Mathisson s New Mechanics : Its Aims and Realisation. W G Dixon, Churchill College, Cambridge, England

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1 Lecture : ims Mathisson s New Mechanics : Its ims Realisation W G Dixon, Churchill College, Cambridge, Engl It gies me great pleasure to be inited to speak at this meeting on the life work of Myron Mathisson I first came upon his work as a graduate student working on equations of motion in general relatiity I had come to this topic through the papers of Papapetrou published in 95 under the title Spinning test-particles in general relatiity, a test particle being a body whose mass is, in some appropriate sense, negligible Papapetrou considered a test particle sufficiently small that only the monopole dipole moments of its energy-momentum tensor T need be considered, a so-called pole-dipole particle He showed that for the purpose of studying its motion, the monopole moment of such a particle could be described by a scalar M the dipole moment by an antisymmetric tensor S, these representing its mass spin, ie angular momentum These were shown to satisfy the equations M s d S S + S R + S S 0 Here s is the proper time along the world line of the body, dx is its four-elocity, denotes the absolute deriatie with respect to s spacetime R is the curature tensor of the The treatment by Papapetrou was extremely non-coariant The quantities M S were constructed from non-coariant parts had to be proed to be a scalar tensor respectiely Similarly the equations as originally deried had to be manipulated into the aboe coariant forms To my mind these aspects were unsatisfactory Indeed it was this unsatisfactory nature of the deriation that led me to study the problem further But the final equations were coariant, so it seemed, new Papapetrou pointed out that in a flat spacetime they agreed with the results of a study of the pole-dipole particle in special relatiity by Mathisson (937 In due course I decided, for completeness, to look up the 937 paper of Mathisson I discoered to my astonishment that not only was the treatment actually in general relatiity but it was also coariant Mathisson deried essentially the same equations of motion as Papapetrou, including the spin-curature interaction term that is arguably the most important discoery of the work Indeed, he went further by considering the quadrupole terms He showed that the equation of motion for the spin then gains additional terms representing a torque exerted by the graitational field, expressed in terms of the quadrupole moment the curature tensor On top of all this, the title of the paper was, in translation from its original German, New Mechanics of Material Systems, a far more all-embracing inspiring title than that of Papapetrou The time was 964 I felt compelled to find out what further work Mathisson had done on this new mechanics I discoered, to my great sadness, that he had died in 940 while

2 still working on this topic I decided to continue Mathisson s programme of work as I understood it to be This came to dominate the next ten years of my life To see the aims of Mathisson s work we need to look at his characterisation of the multipole moments of a body In contrast to Papapetrou, Mathisson treated an extended body defined an infinite set of coariant multipole moments for it We consider a body occupying a world tube of finite spatial extent choose a timelike world line L within this world tube For present purposes L is otherwise arbitrary, but at a later stage we will wish to impose conditions that restrict it to be a suitably defined mass centre Precisely what conditions are best suited for this is in fact one of the big questions of this subject m Mathisson showed that there exists an infinite set of multipole moments, such that T X m m ϕ D ( ϕ + ϕ + m + Ld s L! ϕ m, m, for all symmetric tensor fiel ϕ of compact support Here T is the energy-momentum tensor of the body, which is taken to be symmetric The integral on the left exten oer all 4 space, with DX g d x being the spacetime olume element That on the right is oer the world line L on which s is proper time, denotes coariant differentiation The moments are not uniquely determined by this but they become so if we require in addition that (a they are symmetric on their last two indices, ie those contracted with ϕ, separately are symmetric on all their other indices, ie those contracted with the deriatie operators, (b they are orthogonal to the four-elocity dx of L on all indices except the last two The infinite set of these moments was called by Mathisson the graitational skeleton of the body, or in a later paper its dynamical skeleton If we take ϕ ω ( where ω is an arbitrary ector field of compact support round brackets around indices denote symmetrisation then the left side of the defining equation anishes as a consequence of the conseration equation The moments therefore satisfy L T 0 ( m ϕ + m ϕ + m + L 0! ϕ for all such ϕ Mathisson called this the ariational equation of dynamics, the ariation in question being the ability to ary ω arbitrarily It is a constraint on the graitational skeleton is the central equation of his programme of work Mathisson s new mechanics consists of determining the consequences of this ariational equation, so proiding a description of an extended body its motion in terms of parameters similar to those of Newtonian rigid body mechanics, rather than the description

