L Philpott
email: l.philpott06@imperial.ac.uk

Particle simulations in causal set theory

Keywords: numbers: 04, 60, Bc

Models of particle propagation in causal set theory are investigated through simulations. For the swerves model the simulations are shown to agree with the expected continuum diffusion behaviour. Given the limitations on the simulated causal set size, the agreement is far better than anticipated.

Introduction

Causal set theory is a discrete, Lorentz invariant approach to quantum gravity. For comprehensive reviews of the field see, for example [START_REF] Henson | The causal set approach to quantum gravity[END_REF][START_REF] Dowker | Causal sets as discrete spacetime[END_REF][START_REF] Rafael | Causal sets: Discrete gravity (notes for the valdivia summer school)[END_REF]. The phenomenology of particles in causal set theory was first investigated by Dowker et al. [START_REF] Dowker | Quantum gravity phenomenology, Lorentz invariance and discreteness[END_REF], and later further developed in [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF]. Massive particles propagating in a discrete spacetime are expected to experience small fluctuations in momentum, an effect termed 'swerves'. Several simple classical microscopic models for particle propagation on a causal set were proposed in [START_REF] Dowker | Quantum gravity phenomenology, Lorentz invariance and discreteness[END_REF] and [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF], but it was shown that it was not necessary to choose a specific model to understand the phenomenology in the continuum limit. Dowker et al. demonstrated that any Lorentz invariant Markovian stochastic process on the massive particle state space gives rise to a continuum diffusion equation. In terms of the observers time parameter, 'cosmic time', the swerves diffusion equation is [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF]:

∂ρ ∂t = -p i mγ ∂ i ρ + k∂ a g ab √ g∂ b ρ γ √ g , ( 1
)
where ρ is the probability density for the system to have a certain momentum and position at a given time, p is the particle momentum, m is the particle mass, γ = m 2 + p 2 /m is the standard relativistic factor, and g is the metric on the mass shell, i.e. the hyperboloid in momentum space defined by p µ p µ = -m 2 . k is the diffusion parameter, the one free parameter of the model.

Although no one microscopic model was chosen to derive the diffusion equation, the phenomenological parameter, k, will in reality depend on the properties of the underlying model for particles on a causal set. This paper investigates whether the behaviour of underlying particle models is well described by the continuum limit swerves diffusion equation. As will be shown below, the results are in fact better than expected, the model proposed in [START_REF] Dowker | Quantum gravity phenomenology, Lorentz invariance and discreteness[END_REF] is well approximated by the diffusion equation even far from the continuum limit. This allows the relationship between the microscopic model parameters and the diffusion parameter to be determined.

First it is necessary to review the basic concepts of causal set theory. A causal set is a set C endowed with a binary relation 'precedes', ≺, that satisfies:

(i) transitivity: if x ≺ y and y ≺ z then x ≺ z, ∀x, y, z ∈ C;

(ii) reflexivity: x ≺ x, ∀x ∈ C;

(iii) acyclicity: if x ≺ y and y ≺ x then x = y, ∀x, y ∈ C;

(iv) local finiteness: ∀x, z ∈ C the set {y | x ≺ y ≺ z} of elements is finite. Some definitions will be useful for the following work. Let C be a causal set.

(i) A chain is a totally ordered subset of C.

(ii) A longest chain between two elements x, y ∈ C is a chain whose length is longest amongst chains between those endpoints. There may be more than one longest chain between two elements. The length of the longest chain between elements x, y ∈ C will be denoted d(x, y).

(iii) A link is an irreducible relation: elements x and y are linked if and only if d(x, y) = 1. If two elements x, y ∈ C are linked, it will be denoted x ≺ * y.

(iv) A path is a chain consisting of links.

Particle models

For the purpose of simulations, a causal set can be constructed from a continuum manifold by a process called sprinkling. Points are selected at random from a manifold via a Poisson process in which the probability measure is equal to the spacetime volume measure in fundamental units. The causal order on the points induces the partial order on the elements in the causal set. Construct a causal set by sprinkling into Minkowski spacetime. A massive particle trajectory is taken to be a chain of elements e n in the causal set, i.e. a linearly ordered subset of C. The particle trajectory is constructed iteratively. It is assumed that the trajectory's past determines its future, but that only a certain amount of the past is relevant.

Swerves

First proposed in [START_REF] Dowker | Quantum gravity phenomenology, Lorentz invariance and discreteness[END_REF], this model relies on information about the approximating spacetime. Suppose the particle is currently located 'on' an element e n , with a fourmomentum p n . The next element, e n+1 is chosen such that • e n+1 is in the causal future of e n and within a propertime τ f of e n ,

• the momentum change |p n+1 -p n | is minimized. τ f is the forgetting time of the process. The momentum p n+1 is defined to be proportional to the vector between e n and e n+1 , normalized by the particle mass m.

Intrinsic models

Two models intrinsic to the causal set, i.e. not relying on continuum information, were proposed in [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF] to illustrate the range of possibilities available. Rather than a forgetting time, these models depend on a forgetting number n f . 2.2.1. Model 1 Given a partial particle trajectory . . . e n-1 , e n the next element e n+1 is chosen such that

• d(e n-1 , e n+1 ) ≤ 2n f , • d(e n , e n+1 ) is maximized subject to d(e n , e n+1 ) ≤ n f .
These requirements do not guarantee the existence of a unique e n+1 . There will, however, almost surely be finitely many eligible elements and the trajectory can be constructed by choosing an element uniformly at random from these. Note that this model is slightly different from the first intrinsic model given in [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF], where equalities in the above conditions were given. Model 1 of [START_REF] Philpott | Energy-momentum diffusion from spacetime discreteness[END_REF] does not guarantee the existence of an e n+1 under reasonable conditions.

Model 2

The trajectory is constructed as a path in this model, i.e. d(e n , e n+1 ) = 1 for any e n , e n+1 . Given a partial particle trajectory . . . e n-n f , . . . , e n-1 , e n the next element e n+1 is chosen such that

• d(e n , e n+1 ) = 1,
• d(e n-n f , e n+1 ) + . . . + d(e n-1 , e n+1 ) + d(e n , e n+1 ) is minimized Again this minimization does not necessarily yield a unique e n+1 , in which case the trajectory is constructed by choosing an element uniformly at random from those eligible. Also, if the trajectory has length less than n f the minimization is done over all elements available.

In these models it is assumed that the forgetting parameter τ f or n f is many orders of magnitude greater than the discreteness scale.

Numerical results

Simulations of the particle models given above were developed within the Cactus numerical relativity framework [6], making use of the CausalSets arrangement written by David Rideout. Although the models are entirely general and the code developed can be run in any number of dimensions, due to computational limitations simulations were carried out in 1+1 dimensions. Points were sprinkled into a region of Minkowski spacetime with a Poisson distribution with a mean number of elements N . One additional point was added to each causal set at the origin to give a fixed beginning point, e 1 , for the trajectories. For convenience, the particular frame to which the coordinates of the sprinkled points in Minkowski spacetime refer will be called the embedding frame. For the swerves model the particle was assumed to be initially at rest in the embedding frame, i.e. p 1 = (m, 0). For the intrinsic models the particles were also assumed to be initially close to rest. To construct the first step for the intrinsic model 1, the condition d(e 0 , e 2 ) ≤ 2n f was neglected and e 2 was taken to be the first (in time) element that maximized d(e 1 , e 2 ) ≤ n f . In the case of intrinsic model 2 an additional point was added to the causal set at x = 0, t = 0.25 and the beginning of the trajectory was taken to be a longest chain between the points x = 0, t = 0 and x = 0, t = 0.25, to ensure the particle was initially close to rest. An example trajectory for each of the models is shown in Figure 1. Each model gives fluctuations in momentum, but the amount of fluctuation is clearly model dependent. Note that for a 1+1-dimensional causal set with N = 32768, n f = 20 is roughly equivalent to τ f = 0.1105 if we take d pl = V /N and n f = τ f /d pl .

To investigate how well the diffusion equation approximates the microscopic model, and the relationship between the phenomenological parameter and the underlying model parameters, we focused on the swerves model. For a given causal set and initial position and momentum, the swerves model defines a unique trajectory. To say that the swerves model results in diffusion is, in a sense, saying that the exact underlying causal set is unknown. To simulate this, many different sprinklings into the same region of Minkowski spacetime were generated, and the unique trajectory in each was calculated. For each trajectory, the 'final' position and momentum were determined as the trajectory crossed the t f = 0.95 hypersurface. This gave a distribution in position and momentum that could be compared to that expected from the 1+1 dimensional swerves diffusion equation:

∂ρ ∂t = -p m 2 + p 2 ∂ρ ∂x + k ∂ ∂p m 2 + p 2 m ∂ρ ∂p . (2)
Before comparing the simulation results and the diffusion equation we must note that some trajectories may need to be rejected. If a trajectory is close to the boundary of the region of Minkowski spacetime at any point, it will 'bounce' back and distort the results. For the results shown here, the models parameters were chosen such that there were few such 'invalid' trajectories -any that did occur were removed from the results.

The microscopic model contains three parameters: the forgetting parameter, τ f , a discreteness scale, d pl , and the particle mass, m. Although the discreteness scale does not appear in the trajectory algorithm, the trajectory clearly depends on how many points have been sprinkled into a given volume. For the purposes of this investigation the discreteness length can be defined as d pl = V /N, where V is the volume of the region of 1+1 dimensional Minkowski spacetime and N is the mean number of causal set elements sprinkled.

The particle mass appears in the swerves model algorithm only to normalize the momentum at each step. The trajectory constructed is, in fact, independent of the mass. The final momentum distribution from a collection of trajectories does depend on the mass, but changing the mass only rescales the momentum. Thus, without running any simulations it is possible to determine the dependence of the diffusion parameter, k, on m: examining (2), if m and p are rescaled by a factor α then k must be rescaled by α 2 , therefore k ∼ m 2 .

