What can we learn about GW Physics with an elastic spherical antenna?

A general formalism is set up to analyse the response of an arbitrary solid elastic body to an arbitrary metric Gravitational Wave perturbation, which fully displays the details of the interaction antenna-wave. The formalism is applied to the spherical detector, whose sensitivity parameters are thereby scrutinised. A multimode transfer function is defined to study the amplitude sensitivity, and absorption cross sections are calculated for a general metric theory of GW physics. Their scaling properties are shown to be independent of the underlying theory, with interesting consequences for future detector design. The GW incidence direction deconvolution problem is also discussed, always within the context of a general metric theory of the gravitational field.

A general formalism is set up to analyse the response of an arbitrary solid elastic body to an arbitrary metric Gravitational Wave perturbation, which fully displays the details of the interaction antenna-wave.The formalism is applied to the spherical detector, whose sensitivity parameters are thereby scrutinised.A multimode transfer function is defined to study the amplitude sensitivity, and absorption cross sections are calculated for a general metric theory of GW physics.Their scaling properties are shown to be independent of the underlying theory, with interesting consequences for future detector design.The GW incidence direction deconvolution problem is also discussed, always within the context of a general metric theory of the gravitational field.04.80.Nn, 95.55.Ym

I. INTRODUCTION
The idea of building ultracryogenic spherical Gravitational Wave (GW) antennae seems to be progressively winning adepts, even despite the technological difficulties of various kinds posed by a project like that, which every expert acknowledges.Confidence in its feasibility stems from many years of experience: groups at Stanford, Louisiana State University, Roma and Legnaro (Italy) and Perth (Western Australia) have constructed and operated, at different levels, cryogenic cylindrical bars of the Weber type [1].In particular, a long term strain sensitivity h = 6 × 10 −19 for millisecond bursts has been reported from the bar EXPLORER [2].The new generation ultracryogenic cylinder NAUTILUS, of the Frascati group [3], is beginning operation as these lines are written [4], with an expected sensitivity nearly an order of magnitude better than the above.
Spherical antennae are considered by many to be the natural next step in the development of resonant GW detectors [5][6][7][8][9][10].The reasons for this new trend essentially derive from the improved sensitivity of a sphere -which can be nearly an order of magnitude better than a clyinder having the same resonance frequency, see below and [10]-, and from its multimode capabilities, first recognised by Forward [5] and further elaborated in [7,8].
Although some of the most relevant aspects of detector sensitivity have already received attention in the literature, it seems to me that a sufficiently general and flexible analysis of the interaction detector-GW has not been satisfactorily developed to date.This theoretical shortage has a number of practical negative consequences, too.Traditional analysis, to mention but an example, is almost invariably restricted to General Relativity or scalar-tensor theories of gravity; while it may be argued that this is already very general, any such argument is, as a matter of fact, understating the potentialities actually offered by a spherical GW antenna to help decide for or against any one specific theory of the gravitational field on the basis of experimental observation.
I thus propose to develop in this paper a full fledged mathematical formalism which will enable analysis of the antenna's response to a completely general GW, i.e., making no a priori assumptions about which is the correct theory underlying GW physics (other than, indeed, that it is a metric theory), and also making no assumptions about detector shape, structure or boundary conditions.Considering things in such generality is not only "theoretically nice" -it also brings about new results and a better understanding of older ones.For example, it will be proved that the sphere is the most efficient GW elastic detector shape, and that higher mode absorption cross sections scale independently of GW physics.I will also discuss the direction of incidence deconvolution problem in the context of a general metric theory of gravity.
The paper is organised as follows: section 2 is devoted to the development of the general mathematical framework, leading to a formula in which an elastic solid's response is related to the action of an arbitrary metric GW impinging on it.In section 3 the general equations are applied to the homogeneous spherical body, and a discussion of the deconvolution problem is presented as well.Section 4 contains the description of the sphere's sensitivity parameters, specifically leading to the concept of multimode, or vector, transfer function, and to an analysis of the absorption cross section presented by this detector to a passing by GW. Conclusions and prospects are summarised in section 5, and two appendices are added which include mathematical derivations.

