ1 ...µn µν (Ts , ~η) = 0,
uµ1 (ps )
def
tTµ1 ...µn µ µ (Ts , ~η) = ǫµr11 (u(ps ))...ǫµrnn (u(ps ))qTr1 ...rnA A (Ts , ~η) =
Z
q+ )µ
=
d3 σδz µ1 (τ, ~σ )...δz µn (τ, ~σ )T µ µ [x(~
(Ts ) + ǫµu (u(ps ))σ u ] =
s
= ǫµr11 (u(ps ))...ǫµrnn (u(ps ))
N
X
i=1
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )] p
t̃Tµ1 ...µn (Ts , ~η) = tTµ1 ...µn µν (Ts , ~η)uµ (ps )uν (ps ) =
...rn τ τ
= ǫµr11 (u(ps ))...ǫµrnn (u(ps ))qTr130
(Ts , ~η) =
µ1
µn
= ǫr1 (u(ps ))...ǫrn (u(ps ))
N
q
m2i
m2i + ~κ2i (Ts )
=
Related multipoles are
pµT1 ..µn µ (Ts , ~η ) = tTµ1 ...µn µν (Ts , ~η)uν (ps ) =
= ǫµr11 (u(ps ))...ǫµrnn (u(ps ))ǫµA (u(ps ))qTr1 ...rnAτ (Ts , ~η ) =
= ǫµr11 (u(ps ))....ǫµrnn (u(ps ))
N
X
i=1
[ηir1 (Ts ) − η r1 (Ts )]...[ηirn (Ts ) − η rn (Ts )]
hq
i
m2i + ~κ2i (τ )uµ (ps ) + kir (Ts )ǫµr (u(ps )) ,
uµ1 (ps )pTµ1 ...µn µ (Ts , ~η) = 0,
pTµ1 ...µn µ (Ts , ~η )uµ (ps ) = t̃Tµ1 ...µn (Ts , ~η),
n=0
⇒ pµT (Ts , ~η ) = ǫµA (u(ps ))qTAτ (Ts ) = PTµ ≈ pµs .
(4.3)
The inverse formulas, giving the multipolar expansion, are
T µν [w β (Ts ) + δz β (Ts , ~σ)] = T µν [xs(~q+ )β (Ts ) + ǫβr (u(ps )) σ r ] =
= ǫµA (u(ps ))ǫνB (u(ps ))T AB (Ts , ~σ) =
=
ǫµA (u(ps ))ǫνB (u(ps ))
∞
X
(−1)n
n=0
∂
n
∂σ r1 ...∂σ rn
=
∞
X
(−1)
δ 3 (~σ − ~η (Ts )) =
µ1 ...µn µν
(Ts , ~η)
n tT
n=0
qTr1 ...rnAB (Ts , ~η )
n!
n!
ǫr1 µ1 (u(ps ))...ǫrn µn (u(ps ))
∂n
δ 3 (~σ − ~η (Ts )).
∂σ r1 ...∂σ rn
(4.4)
Note however that, as pointed out by Dixon [25], the distributional equation (4.4) is valid
only if analytic test functions are used, defined on the support of the energy-momentum
tensor.
The quantities qTr1 ...rnτ τ (Ts , ~η), qTr1 ...rnrτ (Ts , ~η) = qTr1 ...rn τ r (Ts , ~η ), qTr1 ...rn uv (Ts , ~η ) are the
mass density, momentum density and stress tensor multipoles with respect to the worldline w µ (Ts ) (barycentric for ~η = ~η̃ = 0).
31
B.
Monopoles.
The monopoles correspond to n = 0 [64] and have the following expression [65] (see
Appendix C for the definition of →α→∞ )
qTAB (Ts , ~η)
=
δτA δτB M
+
δuA δvB
N
X
i=1
→α→∞ δτA δτB
+
N s
X
i=1
δuA δvB N
N
X
i=1
qTτ τ (Ts , ~η )
→c→∞
=
N
X
i=1
N
X
m2i + N
κu κv
p i i
+ (δτA δuB + δuA δτB )κu+ ≈
2
2
mi + ~κi
X
de
γdi γei~πqd · ~πqe +
P1..N −1
γai γbi~πqa · ~πqb
ab
p
,
P
m2i + N de γdi γei~πqd · ~πqe
1..N −1 N
1 X X Nγai γbi
~πqa · ~πqb + O(1/c) =
mi c +
2 ab i=1 mi
2
mi c2 + Hrel,nr + O(1/c),
i=1
qTrτ (Ts , ~η)
=
qTuv (Ts , ~η ) →c→∞
=
qTAA (Ts , ~η)
=
κr+ ≈ 0,
1..N
N
X−1 X
rest − f rame condition (also at the non − relativistic level),
Nγai γbi u v
πqa πqb + O(1/c) =
mi
i=1
ab
1..N
−1
X
−1 u v
kab
πqa πqb
ab
tµT µ (Ts , ~η )
=
N
X
i=1
→α→∞
→c→∞
=
N
X
i=1
N
X
i=1
N
X
i=1
p
+ O(1/c) =
1..N
X−1
kab ρ̇ua ρ̇vb + O(1/c),
ab
m2i
p
m2i + ~κ2i
m2i
P
m2i + N de γdi γei~πqd · ~πqe
1..N −1 N
1 X X Nγai γbi
mi c −
~πqa · ~πqb + O(1/c) =
2 ab i=1 mi
2
mi c2 − Hrel,nr + O(1/c).
(4.5)
where we have exploited Eqs. (5.10), (5.11) of Ref.[2] to obtain the expression in terms of
the internal relative variables.
Therefore, independently of the choice of the world-line w µp
(τ ), in the rest-frame instant
PN
ττ
form the mass monopole qT is the invariant mass M = i=1 m2i + ~κ2i , while the momentum monopole qTrτ vanishes and qTuv is the stress tensor monopole.
32
C.
Dipoles.
The mass, momentum and stress tensor dipoles correspond to n = 1 [66]
qTrAB (Ts , ~η )
N
hX
i
ηir κui κvi
r
uv
p
=
− η (Ts )] +
(Ts ) − η (Ts )qT (Ts , ~η ) +
m2i + ~κ2i
i=1
N
hX
i
A B
A B
r u
r
u
+ (δτ δu + δu δτ )
[ηi κi ](Ts ) − η (Ts )κ+ ,
r
δτA δτB M[R+
(Ts )
r
δuA δvB
i=1
qTrA A (Ts , ~η )
=
ǫrµ11 (u(ps ))tµT1 µ µ (Ts , ~η)
N
X
[ηir − η r ]m2i
p
=
(Ts ).
m2i + ~κ2i
i=1
(4.6)
Rτ
~
The vanishing of the mass dipole qTrτ τ implies ~η(τ )h = ~η̃(τ ) − o dτ1 ~λ(τ
i 1 ) = R+ and
R
τ
(~
q )µ
r
identifies the world-line w µ (τ ) = xs + (τ ) + ǫµr (u(ps )) R+
+ o dτ1 λr (τ1 ) . In the gauge
~ + ≈ ~q+ ≈ ~y+ ≈ 0, where ~λ(τ ) = 0, this is the world-line w µ (τ ) = xs(~q+ )µ (τ ) of the centroid
R
associated with the internal Møller 3-center of energy and, as a consequence of the rest
frame condition, also with the rest-frame internal 3-center of mass ~q+ . Therefore we have
the implications following from the vanishing of the barycentric (i.e. ~λ(τ ) = 0) mass dipole
h
i
r
qTrτ τ (Ts , ~η ) = ǫrµ11 (u(ps ))t̃µT1 (Ts , ~η) = M R+
(Ts ) − η r (Ts ) = 0,
~ + ≈ ~q+ ≈ ~y+ .
⇒ ~η (Ts ) = ~η̃(Ts ) = R
and ~λ(τ ) = 0,
(4.7)
~ + ≈ ~q+ ≈ ~y+ ≈ 0, Eq.(4.7) with ~η = ~η̃ = 0 implies the vanishing of the
In the gauge R
time derivative of the barycentric mass dipole: this identifies the center-of-mass momentumvelocity relation (or constitutive equation) for the system
dqTrτ τ (Ts , ~η) ◦ r
r
= κ+ − M Ṙ+
= 0.
dTs
(4.8)
The expression of the barycentric dipoles in terms of the internal relative variables, when
~ + ≈ ~q+ ≈ 0 and ~κ+ ≈ 0, is obtained by using the results of Appendix C.
~η = ~η̃ = R
33
~ +)
qTrτ τ (Ts , R
~ +)
qTruτ (Ts , R
=
=
→α→∞
→c→∞
~ +)
qTruv (Ts , R
=
=
→α→∞
→c→∞
=
~ +)
qTrA A (Ts , R
=
0,
N
X
ηir κui
i=1
N
−1
X
a=1
N
−1
X
−
=
N
−1
X
a=1
r
r
ρra πau + (η+
− R+
)κu+
u
ρrqa πqa
u
ρra πqa
=
a=1
1..N
X−1
kab ρra ρ̇ub ,
ab
N
X
κu κv
ηir i i −
Hi
i=1
N N
−1
X
X
r
R+
N
X
κu κv
i
i=1
i
Hi
=
N
X
κui κvi
κui κvi
r
r
+ (η+
− R+
)
Hi
Hi
i=1
i=1 a=1
q
P
N
−1 1..N
m2j + N de γdj γej ~πqd · ~πqe
X
√ X
×
c N
(γai − γaj ) p 2
P
mi + N de γdi γei~πqd · ~πqe
a=1
ij
P1..N −1
u v
γbi γciπqb
πqc
bc
ρrqa
PN p 2
P
m
+
N
γ
γ
~
π
·
~
π
qe
k
k=1
de dk ek qd
1..N
N
−1
1..N
−1
X X γai − γaj mj N X
u v
√
ρra
γbi γci πqb
πqc + O(1/c) =
m
m
N
i
ij a=1
bc
PN
N
1..N −1
i
1 X h X γai γbi γci
j=1 mj γaj
u v
√
πqc + O(1/c),
N
−
ρra πqb
m
m
N abc
i
i=1
1
√
N
N
X
γai ρra
(ηir
i=1
−
2
r mi
R+ )
Hi
=
q
P
m2j + N de γdj γej ~πqd · ~πqe
(γai − γaj ) p 2
×
P
mi + N de γdi γei~πqd · ~πqe
a=1
ij
mi
p
ρra
P
PN
2
mk + N de γdk γek ~πqd · ~πqe
k=1
→α→∞
N
−1 X
→c→∞
1..N N
−1
X
X
√
=
r u
R+
κ+
1
√
N
1..N
X
1..N −1
mj N X
γbi γci~πqb · ~πqc + O(1/c) =
m
m
i
ij a=1
bc
P
N
N
1..N
−1
i
√ X h X γai γbi γci
j=1 mj γaj
ρra~πqb · ~πqc + O(1/c).
(4.9)
−
N
N
m
m
i
i=1
N(γai − γaj )ρra
abc
The antisymmetric part of the related dipole pµT1 µ (Ts , ~η) identifies the spin tensor. Indeed,
the spin dipole [67] is
34
[µν]
ν]
rAτ
STµν (Ts )[~η ] = 2pT (Ts , ~η) = 2ǫ[µ
η) =
r (u(ps )) ǫA (u(ps )) qT (Ts , ~
h
i
r
r
µ
ν
ν
µ
= M[R+ (Ts ) − η (Ts )] ǫr (u(ps ))u (ps ) − ǫr (u(ps ))u (ps ) +
+
N
X
i=1
h
i
[ηir (Ts ) − η r (Ts )]κsi (Ts ) ǫµr (u(ps ))ǫνs (u(ps )) − ǫνr (u(ps ))ǫµs (u(ps )) ,
mνu(ps ) (Ts , ~η) = uµ (ps )STµν (Ts )[~η ] = −ǫνr (u(ps ))[S̄sτ r − Mη r (Ts )] =
r
= −ǫνr (u(ps ))M[R+
(Ts ) − η r (Ts )] = −ǫνr (u(ps ))qTrτ τ (Ts , ~η),
⇒
~ +,
⇒ ~η = R
uµ (ps )STµν (Ts )[~η ] = 0,
⇓
barycentric spin f or ~η = ~η̃ = 0, see Eq(2.9),
◦
STµν (Ts )[~η = 0] = Ssµν =
N
i
X
m η r (T ) h µ
◦
q i i s
ǫr (u(ps ))uν (ps ) − ǫνr (u(ps ))uµ (ps ) +
=
2
i=1
1 − ~η˙ i (Ts )
N
i
X
mi ηir (Ts )η̇is (Ts ) h µ
◦
q
+
ǫr (u(ps ))ǫνs (u(ps )) − ǫνr (u(ps ))ǫµs (u(ps )) =
2
i=1
1 − ~η˙ i (Ts )
q
h
i
X
◦
=
ηir (Ts ) m2i + ~κ2i ǫµr (u(ps ))uν (ps ) − ǫνr (u(ps ))uµ (ps ) +
i
rsu
+ ǫ
S̄su ǫµr (u(ps ))ǫνs (u(ps )).
(4.10)
This explains why mµu(ps ) (Ts , ~η ) is also called the mass dipole moment.
~ + ≈ ~q+ ≈ ~y+ ≈ 0 with P µ = M uµ (ps ) =
We find, therefore, that in the gauge R
T
(~
q )µ
(~
q )µ
M ẋs + (Ts ) the Møller and barycentric centroid xs + (Ts ) is simultaneously the Tulczyjew
centroid[34, 46, 47, 71] (defined by S µν Pν = 0) and also the Pirani centroid[45, 72] (defined
(~
q )
by S µν ẋsν+ = 0). In general, lacking a relation between 4-momentum and 4-velocity, they
are different centroids [73].
Note that non-covariant centroids could also be connected with the non-covariant external
center of mass x̃µs and the non-covariant external Møller center of energy.
D.
Quadrupoles and the Barycentric Tensor of Inertia.
