Silvio Zilio
email: zilio@pd.infn.it

Dynamics of a test rigid body from a minimal action principle

Keywords: numbers: 04, 20, Fy

Making use of a minimal action principle, in this work we derive the dynamics of a test rigid body moving in a curved spacetime by means of a parametric invariant lagrangian formalism. In doing so we complete a line of research due to Bailey-Israel and Anandan-Dadhich-Singh. This is accomplished through the following new contributions: by fixing the lagrangian of the system, the elaboration of a complete variational procedure, the formulation of a rigidity constraint and the derivation of conserved quantities, already found, but in a very different form, in other approaches to the problem. The dynamics and the equations obtained are also generalized to all orders in the metric expansion by means of new mathematical tools. Besides, by a selection of an appropriate spatial section of the body world-tube, we obtain the simple Papapetru expression of the canonical momentum which, remaining unchanged to all orders, contributes to some reductions of complexity of the dynamics. Finally, in the quadrupolar approximation, applications of our results are presented in the form of useful observables in the context of ideal tests in general relativity

Introduction

The quest for a relativistic theory of extended bodies is a long-standing problem, having important implications in general relativity and in astrophysics. In the multipole extended body formalism -originally due to Mathisson [START_REF] Mathisson | [END_REF], developed by Papapetru [2], and Taub [3] and culminating in 1974 with the elegant Dixon theory [4] -the dynamics are obtained by integrating the conservation equation ∇ ν T νµ = 0 on the body world-tube and using multipolar expansions, with various ad-hoc choices and settlements (for an historical conceptual review see Dixon [4] (1974)). In order to reduce the mathematical complexity of the Dixon theory, a very different approach to the problem was realized in 1980 by Bailey and Israel [START_REF] Bailey | [END_REF]. In that work the authors studied, by an 'eccentric variational' method, the equations of motion of a collection of identical point-like charged sources, which, in the case of null charge, reduced to a fluid of incoherent matter. Although the Bailey and Israel approach was more simple than the Dixon one, generalizations of the work to other more interesting non-charged matter configurations, having internal interactions, was not given by the authors and, up to now, any realistic modelization of an extended body coming from such an approach remains not carried out. This is due to the fact that the eccentric variational method of Bailey and Israel was essentially an n-body problem, a nonobvious coarse-graining procedure of average being required in order to obtain the dynamics of the collective extended body variables.

On the other hand, a general reduction of complexity of the dynamics of an extended body can be obtained by the notions of test body and rigidity. For example in the Dixon works [4] (1970a and 1979), the interdependence between the energy-momentum tensor and the gravitational self-field of a body involves many complications, so that relevant simplifications descend from the use of the test body approximation. Likewise, although no body can be actually considered as a rigid object, owing to the underlying inertial stress caused by rotation and curvature, in spite of its high abstract content, the notion of rigidity appears as another powerful concept. In the past, different constraints were imposed in order to implement the abstract idea of rigidity in general relativity (for a discussion see [4](1979) and [6]). Nevertheless, in spite of the simplifications introduced by both the notions of test body and rigidity, concrete applications of the Dixon theory have been scarcely studied (see [7] and references therein), because, notwithstanding its elegance, it appears formally intricate and, in some respects, too general.

In the restricted domain of the test body approximation, another attack to the complex dynamics of an extended body was realized by a collection of studies based on classical action principles (see Frenkel [8] and Barut[9] for the special relativistic case of spinning particles in electromagnetic fields and Kunzle [10] and Souriau [11] in Einstein-Maxwell fields). In all these works, the variational procedure was carried out without specifying the functional lagrangian form. The most important result of this approach appeared in 1975 and was again due to Bailey and Israel (BI) [12]. In that work the body was abstractly modelized as a point-like object having a spin angular momentum described by the gyration of an orthonormal tetrad e a ν , defined on the particle world-line, and experiencing an arbitrary unspecified multipole coupling with the external gravitational field. The evolution of a spinning test body in a curved spacetime was then obtained by means of a minimal action principle applied to a parametric invariant lagrangian L, whose general functional form, also in this case, was not specified, the resulting equations of motion being, as a consequence, only the general abstract dynamics obeyed by any model of a spinning multipole particle. No explicit model of test body derived by a minimal action principle 'a la BI' was known until the recent publication of the Anandan, Dadhich and Singh (ADS) work [13], which give some progress. In this paper appears, for the first time, the part of the lagrangian coming from the external coupling to the spacetime. However, as in the BI work, the explicit connection between the internal degrees of freedom of the lagrangian was not identified so that, using a non complete variational approach, ADS partially derived the equations of motion by a minimal action principle applied only on trajectory variation but not on the rotational quantities variation. For clarity, let us resume the main ADS formalism.

In the starting point ADS consider a general action of an extended body

S = ∆ L √ -gd 4 x, (1.1)
the integral being extended over a world tube ∆ in spacetime whose thickness Λ is assumed small compared to the radius of curvature R. The lagrangian L nominally encodes all the properties of a rigid body, vaguely defined as a body "...subject only to external gravitational forces and no other forces." [13]. Besides, the multipole structure of the body is expressed through the following definitions of 2n-multipole moments

t κ1...κnµν = δy κ1 ...δy κn √ -gT µν w ρ dΣ ρ , (1.2)
with δy µ = x µ -γ µ , the 4-velocity γµ = dγ µ /dλ = (1, 0, 0, 0) being the tangent vector of the reference world line γ (λ) of the body. In this expression, the integral domain is extended over an unspecified arbitrary space-like hypersurface identified by the unit normal vector field w ρ . The ADS multipole moments are thus defined by means of normal coordinate systems related each others by linear transformations.

All the final results obtained by ADS in this class of coordinate systems are then inductively extended to a general coordinate system. The progress obtained by the ADS method relies on the identification of every coupling of the gravitational field with the multipole moments (1.2), a question not answered in the BI work. In fact, the core of the ADS method is based on the expansion, around the reference world-line z, of the action (1.1) with respect to the first order metric variation, δg µν = g µν,σ z δy σ + 1 2 g µν,σλ z δy σ δy λ +... and on the use of the canonical definition of the energy-momentum tensor in order to obtain, from (1.2), the multipolar expansion of the action. The final step of the ADS method consists to write down the resulting multipolar expansion of the action in a BI-like formalism

S = γ L [g, δg] dλ + O (δg) 2 , (1.3)
and to extremize this action (δS = 0) in order to derive the equations of motion.

In spite of this progress, the ADS strategy does not find L [g, δg] completely, because two important quantities remain unrelated: the spin tensor and the Ricci rotation coefficients. Furthermore, ADS do not use a parametric invariant formalism, so that they define the canonical momentum up to a unknown term transversal to the four velocity, while the minimal action principle is applied only to the trajectory variation, the spin evolution being derived simply by induction from a corresponding Newtonian scheme. In addition, the following inductive procedure adopted by ADS, crucial in order to obtain a covariant generalization of the Newtonian multipole moments,

g 00,ij z I ij = -2R 0i0j z I ij → g 00,ij z t ij00 = -2R 0i0j z I ij → g αβ,µν z t µναβ = -2R αµβν z I µν u α u β , (1.4)
inevitably does not possess an univocal character, the definition I ij ≡ t ij00 being not used in the same way in all occurrences of I ij . The present paper is devoted to complete this BI-ADS works and to supply to various inadequacies again existing. This is useful because, in concrete applications of the test body dynamics, the BI-ADS approach is more simple and manageable than the Dixon theory. Our new contributions are obtained primarily by finding the explicit dependence of the lagrangian on all the observables and by the elaboration of a complete variational procedure applied on all the variables introduced, successively by the formulation of a mathematical well defined rigidity constraint and the derivation of conserved quantities, similar to those already found, but in a very different formalism, in other approaches to the problem [4] [START_REF] Bailey | [END_REF]. The dynamics and the equations obtained are also generalized to all orders in the metric expansion and this is made possible thanks to a new covariant Taylor expansion of the metric field. Besides, by a spacelike section of the body world-tube, selected orthogonal to the reference world-line, we obtain the simple first order Papapetru [2] expression of the canonical momentum which, remaining unchanged to all orders in our work, contributes to some reductions of complexity of the dynamics with respect to the other approaches (compare the Dixon-Ehlers-Rudolph canonical momentum [4](1979) [6] with our expression (6.5)). Finally, in the quadrupolar approximation, applications of our dynamics are presented as typical examples of quantities relevant in the context of ideal tests in general relativity.

The structure of the present work is the following. Without use of normal coordinates of ADS, in Section 2 we give a new covariant expansion of all quantities involved. This is necessary in order to adopt a clear and manageable BI parametric invariant formalism and in view of generalizations. In Section 3 we re-define the multipole moment tensors in a general coordinates system and, by a generalization of the ADS method, we formally expand the lagrangian to all orders in the metric variation. The abstract structure of the lagrangian is found in Section 4 by adding to the system, as in the BI formalism, internal degrees of freedom represented by the vectors of an orthonormal tetrad, co-rotating with the rigid body, the gyration of which simulates the angular velocity observable. In Section 5, we find univocally the explicit form of the parametric invariant lagrangian, which is formally equivalent to a multipolar particle having the Lorentz subgroup of spatial rotation L rot as internal symmetry. Then, by the assumption of a rigidity constraint, in Section 6 we apply a minimal action principle on a trajectory variation and on a tetrad variation, thus obtaining the equations of motion to all orders and without the use of the ADS Newtonian generalization (1.4). In Section 7 we derive the conserved quantities related to cyclic variables of the lagrangian. Finally, in view of applications, in Section 8 we give examples of these conserved quantities up to quadrupolar expansion, as well explicit illustrations of useful classical-like observables, such as relativistic principal axes of inertia and relativistic ellipsoid of inertia surface, both related to the relativistic analogous of the Euler equations.

Throughout the text, unless otherwise specified, use has been made of natu-ral units: c = 1. The metric signature is +2, with determinant g. The dual of a second-rank antisymmetric tensor T αβ is defined as * T αβ = 1 2 η αβµν T µν , where, in a {x µ } coordinate system, the Levi-Civita tensor is η µ1µ2µ3µ4 = √ -gδ µ1µ2µ3µ4 , with δ µ1µ2µ3µ4 = +1 (-1) for an even (odd) permutation of 0123. As usual βµν will stand for the contraction of the Levi-Civita tensor with the velocity, i.e. η βµνα u α .

Covariant expansion of the metric

In this Section we establish in a covariant way the expansion of a parametric invariant lagrangian describing a small test body immersed in a curved space time. Preliminarily, we covariantly expand the metric tensor g µν on the thin world tube ∆ of the body around a representative world line γ (λ) of the trajectory. While ADS define all multipole moments over an unspecified space-like hypersurface, we chose it orthogonally to γ (λ). To this end we introduce some mathematical tools.

Cover ∆ by a neighborhood W of γ constituted by the union of normal neighborhoods of the points of γ [START_REF] De Felice | Relativity on curved manifolds[END_REF]. Let us call Σ = Σ (λ) the space-like sections of W ∩ ∆ spanned by all space-like geodesics σ X (η) which stem from p = σ X (0) = γ (λ) in a direction X µ orthogonal to the vector γ (λ). More precisely, X µ is the space-like tangent vector to σ X (η) at p whose size X is the measure of the proper length of σ X from p to q = σ X (η)

X = η 0 σX dη, (2.1)
Denoting by X µ (q) the vector which satisfies (2.1) for the unique geodesic of Σ joining p and q, then X µ (q) is a vector at p and a function on Σ q which obeys the orthogonality relation

X µ (q) γµ (p) = 0. (2.2)
It is easy to show that

X µp = - ∂ ∂x µp Ω (p, q) , (2.3)
where Ω (p, q) is the world-function [START_REF] De Felice | Relativity on curved manifolds[END_REF] and x µp = x µ (η p ) are local coordinates for σ X .

The first approach to a covariant Taylor expansion of the metric tensor around x µp seems to tentatively give it over Σ. This is suggested by the fact that there exists a unique tensor which reduces, at the pole p of any normal coordinate system {x µ } centered in p (normal coordinate systems naturally induced on Σ), to the n-fold repeated partial derivative of the metric tensor [START_REF] Veblen | [END_REF] and because the vector X µ (q) appears to play a role similar to that of a coordinate distances δx µ between p and q. Indicating this tensor at p as G µpνpκ1 p ...κn p , in a normal coordinate system we have G µ p ν p κ 1p ...κ np ≡ g µ p ν p ,κ 1p ...κ np . In the following we will give the explicit form of this tensors up to the desired order. Thus the hypothetical Taylor expansion of the metric could be

g µqνq (q) = ∞ n=0 1 n! G µ p ν p κ1 p ...κn p (p) X κ1 p (q) ...X κn p (q) . (2.4)
On the other hand, at the l.h.s. of (2.4) we have a tensor at q, while the r.h.s. appears as a scalar at q and a tensor at p. In order to recover consistency we introduce an operator able to 'transport' a tensorial index from a point to another. We will call it the basis connector.