II. GENERAL MATHEMATICAL FRAMEWORK
In the mathematical model, I shall be assuming that the antenna is a solid elastic body which responds to GW perturbations according to the equations of classical non-relativistic linear Elasticity Theory [11].This is fully justified since, as stressed above, GW induced displacements will be very small indeed, and the speed of such displacements much smaller than that of light for any forseeable frequencies.Although our primary interest is a spherical antenna, the considerations which follow in the remainder of this section have general validity for arbitrarily shaped isotropic elastic solids.
Let u(x, t) be the displacement vector of the infinitesimal mass element sitting at point x relative to the solid's centre of mass in its unperturbed state, whose density distribution in that state is ρ(x).Let λ and µ be the material's elastic Lamé coefficients.If a volume force density f (x, t) acts on such solid, the displacement field u(x, t) is the solution to the system of partial differential equations [11] with the appropriate initial and boundary conditions.A summary of notation and general results regarding the solution to that system is briefly outlined in the ensuing subsection, as they are necessary for the subsequent developments in this paper, and also in future work.

A. Separable driving force
For reasons which will become clear later on, we shall only be interested in driving forces of the separable type or, indeed, linear combinations thereof.The solution to (2.1) does not require us to specify the precise boundary conditions on u(x, t) at this stage, but we need to set the initial conditions.We adopt the following: where ˙≡ ∂/∂t, implying that the antenna is at complete rest before observation begins at t=0.The structure of the force field (2.2) is such that the displacements u(x, t) can be expressed by means of a Green function integral of the form The deductive procedure whereby S(x; t − t ′ ) is calculated can be found in many standard textbooks -see e.g.[12].The result is where and u N (x) are the normalised eigen-solutions to with suitable boundary conditions.Here N represents an index, or set of indices, labelling the eigenmode of frequency ω N .The normalisation condition is (arbitrarily) chosen so that where M is the total mass of the solid, and the asterisk denotes complex conjugation.Replacing now (2.5) into (2.4) we can write the solution to our problem as a series expansion: where Equation (2.9) is the formal solution to our problem; it has the standard form of an orthogonal expansion and is valid for any solid driven by a separable force like (2.2) and any boundary conditions.It is therefore completely general , given that type of force.
Before we go on, it is perhaps interesting to quote a simple but useful example.It is the case of a solid hit by a hammer blow , i.e., receiving a sudden stroke at a point on its surface.Exam of the response of a GW antenna to such perturbation is being used for correct tuning and monitoring of the device [13].If the driving force density is represented by the simple model where x 0 is the surface point hit, and f 0 is a constant vector, then the system's response is immediately seen to be . A hammer blow thus excites all the solid's normal modes, except those perpendicular to f 0 , with amplitudes which are inversely proportional to the mode's frequency.This is seen to be a rather general result in the theory of sound waves in isotropic elastic solids.

B. The GW tidal forces
An incoming GW manifests itself as a tidal force density; in the long wavelength linear approximation [14] it only depends on the "electric" components of the Riemann tensor: where c is the speed of light, and sum over the repeated index j is understood.In (2.13) tidal forces are referred to the antenna's centre of mass, and thus x is a vector originating there.Note that I have omitted any dependence of R 0i0j on spatial coordinates, since it only needs to be evaluated at the solid's centre.The Riemann tensor is only required to first order at this stage [15]: where h µν are the perturbations to flat geometry1 , always at the centre of mass of the detector.The form (2.13) is seen to be a sum of three terms like (2.2) -but this three term "straightforward" splitting is not the most convenient, due to lack of invariance and symmetry.A better choice is now outlined.
An arbitrary symmetric tensor S ij admits the following decomposition: where E (m) ij are 5 linearly independent symmetric and traceless tensors, and ij is a multiple of the unit tensor δ ij .S (S) (t) and S (m) (t) are uniquely defined functions, whose explicit form depends on the particular representation of the E-matrices chosen.A convenient one is the following: The excellence of this representation stems from its ability to display the spin features of the driving terms in (2.13).Such features are characterised by the relations (2.17) where n ≡ x/|x| is the radial unit vector, and Y lm (θ, ϕ) are spherical harmonics [16].Details about the above E-matrices are given in Appendix A. In particular, the orthogonality relations (A6) can be used to invert (2.15): where an asterisk denotes complex conjugation.Note that S (S) (t) = √ 4πS(t)/3, where S(t) ≡ δ ij S ij (t) is the tensor's trace.
We now take advantage of (2.15) to express the GW tidal force (2.13) as a sum of split terms like (2.2): Straightforward application of (2.9) yields the formal solution of the antenna response to a GW perturbation: with the notation of (2.6) and (2.10) applied mutatis mutandi to the terms in (2.20).Equation (2.21) gives the response of an arbitrary elastic solid to an incoming weak GW, independently of the underlying gravity theory, be it General Relativity (GR) or indeed any other metric theory of the gravitational interaction.It is also valid for any antenna shape and any boundary conditions, thus giving the formalism, in particular, the capability of being used to study the response of a detector which is suspended by means of a mechanical device in the laboratory site -a situation of much practical importance.It is therefore very general.Equation (2.21) also tells us that that only monopole and quadrupole detector modes can possibly be excited by a metric GW.The nice thing about (2.21) is that it fully displays the monopole-quadrupole structure of the solution to our fundamental differential equations.
In a non-symmetric body, all (or nearly all) the modes have monopole and quadrupole moments, and (2.21) precisely shows how much each of them contributes to the detector's response.A homogeneous spherical antenna, which is very symmetric, has a set of vibrational eigenmodes which are particularly well matched to the form (2.21): it only possesses one series of monopole modes and one (five-fold degenerate) series of quadrupole modes -see next section and Appendix B for details.The existence of so few modes which couple to GWs means that all the absorbed incoming radiation energy will be distributed amongst those few modes only, thereby making the sphere the most efficient detector, even from the sensitivity point of view.The higher energy cross section per unit mass reported for spheres on the basis of GR [10], for example, finds here its qualitative explanation.The generality of (2.21), on the other hand, means that this excellence of the spherical detector is there independently of which is the correct GW theory.
Before going further, let me mention another potentially useful application of the formalism so far.Cylindrical antennas, for instance, are usually studied in the thin rod approximation; although this is generally quite satisfactory, equation (2.21) offers the possibility of eventually considering corrections to such simplifying hypothesis by use of more realistic eigenfunctions, such as those given in [17,18].Recent new proposals for stumpy cylinder arrays [19] may well benefit from the above approach, too.