The quadrupoles correspond to n = 2 [75]
35
qTr1 r2 AB (Ts , ~η)
=
δτA δτB
+ δuA δvB
+
N
q
X
r1
r2
r1
r2
[ηi (Ts ) − η (Ts )][ηi (Ts ) − η (Ts )] m2i + ~κ2i (Ts ) +
i=1
N
X
(δτA δuB
κu κv
[ηir1 (Ts ) − η r1 (Ts )][ηir2 (Ts ) − η r2 (Ts )] p i2 i 2 (Ts ) +
mi + ~κi
i=1
+
δuA δτB )
N
X
i=1
[ηir1 (Ts ) − η r1 (Ts )][ηir2 (Ts ) − η r2 (Ts )]κui (Ts ),
~+ =
and when the mass dipole vanishes, i.e. ~η = R
~ +)
qTr1 r2 τ τ (Ts , R
=
N
X
i=1
~ +) =
qTr1 r2 uτ (Ts , R
N
X
i=1
~ +) =
qTr1 r2 uv (Ts , R
=
N
X
P
i
p
m2i + ~κ2i /M, we get
r1
r2
(ηir1 − R+
)(ηir2 − R+
)
q
m2i + ~κ2i (Ts ),
r1
r2
(ηir1 − R+
)(ηir2 − R+
)κui ,
r1
r2
)(ηir2 − R+
)p
(ηir1 − R+
i=1
1..N 1..N
X
X−1
1
N
~ηi
ijk
κui κvi
m2i + ~κ2i (Ts )
(γai − γaj ).
ab
(4.11)
Following the non-relativistic pattern, Dixon starts from the mass quadrupole
~ +)
qTr1 r2 τ τ (Ts , R
=
N
X
i=1
[ηir1 ηir2
q
r1 r2
m2i + ~κ2i ](Ts ) − M R+
R+ ,
and defines the following barycentric tensor of inertia
36
(4.12)
r1 r2
Idixon
(Ts )
=
δ r1 r2
X
u
=
→α→∞
N
X
[(δ
~ + ) − q r1 r2 τ τ (Ts , R
~ +) =
qTuuτ τ (Ts , R
T
r1 r2
i=1
1..N
X−1 ~ + )2 − (η r1 − Rr1 )(η r2 − Rr2 ))
(~ηi − R
+
+
i
i
1..N
[~ρqa · ρ~qb δ
+
=
1..N
1..N
X−1 X
r1 r2
−
de
ρrqa1 ρrqb2 ]
mi mj mk
(γai − γaj )(γbi − γbk )~ρqa · ρ~qb δ r1 r2 − ρrqa1 ρrqb2 ] ×
2
Nm
ab
ijk
P −1
P −1
h
γcj γdj ~πqc · ~πqd
γciγdi~πqc · ~πqd N 1..N
1 N 1..N
cd
cd
+
+
× 1+
2
c
2mi
2m2j
P −1
P1..N −1
N
i
γch γdh~πqc · ~πqd γck γdk ~πqc · ~πqd
N cd
1 X N 1..N
2
cd
+
O(1/c
)
=
−
2m2k
m h=1
mh
1..N
X−1
ab
=
m2i + ~κ2i ](Ts )
1 X
(γai − γaj )(γbi − γbk )
N ijk
ab
p
P
m2i + N de γdi γei~πqd · ~πqe
p
P
P
( N
m2h + N de γdh γeh~πqd · ~πqe )2
h=1
s
s
X
X
2
2
mj + N
γdj γej ~πqd · ~πqe mk + N
γdk γek~πqd · ~πqe
de
→c→∞
q
kab [~ρqa · ρ~qb δ r1 r2 − ρrqa1 ρrqb2 ] + O(1/c) =
I r1 r2 [~qnr ] + O(1/c).
(4.13)
Note that in the non-relativistic limit we recover the tensor of inertia of Eqs.(A11).
On the other hand, Thorne’s definition of barycentric tensor of inertia[76] is
37
r1 r2
Ithorne
(Ts )
=
δ r1 r2
X
u
=
→α→∞
→c→∞
+
=
=
N
X
~ + ) − q r1 r2 A A (Ts , R
~ +) =
qTuuA A (Ts , R
T
r1
r2
~ + )2 − (η r1 − R+
m2i (δ r1 r2 (~η i − R
)(ηir2 − R+
))
i
p
(Ts )
2
2
mi + ~κi
i=1
1..N
1..N
X−1 c X
(γai − γaj )(γbi − γbk )
N ijk
ab
q
p
P
P
m2i m2j + N de γdj γej ~πqd · ~πqe m2k + N de γdk γek~πqd · ~πqe p
p
P
P
P
m2i + N de γdi γei~πqd · ~πqe ( N
m2h + N de γdh γeh~πqd · ~πqe )2
h=1
[~ρqa · ρ~qb δ r1 r2 − ρrqa1 ρrqb2 ]
1..N
1..N
X−1 X
mi mj mk
(γai − γaj )(γbi − γbk )~ρqa · ρ~qb δ r1 r2 − ρrqa1 ρrqb2 ] ×
2
Nm
ab
ijk
P −1
P −1
h
γci γdi~πqc · ~πqd N 1..N
1 N 1..N
γcj γdj ~πqc · ~πqd
cd
cd
−
× 1+
+
+
2
c
2mi
2m2j
P −1
P1..N −1
N
i
γch γdh~πqc · ~πqd N cd
γck γdk ~πqc · ~πqd
1 X N 1..N
2
cd
+ O(1/c ) =
−
2m2k
m h=1
mh
1..N
X−1
I
ab
r1 r2
kab [~ρqa · ρ~qb δ r1 r2 − ρrqa1 ρrqb2 ] + O(1/c) =
[~qnr ] + O(1/c).
(4.14)
In this case too we recover the tensor of inertia of Eq.(A11).
Note that the Dixon and Thorne barycentric tensors of inertia differ at the postNewtonian level
r1 r2
r1 r2
Idixon
(Ts ) − Ithorne
(Ts ) =
E.
1..N −1 1..N
1 X X mj mk
(γai − γaj )(γbi − γbk )
c ab ijk Nm2
i P1..N −1 γ γ ~π · ~π
h
ci di qc
qd
r1 r2
r1 r2 N
cd
ρ~qa · ρ~qb δ
− ρqa ρqb
+ O(1/c2).
mi
The Multipolar Expansion.
By further using the types of Dixon’s multipoles analyzed in Appendix D as well as the
◦
consequences of Hamilton equations for an isolated system (equivalent to ∂µ T µν = 0), it turns
out that the multipolar expansion (4.4) can be rearranged [see Eqs.(D11)] in the following
form
38
T µν [xs(~q+ )β (Ts ) + ǫβr (u(ps ))σ r ] = T µν [w β (Ts ) + ǫβr (u(ps ))(σ r − η r (Ts ))] =
ν)
= u(µ (ps )ǫA (u(ps ))[δτA M + δuA κu+ ]δ 3 (~σ − ~η (Ts )) +
+
+
1 ρ(µ
∂
ST (Ts )[~η]uν) (ps )ǫrρ (u(ps )) r δ 3 (~σ − ~η(Ts )) +
2
∂σ
∞
X
(−1)n
n=2
n!
ITµ1 ..µn µν (Ts , ~η)ǫrµ11 (u(ps ))..ǫrµnn (u(ps ))
∂n
δ 3 (~σ − ~η (Ts )),
∂σ r1 ..∂σ rn
(µ ..µ
(4.15)
|µ|µ )ν
JT 1 n−1 n (Ts ), with JTµ1 ..µn µνρσ (Ts )
where for n ≥ 2 and ~η = 0 ITµ1 ..µn µν (Ts ) = 4(n−1)
n+1
being the Dixon 22+n -pole inertial moment tensors given in Eqs.(D10). With this form of
the multipolar expansion, the quadrupole term (n = 2) has the form [see Eq.(D11)]
1
1 5 µ
u (ps ) uν (ps ) qFr1r2 τ τ (Ts , ~η) + [uµ (ps ) ǫνu (u(ps )) + uν (ps ) ǫµu (u(ps ))] qTr1r2 uτ (Ts , ~η) +
2 3
2
h
3 (r1 r2 u1 )u2
(r r u )u
qT
(Ts , ~η) + qT 1 2 2 1 (Ts , ~η) +
+ ǫµu1 (u(ps )) ǫνu2 (u(ps )) qTr1 r2 u1 u2 (Ts , ~η) −
2
i
(r1 r2 u1 u2 )
+ qT
(Ts , ~η ) .
Note that, as said in Appendix D, Eq.(4.15) holds only if the multipoles are evaluated
with respect to world-lines w µ (τ ) = z µ (τ, ~η (τ )) with ~η (τ ) = ~η = const., namely with respect
to one of the integral lines of the vector field zτµ (τ, ~σ ) ∂µ .
For an isolated system described by the multipoles appearing in Eq.(4.15) [this is not
◦
true for those in Eq.(4.4)] the equations ∂µ T µν = 0 [see Eqs.(D4) and (D7)] imply no more
than the following Papapetrou-Dixon-Souriau equations of motion [27, 68, 77, 78] for the
total momentum PTµ (Ts ) = ǫµA (u(ps ))qTAτ (Ts ) ≈ pµs and the spin tensor STµν (Ts )[~η = 0]
dPTµ(Ts ) ◦
= 0,
dTs
dSTµν (Ts )[~η = 0] ◦
[µ
ν]
= 2PT (Ts )uν] (ps ) = 2κu+ ǫ[µ
u (u(ps ))u (ps ) ≈ 0,
dTs
or
d~κ+ ◦
=0,
dTs
dM ◦
=0,
dTs
39
dSsµν ◦
=0.
dTs
(4.16)
V. DIXON’S MULTIPOLES AND RELEVANT CENTROIDS FOR CLOSED AND
OPEN SYSTEMS OF INTERACTING RELATIVISTIC PARTICLES.
In this Section we present new applications of the multipolar expansion to interacting
systems of particles and fields. We first deal with the case of an isolated system of positiveenergy relativistic particles with mutual action-at-a-distance interaction (see Section VIII
of Ref.[2] and Section VI of Ref.[5]); then we deal with the case of an open particle subsystem of an isolated system consisting of N charged positive-energy relativistic particles
(with Grassmann-valued electric charges to regularize the Coulomb self-energies) plus the
electro-magnetic field [5].
A. An isolated System of Positive-Energy Particles with Action-at-a-Distance Interactions.
As said in Section VIII of Ref.[2], in the rest-frame instant form the most general expression of the internal energy for an isolated system of N positive-energy particles with mutual
action-at-a-distance interactions is
Xq
~i )2 + V,
M=
m2i + Ui + (~κi − V
(5.1)
i
~i , V are functions of ~κi · ~κj , |~ηi − ~ηj |, ~κk · (~ηi − ~ηj ). On the other
where all the potentials Ui , V
hand, as shown at the end of Section II, in the free case we have
M(f ree) =
Xq
m2i + ~κ2i =
i
≈ M(f ree) =
Xs
i
q
M2(f ree) + ~κ2+ ≈
m2i + N
X
ab
γai γbi ~πqa · ~πqb .
(5.2)
~ + and ~q+ became interaction dependent, in the interacting case we
Since the 3-centers R
do not know the final canonical
basis ~q+ , ~κ+ , ρ~qa , ~πqa explicitly. For an isolated system,
p
2
however, we have M = M + ~κ2+ ≈ M with M independent of ~q+ ({M, ~κ+ } = 0 in the
internal Poincare’ algebra). This suggests that also in the interacting case the same result
should hold true. Indeed, by its definition, the Gartenhaus-Schwartz transformation gives
ρ~qa ≈ ρ~a , ~πqa ≈ ~πa also in presence of interactions, so that we get
M|~κ+ =0
q
X q
2
2
~
mi + Ui + (~κi − Vi ) + V |~κ+ =0 = M2 + ~κ2+ |~κ+ =
=
i
= M|~κ+=0 =
Xq
i
~ )2 + Ṽ ,
m2i + Ũi + (~κi − Ṽ
i
~ , Ṽ are now functions of ~π · ~π , ~π · ρ~ , ρ~ · ρ~ .
where the potentials Ũi , Ṽ
i
qa
qb
qa
qb
qa
qb
40
(5.3)
A relevant example of this type of isolated system has been studied in Ref.[5] starting
from the isolated system of N charged positive-energy particles (with Grassmann-valued
electric charges Qi = θ∗ θ, Q2i = 0, Qi Qj = Qj Qi 6= 0 for i 6= j) plus the electro-magnetic
field. After a Shanmugadhasan canonical transformation, this system can be expressed
only in terms of transverse Dirac observables corresponding to a radiation gauge for the
electro-magnetic field. The expression of the energy-momentum tensor in this gauge will be
shown in the next Subsection. In the semi-classical approximation of Ref.[5], the electromagnetic degrees of freedom are re-expressed in terms of the particle variables by means of
the Lienard-Wiechert solution in the framework of the rest-frame instant form. In this way
it has been possible to derive the exact semi-classical relativistic form of the action-at-adistance Darwin potential in the reduced phase space of the particles. Note that this form is
independent of the choice of the Green function in the Lienard-Wiechert solution. In Ref.[5]
the associated energy-momentum
tensor for the case
N = 2 [Eqs.(6.48)] is also given. The
p
P2 p 2
1 Q2
2
2
internal energy is M = M + ~κ+ ≈ M = i=1 mi + ~π 2 + Q4π
[1 + Ṽ (~π 2 , ~π · ρρ~ )] where
ρ
~
Ṽ is given in Eqs.(6.34), (6.35) [in Eqs. (6.36), (6.37) for m1 = m2 ]. The internal boost K
~ + = − K~ ≈ ~q+ ≈ ~y+ in the
[Eq.(6.46)] allows the determination of the 3-center of energy R
M
present interacting case.
~ + ≈ ~q+ allows to
The knowledge of the energy-momentum tensor T AB (τ, ~σ ) and of R
apply our formalism to find the barycentric multipoles of this interacting case. It turns out
~ + ≈ ~q+ ≈ ~y+ ≈ 0, all the formal properties studied in the previous
that, in the gauge R
Section (like the coincidence of all the relevant centroids) are reproduced in presence of
mutual action-at-a-distance interactions.
B. Open Subsystem of the Isolated System of N Positive-Energy Particles with
Grassmann-valued Electric Charge plus the Electromagnetic Field.