To determine the dependence of k on τ f , 500 trajectories were evolved for each of 11 values of τ f between 0.03 and 0.1, with m = 1 and d pl ∼ 0.0055, in embedding units. For each value of τ f final position and momentum histograms were calculated. The swerves diffusion equation was numerically evolved for a range of values of k and a best fit value of k was determined for each τ f by minimizing the reduced χ 2 value

χ 2 red = 1 f i (O i -E i ) 2 E i , (3)
where O i is the observed frequency for a momentum bin i, E i is the expected frequency (i.e. that given by the evolution of the diffusion equation), and f is the number of degrees of freedom (here, f = number of data points -1). χ 2 is not a good measure of fit if a significant proportion of the expected frequencies are less than five (see, e.g. [START_REF] Sheskin | Handbook of parametric and nonparametric statistical procedures[END_REF]). To avoid this problem multiple bins were combined where necessary. A reduced χ 2 of order 1 is usually considered to indicate a good fit (see, e.g. [START_REF] Robert | An introduction to error analysis[END_REF]). The momentum diffusion was chosen for the comparison as the position diffusion is driven by the momentum diffusion. Example position and momentum histograms and the corresponding best fit solutions for τ f = 0.04 are shown in Figure 2. Note that τ f = 0.04 has not been chosen for any particular reason, the other values of τ f have equally good fits. It is clear from this figure that the swerves diffusion equation is a very good approximation to the underlying model even though τ f is not many orders of magnitude greater than d pl and the discreteness length is nowhere near the d pl → 0 continuum limit.

Plotting ln k vs. ln τ f , Figure 3(a), for all 11 values of τ f reveals the dependence k ∼ τ -5 f . Dimensional analysis now allows the dependence on the final parameter, d pl , to be determined, but for completeness it was checked through simulations. For four different causal set sizes N = {4096, 8192, 16384, 32768}, 500 trajectories were evolved with fixed τ f and m. Again, best fit values of k were determined for each value of N . Figure 3(b) clearly shows, despite only four data points, the dependence k ∼ d 4 pl . Thus, k ∼ m 2 d 4 pl /τ 5 f for the swerves model. The constant of proportionality can of course be determined: returning to the varying τ f data and working in discreteness units where d pl = 1 it is found that k ≈ 2m 2 /τ 5 f , as shown in Figure 3(c). This relationship will be crucial if and when there is an independent reason for a particular value of τ f . For example, it has been hypothesized that causal set theory will contain a 'nonlocality' scale [START_REF] Rafael | Does locality fail at intermediate length-scales?[END_REF][START_REF] Rafael | Light, Links and Causal Sets[END_REF]. The forgetting parameter that appears in the above particle models may be a measure of this nonlocality scale. Sorkin [START_REF] Rafael | Does locality fail at intermediate length-scales?[END_REF] estimates the nonlocality scale to be of the order 10 -12 cm, or 10 20 in Planck units. If the results above are extrapolated to such large values of τ f , a proton with m = 10 -20 would have a diffusion parameter of order k ∼ 10 -140 . The diffusion parameter for hydrogen molecules (the factor of two mass difference is inconsequential here) has been constrained to be k < 10 -102 [START_REF] Dowker | Quantum gravity phenomenology, Lorentz invariance and discreteness[END_REF]. The above estimate is thus not ruled out by existing constraints, but is also, unfortunately, too small to be currently tested.

Conclusions

There is a considerable focus in quantum gravity phenomenology on violations of Lorentz invariance despite there being no evidence to date that Lorentz invariance is violated. Causal set theory offers a way to investigate Lorentz invariant quantum gravity phenomenology. Earlier work led to a diffusion equation describing the behaviour of massive particles in a discrete spacetime in the continuum limit. Although some underlying models were proposed, no formal connection between the models and the continuum behaviour was made and it was expected that the limitations on the size of causal sets that can be simulated would prohibit any direct demonstration of the diffusion behaviour. The results given here show that microscopic models of particle motion do indeed give rise to diffusion and, moreover, it is not necessary to take the discreteness scale, d pl , to zero for this behaviour to occur. The models discussed here are classical point particle models, and there is no claim that they are the true description of particles in causal set theory. This work demonstrates, however, that we can gain much useful information about the observable consequences of discreteness through simulations, despite computational limitations on causal set size.

  Swerves, N = 32768, τ f = 0.1105, l = 16. Model 1, N = 32768, n f = 20, l = 13. Model 2, N = 32768, n f = 20, l = 163.

Figure 1 .

 1 Figure 1. Example trajectories with causal set size, N = 32768, forgetting parameter τ f /n f and trajectory length l.

Figure 2 .

 2 Figure 2. Histograms for 500 trajectories with τ f = 0.04, N = 32768 and best fit solutions from the swerves diffusion equation, k = 2.8 × 10 -9 . The reduced χ 2 value for the momentum distribution is χ 2 red = 0.59.

Figure 3 .

 3 Figure 3. The relationship between the diffusion parameter and underlying model parameters.

  ln k vs. ln τ f and best fit line -5.02 ln τ f -9.8

		-18
		-19
		-20
	lnk	-21
		-22
		-23
		-24
		1.6	1.8	2	2.2	2.4	2.6	2.8
				lnΤ f inunits ofdpl
		(a) -5.2	-5	-4.8	-4.6	-4.4	-4.2	-4
					lndpl

Acknowledgments

The author would like to thank Fay Dowker for ongoing support, Steven Johnston for helpful discussions, and David Rideout for supplying and providing support for the CausalSets toolkit. This work was supported by Tertiary Education Commission of New Zealand (TAD1939) and the Fund for Women Graduates.


J L Hernández-Pastora
email: jlhp@usal.es

ON THE CALCULATION AND INTERPRETATION OF MSA COORDINATES

Keywords: numbers: 02.00.00, 02.20.Hj, 04.20.Cv, 04.20.-q, 04.20.Jb

The static solutions of the axially symmetric vacuum Einstein equations with a finite number of Relativistic Multipole Moments (RMM) are described by means of a function that can be written in the same analytic form as the Newtonian gravitational multipole potential. A family of so-called MSA (Multipole-Symmetry Adapted) coordinates are introduced to perform the transformation of the Weyl solutions; a procedure for their calculation at any multipole order is given, and the results for a low order are shown.

In analogy with a previous result [10] obtained in Newtonian gravity, the existence of a symmetry of a certain system of differential equations leading to the determination of that kind of multipole solutions in General Relativity is explored. The relationship between the existence of this kind of coordinates and the symmetries mentioned is proved for some cases, and the characterization of the MSA system of coordinates by means of this relationship is discussed.

Introduction

As is known, the description of Newtonian Gravity (NG) in the vacuum involves solutions of the Laplace equation whose general well-behaved solution is a series with arbitrary constants that can be identified with the Multipole Moments (MM) of the source, and these quantities allow us to characterize the specific solutions given by the succession of partial sums of the series.

In contrast, the static solutions of the axially symmetric Einstein vacuum equations describing the gravitational field of a bounded isolated mass distribution in General Relativity (GR) can be described by means of only one metric function, f ≡ g 00 , which satisfies the Ernst equation [1].

We are interested in the following questions: might it be possible to obtain a description of these solutions in GR by means of a function, namely u, with the same behavior as the classical potential? Could we write this function u analytically equal to the Newtonian gravitational series but in terms of Relativistic Multipole Moments (RMM)?

We are concerned with these questions for several reasons, in particular because such a description of the relativistic gravitational solution would recover the benefits of the classical interpretation of the gravitational potential (see [10] for details). Moreover, the Weyl family of solutions depends on arbitrary constants, a n , in principle without any physical criteria to choose one or another solution from them, whereas the function u would allow us to deal, in a very simple form, with the Relativistic Multipole Solutions. This has been the aim of some authors and their works devoted to obtain solutions of the Einstein vacuum equations with a finite number of prescribed RMM [7,9,11,12].

In this work we seek an answer to these questions by introducing a family of coordinate systems referred to as MSA (Multipole-Symmetry Adapted). The possibility of extrapolating the symmetries obtained in NG [10] to GR, as well as characterizing the solutions with a finite number of RMM by means of group-invariant solutions, are the relevant features of these coordinate systems and the reason for their proposed name.

In a work published recently [10], the existence of some kinds of symmetries in NG has been proved, which makes it possible to extract from all solutions of the axially symmetric Laplace equation those with the prescribed Newtonian Multipole Moments. A family of vector fields that are the infinitesimal generators of certain one-parameter groups of transformations can be constructed. These vector fields represent symmetries of certain systems of differential equations whose group-invariant solutions turn out to be the family of axisymmetric potentials related to specific gravitational multipoles.

By introducing these coordinates, the function u, which describes the static and axially symmetric vacuum solution with a finite number of RMM, should satisfy the same system of differential equations as the classical potential in NG, and the symmetries of these equations [10] thus allow us to describe and determine the Multipole Solutions in GR analogously to the Newtonian case.

Since the function u, which is transformed from g 00 , must fulfil the corresponding Ernst equation written in MSA coordinates, the following question arises: is it possible to obtain conditions on the change of coordinates, to choose the suitable gauge, from the extension of the symmetries to the corresponding Ernst equation? In other words, can the symmetry groups obtained for the Laplace equation and the supplementary equation [10] be extrapolated to the Ernst equation in that system of coordinates? And if so, could we establish theorems relating the existence of the symmetry of a system of differential equations to that system of coordinates? We shall see, at least for the Monopole case, that this relationship can be used to determine the MSA radial coordinate explicitly.

We shall try to answer these questions along the work in the following way:

In section 2, the MSA systems of coordinates are defined from the context of the multipole expansion of gravitation introduced by Thorne [2], and the procedure to calculate these coordinates is shown for each set of multipole moments of the desired solution. The procedure first consists of performing a coordinate transformation from Weyl coordinates, preserving the Killing vectors and the asymptotically Cartesian behavior. Second, we introduce a function u by redefining the g 00 metric component and we force this function u to be a solution of the corresponding Ernst equation written in the new system of coordinates. The results are addressed in Appendix B. Some comments about the behavior and interpretation of the coordinates obtained complete this section.

In section 3 we attempt to provide these coordinates with a meaningful interpretation by means of the existence of symmetries of certain differential equations. We recall that the static and axially symmetric vacuum solutions with a finite number of RMM can be described, in these coordinates, as the group-invariant solutions of the same system of differential equations that ad-mits the symmetries obtained for the Newtonian case [10]. Furthermore, we prove two theorems that extend those symmetries to the corresponding Ernst equation written in the MSA system of coordinates for the Monopole and the Monopole-Dipole cases. And these theorems provide a relationship between the existence of these symmetries and the Newtonian-type solutions in GR, at least for these two cases. In subsection 3.2 a possible characterization of these systems of coordinates for each multipolar solution is explored, without taking into account our knowledge of the corresponding set of constants a n for each case. In this sense, for the Monopole case the corresponding MSA coordinates can be obtained as the unique solution of the Ernst equation (and the suitable constraints) with certain boundary condition. Nevertheless, for any other case this procedure fails to provide the uniqueness of the MSA system, since two coordinates must be solved rather than the radial coordinate alone, as is the case only for Spherical symmetry. Finally, in Appendix A the expressions of the RMM in terms of the a n constants and the inverse relation are shown for a solution with a set of arbitrary RMM up to order 10.