III. THE SPHERICAL ANTENNA
To explore the consequences of (2.21) in a particular case, the mode amplitudes u N (x) and frequencies ω N must be specified.From now on I will focus on a homogeneous sphere whose surface is free of tractions and/or tensions; the latter happens to be quite a good approximation, even if the sphere is suspended in the static gravitational field [20].
The normal modes of the free sphere fall into two families: so called toroidal -where the sphere only undergoes twistings which keep its shape unchanged throughout the volume-and spheroidal [21], where radial as well as tangential displacements take place.I use the notation for them, respectively; note that the index N of the previous section is a multiple index {nlm} for each family; l and m are the usual multipole indices, and n numbers from 1 to ∞ each of the l-pole modes.The frequencies happen to be independent of m, and so every one mode (3.1) is (2l+1)-fold degenerate.Further details about these eigenmodes are given in Appendix B.
In order to see what (2.21) looks like in this case, integrals of the form (2.6) ought to be evaluated.It is straightforward to prove that they all vanish for the toroidal modes, the spheroidal modes contributing the only non-vanishing terms; after some algebra one finds where The functions A nl (r), B nl (r) are given in Appendix B, and R is the sphere's radius.To our reassurance, only the monopole and quadrupole sphere modes survive, as seen by the presence of the factors δ l0 and δ l2 in (3.2a) and (3.2b), respectively.The final series is thus a relatively simple one, even in spite of its generality2 : where, it is recalled, Equation (3.4) constitutes the sphere's response to an arbitrary tidal GW perturbation, and will be used to analyse the sensitivity of the spherical detector in the next section.Before doing so, however, a few comments on the antenna's signal deconvolution capabilities, within the context of a completely general metric theory of GWs, are in order.