Consider the bitensor

X µpνq = -∇ ν X µp | x(ηq) . (2.5)
A long calculation shows that this quantity can be expanded as follows (see [START_REF] De Felice | Relativity on curved manifolds[END_REF] for details)

X µpνq = -g µpλp Γ νq λp + O (R X Γ) , (2.6)
where O (R X Γ) is a complicated integral expression of products of the Riemann tensor, of X and Γ, the last being the connector Γ νq λp , the operator causing the parallel transport along the geodesic σ a vector from p to q, i.e.

Vµq = Γ µq λp V λp , (2.7)
Vµq being the parallel to V λp at q. But we are interested in the change of tensors from a point to another caused uniquely by the transformation of the basis between the points, not by curvature. Then, suppose that the chart defining the coordinate system {x} contains a sufficiently wide region U of the body dynamics. Consider in U the flat metric ḡµν satisfying the property ḡµν (p) = g µν (p) and ∂ κ ḡµν (p) = ∂ κ g µν (p). So we have Γκ µν (p) = Γ κ µν (p) but Rµνρσ (q) = 0. The two metrics considered will have by construction the same normal coordinate systems centered on p. By the use of the connector operator Γ λp νq induced by ḡµν and using (2.6) we can define the following basis connector two point tensor

X µpνq = -g µpλp Γ λp νq = -∇ν X µp | x(ηq) , (2.8)
which is an exact operator to all order in the coordinate {x} (X is defined as X with the use of ḡ). Using (normal) cartesian coordinates {x µ } centered on p we have Γ λ p ν q = δ λ p ν p [START_REF] De Felice | Relativity on curved manifolds[END_REF]. By construction, this operator accounts only for the change of tensors caused by an arbitrary basis. This is also evident from the following interpretation of the operator action. The scalar character of X µp (x) at x = x (η q ) and definition (2.5) allow us to write X µpνq = ∂X µp /∂x νq . On the other hand, from the chain rule

Γ µp νq = Γ λr n νq ... Γ λr 1 λr 2 Γ µp λr 1 , (2.9)
with η p < η r1 < η r2 ... < η n < η q we have from (2.9), in the limit n → ∞,

X µp νq = ∞ i=1 ∂X λi-1 ∂x λi , (2.10)
with λ o = µ p and λ ∞ = ν q . This relation can be regarded as an infinite product of changes of coordinate operated from the infinitesimally near points r i-1 to r i of the geodesic σX in the active interpretation point of view. In fact, the variation ∂X λi-1 of a connecting vector X λi-1 of two points r i-1 and r i , on a geodesic straight line σX ( Rµνρσ = 0), due to the (partial) change of the coordinate x λi-1 , is equivalent to the effect of the change from the basis e λi-1 to the basis {e λi }, or equal to ∂x λi , with {x } coordinates of e λi i in the passive point of view. Thus in (2.10) we can use

∂X λi-1 ∂x λi = ∂x λi ∂x λi , (2.11)
from which follows the interpretation. In conclusion, if V µp is a tensor at p, then V νq = V µp X νq µp is the "equivalent" vector at q corrected with the entire basis history change along the trajectory. The operator X µpνq will be used in the the right hand side of (2.4) in order to obtain a tensor at q by saturation of both indexes at p. Finally, taking into account (2.4), a final change is required, in order to consistently use the Taylor theorem. In fact, in its genuine formulation, all expansions and approximations depend rigorously on the variable increment δx µ (x) = x µ -x µ o and on the coordinate position x, not on an arbitrary function of the coordinate position, like X µ (x). Thus, because of the tensorial character of our expansion, we must substitute X µ (x) in (2.4) with X µ (x), the only quantity which reduces, in cartesian coordinates (and without ambiguous mixing of various order of approximation in the performed metric expansions) to a coordinate increment in any point. This problematic aspect appears also in the Dixon formalism but only after the introduction of tensor-valued distributions. In fact, while we choose to Taylor expand directly on the foliation Σ induced on ∆ by X (x) (note that X µ (q) γµ (p) = 0), the Dixon approach employs as a preliminary step a Taylor-like expansion on the tangent space M p at p using the vector X µ ∈ M p but in a formulation that requires the introduction of a tensor-valued distribution of the energy-momentum tensor T µν on M p .

With the use of the above introduced tools the covariant expansion of the metric is thus

g µν = ḡµν + (2.12) ∞ n=1 1 n! G ρ o λ o κ1 o ...κn o X ρ o µ X λ o ν X κ1 o ...X κn o ,
where we have suppressed the subscript q and used the o instead of p. Note that, in a normal coordinate system {x } centered on p, this expansion reduces to the ordinary Taylor expansion or

g µ ν = ∞ n=0 1 n! g µ o ν o ,κ 1o ...κ no δx κ1 o ...δx κn o . (2.13)
The use of the expansion (2.12) is a crucial point in order to have a clear and manageable invariant variational formulation of the multipolar test body dynamics.

Covariant expansion of the lagrangian

In this section we expand the lagrangian in the action (1.1) following the ADS approach, but in a full covariant way. This give the possibility to generalize ADS approach to all order in the metric expansion.

In a general coordinate system let x = x (q) and x o = x (p) = γ (λ). Thus, assuming that the lagrangian L depends on the field variables and on g µν , not on its derivatives, as it happens for all known reasonable matter distributions, we have

S = ∆ L ḡ √ -ḡd 4 x + ∆ ∂L √ -g ∂g µν ḡ δg µν d 4 x + 1 2 ∆ ∂ ∂g ρσ ∂L √ -g ∂g µν ḡ δg ρσ δg µν d 4 x... (3.1)
where δg µν (x) = g µν (x) -ḡµν (x) and L ḡ = L| g(x)=ḡ(x) and so on. In (3.1), L ḡ cannot contains the Riemann tensor or its covariant derivative and so it encodes all special relativistic dynamics. The perturbed lagrangian, obtained by the repeated application of the operator ∂ ∂gµν ( ) ḡ δg µν , describes, through the covariant expansion of the metric, all general relativistic contribution to the dynamics at the order desired.

To write explicitly higher order term, we express (3.1) in a more useful form. By construction X µo µ is a bitensor in the ḡ metric, thus we have ḡµν

= X µo µ X -1 νµo = X µo µ X µoν = ḡµoνo X µo µ X νo ν . (3.2)
In posing δg µν = δg µoνo X µo µ X νo ν with

δg µoνo = ∞ n=1 1 n! G µoνoκ1 o ...κn o X κ1 o ...X κn o , (3.3)
we have

∂L √ -g ∂g µν ḡ δg µν = ∂L ḡ √ -ḡ ∂ḡ µoνo δg µoνo (3.4) and ∂ ∂g ρσ ∂L √ -g ∂g µν ḡ δg ρσ δg µν = (3.5) ∂ ∂ḡ ρoσo ∂L ḡ √ -ḡ ∂ḡ µoνo
δg ρoσo δg µoνo and similarly for higher order terms. We now express the measure in the integrals in a way useful to obtain a parametric invariant action. Let Σ be the space-like surface section induced on ∆ by σX . We can split the invariant proper volume as follows √ -ḡd 4 x = ldλd 3 Σ.

(3.6)

with l = l x (λ) = {-ḡ µν [x (λ)] ẋµ (λ) ẋν (λ)} 1/2
the length factor of a congruence of time-like trajectories x (λ) orthogonal to Σ (λ) at every point x ∈ Σ ((∆, ḡ) is globally flat), with x o (λ) ≡ γ (λ) parametrized in such a way to have

l x = l xo = {-g µν [x o (λ)] ẋµ o (λ) ẋν o (λ)} 1/2 constant in Σ. Thus, posing ldλM κ1 o ...κn o (λ) = (3.7) 2 Σ(λ) L ḡ X κ1 o ...X κn o √ -ḡd 4 x
we define all 2n-multipole moments as follows

M µoνoκ1 o ...κn o (λ) = (3.8) ∂ ∂ḡ µoνo M κ1 o ...κn o (λ)
and, from them, other derived quantities, such as

M ρoσoµoνoκ1 o ...κn o (λ) = (3.9) ∂ ∂ḡ ρoσo M µoνoκ1 o ...κn o (λ)
and so on for other quantities M α1 o ...αm o κ1 o ...κn o with m odd integer. Observe that

M α1 o ...αm o κ1 o ...κn o = (3.10) M (α1 o α2 o )...(αm-1 o αm o )(κ1 o ...κn o ) and M α3 o α4 o ...αm o κ1 o ...κn o = (3.11) ḡα1 o α2 o M α1 o α2 o α3 o α4 o ...αm o κ1 o ...κn o
and similarly for other contractions. By construction, all 2n-multipole moment tensors (3.8) and all the other derived tensors, are space-like in the index κ 1o ...κ no with respect to the reference world-line γ (λ). Note that if L is linear in the metric, only the 2nmultipole moment (3.8) will be employed in the expansion. Higher power dependences of L on the metric will determine the use of the other quantities introduced.

Using this apparatus, turning back to (3.1) and (3.6) we have

S = γ L (λ) dλ = γ [L g (λ) + L δg (λ) (3.12)
+ L δg 2 (λ) + ...]dλ, where

L g = l Σ(λ) L ḡ d 3 Σ (3.13)
is the unperturbed lagrangian in Σ (λ), whose physical content must reproduce the special relativistic dynamics. Analogously,

L δg = l n 1 2n! M µoνoκ1 o ...κn o (3.14) • G µoνoκ1 o ...κn o ,
where n runs over the desired 2n multipolar order. L δg is the first order multipolar expansion of the lagrangian, which encodes the influence of the gravitational field on the dynamics. Here, l ≡ l γ (λ) = [-g µν γµ (λ) γν (λ)] 1/2 is the length factor along γ which reduces to unity in the proper time parametrization. Using the definition of the energy-momentum tensor in (3.8) we have

M µoνoκ1 o ...κn o (λ) = (3.15) Σ(λ) T µν ḡ X µo µ X νo ν X κ1 o ...X κn o d 3 Σ,
Higher order terms of the expansion are

L δg m = l N 1 A N G α 1 ...α M κ 1 ...κ N (3.16) • M α 1 ...α M κ 1 ...κ N ,
where m = M/2 is the order of the metric expansion, N = n + p + ... + q, while A N = n!p!...q!. In (3.16) we have suppressed the subscript o and used the quantity

G α 1 ...α M κ 1 ...κ N = G α 1 α 2 κ 1 ...κ n (3.17) • G α 3 α 4 κ n+1 ...κ n+p ... • G α M -1 α M κ N -q+1 ...κ N
Finally, we use the energy-momentum conservation law to obtain the symmetry properties of the multipole tensors. Consider the quantity ∇µ T µν ḡ X νo ν . In a particular global coordinate system in which

X ν o µ = δ ν µ it becomes ∇µ T µ ν ḡ X ν o ν = 0
X νo ν X µo X κ1 o ...X κn o (3.18) = -(n + 1) X νo ν X µo µ T ν(µ ḡ X κ1 o ...X κn o ) .
Integrating this equality over Σ (λ) with the measure √ -ḡd 4 x we obtain the symmetry properties of the 2n-multipole moments

M νo(µoκ1 o ...κn o ) = 0. ( 3
M α 1 ...(α M κ 1 ...κ N ) = 0. (3.20)
Some brief comments are necessary now. First, note that the expansion of the lagrangian is not limited to the first order in the metric variation, which contributes to a more generality of our equations than to those of ADS. A further observation concern the expression of the multipole moments (3.15), which are only vaguely similar to those of Dixon. This is because in some case Dixon defines multipole moments containing only higher-order information, the monopole moment (n = 0) and the dipole moment (n = 1) being identically vanishing [4](1979). A more important discrepancy lies on the fact that the Dixon resulting expressions of the multipole moments [4](1974 and 1979) are obtained by intergrations on a space-like hypersurface orthogonal to the momentum P µ , while our 3-dimensional sections Σ over which we define our quantities are orthogonal to the reference world line γ µ . This shows that the re-definition of the canonical momentum operated by ADS [13] (up to an unknown term transversal to the four velocity) in order to obtain formally the same dynamical equations of Dixon, is not necessary and is simply conventional, also because ADS use the Pirani S µν u ν = 0 condition [16] instead of the Tulczyjew-Dixon condition [4][17] S µν P ν = 0. As the techniques and the definitions used are different, the present method and the Dixon theory give rise -even though formally identical -only to similar dynamical equations (Section 6). The present multipolar method seems to be more simple and manageable compared to that of Dixon, because of the more simple mathematical tools used, thanks of the cited orthogonality and to a simpler calculability of multipole tensors by virtue of the use of a flat metric ḡ. Besides, by the introduction of the sections Σ we will obtain in Section 6 the simple Papapetru expression of the canonical momentum which, remaining unchanged to all orders, contributes to some reductions of complexity of the dynamics.