The MSA system of coordinates

Definition

In 1980 Thorne introduced a system of coordinates called ACMC (Asymptotically Cartesian and Mass Centered) in the context of Multipole expansions of gravitational radiation [2]. His work presents a definition of Relativistic Multipole Moments (RMM) and shows us how to deduce the RMM of a source from the form of its stationary and asymptotically flat vacuum metric in an ACMC coordinate system.

If the components of the metric are written in the coordinates { t, r, θ, φ} we can read off the RMM from the resulting expressions, other terms, R (n-1) ij (y), called Thorne rests, appearing at the same time that are functions depending on the angular variable y ≡ cos θ in at least one degree lower than those associated with the RMM. For the case of axial symmetry, the g 00 component of any static metric written in that kind of coordinates is:

g 00 = -1 + 2 c 2 ∞ n=0 1 r n+1 M n P n (y) + ∞ n=1 1 r n+1 R (n-1) 00 (y) , (1)
M n being the RMM of order n, and P n (ω) the Legendre polynomial.

There is gauge freedom in the choice of ACMC coordinates preserving the invariance of the first series in (1) and addressing the differences in the metric expansion through the Thorne rests. Among the broad class of coordinate systems of this type, we wish that system to lead to an expansion of the metric in such a way that all the R (n-1) 00 (y) Thorne rests will vanish. We propose that this system of coordinates should be referred to as ACMC-TRF (ACMC-Thorne Rest-Free) in a first step, and then become the so-called MSA (Multipole-Symmetry Adapted) system of coordinates for reasons we shall see in the next section.

Thorne showed that de Donder coordinates are ACMC-to order N for all N, and in [2] he discussed that a de Donder transformation of the coordinates is not necessary. Hence, we do not impose the harmonic condition on our system of coordinates, and we perform a coordinate transformation of the following form

xα = x α + χ α (x i ) ; χ α = cte + O(1/R) , (2)
{x α } being the Weyl spherical coordinates {t, R, ω ≡ cos θ, ϕ}, {x α } = {t, r, y ≡ cos θ, ϕ} being the new system of coordinates, and cte denotes a constant. This keeps the Killing vectors unchanged and maintains the asymptotically flat form of the metric: g αβ = η αβ + O(1/R). Now, the χ α functions must be adjusted so that the metric in the new system of coordinates will satisfy the following condition: (c = 1)

g 00 ≡ -1 + 2u = -1 + 2 N n=0 1 r n+1 M n P n (y) , (3)
where N stands for the number of RMM we wish to consider for our metric. Henceforth, the g 00 component of the metric in this system of coordinates acquires a form that is analytically related to the Newtonian gravitational potential of a classical Multipole Solution, and (3) represents the solution, with a finite number (N+1) of RMM, of the static and axially symmetric Einstein vacuum equations.

Calculation of MSA coordinates for any Multipole Solution

The line element of a static and axisymmetric vacuum metric is as follows:

ds 2 = -e 2Ψ dt 2 + e -2Ψ+2γ dR 2 + R 2 1 -ω 2 dω 2 + e -2Ψ R 2 (1 -ω 2 )dϕ 2 , (4)
{R, ω ≡ cos θ, ϕ} being the Weyl spherical coordinates, and Ψ, γ are metric functions satisfying the following equations 1

Ψ ≡ R 2 Ψ RR + 2RΨ R + (1 -ω 2 )Ψ ωω -2ωΨ ω = 0 γ R = (1 -ω 2 ) R R 2 Ψ 2 R -(1 -ω 2 )Ψ 2 ω -2RωΨ R Ψ ω γ ω = ω R 2 Ψ 2 R -(1 -ω 2 )Ψ 2 ω + 2R(1 -ω 2 )Ψ R Ψ ω , ( 5
)
where the sub-indices denote partial derivation with respect to them. The general solution for an isolated and static source with axial symmetry is given by the following asymptotically flat series (the family of Weyl solutions):

Ψ = ∞ n=0 a n R n+1 P n (ω) , (6)
where coefficients a n are arbitrary constants; any set of those coefficients univocally determines the solution. From the calculation 2 of the RMM of this metric (4) with the function Ψ (6), one can obtain an expression for the coefficients a n in terms of the RMM and hence it is possible to choose these coefficients by neglecting the undesirable RMM. This procedure affords a Multipole Solution having a finite number of Multipole Moments [7]. Some authors have devoted some time to seeking such solutions (Pure Multipole Solutions in GR) [7], [8], [9]. In [7], the M-Q Solution is obtained, as well as 1 Let us note that the integrability condition of the equation for γ is simply the equation for Ψ, and therefore the solution of the Laplace equation univocally identifies the spacetime. 2 We perform the FHP method [3], which allows us to obtain the RMM in terms of coefficients m n involved in the expansion series on the symmetry axis of a conformal Ernst potential. Since that conformal potential is related to the metric function Ψ (6), the coefficients m n can be expressed in terms of the set of coefficients {a n }, and hence the final result provides a relation M n = M n (a n ) (57); the triangular structure of this relation [START_REF] Hernández-Pastora | Relativistic gravitational fields close to Schwarzschild solution[END_REF], [7] allows us to calculate the inverse relation, a n = a n (M n ) (58).

the Quadrupole Solution itself, and more recently in [11], [12] a method has been proposed for obtaining the general terms of the series (the coefficients a n ) that define the Pure 2 N -pole Solutions. In fact, the general term of the series corresponding to the gravitational Dipole and the solutions with Monopole plus any other 2 N -pole moment are written specifically.

In Appendix A, the RMM of the general static and axisymmetric solution (6) are written in terms of the coefficients a n , as well as the inverse relation for a solution with arbitrary RMM up to order 9. The method described in [12] can be used to obtain the general term of the series (6) with any finite set of RMM.

Let us set u ≡ 1 2

(1 + g 00 ); this function u, corresponding to the solution (6) written in Weyl coordinates (u W ), resembles the following expression

u W = 1 2 1 -exp 2 ∞ n=0 a n R n+1 P n (ω) , ( 7
)
and would provide the gravitational solution with a finite number (N + 1) of RMM by substituting the coefficients a n from expression (58).

We want to find a change of coordinates which allows to write this function u, corresponding to a Multipole Solution with a finite number (N +1) of RMM in the following way (3):

u MSA = N n=0 M n r n+1 P n (y) . ( 8
)
We can state the following Lemma: Lemma 1

It is possible to calculate a system of coordinates {r, y} such that:

(E0) u W = u MSA (9)
Proof: First, let us perform the above-mentioned coordinate transformation (2) by assuming the following asymptotically Cartesian behavior of the new coordinates:

r = R 1 + ∞ n=1 f n (ω) 1 R n y = w + ∞ n=1 g n (ω) 1 R n , ( 10
)
and let us suppose that both series in (10) converge at least in a neigbourhood of infinity. Second, we must take into account that the g 00 metric component equals, for the static case, the Ernst potential [1], and hence the function u satisfies the following equation derived from the Ernst equation:

(2u -1)[R 2 u RR + 2Ru R + (1 -ω 2 )u ωω -2ωu ω ] = 2[R 2 u 2 R + (1 -ω 2 )u 2 ω ] . ( 11
)
It is straightforward to calculate the transformation of equation ( 11) by means of an arbitrary change of coordinates from the Weyl system ({R, ω}) to another one ({r, y}), leading to the following expressions:

Au rr + Bu yy + 2Cu ry + Du r + Eu y - 2 2u -1 [Au 2 r + Bu 2 y + 2Cu r u y ] = 0,( 12
)
where A, B, C, D and E are functions of the coordinates {r, y} defined as follows

A(r, y) ≡ LB 1 (r, r) B(r, y) ≡ LB 1 (y, y) C(r, y) ≡ LB 1 (r, y) D(r, y) ≡ LB 2 (r) E(r, y) ≡ LB 2 (y), (13)
LB 1 () and LB 2 () being the Laplace-Beltrami operators with respect to 3dimensional Euclidean metric (with axial symmetry) written in Weyl spherical coordinates; i.e.,

LB 1 (, ) ≡ η ij ∇ i ()∇ j () = R 2 ∂ R ()∂ R () + (1 -ω 2 )∂ ω ()∂ ω () LB 2 () ≡ η ij ∇ ij () = R 2 ∂ 2 RR + 2R∂ R + (1 -ω 2 )∂ 2 ωω -2ω∂ ω . ( 14
)
The equation (E0) means that we force the solution with a finite number (N + 1) of RMM, written in Weyl coordinates (u W ), to be functionally equal to the Newtonian gravitational potential with that number of Multipole Moments written in the MSA system of coordinates (u MSA ). This condition is equivalent to the change of the metric component g 00 by means of the coordinate transformation.

We shall now impose the following condition:

(EI) [ Au rr + Bu yy + 2Cu ry + Du r + Eu y ] u=u MSA = 2 2u -1 [Au 2 r + Bu 2 y + 2Cu r u y ] u=u MSA . ( 15
)
This equation (EI) should be understood in the following way: the function u MSA must be a solution of the differential equation ( 12) obtained from the Ernst equation for u (11) by means of the gauge transformation.

These two conditions, (E0) and (EI), univocally determine the functions f n (ω) and g n (ω) (10), up to any order, in the following way: first, equation (E0) provides a relation between each f n (ω) and g k (ω), for k from 1 to n -1, by developing a power series expansion on the inverse of the radial coordinate R. Second, we substitute u MSA (8) in the differential equation (EI) and the resulting expression can be expanded in power series of 1/R by using the gauge (10). Since we have considered the previously obtained relations between functions f n (ω) and g n (ω) in this expansion, the condition (EI) finally leads to a constructive calculation of those functions up to any order n. This procedure is straightforward and it can be followed up to any order O(1/R n ) and for any set of RMM. In particular, and as a matter of illustration, we have computed the gauge (10) for a solution with the first three RMM (M 0 ,M 1 and M 2 , monopole, dipole and quadrupole moments respectively) up to order O(1/R 10 ); the results obtained are very cumbersome and too long for high orders to be explicitly shown. In Appendix B some of the corresponding functions f n (ω) and g n (ω) of the gauge (10) are included, and complete expressions are supplied in a file to the editor or can be obtained by asking for it to the author.