Let us now consider an open sub-system of the isolated system of N charged positiveenergy particles plus the electro-magnetic field in the radiation gauge. The energymomentum tensor and the Hamilton
on the Wigner hyper-plane of the isolated
P equations
◦
system are, respectively, [~κ+ = i ~κi ; = means evaluated on the equations of motion; to
avoid degenerations we assume that all the masses mi are different]
41
ττ
T (τ, ~σ ) =
N
X
i=1
3
δ (~σ − ~ηi (τ ))
q
~ ⊥ (τ, ~ηi (τ ))]2 +
m2i + [~κi (τ ) − Qi A
N
2
X
∂~ 3
1
~ 2 ](τ, ~σ ) =
Qi δ (~σ − ~ηi (τ )) + B
+ [ ~π⊥ +
2
△
i=1
q
N
X
3
~ ⊥ (τ, ~ηi (τ ))]2 +
=
δ (~σ − ~ηi (τ )) m2i + [~κi (τ ) − Qi A
+
i=1
N
X
i=1
+
rτ
T (τ, ~σ ) =
Qi ~π⊥ (τ, ~σ ) ×
1 2
∂~ 3
~ 2 ](τ, ~σ) +
δ (~σ − ~ηi (τ )) + [~π⊥
+B
△
2
1..N
1 X
∂~
∂~
Qi Qk δ 3 (~σ − ~ηi (τ )) · δ 3 (~σ − ~ηk (τ )),
2 i,k,i6=k
△
△
N
X
δ 3 (~σ − ~ηi (τ ))[κri (τ ) − Qi Ar⊥ (τ, ~ηi (τ ))] +
N
X
δ 3 (~σ − ~ηi (τ ))
i=1
N
X
∂~
~
~σ ),
+ [ ~π⊥ +
Qi δ 3 (~σ − ~ηi (τ )) × B](τ,
△
i=1
rs
T (τ, ~σ ) =
i=1
−
[κri (τ ) − Qi Ar⊥ (τ, ~ηi (τ ))][κsi (τ ) − Qi As⊥ (τ, ~ηi (τ ))]
q
−
2
2
~
mi + [~κi (τ ) − Qi A⊥ (τ, ~ηi (τ ))]
N
2
X
∂~ 3
~ 2] −
δ [ ~π⊥ +
Qi δ (~σ − ~ηi (τ )) + B
2
△
i=1
h1
rs
N
N
r s
X
X
∂~ 3
∂~
− [ ~π⊥ +
Qi δ (~σ − ~ηi (τ ))
~π⊥ +
Qi δ 3 (~σ − ~ηi (τ )) +
△
△
i=1
i i=1
+ B r B s ] (τ, ~σ).
42
(5.4)
◦
~η˙i (τ ) = q
~ ⊥ (τ, ~ηi (τ ))
~κi (τ ) − Qi A
m2i
~ ⊥ (τ, ~ηi (τ )))2
+ (~κi (τ ) − Qi A
X Qi Qk (~ηi (τ ) − ~ηk (τ ))
◦
~κ˙ i (τ ) =
k6=i
4π | ~ηi (τ ) − ~ηk (τ )
|3
,
+ Qi η̇iu (τ )
∂ u
A (τ, ~ηi (τ ))],
∂~ηi ⊥
◦
Ȧ⊥r (τ, ~σ ) = −π⊥r (τ, ~σ),
◦
r
π̇⊥
(τ, ~σ ) = ∆Ar⊥ (τ, ~σ) −
Z
~κ+ (τ ) +
X
i
Qi P⊥rs (~σ )η̇is (τ )δ 3 (~σ − ~ηi (τ )),
~
d3 σ[~π⊥ × B](τ,
~σ ) ≈ 0 (rest − f rame condition).
(5.5)
Let us note that in this reduced phase space there are only either particle-field interactions
or action-at-a-distance 2-body interactions.
µ
r
µ
µ
µ
The particle
q world-lines are xi (τ ) = xo +u (ps ) τ +ǫr (u(ps )) ηi (τ ), while their 4-momenta
~ ⊥ (τ, ~ηi )]2 uµ (ps ) + ǫµ (u(ps )) [κr − Qi Ar (τ, ~ηi )].
are pµi (τ ) = m2i + [~κi − Qi A
r
i
⊥
The generators of the internal Poincaré group are
τ
P(int)
~ (int)
P
r
J(int)
r
K(int)
= M=
N q
X
~ ⊥ (τ, ~ηi (τ )))2 +
m2i + (~κi (τ ) − Qi A
i=1
Z
Qi Qj
1 2
1 X
~ 2 ](τ, ~σ ),
+ d3 σ [~π⊥
+B
+
2 i6=j 4π | ~ηi (τ ) − ~ηj (τ ) |
2
Z
~
= ~κ+ (τ ) + d3 σ[~π⊥ × B](τ,
~σ ) ≈ 0,
=
N
X
i=1
= −
r
(~ηi (τ ) × ~κi (τ )) +
N
X
i=1
~ηi (τ )
q
Z
~ r (τ, ~σ),
d3 σ (~σ × [~π ⊥ ×B]
~ ⊥ (τ, ~ηi (τ ))]2 +
m2i + [~κi (τ ) − Qi A
Z
N 1..N
1 h XX
+
Qi
Qj
d3 σ σ r ~c(~σ − ~ηi (τ )) · ~c(~σ − ~ηj (τ )) +
2
i=1 j6=i
Z
i 1Z
2
3
r
~ 2 )(τ, ~σ ),
d3 σσ r (~π⊥
+B
+ Qi
d σπ⊥ (τ, ~σ )c(~σ − ~ηi (τ )) −
2
43
(5.6)
with c(~ηi − ~ηj ) = 1/(4π|~ηj − ~ηi |) [△ c(~σ) = δ 3 (~σ ), △ = −∂~2 , ~c(~σ ) = ∂~ c(~σ ) = ~σ /(4π |~σ |3 )].
τ
r
Note that P(int)
= q τ τ and P(int)
= q rτ are the mass and momentum monopoles, respectively.
For the sake of simplicity let us consider the sub-system formed by the two particles of
mass m1 and m2 . Our considerations may be extended to any cluster of particles both in
this case and in the case discussed in the previous Subsection. This sub-system is open:
besides their mutual interaction the two particles have Coulomb interaction with the other
N − 2 particles and feel the transverse electric and magnetic fields.
By using the multipoles we will select a set of effective parameters (mass, 3-center of
motion, 3-momentum, spin) describing the two-particle cluster as a global entity subject to
external forces in the global rest-frame instant form. This was the original motivation of the
multipolar expansion in general relativity: actually replacing an extended object (an open
system due to the presence of the gravitational field) with a set of multipoles concentrated
on a center of motion. In the rest-frame instant form it is possible to show that there is no
preferred centroid for an open system, namely different centers of motion may be selected
according to different conventions unlike the case of isolated systems where, in the rest frame
~κ+ ≈ 0, all these conventions identify the same centroid. We will see, however, that there is
a choice which seems preferable due to its properties.
Given the energy-momentum tensor T AB (τ, ~σ ) (5.4) of the isolated system, it would seem
AB
natural to define the energy-momentum tensor Tc(n)
(τ, ~σ) of an open sub-system composed by
a cluster of n ≤ N particles as the sum of all the terms in Eq.(5.4) containing a dependence
on the variables ~ηi , ~κi , of the particles of the cluster. Besides kinetic terms, this tensor
would contain internal mutual interactions as well as external interactions of the cluster
particles with the environment composed by the other N − n particles and by the transverse
electro-magnetic field. There is an ambiguity, however: why attributing just to the cluster
all the external interactions with the other N − n particles (no such ambiguity exists for the
interaction with the electro-magnetic field)? Since we have 2-body interactions, it seems
more reasonable to attribute only half of these external interactions to the cluster and
consider the other half as a property of the remaining N − n particles. Let us remark that
according to the first choice, if we consider two clusters composed by two non-overlapping
AB
AB
AB
, since the
sets of n1 and n2 particles respectively, we would get Tc(n
6= Tc(n
+ Tc(n
1)
2)
1 +n2 )
AB
AB
.
mutual Coulomb interactions between the two clusters are present in both Tc(n1 ) and Tc(n
2)
AB
AB
AB
Instead according to the second choice we would get Tc(n1 +n2 ) = Tc(n1 ) + Tc(n2 ) . Since this
property is important for studying the mutual relative motion of two clusters in actual
cases, we will adopt the convention that the energy-momentum tensor of a n particle cluster
contains only half of the external interaction with the other N − n particles.
Let us remark that, in the case of k-body forces, this convention should be replaced
by the following rule: i) for each particle mi of the cluster and each k-body term in the
energy-momentum tensor involving this particle, we write k = hi + (k − hi ), where hi is
the number of particles of the cluster participating to this particular k-body interaction; ii)
then only the fraction hi /k of this particular k-body interaction term containing mi has to
be attributed to the cluster.
Let us consider the cluster composed by the two particles with mass m1 and m2 .
def AB
The knowledge of TcAB = Tc(2)
on the Wigner hyper-plane of the global rest-frame in44
stant
form allows to find the following 10 non conserved charges [due to Q2i = 0 we have
q
p
~
ηi )
i ·A⊥ (τ,~
~ ⊥ (τ, ~ηi )]2 = m2 + ~κ21 − Qi ~κ√
m2i + [~κi − Qi A
]
i
2
2
mi +~
κi
Mc =
Z
3
Tcτ τ (τ, ~σ )
dσ
=
2 q
X
i=1
~ ⊥ (τ, ~ηi (τ ))]2 +
m2i + [~κi (τ ) − Qi A
2
Qi Qk
1 X X
Q1 Q2
+
=
+
2
4π |~η1 (τ ) − ~η2 (τ )|
2 i=1 k6=1,2 4π |~ηi (τ ) − ~ηk (τ )|2
= Mc(int) + Mc(ext) ,
2 q
X
Mc(int) =
m2i + ~κ2i −
i=1
~c = {
P
Z
Q1 Q2
,
4π |~η1 (τ ) − ~η2 (τ )|2
d3 σ Tcrτ (τ, ~σ)} = ~κ1 (τ ) + ~κ2 (τ ),
Z
J~c = {ǫruv
d3 σ [σ u Tcvτ − σ v Tcuτ ](τ, ~σ )} =
= ~ηi (τ ) × ~κ1 (τ ) + ~η2 (τ ) × ~κ2 (τ ),
~c = −
K
= −
−
Z
d3 σ ~σ Tcτ τ (τ, ~σ) =
2
X
i=1
2
X
q
~ ⊥ (τ, ~ηi (τ ))]2 −
~ηi (τ ) m2i + [~κi (τ ) − Qi A
Qi
i=1
− Q1 Q2
Z
Z
d3 σ ~π⊥ (τ, ~σ ) c(~σ − ~ηi (τ )) −
d3 σ ~σ ~c(~σ − ~η1 (τ )) · ~c(~σ − ~η2 (τ )) −
Z
2
X
1 X
−
Qk
d3 σ ~σ ~c(~σ − ~ηi (τ )) · ~c(~σ − ~ηk (τ )) =
Qi
2 i=1
k6=1,2
~ c(int) + K
~ c(ext) ,
= K
Z
2
q
X
2
2
~
Kc(int) = −
~ηi (τ ) mi + ~κi − Q1 Q2
d3 σ ~σ ~c(~σ − ~η1 (τ )) · ~c(~σ − ~η2 (τ )),(5.7)
i=1
which do not satisfy the algebra of an internal Poincare’ group due to the openess of the
system. Since we work in an instant form of dynamics only the cluster internal energy and
boosts depend on the (internal and external) interactions. Again Mc = qcτ τ and Pcr = qcrτ
are the mass and momentum monopoles of the cluster.
Another needed quantity is the momentum dipole
45
pru
c
=
=
Z
d3 σ σ r Tcuτ (τ, ~σ ) =
2
X
ηir (τ ) κui (τ )
i=1
ur
pru
c + pc =
2
X
i=1
− 2
2
X
i=1
Qi
Z
−
2
X
i=1
Qi
Z
d3 σ c(~σ − ~ηi (τ )) [∂ r As⊥ + ∂ s Ar⊥ ](τ, ~σ ),
[ηir (τ ) κui (τ ) + ηiu (τ ) κri (τ )] −
d3 σ c(~σ − ~ηi (τ )) [∂ r As⊥ + ∂ s Ar⊥ ](τ, ~σ ),
ur
ruv
pru
Jcv .
c − pc = ǫ
(5.8)
The time variation of the 10 charges (5.7) can be evaluated by using the equations of
motion (5.5)
46
2
~κ (τ ) · ~π (τ, ~η (τ ))
X
dMc
i
⊥
i
p
+
=
Qi
2
2
dτ
m
+
~
κ
i
i
i=1
h ~κ (τ )
~ηi (τ ) − ~ηk (τ ) 1 X
~κ (τ ) i
i
p k
·
+
Qk p 2
+
,
2 k6=1,2
m2k + ~κ2k 4π |~ηi (τ ) − ~ηk (τ )|3
mi + ~κ2i
2
~κ (τ )
X
X
ηir (τ ) − ηkr (τ ) ∂Ar⊥ (τ, ~ηi (τ ))
dPcr
i
Qk
·
+
=
Qi p 2
,
dτ
∂~ηi
4π |~ηi(τ ) − ~ηk (τ )|3
mi + ~κ2i
i=1
k6=1,2
2
i
~κ (τ )
h
X
dJ~c
i
~ ⊥ (τ, ~ηi (τ )) −
~ ⊥ (τ, ~ηi (τ )) + ~ηi (τ ) × p ~κi (τ ) · ∂ A
=
Qi p 2
×
A
2
2
2 ∂~
dτ
η
i
mi + ~κi
mi + ~κi
i=1
X
~ηi (τ ) × ~ηk (τ )
−
Qk
,
4π |~ηi (τ ) − ~ηk (τ )|3
k6=i
dKcr
dτ
= −Pcr −
−
+
+
=
+
−
+
2
X
i=1
2
X
Qi ~ηi (τ ) p
~κi (τ )
m2i + ~κ2i (τ )
i
h
X
Qk ~c(~ηi (τ ) − ~ηk (τ )) +
· ~π⊥ (τ, ~ηi (τ )) +
k6=i
Z
~κk (τ )
~κi (τ ) · ~c(~σ − ~ηi (τ )) i
3
p
+
Qk p 2
Qi
c(~
η
(τ
)
−
~
η
(τ
))
−
d
σ
~
π
(τ,
~
σ
)
i
k
⊥
2
2
2
(τ
)
+
~
κ
m
m
+
~
κ
(τ
)
i
i
k
k
i=1
k6=i
Z
2
X
X
~κk (τ )
~ ~c(~σ − ~ηk (τ )) −
Qi
Qk
d3 σ c(~σ − ~ηi (τ )) p 2
·
∂
mk + ~κ2k (τ )
i=1
k6=i
Z
h
i
~κ1 (τ )
~ ~c(~σ − ~η1 (τ )) · ~c(~σ − ~η2 (τ )) +
p
Q1 Q2
d3 σ ~σ
·
∂
m21 + ~κ21 (τ )
h
i
~κ2 (τ )
~ ~c(~σ − ~η2 (τ )) −
~c(~σ − ~η1 (τ )) · p 2
·
∂
m2 + ~κ22 (τ )
Z
2
h
i
X
~κi (τ )
1 X
3
~
p
Qi
Qk
d σ ~σ
· ∂ ~c(~σ − ~ηi (τ )) · ~c(~σ − ~ηk (τ )) +
2 i=1
m2i + ~κ2i (τ )
k6=1,2
h
i
~κk (τ )
~ ~c(~σ − ~ηk (τ )) .