In the next Section we will give the explicit form of L up to the desired order.

The lagrangian structure

The purpose to find the lagrangian functional requires some care about on the order of expansions we will use in the next. The hypothesis of a weak metric variation within the spatial body extension force us to define consistently the spatial size Λ of the body. We designate this quantity as the maximum of X σ in ∆. Thus, for a body moving in a spacetime having radius of curvature R, we adopt the ADS assumption that the spatial size Λ of the body be small compared to R, or Λ/R ≈ O (1). The ADS method is based on a generic assumption about the weakness of the metric but, to our purposes, it is sufficient to assume a Dixon-like criterion, requiring that the induced metric h µν in Σ be weak. Thus Λ/R ≈ O (1) means exactly that the variation of the gravitational field must be small on Σ (what is, actually, the Dixon criterion [4](1979) p.190 and p.196).

The notion of spatial size Λ enables us to define another parameter in the expansion, which accounts for the 'surface velocity' v s of the body, interpreted as the spatial velocity of the points of ∂ Σ as measured by a Fermi-Walker observer on γ. We adopt the constraint that the Fermi-Walker observer will not experience a surface horizon on the region occupied by the body. Being v s ≈ ΛΩ with Ω angular velocity of the body (see the next Section), this condition is equivalent to make all expansions up to the standard (v s /c) 2 , or to consider negligible all quantities of order O (ΛΩ) 3 .

Thus, our goal is to find the explicit form of the lagrangian up to this order. Finally, because we wish to carry out exemplifications up to the quadrupole moments of the body, it is sufficient to use 2 n=1 (see [13] for details) in the multipole expansion (3.14). Thus m = 1 and n = 1, 2 in (3.16). In this case we will consider negligible all the quantities of order O (δg)

2 , (ΛΩ) 3 , (Λ/R) 3 .

The first step amounts to define in a standard way the above mentioned angular velocity Ω. To this end we use an orthonormal tetrad field e a (λ) defined on the representative world-line of the body γ(λ):

η ab e a µ (λ)e b ν (λ) = g µν (λ). (4.1)
Consider the tensor

Ω µν ≡ ėaµ e a ν -2e o[µ ėoν] = ω µν -2u [µ uν] (4.2)
with e µ a dual of the tetrad field and ω µν = ω [µν] , having defined the normalized four velocity as

u µ = γµ l . (4.3)
We have

Ω µ ν u ν = 0. (4.4)
The tensor components of Ω µ ν are the Fermi rotation coefficients and represent the spatial angular velocity tensor of the body, while ω µν is the corresponding spacetime generalization. The spatial angular velocity vector is

Ω µ ≡ * Ω κµ u κ . (4.5)
Note that Ω µ ν fulfills a crucial property: because of the spatial summation on the Lorentz indices

Ω µν = ėα[µ e α ν] -u [µ uν] , α = 1, 2, 3 (4.6)
then Ω µ ν is invariant with respect to the Lorentz subgroup L rot of spatial rotation transformations e a ν = Λ a rotb e b ν , provided that the Lorentz spatial rotation matrix Λ a rotb is constant along the trajectory Λ a rotb ˙= 0. (4.7)

Our main purpose is to derive all the equations of motion, for the trajectory and for the spin of a test body, by means of a parametric invariant action principle δ γ L (λ) dλ = 0, for a total variations of the trajectory γ µ (λ) and of the tetrad field e a (λ) with fixed endpoints. This is just the BI formalism [12]. Thus we write the unperturbed Lagrange functional as

L ḡ ≡ L ḡ [l (λ) , g (λ) , γ(λ), e(λ), ė(λ), M(λ)] , (4.8)
the dot meaning the convective covariant derivative along the world-line. L ḡ must necessarily depend only on quantities defined on the reference trajectory as, e.g., on the multipole moments M µνκ1...κn (λ). As we pointed out, L ḡ does not depend on derivatives of the metric tensor and must reproduce, in the non relativistic limit, the classical physics of a body in a constant gravitational field or in a free falling reference frame. Now we assume a dependence of the lagrangian on the tetrad field e µ a and ėµ a is accomplished through the angular velocity tensor Ω µν

L ḡ = L ḡ [l (λ) , γ (λ) , Ω(λ), M(λ)] . (4.9)
With this choice L ḡ will be invariant under the internal Lorentz subgroup of spatial rotation constant along the trajectory, the physical meaning being immediate: the spatial angular velocity tensor Ω µ ν and the spatial angular velocity vector Ω µ will be obviously unchanged no matter the tetrad e a ν is linked to the body. Another important symmetry is required: the action S = γ L (λ) dλ must be parametrically invariant. This is accomplished provided that L is a 1st-degree functional polynomial in the γµ 's, or

L ≡ ∂L ∂ γµ γµ . (4.10)
Thus, the symmetries described will give the constraints required to construct the covariant model up to the order desired.

The model

First, let us search for the explicit form of the unperturbed lagrangian reproducing, in the non relativistic limit, the dynamical evolution in a constant gravitational field or in a free falling reference frame. Note that in the ADS work the part of the lagrangian corresponding to our L ḡ remains unknown because the internal degrees of freedom are uncorrelated.

We start by decomposing the unperturbed lagrangian as

L ḡ = L M + L S , (5.1)
where

L M = -lM, (5.2)
is the O (ΛΩ) 0 contribution to the lagrangian, M ≡ M µ µ being obtained from (3.15) with n = 0. This lagrangian term manifestly satisfies (4.10). L M is easily interpreted as the mass lagrangian of the body.

For the other coupling we use as a guide the Newtonian analogy. We observe that the O (ΛΩ)

1 contribution to the lagrangian obtained by means of the geometrical object M µκ1 µ is absent by dimensionality. Thus a condition on this tensor will result as a gauge. If we assume the most simple gauge M µκ1 µ = 0, we obtain the usual condition which states that the reference world line γ will be the center of mass of the system as regards to which the subspace Σ is defined and the quadrupolar tensor is referred.

The last task is to find the O (ΛΩ) 2 spin lagrangian contribution L S , which describes the spin of the body. First, we introduce a natural definition of the antisymmetric spin tensor as

S µν ≡ 2M κ [ν Ω µ]κ , (5.3)
with the spin vector being

S µ ≡ * S κµ u κ . (5.4)
Here,

M κ1κ2 ≡ M µ µκ1κ2 (5.5)
and from the spatial character of the multipole moments we have the property

M µ ν γν = 0 (5.6)
and consequently both the spin tensor S µν and the spin vector S µ , like Ω µν and Ω µ , will have spatial character

S µ ν u ν = 0. ( 5.7)
Thus the spin tensor satisfies the Pirani condition [START_REF] De Felice | Relativity on curved manifolds[END_REF]. Finally, the exact form of the rotational energy L S must reproduce the Newtonian limit. Thus we assume

L S = - 1 4 lS µν Ω µν . (5.8)
Note that L S can be written also as

L S = 1 2 lΩ σκ M κλ Ω λ σ (5.9)
and therefore it turns out to be the most simple scalar built out with two tensors, symmetric and antisymmetric, defined along a world line γ. It is easy to see that any other coupling constructed by means of the multipolar tensors and the other variables must be of higher order than O (ΛΩ) 2 or must violate the dimensionality.

Thus, collecting all the definitions, the explicit parametric invariant lagrangian satisfying the imposed symmetries is

L = -l M + 1 4 S µν Ω µν (5.10) + l M,N 1 A N G α 1 ...α M κ 1 ...κ N M α 1 ...α M κ 1 ...κ N .
If we limit ourself to study the quadrupolar expansion, we must restrict to the O (δg) 2 , (ΛΩ) 3 , (Λ/R) 3 approximation. Thus, using the relations [START_REF] Veblen | [END_REF] G µνρ ≡ 0 (5.11)

G µνρσ ≡ - 2 3 R µ(ρσ)ν ,
we get

L δḡ = - 1 6 lM µνρσ R µρσν (λ) ,
(5.12) having used the symmetry property of the quadrupolar tensor. To this order (5.10) becomes

L = -lM - 1 4 lS µν Ω µν (5.13) - 1 6 lM µνρσ R µρσν .
Because the tensor ω µν = ėµ a e aν depends only on the free tetrad lagrangian variables e µ α and ėµ α (α = 1, 2, 3) and on e µ 0 = u µ , then from (4.3) it is a zero degree polynomial in γµ . From (5.3) and (4.2), also S µν is of the same degree, while l is of order one. Thus, from (5.13) we easily verify that the total lagrangian L is of order one and that it satisfies (4.10), as required.

In conclusion we have found the explicit form of the lagrangian of a test tiny body in a curved space time. In the next Section we will impose an analytical constraint on the structure of the multipole moments in order to give rigidity to the body. All the equations of motion will be then derived through a minimal action principle applied on the variation of the trajectory γ µ and on the variation of the rotational variable e a µ .

The Minimal Action Principle

In order to apply the variational procedure to the lagrangian functional (5.10), we consider the covariant variation of the trajectory carried out by the operator

δ γ ≡ δγ κ ∇ κ . (6.1)
Contrarily to the BI approach, we do not assume that the operator δ γ holds the tetrad fixed by parallel propagation, but that it is extended by Fermi-Walker transport in order to preserve its character. Thus δ γ e a µ = 0. Indicating by δ e the operator which acts on the internal lagrangian variables e a µ , the total variation will be described by

δ = δ γ + δ e . (6.2)
• Variations

Applying the total variation to the lagrangian we have

δL = ∂L ∂ γµ δ γµ + ∂L ∂ω µν δω µν + ∂L ∂M δM (6.3) + ∂L ∂M µν δM µν + M,N ∂L ∂M α 1 ..α M κ 1 ...κ N δM α 1 ...α M κ 1 ...κ N + M,N ∂L ∂G α 1 ...α M κ 1 ...κ N δG α 1 ...α M κ 1 ...κ N , with P µ = ∂L ∂ γµ (6.4)
the canonical momentum of the system. In (6.3) we have used the obvious covariant variation δ γ g µν = 0 and assumed the condition δ e g µν = δ e ḡµν = 0 because the geometry is kept fixed (observe that g µν (λ) = ḡµν (λ)).

As a first step, we derive the analytical expression of P µ . From (5.10) we have

P µ = -Lu µ + Ṡνµ u ν . (6.5)
This expression coincides with the simple Papapetru [2] but here it holds to all order. Thanks to our formalism, this Papapetru relation between the moment and the other quantities is simpler than the corresponding non-linear Dixon-Ehlers-Rudolph one (see [4](1979) and [6]).

We consider now the second term in (6.3). A direct calculation yields

∂L ∂ω µν = - l 2 S µν . (6.6)
Note that (6.6) would have been equivalent to a definition of the spin tensor if its explicit form (5.3) were unknown. This happened in the BI approach, where the abstract definition of the spin tensor was indeed performed without any model specification [12]. When applied to our explicit lagrangian, but with an (l) -1 factor in front, the BI like definition correctly gives the spin tensor,

S µν = (l) -1 2e a[µ ∂L ∂ ėν] a = 2 l e a[µ ∂L ∂ω κλ ∂ω κλ ∂ ėν] a = - 2 l ∂L ∂ω µν . (6.7)
Then, with the use of the proper time parametrization λ = τ , the length factor disappears and the two definitions become coincident. The third step will amount to evaluate the dependence on the quadrupolar moments in (6.3). It is easily derived as

∂L ∂M µν = 1 2 Ω µ κ Ω κν l = 1 2 Ω µ Ω ν -Ω 2 h µν l (6.8)
and so on for other higher order variations. Finally, to perform the total variational procedure, we display the explicit form of the variations δ ẋµ , δω µν , δM µν , δM µνρσ , etc. Using (6.1) and (6.2), the variation of the unnormalized four-velocity yields

δ γµ ≡ δ γ γµ = δγ κ (∂ κ γµ + Γ µ κν γν ) = (δγ µ )˙(6.9)
where we have made use of the first order relations The variations of the multipolar tensors are obviously at the core of the rigidity condition. Applying the total variation we have δM µν = δ e µ a e ν b M ab (6.15) = 2M κ(µ δ λ ν) e aκ δe aλ + e µ a e ν b δM ab . For higher order term, using the symmetries properties (3.10), we have

δγ κ ∂ κ γµ = γµ (x ) -γµ (x) ≈ dδγ µ dλ = γκ ∂ κ δγ µ . ( 6
δM α 1 ...α M κ 1 ...κ N = M l=1 δe α l a l M α 1 ...a l ...α M κ 1 ...κ N (6.16) +N δe κ N b M α 1 ...α M κ 1 ...κ N -1 b +e α1 a1 ...e α M a M e κ1 b1 ...e κ N b N δM a 1 ...a M b 1 ...b N .
But what about δM ab or δM a 1 ...a M b 1 ...b N , the variations of the multipolar tensor in the body reference frame, induced by the variation of the trajectory?