Let us make some comments about the good behavior of these coordinates {r, y}: i) We assume that the series of the gauge (10) are convergent, but is not the aim of this work to prove it. Nevertheless, some guidelines can be introduced to show that both series are convergent on the axis (ω = 1) in some neighbourhood of infinity. The complete proof of convergence of the series is a very interesting open problem to be carried out in a future work. For the spherical symmetry case, the convergence of the series of the gauge (10) is proved in Lemma 2. If one considers solutions with more RMM than the Monopole, then we can obtain the convergence on the symmetry axis by an argument as follows:

As can be seen in the expressions of the functions g n (ω) (61), all these functions vanish for ω = ±1. This means that the coordinate y preserves the axial symmetry, since y = ω along the axis. This characteristic of the functions g n (ω) can be derived from the equation (56) because that expression allows us to calculate iteratively any function g k (ω) (with k ≥ 1) in terms of the functions g j (ω) and their derivatives of lower order (for j ≤ k -1); since g 1 (ω) = 0 and g 2 (ω) is proportional to (1 -ω 2 ), this factor (1 -ω 2 ) at the left-hand side of equation ( 56) leads to the conclusion g k (ω = 1) = 0 for all k ≥ 1.

The equation ( 56) represents the condition C = 0 (41) or equivalently LB 1 (r, y) = 0 (53)(a). This constraint is imposed by Theorem 2 to the MSA coordinates for the Monopole-Dipole solution, but nevertheless it is not exclusive for this case; in fact, the MSA coordinates corresponding to any solution with a finite number of RMM fulfill this condition which implies that there are no cross terms in the metric written in MSA coordinates (g ij = 0, for i = j). We can either consider this constraint to the MSA coordinates in addition to the conditions (E0) and (EI) ( 9), ( 15), or use it instead of condition (EI): in fact, it is straightforward to verify that the pair of conditions (E0)-(EI) is equivalent to the pair of conditions (E0)-( 56) in order to determine the functions f n (ω) and g n (ω), i.e., these functions are univocally determined by introducing the functions f n (ω) obtained from the condition (E0), in terms of g n (ω), into the equation ( 56) and this procedure provides the same set of functions previously obtained from conditions (E0) and (EI).

Therefore, by taking ω = 1 as well as y = ω = 1 in ( 7) and ( 8), we can rewrite the equation ( 9) as follows:

Ru W = N n=0 M n R n R r n+1 . ( 16
)
This expression ( 16) is a polynomial equation in R/r with finite coefficients, for fixed R such that |R| ≥ R 0 , if R 0 is the constraint introduced by the convergence 3 , in a neighbourhood of infinity, of the series appearing in the expression (7). Taking the limit R → ∞ in ( 16) one obtains the asymp-

totics lim R→∞ R r = 1, since Ru W = - ∞ n=0
a n R n , (with a 0 = -M 0 ). All finite solutions to a polynomial equation depend continuously on the coefficients (see [12], and references therein for details, where solutions to a general algebraic equation is studied as well as their convergence when these solutions are written as a series), and hence, there is an

R 1 ≥ R 0 such that R r ≥ 1 2 for all R ≥ R 1 . This means that r R = 1 + ∞ n=1 f n (ω = 1) 1 R n ≤ 2 , ( 17
)
and hence, the series converges for R > R 1 , at least in a neighbourhood of infinity.

ii) By taking m 2 = 0 in the expressions appearing in Appendix B, we obtain the gauge corresponding to a solution with only Monopole and Dipole Moments. Obviously, these coordinates are not mass-centered since the Dipole Moment does not vanish. Nevertheless, in spite of this fact, we have considered this solution because, as we will see in section 3, this kind of solution, with a prescribed multipole structure, described by a function u MSA in the same way as the Newtonian potential can be related with the existence of a symmetry of certain system of differential equations in analogy with the case of spherical symmetry. The Monopole-Dipole gauge is useful as a test to implement an additional multipole and its knowledge deserves to generalize to GR the Newtonian symmetries previously obtained [10]. The Monopole-Quadrupole solution could be a more appropriated case to be considered; in fact the equatorial symmetry, (which implies that the coefficients a n and the Multipole Moments M n with odd indexes n vanish), would simplify the gauge.

iii) According to the meaning of the RMM, these coordinates reveal the loss of relevance of high-order multipoles in the description of the solutions at large distances from the source. Moreover, as was previously said, all the functions g n (ω) (61) vanish for ω = ±1. This means that the coordinate y preserves the axial symmetry, since y = ω along the axis. iv) They are not harmonic coordinates because that condition (the de Donder gauge) is not fulfilled by the associated Cartesian coordinate z. 4v) In addition, if we consider the MSA coordinates for the case of spherical symmetry, which can be done by neglecting all RMM 5 in (60)-(61) greater than the monopole (M 0 ), then we observe that all the functions f n (ω = ±1) = 0 for n ≥ 2, (i.e., the coordinate r along the axis orthogonal to the equatorial plane shows a good behavior; in other words, it equals the Cartesian coordinate z, up to a displacement along the axis). These functions f n (ω) and g n (ω), for the case of spherical symmetry, can be written as follows:

f 2n (ω) = M 2n C (-1/2) 2n (ω) , f 2n+1 (ω) = 0 , n ≥ 1, f 1 (ω) = M g 2n (ω) = -M 2n C (-1/2) 2n+1 (ω) , g 2n+1 (ω) = 0 , n ≥ 1, g 1 (ω) = 0, (18) C (-1/2) n
(ω) being Gegenbauer orthogonal polynomials, and henceforth M ≡ M 0 is the Monopole Moment.

In fact, the case of spherical symmetry deserves a more detailed analysis; we can state the following lemma: Lemma 2

The sums of the series appearing in the gauge transformation (10) with the functions f n (ω) and g n (ω) given by ( 18) can be obtained, and the corresponding coordinate r leads to the Standard Radial Coordinate of Schwarzschild.

Proof:

Since the generator function of these polynomials C

(-1/2) n
is known:

r ± ≡ √ 1 ± 2ωλ + λ 2 = ∞ n=0 C (-1/2) n (ω)(∓λ) n , ( 19
)
the following relations hold

∞ n=0 C (-1/2) 2n (ω)λ 2n = 1 2 (r + + r -) - ∞ n=0 C (-1/2) 2n+1 (ω)λ 2n+1 = 1 2 (r + -r -) , ( 20
)
sidered. This condition leads to the following equation for the coordinate z: LB 2 (ẑ) = 0. It is easy to see that LB 2 (ẑ) = cos θ LB 2 (r) + r LB 2 (cos θ) + 2 LB 1 (r, cos θ) and the coordinates {r = r, cos θ = y} given in (10) (60-61) do not satisfy this harmonic condition. 5 As can be seen, expressions (59) are recovered from (60) by taking all RMM equal to zero, except for M 0 . and therefore, by taking λ = M/R, the coordinates {r, y} from (10) for this case are given by

r = M + R 2 (r + + r -) = M(x + 1) y = R 2M (r + -r -) = y p , ( 21
)
where {x, y p } are the prolate spheroidal coordinates [18], [START_REF] Hernández-Pastora | Relativistic gravitational fields close to Schwarzschild solution[END_REF]. This radial coordinate r is easily recognized since it is merely the socalled standard radial coordinate of Schwarzschild and, as is known, the g 00 component of the Schwarzschild metric written in these coordinates is g 00 = -1 + 2M/r, and hence the prescribed form of the metric component ( 3) is recovered by this coordinate system and the relativistic Monopole Solution is described by a function u equal to the spherical Newtonian potential M/r.

In the following section we shall see the reasons why we refer to this system of coordinates as a Multipole-Symmetry Adapted one.

Interpretation and characterization of MSA coordinates

The function u written in a MSA system of coordinates {r, y} should be a solution of equation ( 12), (that is, the meaning of the condition (EI)), but at the same time it is also a solution of the following system of differential equations

0 = r 2 u rr + 2ru r + (1 -y 2 )u yy -2yu y 0 = ∂ N +1 y u , (22)
for any value of N , whenever the function u represents the Multipole Solution with a finite number (N + 1) of RMM (8). Therefore, the symmetries of the system of differential equations ( 22), obtained in NG [10], that allow one to extract from all solutions of the axially symmetric Laplace equation those with the prescribed Newtonian Multipole Moments work identically in the case of using MSA coordinates, but now the quantities M n are the RMM (8). In this sense, the same family of vector fields that are the infinitesimal generators of certain one-parameter groups of transformations can be constructed. These vector fields represent symmetries of the system of differential equations (22) whose group-invariant solutions turn out to be the family of axisymmetric potentials related to specific gravitational multipoles (8).

The question we want to answer now is whether those groups of symmetry exist for the system of differential equations joined by ( 22) and ( 12); if so, we could generalize the above-mentioned results to GR, establishing a relationship between the existence of a certain symmetry and the uniqueness of the solutions of the Einstein equations with a prescribed multipole structure. Let us remark that equation ( 12) is the corresponding Ernst equation for the function u, and hence we could say more appropriately that Einstein equations admit the symmetry. Nevertheless, the existence of these coordinates itself leads to conclude that the static and axisymmetric solutions of the vacuum Einstein equations with a prescribed number (N + 1) of RMM can be described by means of the group-invariant solutions [10] of the system of differential equations ( 22) for each value of N .

Moreover, we wish to know whether the action of these symmetries on the equation ( 12) might provide conditions on the unknown functions f n (ω) and g n (ω) (10) to characterize the MSA coordinates.

Multipole symmetries in GR

Let v = r ∂ ∂r -u ∂ ∂u ( 23
)
be a vector field on an open subset M ⊂ X × U , where X = R 2 is the space representing the independent variables, coordinates {x} = (r, y) being MSA coordinates such that ( 8) is fulfilled for N = 0, and U = R, with the coordinate u that represents the dependent variable. We can state the following theorem: Theorem 1

The system of equations ν (x,

u (n) ) = 0 given by        1 (x, u (n) ) ≡ r 2 u rr + 2ru r + (1 -y 2 )u yy -2yu y = 0 2 (x, u (n) ) ≡ u y = 0 3 (x, u (n) ) ≡ Au rr + Bu yy + 2Cu ry + Du r + Eu y + -2 2u-1 [Au 2 r + Bu 2 y + 2Cu r u y ] = 0, ( 24
)
where 1 is the Laplace equation (with axial symmetry), 2 the so-called supplementary equation [10], and 3 equation ( 12), admits a symmetry group whose infinitesimal generator is v.

Proof:

The prolongation of v acting on the supplementary equation is pr (1) v [ 2 ] = -u y , and the second prolongation of this vector acting on 1 is pr (2) v [ 1 ] = -1 . Therefore, both prolongations are zero whenever the system of equations ν (x, u (n) ) = 0 is fulfilled.