~c(~σ − ~ηi (τ )) · p 2
·
∂
(5.9)
mk + ~κ2k (τ )
hX
Let us remark that, if we have two clusters of n1 and n2 particles respectively, our
definition of cluster energy-momentum tensor implies
Mc(n1 +n2 ) = Mc(n1 ) + Mc(n2 ) ,
~ c(n +n ) = P
~ c(n ) + P
~ c(n ) ,
P
1
2
1
2
J~c(n +n ) = J~c(n ) + J~c(n ) ,
1
2
1
2
~ c(n +n ) = K
~ c(n ) + K
~ c(n ) .
K
1
2
1
2
47
(5.10)
The main problem is the determination of an effective center of motion ζcr (τ ) with world~ + = ~y+ ≡ 0 of the
line wcµ (τ ) = xµo + uµ (ps ) τ + ǫµr (u(ps )) ζcr (τ ) in the gauge Ts ≡ τ , ~q+ = R
q
˙2
isolated system. The unit 4-velocity of this center of motion is uµc (τ ) = ẇcµ (τ )/ 1 − ζ~c (τ )
with ẇcµ (τ ) = uµ (ps ) + ǫµr (u(ps )) ζ̇cr (τ ). By using δ z µ (τ, ~σ ) = ǫµr (u(ps )) (σ r − ζ r (τ )) we can
define the multipoles of the cluster with respect to the world-line wcµ (τ )
Z
r1 ..rn AB
qc
(τ ) = d3 σ [σ r1 − ζcr1 (τ )]..[σ rn − ζcrn (τ )] TcAB (τ, ~σ ).
(5.11)
The mass and momentum monopoles and the mass, momentum and spin dipoles are
respectively
qcτ τ = Mc ,
qcrτ = Pcr ,
qcrτ τ = −Kcr − Mc ζcr (τ ) = Mc (Rcr (τ ) − ζcr (τ )),
r
u
qcruτ = pru
c (τ ) − ζc (τ ) Pc ,
Scµν = [ǫµr (u(ps )) uν (ps ) − ǫνr (u(ps )) uµ(ps )] qcrτ τ + ǫµr (u(ps )) ǫνu (u(ps )) (qcruτ − qcurτ ) =
= [ǫµr (u(ps )) uν (ps ) − ǫνr (u(ps )) uµ(ps )] Mc (Rcr − ζcr ) +
h
i
µ
ν
ruv
v
r u
u r
+ ǫr (u(ps )) ǫu (u(ps )) ǫ Jc − (ζc Pc − ζc Pc ) ,
⇒ mµc(ps ) = −Scµν uν (ps ) = −ǫµr (u(ps )) qcrτ τ .
(5.12)
Let us now consider the following possible definitions of effective centers of motion (many
other possibilities exist)
~ c (τ ), where R
~ c (τ ) is a 3-center of
1) Center of energy as center of motion, ζ~c(E) (τ ) = R
energy for the cluster built by means of the standard definition
~
~ c = − Kc .
R
Mc
(5.13)
It is determined by the requirement that either the mass dipole vanishes, qcrτ τ = 0 or the
mass dipole moment with respect to uµ (ps ) vanishes, mµc(ps ) = 0.
The center of energy seems to be the only center of motion enjoying the simple composition rule
~
~
~ c(n1 +n2 ) = Mc(n1 ) Rc(n1 ) + Mc(n2 ) Rc(n2 ) .
R
Mc(n1 +n2 )
~ c and R
~˙ c (τ ), see Eq.(4.8), is
The constitutive relation between P
48
(5.14)
0 =
dqcrτ τ
= −K̇cr − Ṁc Rcr − Mc Ṙcr ,
dτ
⇓
~ c = Mc R
~˙ c + Ṁc R
~c −
P
2
h
i
X
X
~κi (τ )
−
Qi ~ηi (τ ) p 2
·
~
π
(τ,
~
η
(τ
))
+
Q
~
c
(~
η
(τ
)
−
~
η
(τ
))
+
⊥
i
k
i
k
2
m
+
~
κ
(τ
)
i
i
i=1
k6=i
Z
2
hX
X
~κi (τ ) · ~c(~σ − ~ηi (τ )) i
~κk (τ )
3
p
+
Qi
c(~
η
(τ
)
−
~
η
(τ
))
−
d
σ
~
π
(τ,
~
σ
)
Qk p 2
+
i
k
⊥
mk + ~κ2k (τ )
m2i + ~κ2i (τ )
i=1
k6=i
Z
2
X
X
~κk (τ )
~ ~c(~σ − ~ηk (τ )) −
+
Qi
Qk
d3 σ c(~σ − ~ηi (τ )) p 2
·
∂
mk + ~κ2k (τ )
i=1
k6=i
Z
i
h
~κ1 (τ )
~ ~c(~σ − ~η1 (τ )) · ~c(~σ − ~η2 (τ )) +
p
·
∂
= Q1 Q2
d3 σ ~σ
m21 + ~κ21 (τ )
h
i
~κ2 (τ )
~
p
+ ~c(~σ − ~η1 (τ )) ·
· ∂ ~c(~σ − ~η2 (τ )) −
m22 + ~κ22 (τ )
Z
2
h
i
X
~κi (τ )
1 X
~ ~c(~σ − ~ηi (τ )) · ~c(~σ − ~ηk (τ )) +
p
Qi
Qk
d3 σ ~σ
·
∂
−
2 i=1
m2i + ~κ2i (τ )
k6=1,2
i
h
~κk (τ )
~
· ∂ ~c(~σ − ~ηk (τ )) .
(5.15)
+ ~c(~σ − ~ηi (τ )) · p 2
mk + ~κ2k (τ )
From Eq.(4.10) the associated cluster spin tensor is
Scµν = ǫµr (u(ps )) ǫνu (u(ps )) [qcruτ − qcurτ ] =
h
i
~c × P
~ c )v .
= ǫµr (u(ps )) ǫνu (u(ps )) ǫruv Jcv − (R
(5.16)
2) Pirani centroid ζ~c(P ) (τ ) as center of motion. It is determined by the requirement
that the mass dipole moment with respect to 4-velocity ẇcµ (τ ) vanishes (it involves the
anti-symmetric part of pur
c )
mµc(ẇc ) = −Scµν ẇcν = 0,
⇓
ζ~c(P ) (τ ) =
˙
˙
~ c,
⇒ ζ~c(P ) · ζ~c(P ) = ζ~c(P ) · R
1
~ c · ζ~˙ c(P ) (τ )
Mc − P
h
i
~c − R
~ c · ζ~˙c(P ) (τ ) P
~ c − ζ~˙c(P ) (τ ) × J~c .
Mc R
49
(5.17)
Therefore this centroid is implicitly defined as the solution of these three coupled first
order ordinary differential equations.
3) Tulczyjew centroid ζ~c(T ) (τ ) as center of motion. If we define the cluster 4-momentum
2
~ 2 def
Pcµ = Mc uµ (ps ) + Pcs ǫµs (u(ps )) [Pc2 = Mc2 − P
c = Mc ], its definition is the requirement that
µ
the mass dipole moment with respect to Pc vanishes (it involves the anti-symmetric part of
pur
c )
mµc(Pc ) = −Scµν Pcν = 0,
⇓
ζ~c(T ) (τ ) =
1
~2
Mc2 − P
c
~ c · ζ~c(T ) = P
~c · R
~ c,
⇒P
h
i
2 ~
~
~
~
~
~
Mc Rc − Pc · Rc Pc − Pc × Jc .
(5.18)
Let us show that this centroid satisfies the free particle relation as constitutive relation
~ c = Mc ζ~˙ c(T ) ,
P
⇓
h
i
s
µ
Pcµ = Mc uµ (ps ) + ζ̇c(T
ǫ
(u(p
))
,
s
) s
h
i
Mc
rτ τ
2 ~
~
~
~
~
~
~
qc(T
=
P
R
+
P
·
R
P
+
P
×
J
c
c c
c
c ,
)
~2 c c
Mc2 − P
c
rτ τ
Scµν = [ǫµr (u(ps )) uν (ps ) − ǫνr (u(ps )) uµ (ps )] qc(T
) +
i
h
~ c )v .
+ǫµr (u(ps )) ǫνu (u(ps )) ǫruv Jcv − (ζ~c(T ) × P
(5.19)
˙
~ c /Mc , it turns out that the
If we use Eq.(5.17) to find a Pirani centroid such that ζ~c = P
condition (5.17) becomes Eq.(5.18) and this implies Eq.(5.19).
The equations of motion
¨
~˙ c (τ ) − Ṁc (τ ) ζ~˙ c(T ) (τ ),
Mc (τ ) ζ~c(T ) (τ ) = P
(5.20)
contain both internal and external forces. Notwithstanding the nice properties (5.19) and
(5.20) of the Tulczyjew centroid, this effective center of motion suffers the drawback of not
satisfying a simple composition property. The relation among the Tulczyjew centroids of
clusters with n1 , n2 and n1 + n2 particles respectively is much more complicated of the
composition (5.14) of the centers of energy.
All the previous centroids coincide for an isolated system in the rest-frame instant form
~ c = ~κ+ ≈ 0 in the gauge ~q+ ≈ R
~ + ≈ ~y+ ≈ 0.
with P
50
4) The Corinaldesi-Papapetrou centroid with respect to a time-like observer with 4-velocity
(v)
v (τ ), ζ~c(CP ) (τ ) as center of motion.
µ
mµc(v) = −Scµν vν = 0.
(5.21)
Clearly these centroids are unrelated to the previous ones being dependent on the choice
of an arbitrary observer.
5) The Pryce center of spin or classical canonical Newton-Wigner centroid ζ~c(N W ).
It defined as the solution of the differential equations implied by the requirement
r
s
r
s
rs
{ζc(N
W ) , ζc(N W ) } = 0, {ζc(N W ) , Pc } = δ . Let us remark that, being in an instant form
of dynamics, we have {Pcr , Pcs } = 0 also for an open system.
The two effective centers of motion which look more useful for applications seems to be
the center of energy ζ~c(E) (τ ) and Tulczyjew’s centroid ζ~c(T ) (τ ), with ζ~c(E) (τ ) preferred for
the study of the mutual motion of clusters due to Eq.(5.14).
Therefore, in the spirit of the multipolar expansion, our two-body cluster may be described by an effective non-conserved internal energy (or mass) Mc (τ ), by the world-line
r
wcµ = xµo + uµ (ps ) τ + ǫµr (u(ps )) ζc(E
or T ) (τ ) associated with the effective center of motion
~ c (τ ), with ζ~c(E or T ) (τ ) and P
~ c (τ ) forming a
ζ~c(E or T ) (τ ) and by the effective 3-momentum P
non-canonical basis for the collective variables of the cluster. A non-canonical effective spin
for the cluster in the 1) and 3) cases is defined by
a) case of the center of energy,
~ c (τ ) × P
~ c (τ ),
S~c(E) (τ ) = J~c (τ ) − R
~ c (τ )
~
dζ~c(E)(τ )
dJ~c (τ ) dR
~ c (τ ) − R
~ c (τ ) × dPc (τ ) ,
=
−
×P
dτ
dτ
dτ
dτ
b) case of the T ulczyjew centroid,
~ c (τ ) =
S~c(T ) (τ ) = J~c (τ ) − ζ~c(T ) (τ ) × P
~ c (τ ) · J~c (τ ) P
~ c (τ )
Mc2 (τ ) S~c(E) (τ ) − P
,
=
~ 2 (τ )
Mc2 (τ ) − P
c
~ c (τ )
dζ~c(T )(τ )
dJ~c (τ ) ~
dP
=
− ζc(T ) (τ ) ×
.
dτ
dτ
dτ
(5.22)
Since our cluster contains only two particles, this pole-dipole description concentrated
on the world-line wcµ (τ ) is equivalent to the original description in terms of the canonical
variables ~ηi (τ ), ~κi (τ ) (all the higher multipoles are not independent quantities in this case).
51
Let us see whether it is possible to replace the description of the two body system as an
effective pole-dipole system with a description as an effective extended two-body system by
introducing two non-canonical relative variables ρ~c(E or T ) (τ ), ~πc(E or T ) (τ ) with the following
definitions
1
def
~η1 = ζ~c(E or T ) + ρ~c(E or T ) ,
2
1
def
~η2 = ζ~c(E or T ) − ρ~c(E or T ) ,
2
1 ~
Pc + ~πc(E or T ) ,
2
def 1
~ c − ~πc(E or T ) ,
~κ2 =
P
2
def
~κ1 =
1
ζ~c(E or T ) = (~η1 + ~η2 ),
2
ρ~c(E or T ) = ~η1 − ~η2 ,
~ c = ~κ1 + ~κ2 ,
P
~πc(E or T ) =
1
(~κ1 − ~κ2 ),
2
~ c + ρ~c(E or T ) × ~πc(E or T ) ,
J~c = ~η1 × ~κ1 + ~η2 × ~κ2 = ζ~c(E or T ) × P
⇒ S~c(E or T ) = ρ~c(E or T ) × ~πc(E or T ) .