A. The deconvolution problem
Let us first of all take the Fourier transform of (3.4): This is seen to be where G (S) (ω) and G (m) (ω) are the Fourier transforms of g (S) (t) and g (m) (t), respectively: The δ-function factors are of course idealisations corresponding to infinitely long integration times and infinitely narrow resonance linewidths -but the essentials of the ensuing discussion will not be affected by those idealisations.
If the measuring system were (ideally) sensitive to all frequencies, filters could be applied to examine the antenna's oscillations at each monopole and quadrupole frequency: a single transducer would suffice to reveal G (S) (ω) around the monopole frequencies ω n0 , whilst five (placed at suitable positions) would be required to calculate the five degenerate amplitudes G (m) (ω) around the quadrupole frequencies ω n2 .Once the six functions G (S,m) (ω) would have thus been determined, inverse Fourier transforms would give us the functions g (S,m) (t), and thereby the six Riemann tensor components R 0i0j (t) through inversion of the second equations (2.20), i.e., as an expansion like (2.15) -only with g's instead of S's.Deconvolution would then be complete.
Well, not quite. . .Knowledge of the Riemann tensor in the laboratory frame coordinates is not really sufficient to say the waveform has been completely deconvolved, unless we also know the source position in the sky.There clearly are two possibilities: i) The source position is known ahead of time by some other astronomical observation methods.Let me rush to emphasise that, far from trivial or uninteresting, this is a very important case to consider, specially during the first stages of GW Astronomy, when any reported GW event will have to be thoroughly checked by all possible means.
If the incidence direction is known, then a rotation must be applied to the just obtained quantities R 0i0j (t), which takes the laboratory z-axis into coincidence with the incoming wave propagation vector.A classification procedure must thereafter be applied to the so transformed Riemann tensor in order to see which is the theory (or class of theories) compatible with the actual observations.Such classification procedure has been described in detail in [22].
The spherical antenna is thus seen to have the capability of furnishing the analyst sufficient information to discern amongst different competing theories of GW physics, whenever the wave incidence direction is known prior to detection.
ii) The source position is not known at detection time.This makes things more complex, since the above rotation between the laboratory and GW frames cannot be performed.In order to deconvolve the incidence direction in this case, a specific theory of the GWs must be assumed -a given choice being made on the basis of whatever prior information is available or, simply, dictated by the the decision to probe a particular theory.Wagoner and Paik [7] propose a method which is useful both for GR and BD theory, their idea being simple and elegant at the same time: since neither of these theories predicts the excitation of the m=±1 quadrupole modes of the wave, the source position is determined precisely by the rotation angles which, when applied to the laboratory axes, cause the amplitudes of those antenna modes to vanish; the rotated frame is thereby associated to the GW natural frame.
A generalisation of this idea can conceivably be found on the basis of a detailed -and possibly rather casuisticanalysis of the canonical forms of of the Riemann tensor for a list of theories of gravity, along the following line of argument: any one particular theory will be characterised by certain (homogeneous) canonical relationships amongst the monopole and quadrupole components of the Riemann tensor, g (S,m) (t), and so enforcement of those relations upon rotation of the laboratory frame axes should enable determination of the rotation angles or, equivalently, of the incoming radiation incidence direction.Scalar-tensor theories e.g. have g (±1) (t) = 0 in their canonical forms, hence Wagoner and Paik's proposal for this particular case.Before any deconvolution procedure is triggered off, however, it is very important to make sure that it will be viable.More precisely, since the transformation from the laboratory to the ultimate canonical frame is going to be linear, invariants must be preserved.This means that, even if the source position is unknown, certain theories will forthrightly be vetoed by the observed R 0i0j (t) if their predicted invariants are incompatible with the observed ones.To give but an easy example, if R 0i0j (t) is observed to have a non-null trace R 0i0i (t), then a veto on GR will be readily served, and therefore no algorithm based on that theory should be applied.I would like to make a final remark here.Assume a direction deconvolution procedure has been successfully carried through to the end on the basis of certain GW theory, so that the analyst comes up with a pair of numbers (θ, ϕ) expressing the source's coordinates in the sky.Of course, these numbers will represent the actual source position only if the assumed theory is correct.Now, how do we know it is correct?Strictly speaking, "correctness" of a scientific theory is an asymptotic concept -in the sense that the possibility always remains open that new facts be eventually discovered which contradict the theory-, and so reliability of the estimate (θ, ϕ) of the source position can only be assessed in practice in terms of the consistency between the assumed theory and whatever experimental evidence is available to date, including, indeed, GW measurements themselves.It is thus very important to have a method to verify that the estimate (θ, ϕ) does not contradict the theory which enabled its very determination.Such verification is a logical absurdity if only one measurement of position is available; this happens for instance if the recorded signal is a short burst of radiation, and so two antennas are at least necessary to check consistency in that case.The test would proceed as a check that the time delay between reception of the signal at both detectors is consistent with the calculated (θ, ϕ)3 , given their relative position and the wave propagation speed predicted by the assumed theory.If, on the other hand, the signal being tracked is a long duration signal, then a single antenna may be sufficient to peform the test by looking at the observed Doppler patterns and checking them against those expected with the given (θ, ϕ).
The above considerations have been made ignoring noise in the detector and monitor systems.A fundamental constraint introduced by noise is that it makes the antenna bandwidth limited in sensitivity.As a consequence, any deconvolution procedure is deemed to be incomplete or, rather, ambiguous [23], since information about the signal can possibly be retrieved only within a reduced bandwidth, whilst the rest will be lost.I thus come to a detailed discussion of the sensitivity of the spherical GW antenna in the next section.

IV. THE SENSITIVITY PARAMETERS
I will consider successively amplitude and energy sensitivities; the first leads to the concept of transfer function, while the second to that of absorption cross section.I devote separate subsections to analyse each of them in some detail.