With respect to the third equation of the system, it is straightforward to see that pr (2)

v [ 3 ] = Du r + Eu y + r (A r u rr + B r u yy + 2C r u ry + D r u r + E r u y ) + - 1 2u -1 (Au rr + Bu yy + 2Cu ry + Du r + Eu y ) + - 2r 2u -1 A r u 2 r + B r u 2 y + 2C r u r u y -4 3 . ( 25
)
Since the prolongations of the vector only need to vanish on solutions of the system of equations (24) [START_REF] Olver | Applications of Lie Groups to Differential equations[END_REF], we make use of equations 2 and 3 to obtain the following relation between the derivatives of u

u rr = 2 2u -1 u 2 r - D A u r , ( 26
)
and we substitute this and 2 into (25) to obtain

pr (2) v [ 3 ] = u r rD r - D A (rA r -A) + u 2 r - 2A (2u -1) 2 . ( 27
)
The derivatives u r and u 2 r are related by equations 3 and 1 (with u y = 0) as follows

u r - 2A r = u 2 r 2A 2u -1 , ( 28
)
and therefore the second prolongation of v acting on 3 is

pr (2) v [ 3 ] = u r rD r - D A (rA r -A) - 1 2u -1 (- 2A r + D) . ( 29
)
Equation ( 26) is the Ernst equation for the function u with the constraint 2 . It should therefore be fulfilled by the Monopole Solution, represented by u = M/r, since we have required the system {r, y} to be MSA coordinates for the Monopole case (spherical symmetry), and hence we can state the following relation between the coefficients A and

D D A = 2 r r -M r -2M . ( 30
)
Finally, by using equation ( 30) and its derivative with respect to r, i.e.,

rD r = r D A A r - A r + 2A -2A r -M (r -2M) 2 (31)
in (29), we have that:

pr (2) v [ 3 ] = - 2AM r -2M u r 1 r(2u -1) + 1 r -2M . ( 32
)
This expression is zero if at least one of the following conditions hold:

A = 0 , u r = 0 , u = M/r . ( 33
)
The first condition (A = 0) can obviously be neglected because it implies that equation 3 disappears; the second condition means that u = a + bf (y), a, and b being arbitrary constants and f (y) an arbitrary function of the variable y. Nevertheless, that expression is not a solution of the system (24), except for the case u = a, because u y = 0, and, since u = M/r has been forced to be a solution of 3 , which is a non-linear equation, then the linear combination u = a + M/r is no longer a solution of 3 . Hence, we must finally conclude that pr (2) v [ 3 ] = 0 iff u = M/r, i.e., whenever ν (x, u (n) ) = 0, as is the case, since that is the only solution of system (24).

Let {r, y} be an MSA system of coordinates such that (8) is fulfilled for N = 1. We can then state the following: Theorem 2

The system of equations ν (x, u (n) ) = 0 given by

       1 (x, u (n) ) ≡ r 2 u rr + 2ru r + (1 -y 2 )u yy -2yu y = 0 2 (x, u (n) ) ≡ u yy = 0 3 (x, u (n) ) ≡ Au rr + Bu yy + 2Cu ry + Du r + Eu y + -2 2u-1 [Au 2 r + Bu 2 y + 2Cu r u y ] = 0, ( 34
)
where 1 is the Laplace equation (with axial symmetry), 2 the so-called supplementary equation [10], and 3 equation ( 12), admits a symmetry group whose infinitesimal generator is

v = r ∂ ∂r + y ∂ ∂y -u ∂ ∂u . ( 35
)
Proof:

In [10] the null conditions on the prolongations of the vector field (35) acting on the first two equations of system (34), pr (2) v [ 1 ] = pr (2) v [ 2 ] = 0, whenever these two equations are fulfilled, were satisfied. Now, we explore whether pr (2) v [ 3 ] vanishes for the solutions of the system (34). It is straightforward to calculate (see [10] for details) that the second prolongation of vector (35) acting on equation 3 is

pr (2) v [ 3 ] = u rr (rA r + yA y -A) + u yy (rB r + yB y -B) + u ry (2rC r + 2yC y -2C) + u r (rD r + yD y ) + u y (rE r + yE y ) + u r u y - 4 2u -1 (rC r + yC y ) + 2 2u -2 (2u -1) 2 2C + u 2 r - 2 2u -1 (rA r + yA y ) + 2 2u -2 (2u -1) 2 A + u 2 y - 2 2u -1 (rB r + yB y ) + 2 2u -2 (2u -1) 2 B . ( 36
)
Since pr (2) v [ 3 ] = 0 only needs to hold for the solutions of (34), we can substitute the derivative u rr from equations 3 and 2 ,

u rr = 2 A(2u -1) Au 2 r + Bu 2 y + 2Cu r u y - 2C A u ry - D A u r - E A u y (37)
into (36) as follows

pr (2) v [ 3 ] = u yy (rB r + yB y -B) + u ry 2rC r + 2yC y -2C r A r A + y A y A + u r rD r + yD y + D 1 -r A r A + y A y A + u y rE r + yE y + E 1 -r A r A + y A y A + u 2 r - 2A (2u -1) 2 u r u y - 4 2u -1 (rC r + yC y ) + 2 2u -2 (2u -1) 2 2C + 4C 2u -1 r A r A + y A y A + u 2 y - 2 2u -1 (rB r + yB y ) + 2 2u -2 (2u -1) 2 B + 2B 2u -1 r A r A + y A y A .(38)
By using expression (37) and 2 in equation 1 , the derivative u 2 r should satisfy the following equation

2 2u -1 u 2 r = - 2 Bu 2 y + 2Cu r u y (2u -1)A + 2 C A u ry - 2 r - D A u r + E A + 2y r 2 u y .
(39) By replacing expression (39) and equation 2 in (38) we have that pr (2)

v [ 3 ] = u ry 2rC r + 2yC y -2C r A r A + y A y A - 2C 2u -1 + u r rD r + yD y + D 1 -r A r A + y A y A + 2A r(2u -1) - D 2u -1 + u y rE r + yE y + E 1 -r A r A + y A y A - 2Ay r 2 (2u -1) - E 2u -1 + u r u y - 4 2u -1 (rC r + yC y ) + 4C 2u -1 r A r A + y A y A + u 2 y - 2 2u -1 (rB r + yB y ) + 2B 2u -1 r A r A + y A y A . ( 40
)
Since the coefficients of the various monomials in the first-order and second-order partial derivatives of u in this expression must be equal to zero, we must impose the following conditions on the coefficients of equation 3 and their derivatives:

C = 0 (41) A (rB r + yB y ) = B (rA r + yA y ) (42) rD r + yD y - D A (rA r + yA y ) + 2u -2 2u -1 D = - 2A r(2u -1) (43) rE r + yE y - E A (rA r + yA y ) + 2u -2 2u -1 E = 2Ay r 2 (2u -1) . ( 44
)
The general solution of the system of equations 1 and 2 is as follows

u = c 1 y r 2 + c 2 ry + c 3 + c 4 r . ( 45
)
Since {r, y} is a MSA system of coordinates (for the multipole order considered) we can force equation 3 to possess the Monopole-Dipole Solution (among those from ( 45) ) represented by ū = 1 r M + y r 2 M 1 (M and M 1 being the Monopole and Dipole moments respectively) and so, in addition to (41)-( 44), we have the following condition (C = 0):

A ūrr - 2 2ū -1 ū2 r - 2 2ū -1 B ū2 y + Dū r + E ūy = 0 . ( 46
)
Moreover, this imposed condition implies that the only solution of the system of equations ( 34) is the Monopole-Dipole ū, and henceforth we can replace u by ū in equations ( 41)-( 44), leading to the following expressions:

(a) rν r + yν y = 0 (b) rκ r + yκ y + 2ū -2 2ū -1 κ = - 2 r(2ū -1) (c) rµ r + yµ y + 2ū -2 2ū -1 µ = 2y r 2 (2ū -1) , ( 47
)
where ν ≡ B A , κ ≡ D A and µ ≡ E A . Equation (47.a) requires ν to be an arbitrary function of y/r and the solutions of equations (47.b) and (47.c) for κ and µ respectively are the following functions:

ν = ν(y/r) , κ= - 2r + F 1 (y/r) r 2 (2ū -1) , µ = 2y -F 2 (y/r) r 2 (2ū -1) . ( 48
)
Equivalently, these solutions of the determining equations (47) can be written as follows:

νA = B (2r + F 1 )A = (r 2 -2Mr -2M 1 y)D (2y -F 2 )A = -(r 2 -2Mr -2M 1 y)E . ( 49
)
These functions ν, κ and µ are related by means of condition (46), leading to the following relation between F 1 , F 2 and ν

M 1 F 2 = 2M 2 + 4M 2 1 y r 2 + 8MM 1 y r -2M 2 1 ν + MF 1 + 2M 1 F 1 y r . ( 50
)
As already noted, that radial coordinate r is merely the Schwarzschild standard coordinate, the coordinate y being free of constraints because of the spherical symmetry. The system of coordinates characterized by solving equation ( 54) with boundary conditions (55), can aptly be said to be adapted to the Monopole symmetry group, whose infinitesimal generator is (23), for several reasons. First, we see that, written in these coordinates, the solution does not depend on the angular coordinate. In addition, another feature contributes to characterizing these coordinates: we refer to it as MSA because of the interrelation between the existence of the symmetry and the system of coordinates itself. Second, the function u that describes the relativistic solution with a finite number of RMM acquires the form of the classical Multipole potential, and hence all the conclusions obtained for the Newtonian case can be assumed again for this function, which can be considered as the group-invariant solution of a system of differential equations ( 22) that admits the symmetry.

B)The Monopole-Dipole symmetry

In analogy with the previous case, Theorem 2 allows us to establish a relationship between the existence of a symmetry of a certain system of equations and the MSA coordinates for the Monopole-Dipole Solution. We have seen that the uniqueness of the solution of the system of differential equations (34) can be deduced if the Ernst equation for the function u written in MSA coordinates is required to have a solution with the analytic form of the classical Monopole-Dipole gravitational potential. At the same time, this condition leads to the existence of a symmetry group for that system of equations.

The main goal obtained in the previous case (Monopole) is the calculation of the coordinate r by means of the constraint (54) that Theorem 1 introduces in the transformation of the coordinates; with appropriate boundary conditions (55), the choice of the functions f n (ω) is unique and r is fully determined.