• Relativistic Rigidity Condition Let γ µ (λ, ) be a 1-parameter family of time-like curves with both definitions, of the orthonormal tetrad field e aν (λ, ) and of the multipolar tensor field M α 1 ...α M κ 1 ...κ N (λ, ), extended on them, being an infinitesimal parameter. Then e aν (λ, ) is the Fermi-Walker transported of e aν (λ, 0) along γ µ (λ, ). A slightly varied curve γ µ (λ) with respect to the reference trajectory γ µ (λ) ≡ γ µ (λ, = 0) with endfixed points is given by the smooth arbitrary function of re-parametrization (λ), such that γ µ (λ) ≡ γ µ [λ, (λ)]. Consider an arbitrary variation of the reference curve γ µ (λ)

δγ µ ≡ γ µ (λ) -γ µ (λ) = ∂γ µ ∂ δ . (6.17)
Then

δ γ M a 1 ...a M b 1 ...b N = ∂γ µ ∂ ∇ µ M a 1 ...b N δ = ∂γ µ ∂λ + ∂γ µ ∂ d dλ - dγ µ dλ ∇ µ M a 1 ...a M b 1 ...b N δλ (6.18) M a 1 ...a M b 1 ...b N ˙γ µ -M a 1 ...a M b 1 ...b N ˙γµ δλ,
where we have used ∂γ µ /∂λ ≡ dγ µ /dλ and the subscript in the dot derivative denotes along which curve the covariant convective derivative is computed. The last relation suggests to assume

M a 1 ...a M b 1 ...b N ˙γ µ (λ) = 0, ∀γ µ (λ) , (6.19)
or

δ γ M a 1 ...a M b 1 ...b N = 0. (6.20)
which is equivalent to the total variational condition δM a 1 ...a M b 1 ...b N = 0 (6.21) in (6.16), because of e µ a δ e e aν = 0. All this relations can be obtained from the following relativistic rigidity condition on the 2n-multipole moments

δM a 1 a 2 b 1 ...b N = 0. (6.22) or M a 1 a 2 b 1 ...b N ˙= 0. (6.23)
In fact (6.21) can be easily obtained from (6.23) using (3.11) and the relations δ e ḡa i a j = δ γ ḡa i a j = 0. Formally our relativistic rigidity conditions on the multipole moments coincide with that of Dixon [4]. Contracting (6.21) we have also δM ab = 0 and the condition of the constance of the 'rest mass' δM = 0. The interpretation is straightforward: because of rigidity, in the body reference frame every non inertial stress caused by rotation, or curvature, leaves unchanged all tensors M a 1 ...b N . This considerably simplifies the variational procedure. On the contrary, in the case of a non rigid body, it will be necessary to specify an equation of state describing the internal structure of the body and the evolution of the energy momentum tensor along the trajectory.

• Equations of motion

Applying the action principle to the lagrangian, and using the equations obtained in the previous subsection, after a tedious calculation we have δL = X µ δγ µ + E µa δe aµ + S (6.24)

with

X µ = - ∂L ∂ γµ . + ∂L ∂ω κν R κν µλ γλ + l M,N 1 A N M α 1 ...κ N ∇ µ G α 1 ...κ N (6.25)
and

E µν = - ∂L ∂ω µν . + 2 ∂L ∂ω κ[µ ω κ ν] + (6.26) 2 ∂L ∂M κ[µ M κ ν] + l M,N 1 A N cp G [µ|α 2 ...κ N | M • α 2 ...κ N ν] + l M,N N A N G α 1 ...κ N -1 [µ M α 1 ...κ N -1 • ν]
S being a surface term. In (6.26) cp means the sum over the cyclic permutation of the position of the indexes of the tensors included in the sum (for example

cp A µ νσ B ρκλ = A µ νσ B ρκλ + A νσ µ B κλρ + A σ µ ν B λρκ ).
Because of the independence of the variations δγ µ and δe aµ the minimal action principle gives

X µ = 0, E µν = 0. (6.27)
On the other hand, using in what follows an affine parametrization, from (6.6), (6.8), (4.4) and (6.5) we have

2 ∂L ∂ω κ[µ ω κ ν] + 2 ∂L ∂M κ[µ M κ ν] (6.28) = Ṡκ[ν u µ] u κ = P [ν u µ] .
Thus, with the aid of (6.4) and (6.6) and the previous equations, in an affine parametrization the constraints (6.27) become the equations of motion to all order

Ṗµ = 1 2 R κνσµ S κν u σ + M,N 1 A N M α 1 ...κ N ∇ µ G α 1 ...κ N (6.29) and Ṡµν = 2P [µ u ν] + (6.30) 2 M,N 1 A N cp G [ν • α 2 ...κ N M µ]α 2 ...κ N + 2 M,N N A N G [ν α 1 ...κ N -1 • M |α 1 ...κ N -1 |µ] .
By defining, similarly to Dixon [4], the following tensor

J κ ρν σ = M [κ [νρ] σ] (6.31)
and using M = 1 and N = we have

2 ∂L ∂M [µ|κρσ| M ν]κρσ + 2 ∂L ∂M κρσ[µ M |κρσ|ν] = - 2 3 R [µ|ρσκ| J |κρ|ν]σ , (6.32)
so, in the quadrupolar approximation, the general equations (6.29) and (6.30) become

Ṗµ = 1 2 R κνσ µ S κν u σ + 1 6 J κρσν ∇ µ R νκρσ (6.33) and Ṡµν = 2P [µ u ν] + 4 3 R [µ ρσκ J ν]σκρ . (6.34)
As previously stated, equations (6.33) and (6.34) formally coincide with those of Dixon (apart from a sign convention and the metric signature). The present formalism is suitable in searching for interesting implications when we have to deal with cyclic variables.

Conserved quantities

In this Section we derive the expressions of the conserved quantities coming from the existence to cyclic variables in the lagrangian. We obtain this result to all order in the expansion of the lagrangian. Let x c be a cyclic variable of the metric. Thus

∂ c g µν = 0 Γ λ cσ A σ λ = Γ λ cσ A [σ λ] . (7.1)
Consider the first term in the right hand side of the equation of motion (6.29) having posed µ = c. We have

1 2 R κ νσc S κ ν u σ = 1 2 ∂ σ Γ κ cν + Γ κ ρσ Γ ρ cν -Γ κ ρc Γ ρ νσ S κ ν u σ = 1 2 d dλ Γ κ cν S κ ν + 1 2 Γ ρ cν Γ κ ρσ S κ ν u σ -Γ ν κσ S ρ κ u σ = 1 2 d dλ (Γ κ cν S κ ν ) - 1 2 Γ κ cν D Dλ S κ ν . (7.2)
Using the last term and the equation of motions (6.30) and the last of (7.2) we obtain

1 2 Γ κ cν D Dλ S κ ν = Γ κ cν P κ u ν - M,N 1 A N Γ κ cν Σ M l=1 δ α l κ G α 1 ...α l ...κ N δ ν α l M α 1 ...α l ...κ N - M,N N A N Γ κ cν G α 1 ...κ N -1 κ M α 1 ...κ N ν , (7.3)
α l being summed within the same round brackets. On the other hand the last term in (6.29), in the case of the cyclic component, becomes

M,N 1 A N M α 1 ...κ N ∇ c G α 1 ...κ N = - M,N 1 A N Σ M l=1 M α 1 ...α l ...κ N δ ν α l Γ α l cν G α 1 ...α l ..κ N - M,N N A N M α 1 ...κ N Γ κ cκ 1 G α 1 ...κκ 2 ...κ N . (7.4)
Collecting (7.2) and (7.3), the equations of motion (6.29) give

D Dλ P c = 1 2 d dλ (Γ κ cν S κ ν ) -Γ κ cν P κ u ν (7.5) or d dλ P c + 1 2 Γ κ cν S κ ν = 0 (7.6)
by which we obtain the conserved quantities along the trajectory of the rigid test body corresponding to the cyclic coordinate x c

Q c = P c + 1 2 Γ κ cν S κ ν (7.7)
that, using (6.5), explicitly are

Q c = -Lu c + S c κ uκ + 1 2 Γ κ cν S κ ν . (7.8)
Note that these conserved quantities, not found by BI-ADS, hold to all order in the multipolar expansion.

In order to give concrete examples of our results useful in applications, in the next Section we will investigate some conserved quantities and some properties of the relativistic moment of inertia tensor by the introduction of the principal axes of inertia and of the axial moment of inertia of a rigid test body.

Relativistic dynamics of a spinning rigid test body

In what follows we give some examples of the dynamical evolution of a test rigid body in the quadrupolar approximation.

• Conserved quantities

As an example of (7.8) we consider the conserved quantity E = -1 2 cQ t associated to the cyclic temporal coordinate x t in the Schwarzschild metric in the polar coordinates. In standard units, using c as the speed of light, we have

E = - 1 2 cQ t = E M + E T + SO , (8.1)
with

E M = cLu t (8.2) E T = -cS t κ uκ E SO = - 1 2 cS κλ ∂ κ g tλ .
Consider the field k = (-g tt )

-1 2 ∂ t of static observers in the space time outside the horizon. It will use the tetrad

∂ t = (-g tt ) -1 2 ∂ t (8.3) ∂ r = (g rr ) -1 2 ∂ r ∂ θ = (g θθ ) -1 2 ∂ θ ∂ φ = (g φφ ) -1 2 ∂ φ . Thus we have u t = -1 - v 2 c 2 -1 2 1 - 2m r 1 2 c (8.4)
where v is interpreted as the local velocity of the body as measured by k. Using the quadrupolar expansion (5.13) of the lagrangian, in the weak gravitational region of space-time where r >> 2m = 2 GM c 2 , neglecting the term of O v 3 /c 3 , the first term in (8.2) becomes

E M = Mc 2 + 1 2 Mv 2 -G M M r (8.5) + 1 4 S µν Ω µν + 1 6 M µνρσ R µρσν c 2
having used the standard units redefinitions S µν → cS µν and Ω µν → cΩ µν . In the r.h.s. of (8.5), the first three terms are the barycentric energy contribution of the rigid body. The remaining terms encode effects coming from the internal structure of the body: the spin rotational energy and the gravitational energy due to quadrupolar distribution of the "internal masses" within the small space-like region Σ (τ ) occupied by the body and comoving with it. Thus E M is the mechanical energy. The origin of E T in (8.2) is equally well justified: it is the Thomas precession energy contribution of the spin caused by the non inertial motion of the body.

The last term seems more obscure but it can be rapidly calculated. Using the Schwarzschild metric, equation (5.4) and the tetrad (8.3), we have

E SO = - 1 2c √ -gδ trρσ u ρ S σ ∂ r g tt (8.6) = csinθ u θ S φ -u φ S θ = - m r ω θ S θ + ω φS φ
where we have defined the local angular orbital velocity ω θ = -c θ and ω φ = c φsinθ. By defining the spatial angular momentum vector measured by the observer K at infinity as l = Mr 2 ω and the zero order geodesic potential V = -GM M r we finally have

E SO = 1 M 2 c 2 1 r ∂V ∂r S • l. (8.7)
It is the spin-orbital coupling of gravitational origin: the right hand of equation (8.7) is formally double of the analog quantity in the special relativistic case of an atom immersed in a magnetic field, the factor 2 being of general relativistic origin. Summing all the terms the we have a complete description of the conserved total energy. Analogous examples could be derivable from the angular conserved quantity coming from the cyclic variable x φ in the Schwarschild metric.