A. The transfer function
A widely used and useful concept in linear system theory is that of transfer function [24].It is defined as the Fourier transform of the system's impulse response, or as the system's impedance/admittance, and can be inferred from the frequency response function (3.7).
We recall from the previous section that the sphere is a multimode device -due to its monopole and five-fold degenerate quadrupole modes.It appears expedient to define a multimode or vector transfer function as a useful construct which encompasses all six different modes into a single conceptual block, according to where G (α) (ω) are the six driving terms G (S,m) (ω) given in (3.8).The transfer function is Z (α) (x, ω), and its "vector" character alluded above is reflected by the multimode index α.Looking at (3.7) it is readily seen that As we observe in these formulae, the sphere's sensitivity to monopole excitations is governed by a n /ω n0 , and to quadrupole ones by b n /ω n2 .Closed expressions happen to exist for a n and b n ; using the notation of Appendix B, they are Numerical investigation of the behaviour of these coefficients shows that they decay asymptotically as n −2 : Likewise, it is found that the frequencies ω n0 and ω n2 diverge like n for large n, so that Z (α) (x, ω) drops as ω −3 for large ω.Figures 6 and 7 display a symbolic plot of ω 3 Z (S) (x, ω) and ω 3 Z (m) (x, ω), respectively, which illustrates the situation: monopole modes soon reach the asymptotic regime, while there appear to be 3 subfamilies of quadrupole modes regularly intertwined; the asymptotic regime for these subfamilies is more irregularly reached.Note also the perfectly regular alternate changes of phase (by π radians) in both monopole and each quadrupole family.
The sharp fall in sensitivity of a sphere for higher frequency modes (n −3 ) indicates that only the lowest ones stand a chance of being obervable in an actual GW antenna.I report in Table I the numerical values of the relevant parameters for the first few monopole and quadrupole modes.The reason for the last (fourth) columns will become clear later.