(5.23)
Even if suggested by a canonical transformation, it is not a canonical transformation and it
~ c and J~c
only exists because we are working in an instant form of dynamics in which both P
do not depend on the interactions.
Note that we know everything about this new basis except for the unit vector
ρ~c(E or T ) /|~ρc(E or T ) | and the momentum ~πc(E or T ) . The relevant lacking information can be
ur
extracted from the symmetrized momentum dipole pru
c + pc , which is a known effective
quantity due to Eq.(5.9) having the following expression in terms of the variables (5.23)
pru
c
+
pur
c
+2
2
X
i=1
=
+
2
X
i=1
r
ρc(E or T )
Qi
Z
d3 σ c(~σ − ~ηi (τ )) [∂ r As⊥ + ∂ s Ar⊥ ](τ, ~σ ) =
r
u
u
r
(ηir κui + ηiu κri ) = ζc(E
or T ) Pc + ζc(E or T ) Pc +
u
u
r
πc(E
or T ) + ρc(E or T ) πc(E or T ) .
(5.24)
A strategy for getting this information is to construct a spin frame, which, following
Ref.[2] for the N = 2 case, is defined by Ŝc(E or T ) = S~c(E or T ) /|S~c(E or T ) |, R̂c(E or T ) , V̂c(E or T ) =
= 1. Then
= V̂ 2
R̂c(E or T ) × Sˆc(E or T ) , with Ŝc(E or T ) · R̂c(E or T ) = 0, Ŝ 2
= R̂2
c(E or T )
we get the following decomposition ((Sc(E or T )
52
= |S~c(E or T ) |)
c(E or T )
c(E or T )
ρ~c(E or T ) = ρc(E or T ) R̂c(E or T ) ,
~πc(E or T ) = π̃c(E or T ) R̂c(E or T ) −
ρc(E or T ) = |~η1 − ~η2 |,
Sc(E or T )
V̂c(E or T ) ,
ρc(E or T )
π̃c(E or T ) = ~πc(E or T ) ·
ρ~c(E or T )
,
ρc(E or T )
(5.25)
where ρc(E or T ) is just the relative variable appearing in the Coulomb potential. Eqs.(5.25)
show that only the three variables π̃c(E or T ) and R̂c(E or T ) = ρ~c(E or T ) /ρc(E or T ) are still unknown. Then from Eqs.(5.23) and (5.24) we get
u
u
r
r
u
ρrc(E or T ) πc(E
or T ) + ρc(E or T ) πc(E or T ) = 2 ρc(E or T ) π̃c(E or T ) R̂c(E or T ) R̂c(E or T ) −
r
u
u
r
− Sc(E or T ) R̂c(E or T ) V̂c(E or T ) + R̂c(E or T ) V̂c(E or T ) =
ur
r
u
u
r
= pru
c + pc − (ζc(E or T ) Pc + ζc(E or T ) Pc ) +
Z
2
X
Qi
d3 σ c(~σ − ~ηi (τ )) [∂ r Au⊥ + ∂ u Ar⊥ ](τ, ~σ ) =
+ 2
i=1
def
ru
= Fc(E
or T ) ,
ru
u
Fc(E
or T ) Sc(E or T ) ≡ 0.
(5.26)
But these are three independent equations for π̃c(E or T ) and for the two degrees of freedom
ru
in the unit vector R̂c(E or T ) in terms of the known quantities Fc(E
or T ) , Sc(E or T ) , ρc(E or T ) =
P rr
|~η1 − ~η2 |. For instance we get π̃c(E or T ) = ( r Fc(E or T ) )/2 ρc(E or T ) : due to the transversality
of the vector potential π̃c(E or T ) does not depend on it. In conclusion the external electro~ ⊥ enters only in the determination of the axis R̂c(E or T ) of the spin
megnetic potential A
frame.
This completes the construction of the effective relative variables and of the effective spin
frame using the extra input of the 3-momentum dipole. In this way we get a description
of the two-body cluster as an effective two-body system instead of a pole-dipole system.
However, the weak point of this description of the open system as an extended object is
that, whatever definition of effective center of motion one uses, the symmetrized momentum
ur
dipole pru
c + pc does not depend only on the cluster properties but also on the external
electro-magnetic transverse vector potential at the particle positions, as shown by Eq.(5.8).
As a consequence the spin frame, or equivalently the 3 Euler angles associated with the
internal spin, depends upon the external fields.
If we accept this drawback, it is reasonable that, by taking into account higher multipoles,
it is possible to give a description of a cluster of n ≥ 3 particles in terms of as many effective
n-body systems as effective dynamical body frames following the scheme of Ref.[2].
This would open the possibility to have effective descriptions of two clusters of n1 and
n2 particles, respectively, and to compare it with the effective description of the cluster
composed by the same n1 + n2 particles to find the relation between the three centers of
53
motion of the (n1 + n2 )-, n1 - and n2 - clusters and the relative motion of the two n1 - and n2 clusters. To this end the use of the center of energy as center of motion seems unavoidable
due to the simple composition law (5.14). Whatever choice we adopt, however, it turns
out that the relative motion of the two clusters depends on the external fields besides the
effective parameters of the clusters.
These techniques can be extended to relativistic perfect fluids, if described in the restframe instant form as done in Refs. [1]. Moreover, they are needed for the determination
of the post-Minkowskian approximation to the quadrupole formula for the emission of gravitational waves (re-summation of the post-Newtonian approximations) in the backgroundindependent Hamiltonian linearization of tetrad gravity [79] plus a perfect fluid [80].
54
VI.
CONCLUSIONS.
A relativistic description of open systems like binary stars embedded in the gravitational
field would be an important achievement nowadays in view of the construction of templates
for the gravitational radiation. Even by approximating such description by means of a
multipolar expansion in a way suitable for doing actual calculations, either analytical or
numerical, a big amount of kinematical technical preliminaries is needed anyway. With this
in view, we had in mind to develop methods which could be useful in general relativity with
relativistic perfect fluids as matter, where single or binary stars could be described by open
fluid subsystems of the isolated system formed by the gravitational field plus the fluid in the
rest-frame instant form of either metric or tetrad gravity [79, 80].
To pursue our program, in the present paper we have first of all completed the study
of the relativistic kinematics of the system of N free scalar positive-energy particles in the
rest-frame instant form of dynamics on Wigner hyper-planes, initiated in Ref.[2].
Then, we have evaluated the energy momentum tensor of the system on the Wigner
hyperplane and then determined Dixon’s multipoles for the N-body problem with respect
to the internal 3-center of mass located at the origin of the Wigner hyperplane [81]. For
an isolated system most of the existing definitions of a collective centroid identify a unique
world-line, associated with the internal canonical 3-center of mass. In the rest-frame instant
form these multipoles are Cartesian (Wigner-covariant) Euclidean tensors. While the study
of the monopole and dipole moments in the rest frame gives information on the mass, the
spin and the internal center of mass, the quadrupole moment provides the only (though not
unique) way of introducing the concept of barycentric tensor of inertia for extended systems
in special relativity.
By exploiting the canonical spin bases of Refs.[2, 21], after the elimination of the internal
3-center of mass (~q+ = ~κ+ = 0), the Cartesian multipoles qTr1 ...rnAB can be expressed in
terms of 6 orientational variables (the spin vector and the three Euler angles identifying the
dynamical body frame) and of 6N − 6 (rotational scalar) shape variables, i.e. in terms of the
canonical pairs of a canonical spin basis.
Having completed the discussion of the isolated system of N positive energy free scalar
particles the previous formalism has been applied to an isolated system of N positive-energy
particles with mutual action-at-a-distance interactions. Here again we find a unique worldline describing the collective motion of the system.
On the other hand, in the case of an open n < N particle subsystem of an isolated
system consisting of N charged positive-energy particles plus the electro-magnetic field a
more complex description appears. In the rest frame of the isolated system a suitable
definition of the energy-momentum tensor of the open subsystem allows to define its effective
mass, 3-momentum and angular momentum. However, unlike the case of isolated systems,
each centroid putatively describing the collective centers of motion, gives rise to a different
world-line. Starting from the evaluation of the rest-frame Dixon multipoles of the energymomentum tensor of the open subsystem with respect to various centroids we are given
therefore many candidates for an effective center of motion and for an effective intrinsic
spin. Two centroids (the center of energy and Tulczyjew centroid) seems to be preferred
because of their specific properties. In the case n = 2 it is possible to replace the poledipole description of the 2-particle cluster with a description of the cluster as an extended
system (whose effective spin frame can be evaluated) at the price of introducing an explicit
55
dependence on the action of the external electro-magnetic field upon the cluster.
Finally, by comparing the effective parameters of an open cluster of n1 + n2 particles with
the effective parameters of the two clusters with n1 and n2 particles, it is shown that only
the effective center of energy can in fact play the role of a useful center of motion,
The kinematical concepts we have defined for closed and open N-body systems are enough
for the treatment of relativistic continua like relativistic fluids. In [1] a preliminary extension
to closed relativistic fluids is given.
56
APPENDIX A: NON-RELATIVISTIC MULTIPOLAR EXPANSIONS FOR N
FREE PARTICLES.
In the review paper of Ref.[27] it can be found a study of the Newtonian multipolar expansions for a continuum isentropic distribution of matter characterized by a mass density
~ ~σ ) the moρ(t, ~σ ), a velocity field U r (t, ~σ ), and a stress tensor σ rs (t, ~σ ), with ρ(t, ~σ )U(t,
mentum density. In case the system is isolated, the only dynamical equations are the mass
conservation and the continuum equations of motion, respectively
∂ρ(t, ~σ ) ∂ρ(t, ~σ )U r (t, ~σ)
−
= 0,
∂t
∂σ r
∂ρ(t, ~σ )U r (t, ~σ ) ∂[ρU r U s − σ rs ](t, ~σ ) ◦
−
= 0.
∂t
∂σ s
(A1)
We can adapt this description to an isolated system of N particles in the following way.
The mass density
ρ(t, ~σ ) =
N
X
i=1
satisfies
mi δ 3 (~σ − ~ηi (t)),
(A2)
N
X
∂ρ(t, ~σ )
def ∂
=−
mi ~η˙ i (t) · ∂~η~i δ 3 (~σ − ~ηi (t)) =
[ρU r ](t, ~σ ),
r
∂t
∂σ
i=1
(A3)
while the momentum density [82] is
r
ρ(t, ~σ )U (t, ~σ ) =
N
X
i=1
mi ~η˙ i (t)δ 3 (~σ − ~ηi (t)),
The associated constant of motion is the total mass m =
(A4)
PN
i=1 .
If we define a function ζ(~σ, ~ηi ) concentrated in the N points ~ηi , i=1,..,N, such that
ζ(~σ, ~ηi ) = 0 for ~σ 6= ~ηi and ζ(~ηi, ~ηj ) = δij [83], the velocity field associated to N particles becomes
~ ~σ ) =
U(t,
N
X
~η˙ i (t)ζ(~σ, ~ηi (t)).
(A5)
i=1
The continuum equations of motion are replaced by
N
N
X
X
∂
◦ ∂
r
r
s
3
[ρ(t, ~σ )U (t, ~σ )] = s
mi η̇i (t)η̇i (t)δ (~σ − ~ηi (t)) +
mi η̈ir (t) =
∂t
∂σ i=1
i=1
∂[ρU r U s − σ rs ](t, ~σ )
=
.
∂σ s
def
(A6)
57
◦
For a system of free particles we have ~η¨i (t) = 0 so that σ rs (t, ~σ) = 0. If there are interparticle interactions, they will determine the effective stress tensor.
Let us consider an arbitrary point ~η(t). The multipole moments of the mass density ρ
~ and of the stress-like density ρU r U s with respect to the point
and momentum density ρU
~η (t) are defined by setting (N ≥ 0)
r1 ...rn
m
[~η (t)] =
=
Z
d3 σ[σ r1 − η r1 (t)]...[σ rn − η rn (t)]ρ(τ, ~σ ) =
N
X
i=1
mi [ηir1 (τ ) − η r1 (t)]...[ηirn (t) − η rn (t)],
n=0
m[~η (t)] = m =
N
X
mi ,
i=1
p
r1 ...rn r
[~η (t)] =
=
Z
d3 σ[σ r1 − η r1 (t)]...[σ rn − η rn (t)]ρ(t, ~σ )U r (t, ~σ ) =
N
X
i=1
mi η̇ir (t)[ηir1 (t) − η r1 (t)]...[ηirn (t) − η rn (t)],
n=0
r
p [~η (t)] =
N
X
i=1
pr1 ...rnrs [~η (t)] =
=
Z
mi η̇i (t) =
N
X
i=1
κri = κr+ ≈ 0,
d3 σ[σ r1 − η r1 (t)]...[σ rn − η rn (t)]ρ(t, ~σ )U r (t, ~σ )U s (t, ~σ ) =
N
X
i=1
mi η̇ir (t)η̇is (t)[ηir1 (t) − η r1 (t)]...[ηirn (t) − η rn (t)].
(A7)
The mass monopole is the conserved mass, while the momentum monopole is the total
3-momentum, vanishing in the rest frame.
If the mass dipole vanishes, the point ~η(t) is the center of mass:
r
m [~η (t)] =
N
X
i=1
mi [ηir (t) − η r (t)] = 0 ⇒ ~η (t) = ~qnr .
(A8)
The time derivative of the mass dipole is
dmr [~η (t)]
= pr [~η (t)] − mη̇ r (t) = κr+ − mη̇ r (t).
(A9)
dt
When ~η(t) = ~qnr , from the vanishing of this time derivative we get the momentum-velocity
relation for the center of mass
r
pr [~qnr ] = κr+ = mq̇+
[≈ 0 in the rest f rame].