• Spin dynamics

Now we briefly study the intrinsic rotational dynamics of the body. In a proper time parametrization, (4.5) and (5.3)- (5.6) give

S µ = (M κ κ δ µ ν -M µ ν ) Ω ν . (8.8)
The quantity in round brackets is formally identical to the Newtonian expression of the moment of inertia tensor in cartesian coordinates [START_REF] Goldstein | Classical Mechanics[END_REF]. This suggests the following definition of the relativistic moment of inertia tensor I µν

I µν ≡ (M κ κ g µν -M µν ) . (8.9)
By the use of this quantity, the spin vector has the form

S µ = I µ ν Ω ν . (8.10)
In a general coordinate system and in completely relativistic domain, the rigidity condition (6.19) holds as

ݵν = 2I κ (µ ω ν)κ , (8.11)
which is formally similar to the classical one [START_REF] Goldstein | Classical Mechanics[END_REF], apart from the use of a covariant convective derivative and the occurrence of the spacetime angular velocity tensor ω νκ instead of the space angular velocity tensor Ω νκ . The simple use of the fluid T µν = ρu µ u ν in the definition of multipolar tensors would tell us that (8.9) is indeed a good definition.

In order to go on and to introduce the principal axis and axial moments of inertia of the body, we specialize our quantities in a tetrad. M µν it is a real symmetric tensor, then it is an orthogonal matrix and consequently diagonalizable. It has 10 independent components or degrees of freedom. Consider the tetrad t µ a and the decomposition M µν = M ab t µ a t ν b . M ab is diagonalizable too. Using the L rot symmetry of the model, we operate the ordinary diagonalization procedure with a change of the tetrads t µ a → e µ a , where the use of the symbol e µ a is in order to assign a special character to this tetrad: e µ a is the tetrad which diagonalizes the Lorentz mass tensor M ab and therefore the Lorentz moment of inertia tensor I ab . Thus we have

M µν = 3 a=0 M a e µ a e ν a , (8.12)
with M a a Lorentz vector, with 4 degrees of freedom. Because the four bivectors e µ a e ν a are constructed with the use of a tetrad e ν a , they possess 6 degree of freedom: 16 degrees belonging from the vectors components of the tetrad, reduced to 6 by the imposition of the 10 orthonormal conditions (4.1). Thus the decomposition (8.12) contains the 10 exact degrees of freedom of a symmetric tensor.

But what are the principal axes? It is easy to show that, being identically M 0 = 0, the transformation

M a ≡ 1 2 I a+1 + I a+2 -I a (8.13) a + i = 1, 2, 3 mod (3)
realizes the diagonalization of the moment of inertia tensor

I µν ≡ 3 a=1
I a e µ a e ν a -

1 2 3 a=1 I a u µ u ν . (8.14)
Notice the separation between the spatial and the temporal components. The form of (8.14) suggests that the triad e aµ (a = 1, 2, 3), is the triad of the principal axis of inertia. The diagonal form of (8.14) shows also that (8.13) implicitly defines the axial moments of inertia I a , which behave as a Lorentz spatial vector in the tetrad utilized. Using (1.2) we can invert (8.14) obtaining

I a = Σ(λ) T κ ḡ κ X a+1 2 + X a+2 2 d 3 Σ (8.15) a + i = 1, 2, 3 mod (3)
, from which we easily recover the well known Newtonian expression in the non relativistic limit. Besides, the use of the rigidity condition (8.11) gives, after a short manipulation, the rigidity property of the axial moment of inertia vector (I a )˙= 0. (

With the aid of (8.14) the spatial spin vector, (8.10) takes the form

S µ = 3 a=1
I a e µ a e ν a Ω ν . (8.17)

In the case of the most simple model of a rigid body, the spherical rigid body, it is obvious to define it by imposing

I 1 = I 2 = I 3 ≡ I (8.18)
In this case, being 3 a=1 e µ a e ν a = h µν , (8.17) becomes S µ = IΩ µ . (8.19) showing that in the case of a spherical body, the spatial spin vector S µ and the spatial angular velocity vector Ω µ are aligned, as expected.

We now focus our attention to the spin vector evolution. Using (5.4) and (6.34) it is easy to prove that Ṡµ = S κ uκ u µ + 2 3 R λ ρσκ J κρνσ µ λν (8.20) or that the usual special relativistic Fermi-Wolker transport of the spin is broken by the curvature term. From this evolution equation, using the spin expression (8.10) and the rigidity condition (8.11) after a short calculation we have also

I µν Ων -Ω κ uκ u ν = (8.21) κµσ I σ ν Ω ν Ω κ + 2 3 R ρ κλν J ν ρσ λ .
These are the equations of spin evolution in the language of the moment of inertia tensor. We now translate this equation in a form containing the axial moment of inertia vector. Employing (8.14), (6.31) and saturating the previous equation with the principal axis of inertia e aµ we have (Latin repeated index are not summed) Thus, in the trivial case of flat spacetime, we have Ṡ = LS = 0 and using the rigidity condition İa = 0, equation (8.27) describes the Relativistic Binet ellipsoid in the variables S a . Intersecting this ellipsoid with the spherical surface of the constant modulus of the spin, equation (8.28), we obtain the complete trajectory of the spin spatial vector in the principal axis system of reference, provided we have an initial condition to assign. The outcome of the naive case of zero curvature tensor everywhere can be extended if the spacetime is maximally symmetric, i.e., in the constant scalar curvature case, namely when R µνρσ = R 12 (g µρ g νσ -g µσ g νρ ) . (ρ|κ|σ) . On the other hand, saturating the first two indexes in (3.19) we have M κ κρσ = -2M κ (ρ|κ|σ) or M κ κρσ = 0. Then (5.5) and (8.9)-(8.10) give I µν = 0 and S µ = 0 and therefore a trivial case. Inversely, the weaker condition M κ κρσ = 0 or I µν = 0 and S µ = 0 cannot give (8.36); however, for ordinary matter, such as T µν = ρ ẋµ ẋν , ẋµ a congruence on ∆, it is easy to see that (8.36) will be also true.

I

Conclusions

Applying a minimal action principle, in this work we have completed a line of research due to Bailey-Israel and Anandan-Dadhich-Singh concerning the variational formulation of the dynamics of a test rigid body immersed in a curved spacetime. In particular we have found the dependence of the lagrangian from the internal degrees of freedom and, employing a rigidity constraint, we have derived, by a complete variational procedure, the equations of motion to all orders as well as conserved quantities coming from cyclic variables. The equations obtained give rise to some simplifications on the dynamics of an extended rigid test body in general relativity with respect to the Dixon theory. Finally, in view of applications, we have found explicit expressions of these conserved quantities, of relativistic Euler equations and of some other classical-like observables implied in the dynamics up to quadrupolar order.

. 10 )

 10 The variation of the spacetime angular velocity tensor gives δω µν = δ ėµ a e aν + ėµ a δe aν = (δe µ a )˙e aν +ω µκ e aκ δe aν + R µν κλ δγ κ γλ , (6.11) having employed the double derivative relation δ γ ėµ a = (δ γ e µ a )˙+ R µν κλ e aν δγ κ γλ (6.12) and the commutation equation δ e ėµ a = (δ e e µ a )˙(6.13) between the covariant convective operator γµ ∇ µ and the variational derivative operator δ e . In fact, using δ e e µ o ≡ 0, we have δ e ėµ a = δ e (e κ o l) ∇ κ e µ a + (δ e e µ a )˙= (δ e e µ a )˙. (6.14)

- 3 a=1 S 2 aS 2

 322 a (Ω a )˙-Ω b Ω c (I b -I c ) -2 3 R b αβγ M γ[cα]β = 0, (8.22) c = b + 1 mod (3) = a + 2 mod (3) , a = 1, 2, 3.We call(8.22) the Relativistic Euler Equations because they are formally identical to the classical Newtonian ones in the case of flat spacetime[START_REF] Goldstein | Classical Mechanics[END_REF]. Finally, consider the rotational energy-L S -L S = -1 2 Ω µ Ω ν -Ω 2 h µν M µν . (8.23)Solving (8.9) with respect to the mass tensor,M µν = 1 2 h κλ I κλ g µν -I µν (8.24)and using (8.14), after a straightforward calculation (8.23) takes the form vector (8.17) along the principal axes,S a = I a Ω a ,(8.26)(repeated index are not summed) and collecting (8.25) and (8.26), we obtain hand, for the spin vector in the tetrad components we can write = 1. (8.28)Let us now consider the constraint (4.10) of parametric invariance, or L = P µ u µ . Using (6.5) and (5.7) and the equation of motion (6.34) we have

2

 2 symmetry properties in(6.31) and the rigidity condition. Similarly, from (8.31) we have S same conclusion on the trajectory of the spin previously deduced in the flat manifold.Finally, we can ask whether the most general case in which we can infer the same result is that of a rigid body with hight degree of symmetry or of a body for which of(3.19) and therefore M [κ νρ σ] = 0. Saturating the first two indexes we have M κ κρσ = M κ

  because of the energy-momentum conservation law. Thus in any coordinate system we have ∇µ T µν

	ḡ	X νo	ν = 0. Using (2.8), this relation enables
	us to write
	∇µ T µν ḡ

  .19) These symmetries were not found by ADS, while they are formally identical to that contained in the Dixon work. Because of the commutation of the operators ∇µ and

	∂ ∂ ḡρoσo this relation is easily generalized to the other higher multipolar quantities such
	as (3.11)


Yves Coudière
email: yves.coudiere@univ-nantes.fr

Charles Pierre
email: charles.pierre@univ-pau.fr

Benchmark 3D: CeVe-DDFV, a Discrete Duality Scheme with Cell/Vertex Unknowns

Keywords: finite volume, discrete duality, gradient reconstruction MSC2010: 65-06, 65N08, 65N12

This paper presents numerical results for a 3D Cell and Vertex centered DDFV scheme. The method applies to very general 3D meshes including non conformal ones. Its construction and implementation are presented.

1 Presentation of the scheme

General presentation of DDFV methods

"Discrete Duality" Finite Volumes (DDFV) schemes have been specifically designed for anisotropic and/or heterogeneous diffusion problems working on general meshes: distorted, non-conformal and locally refined. They first were introduced in 2D independently by Hermeline [START_REF] Hermeline | Une méthode de volumes finis pour les équations elliptiques du second ordre[END_REF][START_REF] Hermeline | A finite volume method for the approximation of diffusion operators on distorted meshes[END_REF] and Domelevo and Omnès [START_REF] Domelevo | A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids[END_REF], though the key ideas already appear in the work of Nicolaides [START_REF] Nicolaides | Direct discretization of planar div-curl problems[END_REF].

As originally defined in [START_REF] Domelevo | A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids[END_REF], a 2D DDFV scheme consists in associating a second mesh (the dual mesh) to the original (primal) mesh by building dual cells around each (primal) mesh vertex. Cell and vertex centered scalar data are associated to this double mesh framework (one data per primal and dual cell), whereas a vector data consists in one vector per (primal) mesh edge. To a scalar data is associated a discrete gradient that is a vector data. A gradient reconstruction method is used to define this discrete gradient: precisely using the diamond method [START_REF] Coudière | Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem[END_REF]. A discrete divergence acts on vector data by averaging their normal component on the primal and dual cell boundaries, which procedure is classical for finite volume methods. The key feature is a duality property between the discrete gradient and the discrete divergence operators of Green formula type.

Extensions of DDFV schemes to 3D [START_REF] Pierre | Modelling and simulating the electrical activity of the heart embedded in the torso, numerical analysis and finite volumes methods[END_REF][START_REF] Hermeline | Approximation of 2-D and 3-D diffusion operators with variable full tensor coefficients on arbitrary meshes[END_REF][START_REF] Hermeline | A finite volume method for approximating 3D diffusion operators on general meshes[END_REF][START_REF] Andreianov | Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations[END_REF][START_REF] Coudière | A 3d discrete duality finite volume method for nonlinear elliptic equation[END_REF][START_REF] Andreianov | A gradient reconstruction formula for finite volume schemes and discrete duality[END_REF][START_REF] Coudière | A 3d discrete duality finite volume method for nonlinear elliptic equation[END_REF] are of two types.

CV-DDFV.

The original 2D double mesh framework is conserved, dual cells are built around the primal mesh vertices and scalar data consist in a double set of unknowns associated with the (primal) mesh cells and vertices. CeVeFE-DDFV. Recently Coudière and Hubert [START_REF] Coudière | A 3d duality finite volume method for nonlinear elliptic equations[END_REF][START_REF] Coudière | A 3d discrete duality finite volume method for nonlinear elliptic equation[END_REF] modified the 2D framework by considering a third mesh (triple mesh method), with unknowns associated with cells, faces, edges and vertices of the primal mesh.

The method considered here is of CV-DDFV type, CV holding for Cell and Vertex centered. Two versions have been developed so far.