Nevertheless, we cannot perform the complete determination of the MSA coordinates for the Monopole-Dipole case by using the constraints (53) introduced by Theorem 2. Equation (53.a) means that the new coordinates r and y preserve the orthogonality since they must be asymptotically Cartesian coordinates. This equation also implies that there are no cross terms in the metric written in MSA coordinates (g ij = 0, for i = j). If we substitute the prescribed gauge transformation from Weyl coordinates (10) in constraint (53.a), the corresponding series expansion leads to the following equations

(1 -ω 2 ) k i=1 f i (ω)g k-i (ω) = kg k (ω) - k-1 n=2 (n -1)(k -n)f n (ω)g k-n (ω) , ( 56
)
where g 0 (ω) = ω, the symbol ( ) denotes the derivative with respect to the variable ω, and k is the order of the expansion in the parameter 1/R. This expression allows us to write any function g k (ω) in terms of the functions f n (ω) and g n (ω) of lower order (n < k) and their derivatives. As can be seen, the good behavior of the functions g n (ω) is recovered, since they must be zero along the axis of symmetry (ω = ±1). With the expressions of g k (ω) obtained from (56), we may solve the corresponding equations at each order of the series expansion of the other constraint, (53.b), for the functions f n (ω) alone. But now we do not have a suitable boundary condition to obtain a unique solution of the functions f n (ω) for the complete determination of the coordinate r for this Monopole-Dipole case. We can demand that the limit M 1 = 0 must lead to the same functions as the Monopole case, but this only allows us to determine the arbitrary constants of integration for f 1 (ω) and f 2 (ω), because from the next order onwards the equations involve arbitrary constants, since the functions f n (ω) for odd n are null in the Monopole case.

With this procedure we obtain a family of coordinates that transforms the Ernst equation into another one that admits the function ū, representing the Monopole-Dipole Solution, as a solution. Nevertheless, the coordinate transformation is not unique.

Conclusion

The exterior gravitational field of an isolated and static compact body with axial symmetry is described in GR by means of the Weyl family of solutions, which depends on a set of arbitrary coefficients {a n } whose values univocally determine each specific solution. If one is looking for solutions that are well known and physically meaningful, it is necessary to relate this set of coefficients to the RMM in order to make a suitable selection of them. If we work with an MSA system of coordinates, then the solution that describes the gravitational behavior of compact bodies with a prescribed multipole structure can be constructed by identifying the function u with the Newtonian potential and considering the constants of the classical potential to be exactly the RMM of the solution. The transformation of a solution (Weyl) into another one (MSA) fixes the change of coordinates by requiring the function u to be a solution of the corresponding Ernst equation.

Nevertheless, this procedure needs to know, a priori, the set of coefficients {a n } of the desired Multipole Solution, although the existence and explicit knowledge of these systems of coordinates is relevant enough and they become very useful, at least for the following topics: Application of this work could shed light on study of the influence and relevance of different RMM in the behavior of test particles along geodesics [16] for different sources, since the MSA coordinates provide us with the exact Multipole Solutions in a very simple way. The deviation of the source from the spherical configuration is a very important feature, for example, for describing the fate of the collapse of self-gravitating systems [17]. The calculation of circular geodesics at successive distances from the source can be used to determine its multipole structure. Additionally, some authors have attempted to [START_REF] Aguirregabiria | [END_REF] relate the RMM to the structure of the source by means of quantities defined over the distribution tensor of the source. The existence of MSA coordinates for any Multipole Solution seems to be a very useful tool for achieving these aims in the frame of global stationary axisymmetric solutions of the Einstein equations.

Except for the Monopole case, in which the gauge is well-known (Schwarzschild radial coordinate), the MSA system of coordinates for the other cases are given by means of two series expansions in the inverse radial Weyl coordinate. However, if we work at large distances from the source, the approximate character of the coordinates is negligible and the change of coordinates is completely determined. In addition, the function u is an exact solution with the finite number of desired RMM, and hence, we have defined a family of static and axially symmetric exact vacuum solutions with a prescribed multipole structure in a system of coordinates defined with a suitable order of approximation.

Finally, this work affords another conclusion: two theorems have been proved that allow us to establish a relationship between the existence of a certain symmetry of a system of differential equations (the corresponding Ernst equation for the function u is included among them) and the existence of a system of coordinates in which that function u can be written analytically equal to the Newtonian potential but in terms of the RMM.

This result is relevant in itself, and some implications can be derived from it. In particular, the construction of the MSA system of coordinates for the Monopole case is supported by the proof of Theorem 1, without knowledge of the corresponding set of coefficients {a n }, since the existence of the function u, or the MSA system of coordinates, is equivalent to the satisfaction of the corresponding Ernst equation by that function. This condition requires that the functions involved in the change of coordinates must satisfy some differential equations whose solution is unique for suitable boundary conditions. Accordingly, the existence of a one-parameter group of transformations can be stated, whose infinitesimal generator is (23), which represents a symmetry of the system of equations joined by the Laplace equation with axial symmetry, the supplementary equation and the Ernst equation for the function u written in a system of coordinates adapted to that symmetry (MSA).

For the Monopole-Dipole case, Theorem 2 allows us to establish the same relationship between the existence of the Monopole-Dipole symmetry and the corresponding MSA coordinate system, although unfortunately the characterization of the gauge by means of the conditions provided by the theorem leads to a family of undefined coordinates in terms of arbitrary constants.

Appendix A

The following expressions show the first ten RMM of any Weyl solution in terms of its coefficients a n :

M 0 = -a 0 , M 1 = -a 1 , M 2 = -a 2 + 1 3 a 3 0 , M 3 = -a 3 +
From the above expressions we can extract, at each order, the corresponding coefficient a n in terms of the RMM:

a 0 = -M 0 , a 1 = -M 1 , a 2 = - 1 3 M 3 0 -M 2 , a 3 = -M 1 M 2 0 -M 3 a 4 = - 8 7 M 2 0 M 2 - 1 5 M 5 0 - 6 7 M 0 M 2 1 -M 4 a 5 = - 4 3 M 2 0 M 3 - 12 7 M 0 M 1 M 2 -M 1 M 4 0 - 2 7 M 3 1 -M 5 a 6 = - 20 11 M 1 M 0 M 3 - 25 21 M 4 0 M 2 - 38 21 M 3 0 M 2 1 - 17 11 M 2 0 M 4 - 60 77 M 0 M 2 2 + - 6 7 M 2 1 M 2 - 1 7 M 7 0 -M 6 a 7 = -M 1 M 6 0 - 128 33 M 3 0 M 1 M 2 - 49 33 M 4 0 M 3 - 120 143 M 2 2 M 1 - 18 11 M 3 1 M 2 0 + 6 Appendix B
The following expressions show the functions f n (ω) appearing in (10), for the Monopole case. Since the spherical symmetry only requires to define the radial coordinate, and the function u does not depend on y, the condition (E0) provides by itself this result:

f 1 (ω) = M , f 2 (ω) = - 1 2 M 2 (-1 + ω 2 ) , f 3 (ω) = 0 f 4 (ω) = - 1 8 M 4 (5ω 4 + 1 -6ω 2 ) , f 5 (ω) = 0 f 6 (ω) = - 1 16 M 6 (-35ω 4 -1 + 15ω 2 + 21ω 6 ) , f 7 (ω) = 0 f 8 (ω) = - 1 128 M 8 (5 + 429ω 8 -924ω 6 + 630ω 4 -140ω 2 ) , f 9 (ω) = 0 f 10 (ω) = 1 256 M 10 (7 -2431ω 10 + 6435ω 8 -6006ω 6 + 2310ω 4 -315ω 2 ) (59)
For a more general case, we have calculated the MSA coordinates for the solution having only the Monopole, Dipole and Quadrupole moments, and the functions f n (ω) and g n (ω), up to order 5, are the following7 : denotes the Monopole terms obtained for the previous case (59). (2)

f 1 (ω) = M 0 , f 2 (ω) = 1 2 M 2 0 (1 -ω 2 ) f 3 (ω) = M 3 0 m 1 ω(1 -ω 2 ) + 1 2 m 2 (1 -3ω 2 ) + m 2 1 ω 2 f 4 (ω) = M 4 0 m 2 1 (- 5 42 ω 2 + 1 84 - 3 4 ω 4 ) + m 1 m 2 ω(-3 + 7ω 2 )+ + m 3 1 2 3 ω(1 -4ω 2 ) + m 2 (- 19 28 + 39 14 ω 2 - 5 4 ω 4 ) + f (M ) 4 f 5 (ω) = M 5 0 m 4 1 (- 7 2 ω 2 + 22 3 ω 4 + 1 6 ) + m 2 2 (1 -9ω 2 + 12ω 4 )+ + m 1 (5ω 3 - 7 2 ω 5 - 3 2 ω) + m 3 1 ( 71 21 ω 3 - 13 21 ω) + + m 2 (2ω 2 -3ω 4 ) + m 2 1 (2ω 4 -ω 2 ) + + m 2 1 m 2 (- 3
g 1 (ω) = 0 , g 2 (ω) = -(1 -ω 2 ) 1 2 ωM 2 0 g 3 (ω) = -(1 -ω 2 ) M 3 0 3 m 1 (3ω 2 -1) -2m 2 1 ω + 3ωm 2 g 4 (ω) = -(1 -ω 2 ) M 4 0 168 m 2 1 (126ω 3 + 10ω) + (147ω 3 -63ω)+ + m 2 (210ω 3 -234ω) + m 1 m 2 (-882ω 2 + 126) + m 3 1 (336ω 2 -28) g 5 (ω) = -(1 -ω 2 ) M 5 0 420 m 1 m 2 (945ω 4 + 1566ω 2 -255)+ + m 2 (1260ω 3 -336ω) + m 1 (1890ω 4 -1400ω 2 + 126) + m 3 1 (52 -852ω 2 ) + + m 2 1 (-840ω 3 + 224ω) + m 4 1 (-2464ω 3 + 588ω) + m 2 2 (1512ω -4032ω 3 ) + + m 2 m 2 1 (8652ω 3 -2604ω) (61) 1 f 5 (ω) = M 5 0 m 4 1 (- 7 2 ω 2 + 22 3 ω 4 + 1 6 ) + m 2 2 (1 -9ω 2 + 12ω 4 )+ + m 1 (5ω 3 - 7 2 ω 5 - 3 2 ω) + m 3 1 ( 71 21 ω 3 - 13 21 ω) + m 2 1 (2ω 4 -ω 2 ) + + m 2 (2ω 2 -3ω 4 ) + + m 2 1 m 2 (- 3
g 1 (ω) = 0 g 2 (ω) = -(1 -ω 2 ) 1 2 ωM 2

In[12] a conjecture due to Geroch is proved for the special case of axisymmetric spacetimes, which establishes that given any set of RMM, subject to the appropriate convergence condition, there exists a static solution of Einstein's equations having precisely those moments. The authors in[12] also provide the precise condition on the moments.

Let {x i } ≡ {x, ŷ, ẑ} be Cartesian coordinates associated with the spherical ones {r, θ, φ} as ẑ = r cos θ, x + iŷ = r sin θe i φ; these coordinates are said to be harmonic if {x i } = 0, where denotes the D'Alambert operator with respect to the metric con-

This calculation has been done up to order 9 but the resulting expressions are cumbersome to show here explicitly. The reader can obtain them by sending an e-mail to the author, or alternatively by asking for a file containing these expressions to the editor.