The mass quadrupole is
58
(A10)
N
X
rs
m [~η(t)] =
− mη (t)η (t) − η (t)m [~η(t)] + η (t)m [~η(t)] ,
mi ηir (t)ηis (t)
r
i=1
s
r
s
s
r
(A11)
so that the barycentric mass quadrupole and tensor of inertia are, respectively
rs
m [~qnr ] =
N
X
i=1
I rs [~qnr ] = δ rs
r s
mi ηir (t)ηis (t) − mqnr
qnr ,
X
u
=
=
X
2
r s
mi [δ rs~ηi2 (t) − ηir (t)ηis (t)] − m[δ rs ~qnr
− qnr
qnr ] =
i=1
1...N
X−1
a,b
⇒
muu [~qnr ] − mrs [~qnr ] =
kab (~ρa · ρ~b δ rs − ρra ρsb ),
mrs [~qnr ] = δ rs
N
−1
X
a,b=1
kab ρ~a · ρ~b − I rs [~qnr ].
(A12)
The antisymmetric part of the barycentric momentum dipole gives rise to the spin vector
in the following way
rs
p [~qnr ] =
N
X
mi ηir (t)η̇is (t)
i=1
S
u
−
r s
qnr
p [~qnr ]
=
N
X
i=1
N
−1
X
1 urs rs
(~ρa × ~πqa )u .
= ǫ p [~qnr ] =
2
a=1
r s
ηir (t)κsi (t) − q+
κ+ ,
(A13)
The multipolar expansions of the mass and momentum densities around the point ~η(t)
are
ρ(t, ~σ ) =
∞
X
mr1 ....rn [~η]
n=0
r
ρ(t, ~σ )U (t, ~σ ) =
n!
∞
X
pr1 ....rnr [~η ]
n=0
n!
∂n
δ 3 (~σ − ~η(t)),
∂σ r1 ...∂σ rn
∂n
δ 3 (~σ − ~η (t)).
r
r
n
1
∂σ ...∂σ
Finally, for the barycentric multipolar expansions we have
59
(A14)
1 X uu
∂2
1
I [~qnr ]) r s δ 3 (~σ − ~qnr ) +
ρ(t, ~σ ) = mδ 3 (~σ − ~qnr ) − (I rs [~qnr ] − δ rs
2
2
∂σ ∂σ
u
∞
X
mr1 ....rn [~qnr ]
∂n
δ 3 (~σ − ~qnr ),
r1 ...∂σ rn
n!
∂σ
n=3
i ∂
h1
δ 3 (~σ − ~qnr ) +
ρ(t, ~σ )U r (t, ~σ ) = κr+ δ 3 (~σ − ~qnr ) + ǫrsu S u + p(sr) [~qnr ]
2
∂σ s
∞
X
∂n
pr1 ....rnr [~qnr ]
δ 3 (~σ − ~qnr ).
+
r1 ...∂σ rn
n!
∂σ
n=2
+
60
(A15)
APPENDIX B: SYMMETRIC TRACE-FREE TENSORS.
In the applications to gravitational radiation, irreducible symmetric trace-free Cartesian
tensors (STF tensors) [76, 84, 85, 86] are needed instead of Cartesian tensors. While
a Cartesian multipole tensor of rank l (like the rest-frame Dixon multipoles) on R3 has 3l
components, 21 (l+1)(l+2) of which are in general independent, a spherical multipole moment
of order l has only 2l + 1 independent components. Even if spherical multipole moments
are preferred in calculations of molecular interactions, spherical harmonics have various
disadvantages in numerical calculations: for analytical and numerical calculations Cartesian
moments are often more convenient (see for instance Ref.[87] for the case of the electrostatic
potential). It is therefore preferable using the irreducible Cartesian STF tensors[88, 89]
(having 2l + 1 independent components if of rank l), which are obtained by using Cartesian
spherical (or solid) harmonic tensors in place of spherical harmonics.
Given an Euclidean tensor
Ak1 ...kI on R3 , one defines the completely symmetrized tensor
P
Sk1 ..kI ≡ A(k1 ..kI ) = I!1 π Akπ(1) ...kπ(I) . Then, the associated STF tensor is obtained by
removing all traces ([I/2] = largest integer ≤ I/2)
[I/2]
(ST F )
Ak1 ...kI
=
X
an δ(k1 k2 ...δk2n−1 k2n Sk2n+1 ...kI )i1 i1 ...jnjn ,
n=0
an ≡ (−1)n
For instance (Tabc )ST F ≡ T(abc) −
1
5
h
l!(2l − 2n − 1)!!
.
(l − 2n)!(2l − 1)!!(2n)!!
i
δab T(iic) + δac T(ibi) + δbc T(aii) .
61
(B1)
APPENDIX C: THE GARTENAHUS-SCHWARTZ TRANSFORMATION.
In Ref.[2] we defined canonical internal relative variables with respect to the internal
3-center of mass ~q+ by exploiting a Gartenhaus-Schwartz canonical transformation. The
canonical generator of this transformation is
G = ~q+ · ~κ+ ,
(C1)
so that the finite transformation, depending on a parameter α, on a generic function F on
the phase space is
Z α
F (α) = F +
dα {F (α), G(α)}.
(C2)
0
In particular we have
lim ~κ+ (α) = 0,
lim ~q+ (α) = ∞.
α→∞
α→∞
(C3)
As said in Section II, if we define the canonical transformation (2.19), then the quantities
~πqa = lim ~πa (α),
ρ~qa = lim ρ~a (α),
α→∞
α→∞
(C4)
are well defined and the transformation
~ηi , ~κi → ~κ+ , ~q+ , ρ~qa , ~πqa ,
(C5)
r
s
{q+
, κs+ } = δ rs , {ρrqa , πqb
} = δ rs δab ,
(C6)
is a canonical transformation
as said in Eq.(2.18).
The quantities ρ~qa , ~πqa are the searched internal relative variables: they describe the
system after the gauge fixing ~q+ ≈ 0, ~κ+ ≈ 0. We have also
~κ+ ≈ 0 ⇒ ρ~qa ≈ ρ~a , ~πqa ≈ ~πa .
(C7)
Thanks to these results, we can calculate a function F independent of ~q+ on the phase
space, under the constraint ~κ+ ≈ 0, by simply performing the limit
F |~κ+ ≈0 (~ρqa , ~πqa ) = lim F (α).
α→∞
(C8)
This method is applied in Section IV for calculating the multipoles after the gauge fixing
~q+ ≈ 0, ~κ+ ≈ 0. These multipoles depend on ~κi , so that (see Ref.[2])
lim ~κi (α) =
√
α→∞
N
N
−1
X
a=1
~ + ), so that
and on (~ηi − R
62
γai~πqa ,
(C9)
q
m2j + ~κ2j
X
(~ηi − ~ηj ) P p 2
=
mk + ~κ2k
k
j
q
m2j + ~κ2j
XX√
.
=
N(γai − γaj )~ρa P p 2
mk + ~κ2k
k
a
j
~ +) =
(~ηi − R
(C10)
Then using the (C9) and (C4) we have
~ + (α)) =
lim (~ηi (α) − R
α→∞
XX√
a
j
q
P
m2j + N ab ~πqa · ~πqb γaj γbj
N (γai − γaj )~ρqa P p 2
. (C11)
P
mk + N ab ~πqa · ~πqb γak γbk
k
The following notation is used to denote the limits (C8)
F →α→∞ F |~κ+ ≈0 (~ρqa , ~πqa ).
(C12)
For example Eqs. (C9) and (C11) become
~κi →α→∞
~ + ) →α→∞
(~ηi − R
√
N
N
−1
X
γai~πqa ,
a=1
XX√
j
a
q
P
m2j + N ab ~πqa · ~πqb γaj γbj
. (C13)
N(γai − γaj )~ρqa P p 2
P
mk + N ab ~πqa · ~πqb γak γbk
k
The closed form (2.20) - (2.22) of the canonical transformation (C5) was not given in
Ref.[2], but it can derived from the following two equations of that paper [its Eq.(5.13) and
(5.24)]
63
N
1 X
γai~κi (α),
~πa (α) = √
N i=1
~πqa
N
1 X
= ~πa (∞) = √
~κi (∞) =
N i=1
~n+
[(Msys − M)~n+ · ~πa − |~κ+ |Ha ] =
= ~πa +
M
q
def
= ~πa − q
Ha
~κ+
2
Msys
− ~κ2+
[Ha −
Msys −
2 −~
Msys
κ2+
~κ2+
~κ+ · ~πa ] ≈ ~πa ,
N
N
q
1 X
1 X
γai Hi = √
γai m2i + ~κ2i ,
= √
N i=1
N i=1
~κi (∞) =
√
N
N
−1
X
γai~πqa ,
a=1
H(rel)i
v
u
1..N
u
X−1
= Hi (∞) = tm2i + N
γai γbi~πqa · ~πqb ,
ab
Msys =
N
X
Hi =
i=1
=
N
X
q
M2 + ~κ2+ ≈ H(rel) = HM (∞) = M =
Hi (∞) =
i=1
v
N u
X
u
i=1
tm2 + N
i
1..N
X−1
ab
γai γbi~πqa · ~πqb .
(C14)
def
ρ~qa = ρ~a (∞) = ρ~a −
N N
−1
i
X
X
Hi h |~κ+ |~κj (∞)
Msys
√ + ( √ − 1)~n+ ~n+ · ρ~b =
−
γaj (γbi − γbj )
Msys Hj (∞) Π
Π
i,j=1 b=1
= ρ~a −
N N
−1
X
X
i,j=1 b=1
γaj (γbi − γbj )
Hi
~κj (∞)
√ ~κ+ · ρ~b ≈ ρ~a .
Msys Hj (∞) Π
64
(C15)
APPENDIX D: MORE ON DIXON’S MULTIPOLES.
In this Appendix we shall consider multipoles with respect to the origin, i.e. with ~η = 0
def
[we use the notation tTµ1 ...µn µν (Ts ) = tTµ1 ...µn µν (Ts , 0)] and we shall give some of their properties
following Ref.[25]. The proofs of these results are identical to those given in Ref.[25] after
the following kinematical modifications of Eqs.(3.6) and (3.7) of that paper:
i) let W be the world-tube containing the compact support of the isolated system and
w µ (τ ) = z µ (τ, ~η(τ )) a time-like world-line inside it, with tangent vector ẇ µ (τ ) = zτµ (τ, ~η(τ ))+
zrµ (τ, ~η (τ )) η r (τ ), used to evaluate the multipoles (see Section IV);
ii) the Wigner hyper-planes ΣW τ of the rest-frame instant form are in general not orthogonal to the world-line (differently fromR the hyper-surfaces
p
R Σ(s)
R of3 Ref.[25]);
4
iii) Eq.(3.6) is replaced with
d z f (z) =
dτ Στ d σ g(τ, ~σ) f (z(τ, ~σ )) =
R
R
dτ Στ d3 Σµ zτµ (τ, ~σ ) f (z(τ, ~σ)) (see before Eq.(2.1) for the notations);
q
we
iv)
since
lµ (τ, ~σ )
=
[lρ zAρ zµA ](τ, ~σ )
=
[ γg zµτ ](τ, ~σ ),
p
R
R
d3 σ γ(τ, ~σ ) lµ (τ, ~σ ) f (z(τ, ~σ ))
d3 Σµ f (z(τ, ~σ ))
=
get
=
Στ
Στ
p
R
R
d
3
τ
3
=
d σ g(τ, ~σ) zµ (τ, ~σ ) f (z(τ, ~σ )), so that we have dτ Στ d Σµ f (z(τ, ~σ ))
p
R
RΣτ 3 ∂ p
3
τ
A
d σ ∂A [ g(τ, ~σ) zµ (τ, ~σ) f (z(τ, ~σ ))] after
d σ ∂τ [ g(τ, ~σ) zµ (τ, ~σ ) f (z(τ, ~σ))] =
Στ
Στ
p
R
r
3
having added Στ d σ ∂r [ g(τ, ~σ) zµ (τ, ~σ ) f (z(τ, ~σ))] = 0;
√
∂z β ∂z γ ∂z δ
σ ) = 0, we get that
v) since ∂σ∂A [ g zµA ](τ, ~σ ) = ∂σ∂A [ 3!1 ǫABCD ǫµβγδ ∂σ
D ](τ, ~
R
RB ∂σC3 ∂σp
σ ))
d
3
Eq.(3.7) is replaced by dτ Στ d Σµ f (z(τ, ~σ )) = Στ d σ g(τ, ~σ) zµA (τ, ~σ) ∂f (z(τ,~
=
∂σA
p
R
R
∂f
(z(τ,~
σ
))
∂f
(z(τ,~
σ
))
d3 σ γ(τ, ~σ) lν (τ, ~σ ) zτν (τ, ~σ ) ∂z µ = Στ d3 Σν zτν (τ, ~σ) ∂z µ ;
Στ
vi) as a consequence the results quoted in this Appendix hold if the multipoles are taken
with respect to world-lines w µ (τ ) = z µ (τ, ~η ) such that ~η (τ ) = ~η = const. (they are the
integral lines of the vector field zτµ (τ, ~σ ) ∂µ ), so that ẇ µ (τ ) = zτµ (τ, ~η ).
As shown in Ref.[25], if a field has a compact support W on the Wigner hyper-planes ΣW τ
and if f (x) is a C ∞ complex-valued scalar function on Minkowski space-time with compact
support [90], we have
<T
µν
,f > =
=
=
=
=
Z
d4 xT µν (x)f (x) =
Z
Z
dTs d3 σf (xs + δxs )T µν [xs (Ts ) + δxs (~σ )][φ] =
Z
Z
Z
d4 k ˜
3
dTs d σ
f(k)e−ik·[xs(Ts )+δxs (~σ)] T µν [xs (Ts ) + δxs (~σ )][φ] =
(2π)4
Z
Z
Z
d4 k ˜
−ik·xs (Ts )
d3 σT µν [xs (Ts ) + δxs (~σ )][φ]
f(k)e
dTs
(2π)4
∞
X
(−i)n
[kµ ǫµu (u(ps ))σ u ]n =
n!
n=0
Z
Z
∞
X
d4 k ˜
(−i)n
−ik·xs (Ts )
dTs
f
(k)e
kµ1 ...kµn tTµ1 ...µnµν (Ts ),
(D1)
(2π)4
n!
n=0
65
If, in particular, f (x) is analytic on W [25] [91], we get
µν
< T ,f > =
T
µν
(x) =
Z
∞
X
1 µ1 ...µn µν
∂ n f (x)
dTs
|x=xs(Ts ) ,
tT
(Ts ) µ1
n!