(A) A first 3D construction was introduced by Pierre in [START_REF] Pierre | Modelling and simulating the electrical activity of the heart embedded in the torso, numerical analysis and finite volumes methods[END_REF] for anisotropic and/or heterogeneous diffusion problems. The dual cells here do not form a mesh in the classical sense: they recover the domain twice. (B) A second version, independently introduced by Hermeline [START_REF] Hermeline | A finite volume method for approximating 3D diffusion operators on general meshes[END_REF] and Andreianov & al. [START_REF] Andreianov | Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations[END_REF][START_REF] Andreianov | A gradient reconstruction formula for finite volume schemes and discrete duality[END_REF] differs from the previous one by the dual cell definition that here form a partition of Ω .

For both versions, in presence of heterogeneity, auxiliary (locally eliminated) data are added relatively to faces, as presented in [START_REF] Coudiere | 2D/3D DDFV scheme for anisotropicheterogeneous elliptic equations, application to electrograms simulation from medical data[END_REF][START_REF] Hermeline | A finite volume method for approximating 3D diffusion operators on general meshes[END_REF]. In case of complex meshes, involving face shapes other than triangles or quadrangles, this local elimination procedure is made difficult enforcing to consider auxiliary data as real unknowns inside the algorithm, which drastically increases the problem size. We first emphasize the similarities between (A) and (B). These two versions are based on the same definition of the discrete gradient. They also induce comparable discrete duality properties. Indeed, after a careful examination of these duality properties in [START_REF] Pierre | Modelling and simulating the electrical activity of the heart embedded in the torso, numerical analysis and finite volumes methods[END_REF] and in [START_REF] Andreianov | Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations[END_REF] it turns out that they do involve exactly the same stiffness and mass matrices. As a result, between these two versions, only the averaging of the source terms on the dual cells will differ.

In this paper, version (A) will be considered without auxiliary data on the mesh faces. The fifth test case, including heterogeneity and thus necessitating these auxiliary unknowns per face center, will not be treated here because of a lack of time.

CV-DDFV version (A), discrete duality

Let the domain Ω ⊂ R 3 be a connected open subset, its boundary is assumed to be polyhedral. Let M be a (general) mesh of Ω , possibly non conformal, and whose (primal) cells (resp. faces) are general polyhedral (resp. polygonal). The set of cells, faces and vertices of M are respectively denoted C , F and V . To any vertex v ∈ V is associated a dual cell v ⋆ and to any face f ∈ F is associated a diamond cell D f . Diamond cells form a partition of Ω , whereas dual cells intersect and recover Ω exactly twice.

A vector data is a piecewise constant vector function on the diamond cells. A scalar data is provided by one scalar per cell and per vertex of M . The space of vector data is denoted Q h and the space of scalar data F h . A discrete function is obtained by supplementing a scalar data with one scalar data per boundary face. The space of discrete functions is denoted U h . As developed in Sec. 1.3, u h ∈ U h will be interpreted as a function defined on the diamond cell boundaries:

∂ D := f ∈F ∂ D f , u h : ∂ D -→ R, (1)
that moreover is continuous and piecewise affine on the diamond cell faces. Two discrete operators will be defined, ∇h : U h -→ Q h and div h : Q h -→ F h . that satisfy the discrete duality property (see [START_REF] Pierre | Modelling and simulating the electrical activity of the heart embedded in the torso, numerical analysis and finite volumes methods[END_REF])

Ω ∇h u h • q h dx = -u h , div h q h + ∂ Ω u h q h • n ds (2)
for any functions u h ∈ U h andq h ∈ Q h , with n the unit normal on ∂ Ω pointing outside Ω , and the pairing:

u h , div h q h = 1 3 ∑ c∈C u c div c q h |c| + 1 3 ∑ v∈V u v div v q h |v ⋆ |. (3)
Here, |c| and |v ⋆ | are the volumes of the primal and dual cells c and v ⋆ , u c and div c q h are the values associated to the cell c of the two scalar data u h and div h q h , and similarly u v and div v q h are the values associated to the vertex v of the two scalar data u h and div h q h . In (2) the two integrals are well defined. The first integral is an L 2 product on Ω since both q h and ∇h u h are piecewise constant vector functions on the diamond cells.The second integral is an L 2 product on ∂ Ω : q h is piecewise constant on the boundary faces and its normal component q h • n also, moreover ∂ Ω ⊂ ∂ D defined in (1) and so u h has a restriction to ∂ Ω that is continuous.

Dual and diamond cells

A center x c (resp. x f ) is associated to each cell c ∈ C (resp. f ∈ F ). Diamond cells. Let f ∈ F . In case f ⊂ ∂ Ω then f is the interface between two cells c 1 , c 2 ∈ C : f = c 1 ∩ c 2 .
Denoting x i the center of c i , then D f is the union of the two pyramids with apex x i and with base f as depicted on Fig. 1. In case f ⊂ ∂ Ω , then f = ∂ Ω ∩ c for one cell c ∈ C . In this case D f is the pyramid with apex x c and base f . Still in this cases, f can be considered as a degenerated (flat) pyramid of apex its own center x f and base f . Thus, in all cases, D f is the union of two pyramids, and its boundary can be partitioned into triangles. The vertices of these triangles either are: cell centers, vertices or boundary face centers of M . As a result providing a scalar value to each cell, vertex and boundary face of M defines a unique continuous piecewise affine function u h : ∂ D → R, with ∂ D defined in [START_REF] Andreianov | A gradient reconstruction formula for finite volume schemes and discrete duality[END_REF]. This is precisely the lift from the discrete function in U h presented in Sec. 1.2 into continuous piecewise affine functions on ∂ D in (1).

x 2 D f c 1 c 2 x 1 f = c 1 ∩ c 2 x 1 c 1 x e x f x 2 v 1 v 2 e 2 c 2
Fig. 1 Left: diamond cell for an internal triangular face f . Right: dual cell construction.

Dual cells.

Let v ∈ V and consider a cell c ∈ C and a face f ∈ F so that v is a vertex of f and f is a face of c. This configuration is denoted by v ≺ f ≺ c. To a triple (v, f , c) so that v ≺ f ≺ c is associated an element T v, f ,c . The dual cell v ⋆ then is defined as v ⋆ = f ,c: v≺ f ≺c T v, f ,c . Let us eventually define the element T v, f ,c , as depicted on Fig. 1. Introduce w 1 and w 2 the two vertices of f such that [v, w 1 ] and [v, w 2 ] are two edges of f . Then T v, f ,c is the union of the two tetrahedra vx c x f w i for i =1, 2. As one can see, for a fixed face f and a fixed cell c such that f ⊂ ∂ c, considering all elements T v, f ,c for all the vertices v of f recovers exactly twice D f ∩ c. As a result the dual cells recover the whole domain exactly twice:

∑ v∈V |v ⋆ | = 2|Ω |.

Discrete operators

The discrete divergence is classically defined by averaging the normal component of q h ∈ Q h on the primal and dual cells, for all c ∈ C and all v ∈ V :

div c q h = 1 |c| ∂ c q h • nds , div v q h = 1 |v ⋆ | ∂ v ⋆ q h • nds, (4)
for n the unit normal on ∂ c (resp. ∂ v ⋆ ) pointing outside c (resp. v ⋆ ). This definition is well posed since the discontinuity set of q h ∈ Q h has a zero 2-dimensional measure intersection with the primal and dual cell boundaries. The discrete gradient is defined as follows. Let u h ∈ U h , for all f ∈ F :

∇ f u h = 1 |D f | ∂ D f u h nds, (5)
where ∇ f u h is the (vector) value of ∇h u h on D f and for n the unit normal on ∂ D f pointing outside D f . In practice, definition (5) can always be reformulated in terms of data differences as in the 2D case where (see e.g. [START_REF] Delcourte | Discrete-duality finite volume method for second order elliptic problems[END_REF]):

∇ f u h = (u c 1 -u c 2 )N f + (u v 1 -u v 2 )M f ,
f is a mesh interface (edge), c 1 and c 2 the two cells on each side of f , v 1 and v 2 the two vertices of f and N f , M f two vectors. We refer to [START_REF] Pierre | Modelling and simulating the electrical activity of the heart embedded in the torso, numerical analysis and finite volumes methods[END_REF][START_REF] Coudiere | 2D/3D DDFV scheme for anisotropicheterogeneous elliptic equations, application to electrograms simulation from medical data[END_REF] for similar expansions in 3D.

The scheme

The linear diffusion problem -div(K∇u) = f is considered together with a Dirichlet boundary condition u |∂ Ω = g. The tensor K is discretized into K h by averaging K on each diamond cells and the source term f is discretized as f h ∈ F h by averaging f over each primal and dual cells. The problem reads: find u h ∈ U h such that

∀ c ∈ C : div c (K h ∇h u h ) = f c , ∀ v ∈ V , v ∈ ∂ Ω : div v (K h ∇h u h ) = f v (6) ∀ v ∈ V , v ∈ ∂ Ω : u h (v) = g(v) , ∀ f ∈ F , f ⊂ ∂ Ω : u h (x f ) = g(x f ), (7)
To solve (6) (7), we split U h = U h,0 ⊕ B where U h,0 is the subset of discrete functions equal to zero on ∂ Ω . Then u h decomposes as u h = u 0 + ũ, where ũ ∈ B is uniquely determined by [START_REF] Delcourte | Discrete-duality finite volume method for second order elliptic problems[END_REF]. Now u 0 ∈ U h,0 satisfiesdiv h (K h ∇h u 0 ) = f h + div h (K h ∇h ũ) := fh for all primal cells and all interior vertices. This is a square linear system equivalent with: find u 0 ∈ U h,0 so that for all v ∈ U h,0 we have:

-div h (K h ∇h u 0 ), v = fh , v
With the help of the discrete duality property (2) it is also equivalent with finding u 0 ∈ U h,0 so that for all v ∈ U h,0 :

Ω K h ∇h u 0 • ∇h vdx = fh , v .
In practice, introducing the stiffness matrix S associated to the discrete tensor K h , this problem is rewritten as the square positive symmetric linear system

SU 0 = F, (8)
with U 0 (resp. F) the vector formed by the values of u 0 (resp. fh ) at the cell centers and interior vertices. The stiffness matrix S has the coefficients S i j = Ω K h ∇h w i • ∇h w j dx, with w i ∈ U h,0 the base function having value 1 at one cell center or interior vertex and 0 everywhere else. This matrix is clearly symmetric and positive.

Numerical results

The cell centers as well as the face centers are set to their iso-barycenter.

Let us first define the data (source term f and anisotropy tensor K) discretization. Primal, dual and diamond cells are partitioned using a single set of tetrahedra of type E = x c x f v 1 v 2 , with c ∈ C , f a face of c and v 1 , v 2 two vertices of f forming one of its edges. To form the scalar data f h , f is averaged on the tetrahedra E partitioning each primal and dual cells whereas the discrete tensor K h is obtained by averaging K on the tetrahedra E partitioning the diamond cells. Averaging is made by the mean of Gaussian quadrature on each tetrahedra E using a 15 points quadrature formula of order 5, see e.g. [START_REF] Jinyun | Symmetric gaussian quadrature formulae for tetrahedronal regions[END_REF]. Assembling the discrete source term f h and tensor K h requires one loop on the mesh faces.

The stiffness matrix S in (8) also is assembled using a loop on the mesh faces. Precisely two base functions w i and w j have a non zero interaction (i.e. S i j = Ω K h ∇h w i • ∇h w j dx = 0) in case they are associated to two vertices of a same diamond D f .