B. The absorption cross section
Let us calculate now the energy of the oscillating sphere.We first define the spectral energy density at frequency ω, which is naturally given by4 TABLE I. First few monopole (left) and quadrupole (right) sphere parameters, for a σ= 0.33 material.First and second columns on either side of the central line number the modes and give the corresponding eigenvalue; rows are intertwined in order of ascending frequency, which is proportional to kR -see (B6) below.Third columns contain the an and bn coefficients defined in equations (3.3a) and (3.3b), respectively; the fourth columns display the cross section ratios (k10a1/kn0an) 2 and (k12b1/kn2bn) 2 for higher frequency modes, respectively, taking as reference the lowest in each familiy -cf.equations ( 4 and can be easily evaluated: The energy at any one spectral frequency ω nl is obtained by integration of the spectral density in a narrow interval around ω = ±ω nl : In particular, The sensitivity parameter associated with the vibrational energy of the modes is the detector's absorption cross section, defined as the energy it absorbs per unit incident GW spectral flux density, or where Φ(ω) is the number of joules per square metre and Hz carried by the GW at frequency ω as it passes by the antenna.Thus, for the frequencies of interest, These quantities have very precise values, but such values can only be calculated on the basis of a specific underlying theory of the GW physics.In the absence of such theory, neither Φ(ω) nor G (S,m) (ω) can possibly be calculated, since they are not theory independent quantities.To date, only GR calculations have been reported in the literature [7,9,10].As I will now show, even though the fractions in the rhs of (4.10) are not theory independent, some very general results can still be obtained about the sphere's cross section within the context of metric theories of the gravitational interaction.To do so, it will be necessary to go into a short digression on the general nature of weak metric GWs.
No matter which is the (metric) theory which happens to be the "correct one" to describe gravitation, it is beyond reasonable doubt that any GWs reaching the Earth ought to be very weak.The linear approximation should therefore be an extremely good one to describe the propagating field variables in the neighbourhood of the detector.In such circumstances, the field equations can be derived from a Poincaré invariant variational principle based on an action integral of the type where the Lagrangian density L is a quadratic functional of the field variables ψ A (x) and their space-time derivatives ψ A,µ (x); these variables include the metric perturbations h µν , plus any other fields required by the specific theory under consideration -e.g. a scalar field in the theory of Brans-Dicke, etc.The requirement that L be quadratic ensures that the Euler-Lagrange equations of motion are linear.
The energy and momentum transported by the waves can be calculated in this formalism in terms of the components τ µν of the canonical energy-momentum tensor5 The flux energy density, or Poynting, vector is given by where ˙≡ ∂/∂t.Any GW hitting the antenna will be seen plane, due to the enormous distance to the source.If k is the incidence direction (normal to the wave front), then the fields will depend on the variable ct −k•x, so that the GW energy reaching the detector per unit time and area is where x is the sphere's centre position relative to the source -which is fixed , and so its dependence can be safely dropped in the lhs of the above expression.The important thing to note in equation (4.14) is that it tells us that φ(t) can be written as a quadratic form in the time derivatives of the fields ψ A .As a consequence, the spectral density Φ(ω), defined by can be ascertained to factorise as where Φ 0 (ω) is again a quadratic function of the Fourier transforms Ψ A (ω) of the fields ψ A .On the other hand, the functions G (S,m) (ω) in (4.10) which, it is recalled, are the Fourier transforms of g (S,m) (t) in (2.20), contain second order derivatives of the metric fields h µν , and therefore of all the fields ψ A as a result of the theory's field equations.Since we are considering plane wave solutions to those equations, all derivatives can be reduced to time derivatives -just like in (4.14) above.We can thus write with Ψ (S,m) (ω) suitable linear combinations of the Ψ A (ω). Replacing the last two equations into (4.10) and manipulating dimensions expediently, we come to the remarkable result that where v 2 t ≡ (2+2σ) −1 v 2 s , v s being the speed of sound in the detector's material, and σ its Poisson ratio; G is the Gravitational constant.The "remarkable" about the above is that the coefficients K S (ℵ) and K Q (ℵ) are independent of frequency: they exclusively depend on the underlying gravitation theory, which I symbolically denote by ℵ.To see that this is the case, it is enough to consider a monochromatic incident wave: since the coefficients K S (ℵ) and K Q (ℵ) happen to be invariant with respect to field amplitude scalings, this means they will only depend on the amplitudes' relative weights, i.e., on the field equations' specific structure.
By way of example, it is interesting to see what the results for General Relativity (GR) and Brans-Dicke (BD) theory are.After somehow lengthy algebra it is found that In the latter formulae, Ω is the usual Brans-Dicke parameter ω [27], renamed here to avoid confusion with frequency, and k is a dimensionless parameter, generally of order one, depending on the source's properties [28].As is well known, GR is obtained in the limit Ω → ∞ of BD [15]; the quoted results are of course in agreement with that limit.
Incidentally, an interesting consequence of the above equations is this: though not explicitly shown in this paper (see, however, reference [7]), the presence of a scalar field in the theory of Brans and Dicke causes not only the monopole sphere's modes to be excited, but also the m=0 quadrupole ones; what we see in equations (4.20) is that precisely 5/6 of the total energy extracted from the scalar wave goes into the antenna's monopole modes, whilst there is still a remaining 1/6 which is communicated to the quadrupoles, independently of the values of Ω and k6 .This somehow non-intuitive result finds its explanation in the structure of the Riemann tensor in BD theory, in which the excess R 0i0j with respect to General Relativity happens not to be proportional to the scalar part Equations (4.18) show that, no matter which is the gravity theory assumed, the sphere's absorption cross sections for higher modes scale as the successive coefficients (k n0 a n ) 2 and (k n2 b n ) 2 for monopole and quadrupole modes, respectively.In particular, the result quoted in [10] that cross section for the second quadrupole mode is 2.61 times less than that for the first, assuming GR, is in fact valid, as we now see, independently of which is the (metric) theory of gravity actually governing GW physics.The fourth columns in Table I display these scaling properties.It is seen that the drop in cross section from the first to the second monopole mode is as high as 6.46.It should however be stressed that the frequency of such mode would be over 4 kHz for a (likely) sphere whose fundamental quadrupole frequency be 900 Hz [10].Note finally the asymptotic cross section drop as n −2 for large n -cf.equation (4.4) and the ensuing paragraph.