∂x ...∂xµn
n=0
⇓
∞
X
(−1)n
n=0
n!
∂n
∂xµ1 ...∂xµn
Z
dTs δ 4 (x − xs (Ts ))tTµ1 ...µnµν (Ts ).
(D2)
For a N particle system this equation may be rewritten as Eq.(4.4).
On the other hand, for non-analytic functions f (x) we have
N
X
1 µ1 ...µn µν
∂ n f (x)
< T ,f > =
dTs
|x=xs(Ts ) +
tT
(Ts ) µ1
n!
∂x ...∂xµn
n=0
Z
Z
∞
X
(−i)n
d4 k ˜
−ik·xs (Ts )
f
(k)e
kµ1 ...kµn tTµ1 ...µnµν (Ts ), (D3)
+
dTs
(2π)4
n!
n=N +1
µν
Z
and, as shown in Ref.[25], from the knowledge of the moments tTµ1 ...µn µ (Ts ) for all n > N, we
can get T µν (x) and, therefore, all the moments with n ≤ N.
In the case of N free particles and more in general for isolated systems, the Hamilton
def
equations [92] for the multipoles (4.3), with pTµ1 ...µn µ (Ts ) = pTµ1 ...µn µ (Ts , ~0), imply
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
If we define for n ≥ 1
66
n ≥ 1.
(D4)
def
(µ ...µn µ)
bTµ1 ...µn µ (Ts ) = pT 1
(Ts ) =
µ)
= ǫr(µ1 1 (u(ps ))....ǫµrnn (u(ps ))ǫA (u(ps )) qTr1 ..rnAτ (Ts ),
def
(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 ))]qTr1 ..rnAτ (Ts ),
(µ ...µn µ)
cT 1
[µν]
STµν = 2pT
(Ts ) = 0,
= 2cµν
T ,
1
(r ...r r)τ
uµ (ps )qTr1 ...rnτ τ (Ts ) + ǫµr (u(ps ))qT 1 n (Ts ),
n+1
n
ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))cTµ1 ...µn µ (Ts ) =
uµ (ps )qTr1 ...rnτ τ (Ts ) +
n+1
(r ...r r)τ
+ ǫµr (u(ps ))[qTr1 ...rnrτ (Ts ) − qT 1 n (Ts )],
(D5)
ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))bTµ1 ...µn µ (Ts ) =
and for n ≥ 2
(µ ...µ )(µν)
def
dTµ1 ...µn µν (Ts ) = dT 1 n
(Ts ) = tTµ1 ...µn µν (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
i
n+2
ν)
ν)
ǫr(µ1 1 ..ǫµrnn ǫµA ǫB (u(ps ))
+ ǫr(µ1 1 ...ǫµrnn ǫB ǫµA +
n
r1 ..rn AB
qT
(Ts ),
(µ ...µn µ)ν
dT 1
ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))dTµ1 ...µn µν (Ts ) =
+
+
−
+
(Ts ) = 0,
n−1 µ
u (ps )uν (ps )qTr1 ...rn τ τ (Ts ) +
n+1
1 µ
[u (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]
n
(r ...r r)τ
[(n − 1)qTr1 ...rn rτ (Ts ) + qT 1 n (Ts )] +
ǫµs1 (u(ps ))ǫνs2 (u(ps ))[qTr1 ...rns1 s2 (Ts ) −
n + 1 (r1 ...rns1 )s2
(r ...r s )s
(qT
(Ts ) + qT 1 n 2 1 (Ts )) +
n
(r ...r s s )
qT 1 n 1 2 (Ts )],
(D6)
67
then Eqs.(D4) may be rewritten in the form
1) n = 1
(µν)
tµν
T (Ts ) = tT
◦
(Ts ) = pµT (Ts )uν (ps ) +
1 d
(STµν (Ts ) + 2bµν
T (Ts )),
2 dTs
⇓
d µν
b (Ts ) = Muµ (ps )uν (ps ) +
dTs T
(µ
ν)
+ κr+ [u(µ (ps )ǫν)
r (u(ps )) + ǫr (u(ps ))u (ps )] +
◦
(µ
ν)
tµν
T (Ts ) = pT (Ts )u (ps ) +
ν)
ǫ(µ
r (u(ps ))ǫs (u(ps ))
N
X
i=1
κui κvi
p
m2i + ~κ2i
,
d µν
◦
[µ
ν]
S (Ts ) = 2pT (Ts )uν] (ps ) = 2κr+ ǫ[µ
r (u(ps ))u (ps ) ≈ 0,
dTs T
(ρµ)ν
2) n = 2 [identity tρµν
= tT
T
(ρµ)ν
2tT
◦
µ)ν
µ)ν
(ρν)µ
+ tT
(Ts ) = 2u(ρ (ps )bT (Ts ) + u(ρ (ps )ST (Ts ) +
⇓
◦
ρ(µ
µν
ρ
ν)
tρµν
T (Ts ) = u (ps )bT (Ts ) + ST (Ts )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 ...µn µν (Ts ) = dTµ1 ...µn µν (Ts ) + u(µ1 (ps )bT2 n (Ts ) + 2u(µ1 (ps )cT2 n
(Ts ) +
2 µ ...µ (µ
1 µ1 ...µn µν
2 µ ...µ (µν)
d
= cT1 n (Ts )uν) (ps ) +
[
bT
(Ts ) + cT1 n (Ts )].(D7)
n
dTs n + 1
n
This allows to rewrite < T µν , f > in the form[25]
µν
h
d4 k ˜
ν)
ρ(µ
−ik·xs (Ts )
f
(k)e
u(µ (ps )pT (Ts ) − ikρ ST (Ts )uν) (ps ) +
(2π)4
∞
i
X
(−i)n
+
kρ1 ...kρn ITρ1 ...ρnµν (Ts ) ,
(D8)
n!
n=2
< T ,f > =
Z
dTs
Z
with
68
(µ1 ...µn )(µν)
ITµ1 ...µn µν (Ts ) = IT
−
+
=
+
−
−
2
µ ...µ )(µν)
u(µ1 (ps )cT2 n
(Ts ) +
n−1
2 µ1 ...µn (µ
cT
(Ts )uν) (ps ) =
n
h
n + 1 (µ1 µn µ) ν
ǫr1 ...ǫrn ǫA ǫB +
ǫµr11 ...ǫµrnn ǫµA ǫνB −
n
n+2
i
ν)
ν)
ǫr(µ1 1 ...ǫµrnn ǫB ǫµA +
ǫr(µ1 1 ...ǫµrnn ǫµA ǫB (u(ps ))
n
qTr1 ..rn AB (Ts ) −
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
i
2 µ1 µn (µ
(µ)
ǫr1 ...ǫrn ǫA − ǫr(µ1 1 ...ǫµrnn ǫA uν) (ps ) (u(ps ))
n
qTr1 ..rn Aτ (Ts ),
(µ1 ...µn µ)ν
IT
ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))ITµ1 ...µn µν (Ts ) =
+
+
−
+
def
(Ts ) = dTµ1 ...µn µν (Ts ) −
(Ts ) = 0,
n+3 µ
u (ps )uν (ps )qTr1 ...rnτ τ (Ts ) +
n+1
1 µ
[u (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]qTr1 ...rnrτ (Ts ) +
n
ǫµs1 (u(ps ))ǫνs2 (u(ps ))[qTr1 ...rns1 s2 (Ts ) −
n + 1 (r1 ...rn s1 )s2
(r ...r s )s
(qT
(Ts ) + qT 1 n 2 1 (Ts )) +
n
(r1 ...rn s1 s2 )
qT
(Ts )].
(D9)
For a N particle system, Eq.(D8)implies Eq.(4.15).
Finally, a set of multipoles equivalent to the ITµ1 ..µn µν is [93]:
69
f or
n≥0
(µ1 ...µn )[µν][ρσ]
JTµ1 ...µnµνρσ (Ts ) = JT
def
µ ...µn [µ[ρν]σ]
(Ts ) = IT 1
(Ts ) =
µ ...µ [µ[ρν]σ]
= tT1 n
(Ts ) −
h
1
ν]µ ...µ [ρσ]
u[µ (ps )pT 1 n (Ts ) +
−
n+1
i
σ]µ1 ...µn [µν]
+ u[ρ(ps )pT
(Ts ) =
i
h
µ1
µn [µ [ρ ν] σ]
= ǫr1 ..ǫrn ǫr ǫs ǫA ǫB (u(ps ))qTr1 ..rn AB (Ts ) −
1 h [µ
σ]
[ρ
−
u (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
))
s
s
s
r
s
A
ǫµr11 (u(ps ))...ǫµrnn (u(ps ))qTrr1 ..rn sAτ (Ts ),
(n + 4)(3n + 5)
linearly independent components,
µ ...µn−1 (µn µν)ρσ
uµ1 (ps )JTµ1 ...µnµνρσ (Ts ) = JT 1
(Ts ) = 0,
f or n ≥ 1,
4(n − 1) (µ1 ...µn−1 |µ|µn )ν
J
(Ts ),
f or n ≥ 2,
n+1 T
h
i
r1 ..rn AB
[ρ ν] σ]
ǫrµ11 (u(ps ))....ǫrµnn (u(ps ))JTµ1 ...µnµνρσ (Ts ) = ǫ[µ
ǫ
ǫ
ǫ
(Ts ) −
r s A B (u(ps ))qT
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
))
qTrr1 ..rn sAτ (Ts ).
s
s
s
r
s
A
ITµ1 ...µn µν (Ts ) =
(D10)
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 the energy-momentum tensor together with
the monopole PTµ and the spin dipole STµν .
◦
As shown in Section 5 of Ref.[25], the equations of motion ∂µ T µν = 0 do not imply equations of motion for the multipoles ITµ1 ...µn µν (n ≥ 2) or JTµ1 ...µn µνρσ (n ≥ 0), but only Eqs.(4.16)
for the multipoles PTµ and STµν [94]. Instead the multipoles tTµ1 ...µnµν and pTµ1 ...µn µ have non
trivial equations of motion.
When all the multipoles JTµ1 ..µn µνρσ are zero (or negligible) one speaks of a pole-dipole
system.
On the Wigner hyperplane, the content of these 2n+2 -pole inertial moment tensors is replaced by the Euclidean Cartesian tensors qTr1 ...rn τ τ , qTr1 ...rnrτ , qTr1 ...rnrs . As shown in Appendix
70
B, we can decompose these Cartesian tensors in their irreducible STF (symmetric trace-free)
parts (the STF tensors).
Thus the multipolar expansion (4.4) may be rewritten as
q+ )β
T µν [x(~
(Ts ) + ǫβr (u(ps ))σ r ] = T µν [w β (Ts ) + ǫβr (u(ps ))(σ r − η r (Ts ))] =
s
ν)
= u(µ (ps )ǫA (u(ps ))[δτA M + δuA κu+ ]δ 3 (~σ − ~η (Ts )) +
+
+
1 ρ(µ
∂
ST (Ts )[~η ]uν) (ps )ǫrρ (u(ps )) r δ 3 (~σ − ~η(Ts )) +
2
∂σ
∞
X
(−1)n h n + 3
n=2
n!
n+1
uµ (ps )uν (ps )qTr1 ...rnτ τ (Ts , ~η) +
1 µ
[u (ps )ǫνr (u(ps )) + uν (ps )ǫµr (u(ps ))]qTr1 ...rnrτ (Ts , ~η ) +
n
+ ǫµs1 (u(ps ))ǫνs2 (u(ps ))[qTr1 ...rns1 s2 (Ts , ~η ) −
i
n + 1 (r1 ...rn s1 )s2
(r1 ...rn s2 )s1
(r1 ...rn s1 s2 )
(qT
(Ts , ~η) + qT
(Ts , ~η)) + qT
(Ts , ~η)]
−
n
∂n
δ 3 (~σ − ~η (Ts )),
∂σ r1 ..∂σ rn
+
(D11)
leading to Eq.(4.15).
For open systems, subsystems of an isolated system like in Section V, we have
◦
∂ν T µν = F µ 6= 0, with F µ an external force. As shown in Ref.[25] for the case in which
◦
F µ = −F µν Jν (∂µ J µ =0) is the Lorentz force, in this case the multipolar expansion (4.15)
is still valid, while the equations of motion (4.16) become (P µ and Ts are the conserved
4-momentum and the rest-frame time of the global isolated system)
Z
dPcµ(Ts ) ◦
=
d3 σ F µ (Ts , ~σ ),
dTs
Z
dScµν (Ts )[~η = 0] ◦
[µ
ν]
= 2 pc (Ts ) u (P ) − d3 σ σ r [ǫµr (u(P )) F ν (Ts , ~σ) − ǫνr (u(P )) F µ(Ts ,(D12)
~σ )].
dTs
71
[1] L.Lusanna and D. Nowak-Szczepaniak, Int.J.Mod.Phys. A15, 4943 (2000) (hep-th/0003095).
D.Alba and L.Lusanna, Eulerian Coordinates for Relativistic Fluids: Hamiltonian Rest-Frame
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[2] D.Alba, L.Lusanna and M.Pauri, J.Math.Phys. 43, 1677 (2002) (hep-th/0102087).
[3] L.Lusanna, Int.J.Mod.Phys. A4 (1997) 645.
[4] D.Alba and L.Lusanna, Int.J.Mod.Phys A13, 2791 (1998) (hep-th/9705155).
[5] H.Crater and L.Lusanna, Ann.Phys. (NY) 289, 87 (2001) (hep-th/0001046).
[6] L.Lusanna, Towards a Unified Description of the Four Interactions in Terms of DiracBergmann Observables, invited contribution to the book of the Indian National Science
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and Dirac’s Observables, talk given at the Conf. Constraint Dynamics and Quantum Gravity
99, Villasimius 1999 (gr-qc/9912091).The Rest-Frame Instant Form of Dynamics and Dirac’s
Observables, talk given at the Int.Workshop Physical Variables in Gauge Theories, Dubna
1999.
[7] Such hyper-surfaces are the leaves of a foliation of Minkowski space-time, associated with one
among its 3+1 splittings, and are equivalent to a congruence of time-like accelerated observers.