Let us now define the L 2 , H 1 and energy errors reported in the following tables as erl2, ergrad and ener respectively. Let u h denote the discrete solution of one of the test case, and u the solution of the associated continuous problem. The discrete function u h is lifted to a function u h ∈ L 2 (Ω ) as follows. Consider a face f , u h provides a value at each vertex of D f and also at the face center x f in case of a boundary face. In case of an interior face, a supplementary value u f is computed at x f as u f = (∑ n i=1 u v i )/n where the v i are the n vertices of f , which definition is consistent since x f is the iso-barycenter of f . With these additional values, scalars are available for every vertices of the tetrahedra E that partition Ω : this defines a unique function u h by P 1 interpolation, which then is continuous piecewise affine on Ω . We define:

erl2 2 = Ω |u h -u| 2 dx Ω |u| 2 dx .
The discrete vector data ∇h u h is a piecewise constant vector function on the diamond cells. Therefore ∇h u h is an L 2 functions on Ω and the H 1 and energy errors reported in the following tables are defined as:

ergrad 2 = Ω | ∇h u h -∇u| 2 dx Ω |∇u| 2 dx , ener 2 = Ω K( ∇h u h -∇u) • ( ∇h u h -∇u) Ω K∇u • ∇udx .
• Test 1 Mild anisotropy, u(x, y, z) = 1 + sin(πx) sin π y + 1 2 sin π z + 1 3 min = 0, max = 2, Tetrahedral meshes i nu nmat umin uemin umax uemax normg 1 2187 21287 1.34E-02 1.53E-02 1.99E+00 1.99E+00 1.80E+00 2 4301 44813 3.24E-03 6.84E-03 1.99E+00 1.99E+00 1.80E+00 3 8584 94088 8.78E-03 7.44E-03 2.00E+00 1.99E+00 1.80E+00 4 17102 195074 4.74E-03 5.52E-03 2.00E+00 2.00E+00 1.80E+00 5 34343 405077 5.90E-04 1.49E-03 2.00E+00 2.00E+00 1.80E+00 6 69160 838856 1.30E-03 6.19E-04 2.00E+00 2.00E+00 1.80E+00 i nu erl2 ratiol2 ergrad ratiograd ener ratioener 1 2187 1.39E-02 -1.85E-01 -1.80E-01 -2 4301 8.80E-03 2.04E+00 1.48E-01 1.01E+00 1.44E-01 9.89E-01 3 8584 5.64E-03 1.93E+00 1.18E-01 9.73E-01 1.15E-01 9.97E-01 4 17102 3.61E-03 1.94E+00 9.36E-02 1.01E+00 9.10E-02 1.01E+00 5 34343 2.26E-03 2.01E+00 7.43E-02 9.92E-01 7.24E-02 9.81E-01 6 69160 1.42E-03 2.00E+00 5.87E-02 1.01E+00 5.70E-02 1.02E+00

• Test 1 Mild anisotropy, u(x, y, z) = 1 + sin(πx) sin π y + 1 2 sin π z + 1 3 min = 0, max = 2,

Comments

The linear system (8) to be solved is symmetric and positive: a Conjugate Gradient algorithm has been applied, together with a basic Jacobi preconditioner. The sparsity pattern of the stiffness matrix is not compact, especially for matrix lines corresponding to vertex nodes. The stiffness matrix lines corresponding to cell nodes have 1 + n f + n s nonzero terms with n f and n v the number of faces and vertices of the considered cell; for a tetrahedra 1 + n f + n v = 9. The maximum principle is not fulfilled by DDFV schemes. In practice it has been violated only once for test one on Voronoï meshes and more significantly on test 3. Meanwhile no oscillation phenomena are observed. Expected order 2 convergence on erl2 is observed for all tests excepted test 2. Order 1 convergence is observed for ergrad and ener on all tests.

  33E+00 5.42E+00 1.62E+03 3 9229 173951 1.66E-01 -5.76E-06 5.33E+00 5.42E+00 1.62E+03 4 21156 412240 1.25E-01 -7.39E-06 5.33E+00 5.42E+00 1.62E+03 5 44420 882520 9.37E-02 -6.93E-06 5.34E+00 5.42E+00 1.62E+03 6 82335 1654893 7.48E-02 -6.94E-06 5.35E+00 5.42E+00 1.62E+03 7 145079 2937937 5.80E-02 -8.05E-06 5.36E+00 5.42E+00 1.38E-03 2.29E+00 1.22E-01 1.18E+00 1.21E-01 1.16E+00 3 9229 7.45E-04 2.19E+00 8.57E-02 1.25E+00 8.56E-02 1.24E+00 4 21156 5.53E-04 1.08E+00 6.56E-02 9.71E-01 6.55E-02 9.72E-01 5 44420 3.77E-04 1.55E+00 5.14E-02 9.85E-01 5.13E-02 9.83E-01 6 82335 2.44E-04 2.11E+00 4.18E-02 1.01E+00 4.17E-02 1.01E+00 7 145079 1.83E-04 1.53E+00 3.51E-02 9.27E-01 3.50E-02 9.26E-01


Alain Astier

Marie-Geneviève Porquet

High-spin states in the five-valence-particle nucleus 213 Po

Keywords: numbers: 25.70.Hi, 27.80.+w, 23.20.-g, 21.60.Cs

Excited states in 213 Po have been populated using the 18 O + 208 Pb reaction at 85 MeV beam energy and studied with the Euroball IV γ-multidetector array. The level scheme has been built up to ∼2.0 MeV excitation energy and spin I ∼ 25/2 from the triple γ-coincidence data. Spin and parity values of several yrast states have been assigned from the γ-angular properties. The configurations of the yrast states are discussed using results of empirical shell-model calculations and by analogy with the neighbouring nuclei. The spin and parity values of several low-spin states of 213 Po previously identified from the β-decay of 213 Bi are revised.

I. INTRODUCTION

213 Po is a five-particle nucleus with two protons and three neutrons outside the doubly-magic core, 208 Pb. Thus one can expect its first excited states to come from a few spherical configurations mixed by the neutron-proton interactions. Some states issued from pure π 2 configurations on the one hand and from pure ν 3 configurations on the other hand have been already identified in neighbouring nuclei: (i) the first excited states of 210 Po is a textbook example of residual interaction in a two-proton configuration, (ii) the properties of the three-neutron nucleus, 211 Pb, are well accounted for using shell model (SM) calculations [1].

It is worth pointing out that the SM predictions of nuclei lying above 208 Pb are not numerous. For instance, large-scale SM calculations have been performed for the N = 126 isotones, from Z = 84 to Z = 94, in the full Z = 82-126 proton model space using the modified Kuo-Herling interaction [2] and semi-empirical shell model has been used to describe several nuclei close to 208 Pb [3]. Moreover the ground-state configuration of three oddodd nuclei, 210 Bi (π 1 ν 1 ), 212 Bi (π 1 ν 3 ), 212 At (π 3 ν 1 ), have been recently predicted using effective interactions derived from the CD-Bonn nucleon-nucleon potential [4], but there is no calculation dealing with many more valence nucleons. Indeed when adding a lot of protons and neutrons to the the doubly-magic core, 208 Pb, we reach the well-known region of octupole-deformed nuclei [5], which is ouside the scope of SM calculations because of a too large valence space. On the contrary, the study of the heavy Po nuclei with only two protons and many neutrons outside 208 Pb allows us to investigate both the effective neutron-proton interactions and the configuration mixings, as these nuclei remain almost spherical.

In the present paper, we report on high-spin states in 213 Po, which are identified for the first time. The level scheme has been built up to ∼2.0 MeV excitation energy using high-fold γ coincidences and γ-ray anisotropies have been measured to assign the spin values. The yrast states of 213 Po are discussed using results of empirical shell-model calculations and by analogy with the neighbouring nuclei.

II. EXPERIMENTAL METHODS AND DATA ANALYSIS

We have used the 18 O+ 208 Pb reaction, the 18 O beam with an energy of 85 MeV being provided by the Vivitron tandem of IReS (Strasbourg). A 100 mg/cm 2 selfsupporting target of 208 Pb was employed, which was thick enough to stop the recoiling nuclei. Due to this large thickness, the 18 O beam was stopped in the Pb target too and the incident energy covered a large range of values, from 85 MeV down to the Coulomb barrier. The de-exciting γ-rays were recorded with the EUROBALL IV array consisting of 71 Compton-suppressed Ge detectors [6] (15 cluster germanium detectors placed in the backward hemisphere with respect to the beam, 26 clover germanium detectors located around 90 • , 30 tapered single-crystal germanium detectors located at forward angles). Each cluster detector is composed of seven closely packed large-volume Ge crystals [7] and each clover detector consists of four smaller Ge crystals [8].

Events were recorded on tape when at least 3 unsuppressed Ge detectors fired in prompt coincidence. In this way, a set of ∼ 4 × 10 9 three-and higher-fold events were available for subsequent analysis. The main objective of the experiment was actually the study of the fusion-fission channel which leads to the production of the high-spin states of ∼ 150 fragments, mainly located on the neutron-rich side of the valley of stability [9]. In addition, various transfer reactions have produced several Po isotopes. First of all, 210,211,212,214 Po have been identified thanks to their known γ-ray yrast cascades [START_REF]ENSDF data base[END_REF]. Moreover many new γ lines emitted by the two even-even 212,214 Po isotopes have been found to be strong enough in our data set to be precisely analyzed [START_REF] Astier | [END_REF][12][13]. Then we have looked for the γ lines emitted by 213 Po.

Various procedures have been used for the offline analysis in order to fully characterize the excited levels of 213 Po (excitation energy, spin and parity values). Both multi-gated spectra and three-dimensional 'cubes' have been built and analyzed with the Radware package [14], starting from the X-rays of Po in order to identify the new transitions and to build the level scheme.

III. EXPERIMENTAL RESULTS

Prior to this work, the excited states of 213 Po were studied from the β-decay of 213 Bi [15]. All the spin values assigned to these states are 7/2, 9/2 or 11/2, because of the properties of the ground state of 213 Bi (I π =9/2 -). These excited states are not expected to be populated by a reaction induced by heavy ions, since their spin values are very close to the one of the 213 Po ground state (I π =9/2 + ). The last study of the β-decay of 213 Bi [16] has led to the identification of a new state, at 868.0 keV, which mainly de-excites towards the ground state. Its spin value, I=13/2, was proposed on the basis of the comparison with results of semi-empirical shell-model calculations. In that case, the 868.0 keV γ-ray should be observed in the present work, as the first member of the yrast cascade.

In order to identify all the transitions depopulating the yrast states of 213 Po, we have first looked into spectra gated by the K α1 X-ray1 of Po. The spectrum of γ-rays in coincidence with two K α1 X-ray of Po displays, in addition to the transitions emitted by 212 Po [START_REF] Astier | [END_REF]12], 214 Po [13], as well as 210,211 mutual coincidence (see, for instance, the three spectra of Fig. 1), but they are not in coincidence with the transitions emitted by the other Po isotopes. Thus they must belong to the yrast structure of 213 Po, its population being expected in the reaction used in this work since all the neighbouring isotopes are observed. Then all the doublegated spectra involving these three transitions, the K α1 X-rays, as well as all the weaker lines as those observed in the spectra of Fig. 1, have been analyzed. The level scheme built from all the observed coincidence relationships is shown in Fig. 2. We have gathered in Table I the properties of the transitions assigned to 213 Po from this work. The R ADO angular anisotropy with respect to the beam axis has been analyzed to determine the multipole order of the most intense γ-rays, as done for the γ-rays of 212,214 Po obtained in the same experiment [START_REF] Astier | [END_REF][12][13]. The results, given in Table I, allow us to determine the spin and parity values of the first yrast states. The I π values of the 1504 keV and 1619 keV states, given in parenthesis, will be discussed in the following section.

213
As regards the 289 keV transition, we have extracted its internal conversion coefficient using its intensity imbalances measured in spectra in double coincidence with the 146 keV transition and either the 423 keV or the 646 keV transition. The obtained value, α tot =0.15 (5), corroborates the E2 multipolarity for the 289 keV transition.

Lastly, we have estimated the cross section of the production of 213 Po in this 18 O+ 208 Pb reaction, σ ∼ 0.3 mb. This is well lower than the value we have obtained for 212 Po, σ ∼ 10-20 mb [12] and about three times less than obtained for 214 Po, σ ∼ 0.5-1 mb [13]. At bombarding energies around and lower than 85 MeV, the 18 O projectiles may break up into 14,12 C + He. Using the kinematics of the exit channel leading to 212 Po, we have shown [12] that the heavy partner of the break up, 14 C, is emitted backward, implying that the production of 212 Po involves the fusion of the light partner, 4 He, and the 208 Pb target2 . Thus we can infer that 214 Po is produced when the break up leads to 12 C as the heavy fragment. In such a scenario, excited states of 213 Po are likely produced from the levels of 214 Po having the highest energies and decaying by neutron emission. Noteworthy is the fact that the abovementioned transition at 868 keV has not been observed in our data set, it is correlated neither to the X-rays of Po nor to the transitions newly assigned to 213 Po. Thus we do not confirm the previously proposed location of the 13/2 + yrast state at 868 keV [16]. A possible assignment of the 868 keV state observed in the β-decay of 213 Bi will be discussed in the following section.