Acknowledgments

This work was partially supported by the Spanish Ministry of Education and Science under Research Project No. FIS 2006-05319, and the Consejería de Educación of the Junta de Castilla y León under the Research Project Grupo de Excelencia GR234. This work would not have been possible without the ideas provided by Professor Jesús Martín, whom I wish to thank for helpful discussions spread out over a good number of years. I wish to thank Professor Jesús Martín as well as Dr. Marc Mars for their helpful discussions and comments on the work.

By using the expression of F 2 obtained from the equation (50), and by using the first equation of (49) we can write the following expression:

(51) Finally, by substituting expression (51), and the function F 1 obtained from the second equation of (49), into the third equation of (49) we have that:

We must therefore say that the second prolongation of the vector v acting on 3 vanishes whenever the system of equations ( 34) is fulfilled, iff condition (52) holds, and hence the proof of this theorem can be concluded since this condition (52) is equivalent 6 to saying that the Monopole-Dipole function ū is a solution of 3 , that is, the assumption from the beginning of the theorem if coordinates used {r, y} are MSA coordinates.

The relevance of the theorem comes from the relationship that can be established between the existence of the symmetry and the gauge of coordinates that provides the Newtonian form of the Monopole-Dipole Solution.

If we recall definitions [START_REF] Olver | Applications of Lie Groups to Differential equations[END_REF][START_REF] Hernández-Pastora | Relativistic gravitational fields close to Schwarzschild solution[END_REF], then the last expression (52), in addition to the equation (41)(C = 0), provides the following explicit conditions in the coordinate transformation:

6 Note that equation ( 46) is exactly equal to expression (52).

20

Note that if we take M 1 = 0 in the above expression, we obtain condition (30), which must be used for the determination of the coordinates in the Monopole case. We shall discuss these results in the following section.

Characterization of the MSA systems of coordinates

A)The Monopole Solution

From theorem 1, one can conclude that a symmetry of the system of differential equations ( 24) exists iff we force equation ( 26) to posses a solution of the Monopole type u = M/r; in other words, equation (30) must be satisfied. Equation ( 26) is the Ernst equation for the function u with the constraints given by the other equations of the system (24), and it should be taken into account that the system of MSA coordinates that we are using allows us to characterize the relativistic Monopole Solution with a function u written as the Newtonian Monopole. Therefore, since the only solution of system (24) is the Monopole Solution, we can state that this system of differential equations admits a symmetry group that can be related to the uniqueness of the solution of the system by means of the existence of MSA coordinates.

Furthermore, condition (30) allows us to determine the gauge of coordinates in which the relativistic solution having only the Monopole Moment is given by a function with the same analytic form as the classical Monopole potential in NG. We proceed to do this in the following way. First, we substitute the coordinate transformation (10) in condition (30), which becomes a constraint over the coordinate transformation, taking into account definitions [START_REF] Olver | Applications of Lie Groups to Differential equations[END_REF][START_REF] Hernández-Pastora | Relativistic gravitational fields close to Schwarzschild solution[END_REF]

Second, we solve the corresponding differential equations that appear at each order in the power series expansion, and the uniqueness of the solution is provided by the following boundary conditions:

which implies that all functions f n (ω) vanish for all n > 1 along the axis orthogonal to the equatorial plane.

- In what follows we show the functions f n (ω) and g n (ω), up to order 9, corresponding to the MSA coordinates for the solution having only the Monopole, Dipole and Quadrupole moments. We use the following notation for the dimensionless parameters m n and the Monopole terms f (ω) being Gegenbauer orthogonal polynomials.

+ M 4 0 m 2 1 (-


A Dari

¶ F Travasso

¶ H Vocca

L Gammaitoni

Breaking strength tests on silicon and sapphire bondings for Gravitational Wave detectors

The realization of third generation gravitational wave interferometric detectors is under way. An important improvement in detector sensitivity can be obtained with the reduction of thermal noise by reaching a cryogenic stage. Two materials have been identified to perform efficiently at low temperature: silicon and sapphire. In this work the breaking strength of the silicate bonding that glues cylinders of silicon (Si) and sapphire (Al 2 O 3 ) has been investigated. All material combinations, flatness quality and orientations have been tested over time. Breaking strength between Si-Si and Si-Al 2 O 3 shows a value similar to that of fused silica, while Al 2 O 3 -Al 2 O 3 does not seem to be a promising combination.

Introduction

Almost all the GW interferometric detectors (LIGO, TAMA and Virgo) have been built by suspending their mirrors by metallic wires. At present, the Virgo mirror suspensions are made of a specific kind of steel (C85). It is foreseen that in the very near future a monolithic fused silica suspension setup will replace the present one, as just realized in GEO600. In fact, thermal noise investigations for GW detectors in the low frequency range (50 -200Hz) have underlined the need of a monolithic suspension stage [1]. At this stage, noise is composed of three different contributions: clamping losses, material internal loss angle and geometrical dilution factor. The best solution to the first contribution is to obtain a monolithic structure where frictional losses can be minimized [2]. Furthermore, it is well known that, using fused silica (FS) wires, it is possible to significantly reduce the thermal noise that affects suspensions and optics at room temperature [3,4,5,6,[START_REF] Zener | Elasticity and Anelasticity of Metals[END_REF][START_REF] Amico | [END_REF] .

This work is in the way of recent studies that are in progress for 3rd generation GW detector sensitivity improvements [9]. In fact, there are fundamental limitations that occurs at low frequency: the seismic noise, the related gravitational gradient noise and the aforementioned thermal noise. Attention will be focalized on the last kind of noise. According to the fluctuation -dissipation theorem, the thermal noise intensity directly depends on the temperature [2]; thus a proposed solution for 3rd generation interferometers is to decrease thermal fluctuations by working at cryogenic temperature.

In these conditions, FS fibres are not a good candidate; in fact, the dissipation angle of FS shows a (Debye) peak at low temperature and the low thermal conductivity of FS does not allow an efficient extraction of the heat deposited by the laser on the mirror.

The present goal is to understand if it is possible to built a monolithic suspension system also in the cryogenic regime with a lower contribution to thermal noise and with low mechanical losses. Recent works [10,11,[START_REF] Rowan | Proc. of the 2nd E. Amaldi Conference on Gravitational Waves and their Detection[END_REF] proposed crystalline materials as good candidate to realize high performance suspensions at low temperature; in particular silicon and sapphire have been taken into consideration. Crystalline silicon is a good candidate thanks to its high thermal conductivity (k(300 K) = 1.48 × 10 2 W m -1 K -1 ). Thanks to this property in a fibre, the thermoelastic peak is moved toward higher frequencies, where the pendulum thermal noise is not dominant [START_REF] Amico | [END_REF]. In addition, the large thermal conductivity at low temperature permits the extraction of a large fraction of thermal energy from the interferometer mirror [START_REF]Material Properties Database (MPDB)[END_REF][START_REF] Swenson | [END_REF]. In the same manner, sapphire has been chosen because it has shown a high quality factor, a high thermal conductivity and a small thermal expansion coefficient at cryogenic temperature. These characteristics drastically reduce the effects of thermal noise, thermal lensing and thermoelastic noise [16,17].

In this work, the characterization of silicon and sapphire is performed to verify their silicate bonding strength. Si and Al 2 O 3 bonding for different factors and conditions are of strong importance for all the scientific community involved in this field. In this paper first results on the bending stress measurements on the silicate bonding at room temperature are presented. In the near future, cryogenic measurements will be considered.

The hydroxide-catalysis bonding

Bonding may occur between almost flat surfaces of different materials if a silicatelike network can be created between the surfaces, as discussed by Gwo [START_REF] Gwo | Unites States Patent no US 6 284 085 B1[END_REF][START_REF] Gwo | Unites States Patent no US[END_REF]. It is important to use a bonding method which may easily and inexpensively perform either at room temperature and over a broad temperature range. The chemistry of hydroxide-catalysis bonding can be considered to take place in the following steps: hydration and etching, polymerization and dehydration [START_REF] Iler | The Chemistry of Silica[END_REF]. In absence of any catalyst, the activation energy of the dehydration process ranges from few eV to several eV , depending on the material. Using a hydroxide catalyst, the energy can be lowered in some cases to less than few eV , according to the precise temperature at which the reaction occurs. Therefore, the hydration and dehydration processes can be favorable also at room temperature through hydroxide catalysis. In our particular case the catalyst is potassium hydroxide. A solution concentration of 1 : 250 (molecular ratio KOH : H 2 O) has been used.

Moreover it is preferable to choose those materials that can form and/or be chemically linked to a silicate-like network. Silicon has been tested among those materials capable of forming silicate-like network by themselves, while sapphire has been chosen among those materials that can be linked to a silicate-like network.

It is widely accepted [START_REF] Tong | [END_REF] that silicon surface hydrophilicity, due to hydroxyl groups and adsorbed water, is the principal element in bonding processes between silicon surfaces. The hydrophilic nature of the silicon surface can be obtained by wet oxidizing agents (aqueous solution, cleaning solution). At room temperature and relative humidity of 50 %, hydrophilic surface is undoubtedly covered by a few layers of water molecules attached to the OH groups via hydrogen bonds. Brought into contact, the surfaces adhere. At this point dehydration of the Si -OH groups, caused by KOH, starts and siloxane bridges (Si -O -Si) are formed. These bridges, which connect the silicate chains in solution, are responsible for the construction of the solid interface between the two bonding surfaces. The result is the rigid bonding, shown in Fig. 1. In these kind of samples that form a silicate-like network, such as silicon, the bonding is a relatively thick three-dimensional one, harder than those materials that cannot form the silicate-like bond themselves [START_REF] Gwo | Unites States Patent no US[END_REF]. The maximum bond strength should be achieved after some weeks: the exact time it takes depends on temperature, on the concentration of the aqueous solution used, on the relative humidity of the environment and on the degree of mismatch of the surfaces of the bounded pieces. It is not important that material substrates are soluble in the bonding solution, but they must have or can generate hydroxyl groups at the interface.

In Fig. 2, two SEM/EDX (scanning electron microscope/energy dispersive x-ray) spectra of respectively silicon-to-silicon and silicon-to-sapphire bonds are reported. In the first case, elements that characterise the silicon surface are Si, but also oxygen and potassium. The last two come from the bonding solution. The same conclusion holds for the silicon-to-sapphire bonding, where O, K and Al exist. The presence of oxygen confirms the hydrophilic nature of silicon and the possibility to form siloxane groups. However, after only a few hours the bond has sufficient strength to allow the pieces to be handled: initially a hydrogen bond is formed, subsequently surfaces are connected by siloxane bridges.