[8] P.A.M.Dirac, Can.J.Math. 2, 129 (1950); Lectures on Quantum Mechanics, Belfer Graduate
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[11] T.D.Newton and E.P.Wigner, Rev.Mod.Phys. 21, 400 (1949). G.Lugarini and M.Pauri, Nuovo
Cimento 47, 299 (1967).
[12] C.Møller, Ann.Inst.H.Poincaré 11, 251 (1949); The Theory of Relativity (Oxford Univ.Press,
Oxford, 1957).
[13] A.D.Fokker, Relativiteitstheorie (P.Noordhoff, Groningen, 1929), p.171.
[14] M.H.L.Pryce, Proc.Roy.Soc.(London) 195A, 6 (1948).
[15] G.N.Fleming, Phys.Rev. 137B, 188 (1965); 139B, 963 (1965).
[16] ~zs /ǫs is the classical analogue of the Newton-Wigner 3-position operator, only covariant under
the Euclidean subgroup of the Poincaré group.
[17] For instance ~
q+ ≈ 0 implies that the internal 3-center of mass is put in the centroid xµs (τ ) =
z µ (τ, ~σ = 0), origin of the 3-coordinates on the Wigner hyper-planes.
[18] S.Gartenhaus and C.Schwartz, Phys.Rev. 108, 842 (1957).
[19] R.G.Littlejohn and M.Reinsch, Rev.Mod.Phys. 69, 213 (1997).
[20] See Ref.[2] for a review of this approach used in molecular physics for the definition and study
of the vibrations of molecules.
[21] D.Alba, L.Lusanna and M.Pauri, J.Math.Phys. 43, 373 (2002) (hep-th/0011014).
[22] We adhere to the definitions used in Ref.[19]; in the mathematical literature our left action is
a right action.
[23] Their generators are the center-of-mass angular momentum Noether constants of motion.
72
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
Their generators are not constants of motion.
W.G.Dixon, J.Math.Phys. 8, 1591 (1967).
See Ref.[27] for the definition of Dixon’s multipoles in general relativity.
W.G.Dixon, Proc.Roy.Soc.London A314, 499 (1970) and A319, 509 (1970); Gen.Rel.Grav.
4, 199 (1973); Phil. Trans.Roy.Soc.London A277, 59 (1974); Extended Bodies in General Relativity: Their Description and Motion, in Isolated Systems in General Relativity, ed.J.Ehlers
(North Holland, Amsterdam, 1979).
~ = {br } on the Wigner hyper-planes
On it we use the notation A = (τ, r). The 3-vectors B
◦
◦
are Wigner spin-1 3-vectors. ǫµr (u(ps )) = Lµ r (ps , ps ) and ǫµτ (u(ps )) = uµ (ps ) = Lµ o (ps , ps ) are
the columns of the standard Wigner boost for time-like Poincaré orbits. See Appendix B of
Ref.[2].
Obviously each choice χ
~ (τ ) leads to a different set of conjugate canonical relative variables.
As already said, they describe an internal center-of-mass 3-variable ~σcom defined inside the
Wigner hyperplane and conjugate to ~κ+ ; when the ~σcom are canonical variables they are
denoted ~q+ .
W.Beiglböck, Commun.Math.Phys. 5, 106 (1967).
See Refs.[33, 34] for the definition of this concept in general relativity. By using the interpretation of Ref.[33], also the special relativistic limit of the general relativistic Dixon centroid
of Ref.[27] gives the centroid xµs (τ ): it coincides with the special relativistic Dixon centroid of
Ref.[25] defined by using the conserved energy momentum tensor, as we shall see in Section
IV.
I.Bailey and W.Israel, Ann.Phys. (N.Y.) 130, 188 (1980).
J.Ehlers and E.Rudolph, Gen.Rel.Grav. 8, 197 (1977).
It describes a point living on the Wigner hyper-planes and has the covariance of the little
group O(3) of time-like Poincaré orbits, like the Newton-Wigner position operator.
G.Longhi and L.Lusanna, Phys.Rev. D34, 3707 (1986).
It implies λ(τ ) = −1 and identifies the time parameter τ with the Lorentz scalar time of the
center of mass in the rest frame, Ts = ps · x̃s /Msys ; Msys generates the evolution in this time.
See Refs.[39] for a different derivation of this result.
J.M.Pons and L.Shepley, Class.Quant.Grav. 12, 1771 (1995) (gr-qc/9508052). J.M.Pons,
D.C.Salisbury and L.Shepley, Phys.Rev. D55, 658 (1997) (gr-qc/9612037).
As in every instant form of dynamics, there are four independent Hamiltonians pos and Jsoi ,
functions of the invariant mass Msys ; we give also the expression in the basis Ts , ǫs , ~zs , ~ks .
This internal Poincaré algebra realization should not be confused with the previous external
one based on S̃sµν ; Π and W 2 are the two non-fixed invariants of this realization.
The external spin coincides with the internal angular momentum due to Eqs.(A11) of Ref.[2].
~ ≈ 0 is implied by the natural gauge fixing ~q+ ≈ 0.
As we shall see in the next Section, K
M.Pauri and G.M.Prosperi, J.Math.Phys. 16, 1503 (1975).
F.A.E.Pirani, Acta Phys. Polon. 15, 389 (1956). S.Ragusa and L.Bailyn, Gen.Rel.Grav. 27,
163 (1995). I.B.Khriplovich and A.A.Pomeranski, Phys.Lett. A216, 7 (1996).
W.Tulczyjew, Acta Phys.Polon. 18, 393 (1959).
W.G.Dixon, Nuovo Cimento 34, 317 (1964).
See Ref. [49] for the application of these methods to find the center of mass of a configuration
of the Klein-Gordon field after the preliminary work of Ref.[50] on the center of phase for a
real Klein-Gordon field.
L.Lusanna and M.Materassi, Int.J.Mod.Phys. A15, 2821 (2000) (hep-th/9904202).
73
[50] G.Longhi and M.Materassi, Int.J.Mod.Phys. A14,
J.Math.Phys. 40, 480 (1999)(hep-th/9803128).
µ
P
PN
ẋo
µ 2
√ẋi 2 )2 > 0.
[51] poi = mi √i 2 > 0, ǫ( N
i=1 pi ) = ǫ( i=1 mi
ẋi
3387
(1999)
(hep-th/9809024);
ẋi
[52] In this way we are sure to have imposed the restriction to positive energy particles and to
have excluded the other 2N − 1 branches ofqthe total mass spectrum.
P
[53] Namely we get PTµ = pµs and Msys = N
m2i + ~κ2i (τ ).
i=1
[54] Previous studies of multipole theories of point particles can be found in Refs.[46, 47, 55, 56,
57, 58, 59, 60].
[55] M.Mathisson, Acta Phys.Polon. 6, 163 (1937).
[56] J.Lubanski, Acta Phys.Polon. 6, 356 (1937).
[57] W.G.Dixon, Nuovo Cimento 38, 1616 (1965).
[58] H.Hönl and A.Papapetrou, Z.Physik 112, 512 (1939).
[59] H.Hönl and A.Papapetrou, Z.Physik 114, 478 (1939).
[60] A.H.Taub, J.Math.Phys. 5, 112 (1964).
[61] See this paper for the previous definitions given by Bielecki, Mathisson and Weyssenhof[46,
55, 62, 63].
[62] A.Bielecki,
M.Mathisson
and
J.W.Weyssenhoff,
Bull.Intern.Acad.Polon.Sci.Lett.,
Cl.Sci.Math.Nat. Se’r. A: Sci.Math. p.22 (1939).
[63] M.Mathisson, Proc.Cambridge Phil.Soc. 36, 331 (1940).
[64] tµν
η ) = ǫµA (u(ps ))ǫνB (u(ps ))qTAB (Ts , ~η), pµT (Ts , ~η) = ǫµA (u(ps ))qTAτ (Ts , ~η ), t̃T (Ts , ~η ) =
T (Ts , ~
qTτ τ (Ts , ~
η ).
[65] They are ~η independent; see Appendix C of Ref.[2] and Appendix A for the non-relativistic
limit.
[66] tµT1 µν (Ts , ~
η ) = ǫµr11 (u(ps ))ǫµA (u(ps ))ǫνB (u(ps ))qTr1 AB (Ts , ~η ), t̃µT1 (Ts , ~η ) = ǫµr11 (u(ps ))qTr1 τ τ (Ts , ~
η ),
µ1 µ
µ1
µ
r1 Aτ
pT (Ts , ~
η ) = ǫr1 (u(ps ))ǫA (u(ps ))qT (Ts , ~η ).
[67] See Ref.[25] for the ambiguous definitions of pµT , STµν given by Papapetrou, Urich and Papapetrou, and by B.Tulczyjev and W.Tulczyjev[68, 69, 70].
[68] A.Papapetrou, Proc.Roy.Soc.(London) A209, 248 (1951).
[69] W.Urich and A.Papapetrou, Z.Naturforsch. 10a, 109 (1955).
[70] B.Tulczyjew and W.Tulczyjew, in Recent Development in General Relativity (Pergamon Press,
New York, 1962), p.465.
[71] Defined by STµν (Ts )[~
η ]uν (ps ) = 0, namely with Ssor = 0 in the momentum rest frame.
[72] Defined by STµν (Ts )[~
η ]ẋs ν = 0, namely with Ssor = 0 in the instantaneous velocity rest frame.
[73] For instance, the so-called background Corinaldesi-Papapetrou centroid[74] is defined by the
condition STµν (Ts )[~
η ]vν = 0, where v µ is a given fixed unit 4-vector.
[74] E.Corinaldesi and A.Papapetrou, Proc.Roy.Soc.London 209, 259 (1951). B.M.Baker and
R.F.O’Connell, Gen.Rel.Grav. 5, 539 (1974).
[75] tTµ1 µ2 µν
=
ǫµr11 (u(ps ))ǫµr22 (u(ps ))ǫµA (u(ps ))ǫνB (u(ps ))qTr1 r2 AB ,
tTµ1 µ2 µ µ
=
µ1 µ2
µ1
µ2
µ1 µ2 µ
µ1
µ2
µ
r1 r2 A
r1 r2 τ τ
=
t̃T
= ǫr1 (u(ps ))ǫr2 (u(ps ))qT
, pT
ǫr1 (u(ps ))ǫr2 (u(ps ))ǫA (u(ps ))qT
A,
ǫµr11 (u(ps ))ǫµr22 (u(ps ))ǫµA (u(ps ))qTr1 r2 Aτ .
[76] K.S.Thorne, The Theory of Gravitational Radiation: an Introductory Review in Gravitational
Radiation, eds. N.Deruelle and T.Piran, 1982 NATO-ASI School at Les Houches (North Holland, Amsterdam, 1983); Rev.Mod.Phys. 52, 299 (1980).
[77] J.M.Souriau, C.R.Acad.Sc.Paris 271, 751, 1086 (1970); 274, 1082 (1972). Ann.Inst.H.Poincaré
20, 22 (1974).
74
[78] R.Schattner, Gen.Rel.Grav. 10, 377 and 395 (1978). Ann.Inst.H.Poicaré 40, 291 (1984).
J.Madore, Ann.Inst.H.Poicaré XI, 221 (1969).
[79] L.Lusanna, The Rest-Frame Instant Form of Metric Gravity, Gen.Rel.Grav. 33, 1579 (2001)
(gr-qc/0101048).
L.Lusanna and S.Russo, A New Parametrization for Tetrad Gravity, Gen.Rel.Grav. 34, 189
(2002)(gr-qc/0102074).
R.De Pietri, L.Lusanna, L.Martucci and S.Russo, Dirac’s Observables for the Rest-Frame
Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge, Gen.Rel.Grav.
34, 877 (2002) (gr-qc/0105084).
[80] J.Agresti, R.DePietri, L.Lusanna and L.Martucci, Hamiltonian Linearization of the RestFrame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge: a Radiation Gauge for Background-Independent Gravitational Waves in a Post-Minkowskian Einstein
Space-Time (gr-qc/0302084).
J.Agresti, C.Bambi, R.DePietri and L.Lusanna, Hamiltonian Linearization of Tetrad Gravity in the Radiation Gauge and Background- Independent Gravitational Waves in PostMinkowskian Space-Times in Presence of a Perfect Fluid, in preparation.
[81] So that the origin is also the Fokker-Pryce center of inertia and Pirani and Tulczyjew centroids
of the system.
[82] This can be taken as the definitory equation for the velocity field, even if strictly speaking we
do not need it in what follows.
[83] It is a limiting concept deriving from the characteristic function of a manifold.
[84] R.Sachs, Proc.Roy.Soc.London A264, 309 (1961), especially Appendix B.
[85] F.A.E.Pirani, in Lectures on General Relativity, eds.A.Trautman, F.A.E.Pirani and H.Bondi
(Prentice-Hall, Englewood Cliffs, 1964).
[86] L.Blanchet and T.Damour, Phil.Trans.R.Soc.London A320, 379 (1986).
[87] K.Hinsen and B.U.Felderhof, J.Math.Phys. 33, 3731 (1992).
[88] I.M.Gel’fand, R.A.Minlos and Z.Ya.Shapiro, Representations of the Rotation and Lorentz
Groups and Their Applications (Pergamon, New York, 1963).
[89] J.A.R.Coope, R.F.Snider and F.R.McCourt, J.Chem.Phys. 43, 2269 (1965); J.Math.Phys. 11,
1003 (1970). J.A.R.Coope, J.Math.Phys. 11, 1591 (1970).
R
[90] So that its Fourier transform f˜(k) = d4 xf (x)eik·x is a slowly increasing entire analytic funco
3
tion on Minkowski space-time (|(xo + iy o )qo ...(x3 + iy 3 )q3 f (xµ + iy µ )| < Cqo ...q3 eao |y |+...+a3|y | ,
a > 0, qµ positive integers for every µ and Cqo ...q3 > 0), whose inverse is f (x) =
R µ d4 k
−ik·x .
˜
(2π)4 f (k)e
[91] See this paper for related results of Mathisson and Tulczyjev[46, 55, 62, 63].
◦
[92] In Ref.[25] this is a consequence of ∂µ T µν = 0.
def
[93] A[µ[ρν]σ] = 41 (Aµνρσ − Aνρµσ − Aµσνρ + Aνσµρ ).
[94] The so called Papapetrou-Dixon-Souriau equations given in Eq.(4.16).
75