V. CONCLUSION
The main purpose of this paper has been to set up a sound mathematical formalism to tackle with as much generality as possible any questions related to the interaction between a resonant antenna and a weak incoming GW, with much special emphasis on the homogeneous sphere.New results have been found along this line, such as the scaling properties of cross sections for higher frequency modes, or the sensitivity of the antenna to arbitrary metric GWs; also, new ideas have been put forward regarding the direction deconvolution problem within the context of an arbitrary metric theory of GW physics.Less spectacularly, the full machinery has also been applied to produce independent checks of previously published results.
The whole investigation reported herein has been developed with no a priori assumptions about any specific (metric) theory of the GWs, and is therefore very general."Too general solutions" are often impractical in science; here, however, the "very general" appears to be rather "cheap", as seen in the results expressed by the equations of section 3 above.An immediate consequence is that solid elastic detectors of GWs (and, in particular, spheres) offer, as a matter of principle, the possibility of probing any given theory of GW physics with just as much effort as it would take, e.g., to probe General Relativity: the vector transfer function of section 4 supplies the requisite theoretical vehicle for the purpose.
An important question, however, has not been considered in this paper.This is the transducer problem: the sphere's oscillations can only be revelaed to the observer by means of suitable (usually electromechanical) transducers.These devices, however, are not neutral , i.e., they couple to the antenna's motions, thereby excercising a back action on it which must be taken into consideration if one is to correctly interpret the system's readout.Preliminary studies and proposals have already been published [8], but further work is clearly needed for a more thorough understanding of the problems involved.
Progress in this direction is currently being made -which I expect to report on shortly.The formalism developed in this paper provides basic support to that further work.
Equations (2.16) in the text are the matrix representation of the above tensors in the Cartesian basis e x , y , e z , except that they are multiplied by suitable coefficients to ensure that the conditions where n ≡ x/|x| is the radial unit vector, hold.They are arbitrary, but expedient for the calculations in this paper.The following orthogonality relations can be easily established: with the indices m,m ′ running from −2 to 2, and with an understood sum over the repeated i and j.It is also easy to prove the closure properties Equations ( A6) and (A7) constitute the completeness equations of the E-matrix basis of Euclidean symmetric tensors.

APPENDIX B:
This Appendix is intended to give a rather complete summary of the frequency spectrum and eigenmodes of a uniform elastic sphere.Although this is a classical problem in Elasticity Theory [29], some of the results which follow have never been published so far.Also, its scope is to serve as reference for notation, etc., in future work.
The uniform7 elastic sphere's normal modes are obtained as the solutions to the eigenvalue equation with the boundary conditions that its surface be free of any tensions and/or tractions; this is expressed by the equations [11] where R is the sphere's radius, n the outward normal, and σ ij the stress tensor with u ij ≡ 1 2 (u i,j + u j,i ), the strain tensor, and λ, µ the Lamé coefficients [11].Like any differentiable vector field, u(x) can be expressed as a sum of an irrotational vector and a divergence-free vector, say; on substituting this into equation (B1), and after a few easy manipulations, one can see that where Now the irrotational component can generically be expressed as the gradient of a scalar function, i.e., while there are two linearly independent divergence-free components which, as can be readily verified, are u div−free (x) = Lψ (1) (x) , and u where L ≡ −ix×∇ is the "angular momentum" operator, cf.[16], and ψ (1) and ψ (2) are also scalar functions.If (B7) and (B8) are now respectively substituted in (B5), it is found that φ(x), ψ (1) (x), and ψ (2) (x) satisfy Helmholtz equations: where ψ(x) stands for either ψ (1) (x) or ψ (2) (x).Therefore in order to ensure regularity at the centre of the sphere, r=0.Here, j l is a spherical Bessel function -see [30] for general conventions on these functions-, and Y lm a spherical harmonic [16].Finally thus, where C 0 , C 1 , C 2 are three constants which will be determined by the boundary conditions (B2) (the denominators under them have been included for notational convenience).After lengthy algebra, those conditions can be expressed as the following system of linear equations: where There are clearly two families of solutions to (B12): i) Toroidal modes.These are characterised by The frequencies of these modes are independent of λ, and thence independent of the material's Poisson ratio.Their amplitudes are with and C 1 (n, l) a dimensionless normalisation constant determined by the general formula (2.8); k nl R is the n-th root of the first equation (B14) for a given l.
ii) Spheroidal modes.These correspond to det and C 1 = 0.The frequencies of these modes do depend on the Poisson ratio, and their amplitudes are where A nl (r) and B nl (r) have the somewhat complicated form with accents denoting derivatives with respect to implied (dimensionless) arguments, and C(n, l) a new normalisation constant.It is understood that q nl and k nl are obtained after the (transcendental) equation (B17) has been solved for ω -cf.equation (B6).
In actual practice equations (B14) and (B17) are solved for the dimensionless quantity kR, which will hereafter be called the eigenvalue.In view of (B6), the relationship between the latter and the measurable frequencies (in Hz) is given by It is more useful to express the frequencies in terms of the Poisson ratio, σ, and of the speed of sound v s in the selected material.For this the following formulas are required -see e.g.[11]: where Y is the Young modulus, related to the Lamé coefficients and the Poisson ratio by Equation (B24) provides a suitable transformation formula from abstract number eigenvalues (kR) into physical frequencies ν, for given material's properties and sizes.
Tables III and II respectively display a set of values of (kR) for toroidal and spheroidal modes.While GWs can only couple to quadrupole and monopole modes, it is important to have some detailed knowledge of analytical results, as the sphere's frequency spectrum is rather involved.It often happens, both in numerical simulations and in experimental determinations, that it is very difficult to disentangle the wealth of observed frequency lines, and to correctly associate them with the corresponding eigenmode.Complications are further enhanced by partial degeneracy lifting found in practice (due to broken symmetries), which result in even more frequency lines in the spectrum.Accurate analytic results should therefore be very helpful to assist in frequency identification tasks.In Figures 1 and 2 a symbolic line diagramme of the two families of frequencies of the sphere's spectrum is presented.Spheroidal eigenvalues have been plotted for the Poisson ratio σ=0.33.Although only the l=0 and l=2 spheroidal series couple to GW tidal forces, the plots include other eigenvalues, as they can be useful both in bench experiments -cf.equation (2.12) above-and for vetoing purposes in a spherical antenna.
Figures 3, 4 and 5 contain plots of the first three monopole and quadrupole functions T nl (r), A nl (r) and B nl (r), always for σ=0.33.T n0 (r) and B n0 (r) have however been omitted; this is because they are multiplied by an identically zero angular coefficient in the amplitude formulae (B15) and (B18).Indeed, monopole vibrations are spherically symmetric, i.e., purely radial.