Experimental bonding procedures

In order to do the programmed studies, a number of silicon and sapphire cylinders have been used with different crystalline orientations and flatness as listed in Table 1. The bonding strength is expected to depend on the interface mismatch.

Samples are commercially available from the Impex High-Tech GmbH company, with both sides polished to three different optical flatness of approximately λ/10, λ/7 and λ/4, where λ = 632.8 nm. In order to successfully bond cylinders, fundamental steps have to be followed:

• surface preparation, because a high level of cleaned area is demanded • bonding procedure, both at room temperature and in the air

Surface preparation

To allow the growth of a good bond it is essential that surfaces are rigorously cleaned to eliminate contamination as much as possible; this operation has an impact on the silicon surface chemistry and topography and, consequently, on the mismatch at the interface [22]. Firstly, the sample is cleaned with isopropanol, then it is dipped in a cleaning solution (whose mixture is 5:1, H 2 SO 4 (96 %):K 2 Cr 2 O 7 ), rinsed with water and dried with gaseous nitrogen. The surface preparation is, hence, completed. The bonding process is carried out under a laminar flow in a Class 100 clean environment at room temperature (T = 23 ± 5 o C). 1. Samples considered for the breaking strength analysis. For each combination, orientation and flatness, the number of successful bonded samples over the number of total available samples has been reported, respectively waiting 1, 2 or 3 weeks before the breaking strength test. The diameter of all samples is 5 mm.

Bond combination Flatness

Bonding procedure

This step follows immediately after surface preparation. The first cylinder is placed into a teflon fixture to insure immobilization of the sample. Typically a 0.5 to 0.8 µl/cm 2 drop of potassium hydroxide solution is applied to the upper surface using a micropipette. Then the second sample is lowered onto the first one where the bonding solution has been applied. This small quantity of solution spreads out eventually across the entire contact region. Pieces are carefully aligned and minor adjustments could be made within the first minutes. This solution will react with the surfaces extracting the silicates necessary to obtain the bond. A strong, rigid, very thin bond thus forms between the pieces in contact. This procedure has been tested for FS samples and seems to work well using crystalline silicon. It has been noticed that the settling time for such bonds can be within of hours or days [23]. The samples are held in position until dehydration starts and, using the micropipette, some drops of potassium hydroxide solution are added to decrease imperfections in the bonding growth for the first five hours.

Experimental set up and results

After the sample preparation and after waiting few weeks for the bonding formation, the breaking strength has been measured. A test bench has been realized to measure the strength of the bonding in operative conditions. The force is applied at 0.05 cm from the sample interface. This configuration allows the only bending stress measurement. The facility consists of two parallelepipeds vertically translated. The bonded cylindric block (characterized by different parameters, see Table 1) is inserted between the two sample holders and then is stressed with the mechanical apparatus, which applies a progressive force close to the bonding surface. The sample holder is illustrated in Fig. 3. A load cell is used to measure the stress required to break the bonded samples. Measurements show that fractures frequently occur in the bonded interface because the bonding strength is weaker than the material strength. Fig. 4 represents a Si -Si sample after the breaking strength test. In this case the fracture started from the sample interface (starting point in figure) and ended at the bulk (see first cylinder bulk in figure).

For these measurements, fifteen bonded samples (that means five for each selected time interval) for each orientation and flatness quality (λ/k with k = 10; 7; 4) have been studied (see Table 1). The procedure has been repeated for three different surface combinations: silicon-silicon, silicon-sapphire and sapphire-sapphire.

Results of the bonding strength are reported in figures 5, 6, 7 and 8, where the stress is obtained dividing the force applied by the bonding area. Tests have shown that the bond reaches its best strength value just after a week. In Figs. 5, 6 and 7, each value has been obtained by taking a statistics from five samples. In the analysis, only successfully bonded samples were considered (see Table 1). Sometimes the waiting time before the breaking strength tests is not sufficient for the bond formation and a poor statistics occurs: this mainly depends on the surface preparation and the bonding procedure. This is responsible for the large error bars that occur in our plots.

In Fig. 5, increasing time interval, samples with a λ/4 flatness appear worse than λ/10 and λ/7, as supposed in [24]. In figures 5, 6 and 7, the breaking strength values seem constant within the statistical fluctuation of the measured quantities. It shows that results are independent of the settling time before breaking the samples and of the flatness quality of the cylinder surface. From this evidence, all measurements have been summarized in Fig. 8. Fig. 8 presents the breaking strength behavior with a λ/10 flatness versus all the material pairing combinations. It is clearly visible that the silicate bonding between Al 2 O 3 -Al 2 O 3 has the minimum breaking strength value. This is explained with the difficulty in creating a silicate bonding among such surfaces, as expected by Gwo [START_REF] Gwo | Unites States Patent no US 6 284 085 B1[END_REF][START_REF] Gwo | Unites States Patent no US[END_REF]: the mean value is 1.5±0.8 M P a. Instead a mean value of about 8.7±3.7 M P a is obtained for samples with a flatness of λ/10 and silicon material in at least one of the two bonded cylinders. This behaviour is probably due to the formation of networks between surfaces, as explained in the previous paragraphs. Moreover, the obtained value is comparable to the FS one [6]. Each point is obtained by averaging over all the samples of three selected times (from between 10 to 15 depending on the unbroken samples at the moment of test).

Conclusions

The silicate bonding breaking strength of Si and Al 2 O 3 samples with various flatness quality and crystal orientation has been investigated. Our results are compared to the previously published by van Veggel et al. in [25] and by Suzuki et al. in [26]. No significant dependence of the two material combinations on the bonding time or on the crystal orientation appears, while the Si -Si combination shows a slight dependence of the breaking strength on the flatness quality (at least between λ/10 or λ/7 and λ/4).

As measured in previous papers for fused silica samples [6], also for Si -Si and Si -Al 2 O 3 bondings, the breaking strength shows a behaviour that can be considered promising for the realization of monolithic suspensions in future cryogenic GW detectors. In Fig. 8, the breaking strength, obtained for the Si -Si and Si -Al 2 O 3 combinations with a λ/10 flatness quality, provides a mean value of 8.7 ± 3.7 M P a, as just presented in [27]. Comparing this measure with the Si-Si bonding value in [25], it is visible a small improvement even if the error is large. This difference is probably due to the distinct silicate bonding procedure used and to the sample geometry. For the Al 2 O 3 -Al 2 O 3 bondings the performance is not good. As supposed by Gwo [START_REF] Gwo | Unites States Patent no US[END_REF], in this case the probability of generating through hydroxyl groups is very low and the breaking strength mean value is 1.5 ± 0.8 M P a. Although our tests seems not too promising, previous results open a new possibility to sapphire in future GW detectors, suggesting a waiting time of several years for the formation of the bonding [26]. These preliminary room temperature measurements seem promising enough to continue the study with other tests at cryogenic temperatures on silicon-to-silicon or on silicon-to-sapphire bondings.

In conclusion these results and the possibility of Si and Al 2 O 3 to reduce thermal noise seem to indicate a positive perspective for the future of cryogenic detectors. However, several studies have to be considered in order to completely propose a realistic solution for 3rd GW detectors.

Figure 1 .

 1 Figure 1. Model of room temperature bonding between two hydrophilic silicon surfaces.

Figure 2 .

 2 Figure 2. SEM/EDX spectra of Si-Si (random -random) sample with λ/10 flatness and Si -AL 2 O 3 (random -random) sample with λ/4 flatness.

Figure 3 .

 3 Figure 3. Breaking strength test facility (three-dimensional and front views, not to scale) The sample diameter is 5 mm.

Figure 4 .

 4 Figure 4. Broken bond between two silicon cylinders with a (111-111) orientation and a λ/4 flatness after a curing time of a week: scanning electron microscope image.

Figure 5 .

 5 Figure 5. Bonding strength for Si-Si (111-111) combination: three different flatness qualities are reported for each selected time (first, second and third week).

Figure 6 .

 6 Figure 6. Bonding strength for Si -Al 2 O 3 (random -random) combination: three different flatness qualities are reported for each selected time (first, second and third week).

Figure 7 .

 7 Figure 7. Bonding strength for Al 2 O 3 -Al 2 O 3 (C-C) combination: three different flatness qualities are reported for each selected time (first, second and third week).

Figure 8 .

 8 Figure 8. Breaking strength for all combinations and orientations of the λ/10 flatness.Each point is obtained by averaging over all the samples of three selected times (from between 10 to 15 depending on the unbroken samples at the moment of test).

  Table

			Orientation Number of succesfully bonded samples
				1st week 2nd week	3rd week
	Si-Si	λ/10	rand-rand	5/7	5/7	3/8
	Si-Si	λ/7	rand-rand	0/0	4/5	5/5
	Si-Si	λ/4	rand-rand	0/0	2/5	3/5
	Si-Si	λ/10	100 -100	5/5	5/5	5/5
	Si-Si	λ/7	100 -100	5/5	5/5	5/5
	Si-Si	λ/4	100 -100	4/5	4/5	3/5
	Si-Si	λ/10	111-111	5/5	5/5	5/5
	Si-Si	λ/7	111-111	5/5	5/5	5/5
	Si-Si	λ/4	111-111	5/5	5/5	2/5
	Si-Al 2 O 3	λ/10	rand-rand	3/5	4/5	4/5
	Si-Al 2 O 3	λ/7	rand-rand	5/5	5/5	5/5
	Si-Al 2 O 3	λ/4	rand-rand	5/5	5/5	4/5
	Si-Al 2 O 3	λ/10	100 -C	5/5	5/5	5/5
	Si-Al 2 O 3	λ/7	100 -C	5/5	5/5	5/5
	Si-Al 2 O 3	λ/4	100 -C	5/5	5/5	5/5
	Al 2 O 3 -Al 2 O 3	λ/10	rand-rand	4/5	2/5	5/5
	Al 2 O 3 -Al 2 O 3	λ/7	rand-rand	4/5	4/5	2/5
	Al 2 O 3 -Al 2 O 3	λ/4	rand-rand	0/5	5/5	5/5
	Al 2 O 3 -Al 2 O 3	λ/10	C -C	4/5	4/5	4/5
	Al 2 O 3 -Al 2 O 3	λ/7	C -C	0/5	2/5	3/5
	Al 2 O 3 -Al 2 O 3	λ/4	C -C	3/5	3/5	4/5

Acknowledgments

This research has been supported by the European Gravitational Observatory (EGO) and the Istituto Nazionale di Fisica Nucleare INFN under the Virgo Project.