IV. DISCUSSION

213 Po is a five-particle nucleus with two protons and three neutrons outside the doubly-magic core, 208 Pb. Thus one can expect that its first excited states come from the (πh 9/2 ) 2 ⊗ (νg 9/2 ) 3 shell-model (SM) configuration. The states issued from elementary SM configurations such as (νg 9/2 ) 3 and (πh 9/2 ) 2 ⊗ (νg 9/2 ) 1 can be easily computed using the two-body empirical residual energies and the coefficients of fractional parentage (CFP) corresponding to the case of three nucleons [START_REF] Talmi | Simple Models of Complex Nuclei[END_REF]. For that purpose, the residual energies are extracted from the energies of the (νg 9/2 ) 2 multiplet (known in 210 Pb), the (πh 9/2 ) 2 multiplet (known in 210 Po), and the (πh 9/2 ) 1 ⊗ (νg 9/2 ) 1 multiplet (known in 210 Bi). The re- sults of the two calculations are given in Figs. 3(a) and 3(b). The two groups of yrast states (drawn in green and blue) display similar features, those of the (νg 9/2 ) 3 configuration being slightly more compressed in energy. Moreover the (πh 9/2 ) 2 ⊗ (νg 9/2 ) 1 configuration leads to the maximum spin, 25/2 + , the corresponding level is expected to be isomeric as it is located closely above the 21/2 + state. These predicted states compare very well with the results measured in two nuclei having these two elemen-tary SM configurations, 211 Pb and 211 Po respectively (see Figs. 3(c) and 3(d)). 211 Pb exhibits two 11/2 + states close in energy, their wave-functions are likely very mixed, one component from the (νg 9/2 ) 3 configuration and the other from the (νg 9/2 ) 2 ⊗(νi 11/2 ) 1 one, the νi 11/2 orbital being located just above the νg 9/2 one. On the other hand, only one 11/2 + state was observed in 211 Po, its spectroscopic factor measured in the 211 Po(d,p) reaction indicates that this state mainly comes from the (πh 9/2 ) 2 ⊗ (νi 11/2 ) 1 configuration [START_REF]ENSDF data base[END_REF], that means that another 11/2 + state remains to be identified. As regards the yrast states, those of 211 Pb are more compressed in energy than those of 211 Po. Moreover the 25/2 + state of 211 Po is a long-lived isomeric state (T 1/2 = 25.2(6) s) deexciting through α-decay [START_REF]ENSDF data base[END_REF], implying that the 21/2 + state is definitely higher in energy than the 25/2 + state in this nucleus.

At this point, we can surmise that the (πh 9/2 ) 2 ⊗ (νg 9/2 ) 3 configuration of 213 84 Po 129 would display the following features:

• the lowest excitation mainly comes from the (νg 9/2 ) 3 configuration, that gives the 13/2 + 1 , 17/2 + 1 , and 21/2 + 1 states, • the 25/2 + state due to the (πh 9/2 ) 2 ⊗(νi 11/2 ) 1 configuration is no longer a long-lived isomeric state decaying by α emission, as the 21/2 + 1 state is located below it. This is in good agreement with the experimental result (see Fig. 2), provided that the 146 keV transition is E2. It is worth recalling that we have got no direct information about its multipolarity, the only peculiar property of this γ-ray is the fact that no transition could be placed above the 1504 keV state. That rules out the M1 case, as explained now. When taking into account the large value of the internal conversion coefficient (α tot (146, M1) = 3.8), the total intensity of the 146 keV transition would amount to 40 [START_REF]ENSDF data base[END_REF]. Assuming that the intensity of a transition populating the 1504 keV state is only 50% of its total decay, such a transition would give a line having an height around 700 counts in the spectrum of γ-rays in double coincidence with two yrast transitions, at variance with the observation (see the spectra in Fig. 1).

If the half-life of the 1504 keV state is more longer than the time window used to delimit a coincidence event (300 ns in our experiment), the non-observation of the population of this state is obvious. Such a long halflife naturally occurs in case of an E2 transition with a low energy such as 146 keV, provided that the transition probability is slightly hindered: if B(E2) = 0.1 W.u., we obtain T 1/2 (146 keV, E2) = 450 ns. When assuming that the main configuration of the 21/2 + 1 state of 213 Po is not the same than that of the 25/2 + 1 state [(νg 9/2 ) 3 versus (πh 9/2 ) 2 ⊗ (νi 11/2 ) 1 ], we expect a very hindered E2 transition probability. For instance, the yrast-state configurations of 94 42 Mo 52 are known to shift from proton excitations to neutron ones, that gives B(E2; 8 + 1 → 6 + 1 ) = 0.005 W.u. [START_REF]ENSDF data base[END_REF]. In summary, the yrast states of 213 Po identified in the present work are drawn in the right part of Fig. 4 using the same color code as those of Fig. 3 (green and blue). [16]) are discussed in the text. The color code is the same as that of Fig. 3.

As displayed in Fig. 3(a), the (νg 9/2 ) 3 configuration gives rise to several low-spin states which may be observed in the β-decay of the 9/2 -ground state of 213 83 Bi [16]. Moreover, as in 211 Pb, the 11/2 + state of the (νg 9/2 ) 3 configuration can be strongly mixed to the state from the νi 11/2 orbital, leading to two 11/2 + states having similar properties. To suggest the new spin values of the states measured in the β-decay which are drawn in the left part of Fig. 4, we made use of the following arguments:

• three states are directly populated by the βdecay, with similar logf t values (∼6.2): the 9/2 + ground state and the excited states at 440 keV and 1100 keV [16].

• the β-decay of the 9/2 -ground state of 211 85 At populates two states of 211 Po 127 with similar logf t values (∼5.9): the νg 9/2 ground state and the νi 11/2 excited state at 687 keV [START_REF]ENSDF data base[END_REF].

• the 440 keV and 1100 keV states have similar decay schemes: one branch to the ground state and another one to the first excited state at 293 keV. Moreover the 1100 keV state decays to the 440 keV one.

• the first excited state of the (νg 9/2 ) 3 configuration is expected to have I π = 7/2 + (see Fig. 3(a)).

Thus, combining results on the yrast states obtained in the present work and those on lower-spin states coming from the 213 Bi β-decay [16], most of the states of the (νg 9/2 ) 3 configuration can be identified. As said above, two 11/2 + states display similar features, that occurs when two unperturbed states are very close in energy, giving a maximum mixing of the two states (here, [(νg 9/2 ) 3 ] 11/2 and [(νg 9/2 ) 2 (νi 11/2 ) 1 ] 11/2 ). Then we can assume that the energy of the 11/2 + states of the two pure configurations is around half the sum of the two measured energies, i.e. 770 keV. This is in very good agreement with the predicted energy of the 11/2 + state of the (νg 9/2 ) 3 configuration, 788 keV (see Fig. 3(a)). As for the νi 11/2 orbital, it gives rise to many excited states having spin values in the range 1/2 -27/2, from the breaking of either the neutron pair or the proton pair of the (πh 9/2 ) 2 (νg 9/2 ) 2 (νi 11/2 ) 1 configuration. In any case, the yrast states lying above 1357 keV (the 21/2 + state from the (νg 9/2 ) 3 configuration) likely involve the νi 11/2 orbital.

V. SUMMARY

The 18 O + 208 Pb reaction at 85 MeV beam energy was used to populate the high-spin states of 213 Po. Their γ-decays were investigated using the Euroball IV array. The level scheme was built up to ∼ 2.0 MeV excitation energy and spin I ∼ 25/2 from the triple γ-coincidence data. Spin and parity values of several yrast states were assigned from the γ-angular properties. Using results of empirical shell-model calculations and analogy with the neighbouring nuclei, the main configurations of the yrast states are proposed. The first excitations come from the three-neutron configurations, particularly the (νg 9/2 ) 3 one. The spin and parity values of several states previously known from the β-decay of 213 Bi were revised in order to account for the low-spin states expected in the (νg 9/2 ) 3 configuration. Shell model calculations using a realistic effective interaction derived from the CD-Bonn nucleon-nucleon potential have been recently performed in nuclei close to 208 Pb [4], giving results in very good agreement with experiments. It would be worth extending them to nuclei having more valence nucleons, such as the five-particle nucleus, 213 Po.

FIG. 1 :

 1 FIG. 1: Spectra of γ-rays in double coincidence with (a) the first transition of 213 Po (646 keV) and the Kα1 X-ray of Po, (b) the 289 keV and 423 keV transitions, and (c) the 289 keV and 646 keV transitions.

TABLE I :

 I Properties of the γ transitions assigned to 213 Po from this work. Eγ a Iγ b c RADO d Ei E f Intensities measured in this experiment (i.e. with the requirement that a minimum of three unsuppressed Ge detectors fired in prompt coincidence) are normalized to the total intensity of the γ transitions populating the 1068 keV state. c The number in parenthesis is the error in the last digit. d R ADO =Iγ (39.3 • )/ Iγ (76.6 • )

				I π i	I π f
	keV		keV keV
	146.2 8(2)	1504 1357 (25/2 + ) 21/2 +
	261.7 4.8(14)	1619 1357 (23/2 + ) 21/2 +
	289.0 60(10) 1.3(2) 1357 1068 21/2 + 17/2 +
	344.5 15(5)	1413 1068		17/2 +
	398.1 3.5(12)	2017 1619		(23/2 + )
	422.8 100 1.18(10) 1068 646 17/2 + 13/2 +
	645.6	-	1.25(10) 646 0	13/2 +	9/2 +
	711.2 24(6)	1780 1068		17/2 +
	a Uncertainties in transition energies are typically between 0.1 and
	0.5 keV.

b

  The yrast states of the two nuclei are drawn in green and blue, respectively. The wave functions of the 11/2 + states (drawn in magenta) contain a large component of the νi 11/2 orbital located just above the νg 9/2 one.

											(a)	(b)
	816 642 878	788 1048 5/2 + 3/2 + 9/2 +	11/2 + 15/2 +	744 1076 1219	13/2 + 17/2 + 21/2 +	13/2 + 17/2 + 21/2 + 25/2 +	890 1156 1226 1238	11/2 + 15/2 + 19/2 + 23/2 +	704 1108 1233 1357	9/2 + 5/2 + 7/2 + 3/2 + 1/2 + 9/2 +	747 975 1047 1421 1175 1270
				447	7/2	+
								0	9/2	+	9/2	+	0
					(νg 9/2 )	3				(νg 9/2 ) 1 (πh 9/2 ) 2
											(c)	17/2 + 25/2 +	1428 1463	15/2 +	1459	1/2 + 3/2 +	1385 1421 (d)
								1056 1193	17/2 21/2	+ +	13/2	+	1181	5/2 + 7/2 + 9/2 +	1051 1122 1161
	762 815 598	9/2 3/2 5/2	+ + +	643 894	11/2 11/2	+ +	734	13/2 +	11/2 +	687
				439	7/2	+
								0	9/2	+	9/2	+	0
					211 82 Pb 129				211 84 Po 127

FIG. 3: (color on line) Top: Theoretical states belonging to the spherical (νg 9/2 ) 3 configuration (a) and (πh 9/2 ) 2 ⊗ (νg 9/2 ) 1 configuration (b), computed using empirical interactions extracted from the two-nucleon neighbouring nuclei ( 210 Pb, 210 Po, 210 Bi). The yrast states of the two sets are drawn in green and blue, respectively. Bottom: First excited states measured in the three-neutron nucleus, 211 Pb [1, 19] (c), and in the two-proton one-neutron nucleus, 211 Po [20, 21] (d).

  Right: High-spin levels of 213 Po identified in the present work. Left: Low-spin states measured in the β-decay of the 9/2 -ground state of 213 Bi, the spin values (at variance with those assigned in ref.

		β-decay of	213 Bi			18 O +	208 Pb reaction
								25/2 +	1504
							1357	21/2	+
	1003 1046		1100	11/2	+	1068	17/2	+
	868	9/2 +
	600	5/2	+				646	13/2	+
			440	11/2 +
			293	7/2	+
			0					9/2 +
					213 84 Po 129
	FIG. 4: (color on line)

Low-energy transitions, such as the X-rays of Po, are clearly observed in our work since we used a very low threshold for triggering the CFD discriminator of each Ge channel. This was allowed by the VXI electronic cards of the Euroball array. For that purpose, the lower thresholds of

the 239 channels were carefully checked at the beginning of the data acquisition.

The fusion of the heavy partner, 14 C, and the target was proposed to be the main mechanism leading to the production of Ra isotopes in the 18 O+ 208 Pb reaction at 92 MeV[17].

Acknowledgments

The Euroball project was a collaboration between France, the United Kingdom, Germany, Italy, Denmark and Sweden. We are very indebted to our colleagues involved in the EB-02/17 experiment devoted to the fission fragments, in which the present data on 213 Po were recorded. We thank the crews of the Vivitron, as well as M.-A. Saettle for preparing the Pb target, P. Bednarczyk, J. Devin, J.-M. Gallone, P. Médina and D. Vintache for their help during the experiment.