List of Figures
Figure 1 The homogeneous sphere spheroidal eigenvalues for a few multipole families.Only the l=0 and l=2 families couple to metric GWs, so the rest are given for completeness and non-directly-GW uses.Note that there are fewer monopole than any other l-pole modes.The lowest frequency is the first quadrupole.The diagramme corresponds to a sphere with Poisson ratio σ=0.33.Frequencies can be obtained from the plotted values through equation (B6) for any specific case.
Figure 2 The homogeneous sphere toroidal eigenvalues.None of these couple to GWs, but knowledge of them can be useful for vetoing purposes.These eigenvalues are independent of the material's Poisson ratio.To obtain actual frequencies from plotted values, use (B6).The lowest toroidal eigenvalue is kR = 2.5011, with l=2, and happens to be the absolute minimum sphere's eigenvalue.Compared to the spheroidal kR = 2.6497, also with l=2, its frequency is 5.61% smaller.Note also that there are no monopole toroidal modes.A common feature to these radial functions (also in the two previous Figures) is that they present a nodal point at the origin (r = 0), while the sphere's surface (r/R = 1) has a non-zero amplitude value, which is largest (in absolute value) for the lowest n in each group.

Figure 6
The scalar component Z (S) (x, ω) of the multimode transfer function, (4.2a).The diagramme actually displays ω 3 Z (S) (x, ω), so asymptotic behaviours are better appreciated.It is given in units of µ/ρR, and a factor (π/i) u n00 (x), the eigenmode amplitude, has been omitted, too.δ-function amplitudes are symbolically taken as 1.Note that the asymptotic regime, given by equation (4.4), is quickly reached.

Figure 7
The quadrupole component Z (m) (x, ω) of the multimode transfer function, (4.2b).The same prescriptions of Figure 6 apply here; the plot is therefore independent of the value of m.Note the presence of three subfamilies of peaks; asymptotic regimes are reached with variable speed for these subfamilies, and less rapidly than for monopole modes, anyway.

Figure 5
Figure5First three toroidal quadrupole radial functions T n2 (r) (n = 1, 2, 3), equation (B16).A common feature to these radial functions (also in the two previous Figures) is that they present a nodal point at the origin (r = 0), while the sphere's surface (r/R = 1) has a non-zero amplitude value, which is largest (in absolute value) for the lowest n in each group.

TABLE II .
List of a few spheroidal eigenvalues, ordered in columns of ascending harmonics for each multipole value.Spheroidal eigenvalues depend on the sphere's material Poisson ratio -although this dependence is weak.In this table, values are given for σ = 0.33.Note that the table contains all eigenvalues less than or equal to 11.024 yet is not exhaustive for values larger than that one; this would require to stretch the table horizontally beyond l = 10 -see FigureIfor a qualitative inspection of trends in eigenvalue progressions.

TABLE III .
List of a few toroidal eigenvalues, ordered in columns of ascending harmonics for each multipole value.Unlike spheroidal eigenvalues, toroidal eigenvalues are independent of the sphere's material Poisson ratio.Note that the table contains all eigenvalues less than or equal to 12.866 yet is not exhaustive for values larger than that one; this would require to stretch the table horizontally beyond l = 11 -see FigureIIfor a qualitative inspection of trends in eigenvalue progressions.