3 proided by the energy-momentum tensor which correspon more to that of Newtonian continuum mechanics Howeer, he was only able to work with this equation by truncating the series after the first few terms, the justification being that successiely higher moments should hae less less effect on the motion if the graitational field aries only slowly across the body The truncated equation then lea both to restrictions on the form of the moment tensors to differential equations of motion that they must satisfy When only the first two moments are retained, he found that they are determined by a scalar M an antisymmetric tensor S such that where ( m p + ( ( n ( ( m S + n S p M + n These satisfy the equations of motion gien aboe in connection with Papapetrou s work With use of the auxiliary ector p these equations of motion can also be written in the form p S S R, where square brackets around indices denote antisymmetrisation Indeed the equations in this form imply that p has the form gien aboe, where M is now defined by M p, so that alternatiely p S can be regarded as giing the primary description with M being the auxiliary ariable This is perhaps a more elegant iew of the results Mathisson identified n with the static mass dipole moment of the body, which anishes in Newtonian mechanics when taken about the centre of mass He therefore took n 0 as the condition to characterise L as the world line of the centre of mass He did this before he exploited the ariational equation, so that his equations of motion were only deried for this special case This is why I said that Mathisson s equations were only essentially the same as those of Papapetrou, as Papapetrou did not impose this condition He was aware that in fact it does not determine L uniquely but he offered no alternatie of general applicability In a paper of 959, Tulczyjew identified p as defined aboe as the momentum of the body he then adopted p S 0 as the appropriate condition He showed that in a flat spacetime this does determine a unique L He then adopted it in a cured spacetime owing to the lack of another definition If the aboe expressions for m m are put back into the defining relation for the moments, the terms inoling n combine into a single deriatie that integrates to zero We could therefore use instead the simpler expressions ( m p m p S [ ] ( which no longer satisfy the condition (b This is a first indication that the orthogonality condition (b is not the most appropriate addition to (a to ensure uniqueness of the moments S 3

4 We shall see in the next lecture that modification of this orthogonality condition is one of the key steps towar soling the ariational equation exactly Let us now take a step backwar, to get an oerall iew Mathisson s approach to equations of motion falls into two distinct parts The first part is showing that moments exist which satisfy the defining equation any supplementary conditions such as (a (b aboe The second part is exploiting the consequences of the ariational equation In his 937 paper his treatment of the ariational equation was approximate, in that it required a truncation In contrast his existence proof for the moments dealt with the infinite set of moments without approximation Howeer, it left the uniqueness question unanswered, as the orthogonality condition (b was imposed only on the moments retained in the truncated ariational equation n improed more detailed proof was gien by Bielecki, Mathisson Weyssenhoff in 939 This imposed both (a (b from the outset proed both the existence uniqueness of the moments Or at least it did so subject to one explicit proiso, namely that infinite series conerge functions are analytic to whateer extent is required This proiso is important, as the proof requires the moments at any point z L to be determined by the alue of T on a hypersurface through z that has some freedom in its specification This implies some form of analyticity of T that is not physically reasonable It is permissible to require ϕ to be analytic as this is just an auxiliary field introduced for conenience, but the analyticity must not apply also to T The problem originates not in the proof but in the moment definitions themseles Mathisson studied this issue again in 940, this time in the simpler case of special relatiity, in a paper simply titled The Variational Equation of Relatiistic Dynamics He was there able to gie expressions for the moments as explicit integrals of T oer hyperplanes orthogonal to the chosen world line L These were shown to satisfy the defining equations but they were not deduced from it Indeed, the flawed nature of the defining equations makes this task impossible These problems suggest that it may be preferable to abon the Mathisson approach instead seek a coariant ersion of the Papapetrou method Seek a set of multipole moments defined from the outset as explicit tensor-alued integrals oer cross-sections of the body study their properties directly This would by-pass the difficulties associated with Mathisson s moment defining equation leae only a study of the consequences for these moments of the energy-momentum conseration equation This study would correspond to the soling of Mathisson s ariational equation One would expect to hae to retain only the first few moments in this study, but that would correspond to Mathisson s need to truncate the ariational equation This approach was adopted by Tulczyjew Tylczyjew (96 later, with different definitions, by myself in Dixon (964 But once one sees how to construct such tensor-alued integrals it becomes clear that there is an infinite number of possibilities to choose from, no one choice being more natural, in some sense, than the rest Perhaps it does not matter which we choose, as the quantities such as p S that appear in the final equations of motion are constructs from these moments rather than being moment tensors themseles But the goal must be to aoid the need to neglect the higher moments, as this truncation itself is fundamentally flawed We hae only to look at Papapetrou s original form for the equations of motion of a pole-dipole particle to see that een in a flat spacetime the dipole construct S enters the equation goerning the monopole construct M the elocity If higher moments were retained we would expect further contributions to this equation We hae seen that p proides a preferable description of the 4

5 monopole structure in that the spin S then only occurs in the monopole equation of motion in combination with the curature tensor But what happens to this if we retain higher moments? It is not een obious that analogues of p S will continue to exist So how might we aoid truncation We can hope that by a judicious choice of our original moment definitions the consequences of the energy-momentum conseration equation might be sufficiently systematic that we can hle them to all orders But we need a guide to help towar this judicious choice There really is only one guide aailable It is to return to Mathisson s approach, with its implicit definition of the moments through a defining equation, to use the ariational equation as our guide That equation has a deceptie simplicity Our goal is to find our way through that deception So in the 960 s I was led back to Mathisson s new mechanics of 937 to proide the way forward The aims were two-fold To remoe all assumptions of conergence analyticity from the statement of the moment defining equation its associated existence uniqueness proof To sole the ariational equation exactly, only then to truncate the result to proide equations of motion to any desired order of approximation I regard my achieement of these two aims as the realisation of Mathisson s new mechanics Realisation of ( led to explicit unique expressions for the moment tensors as integrals of T Realisation of ( led to exact analogues of p S that hae a natural identification as the momentum spin of the body This enables a third aim to be added 3 To show that the Tulczyjew condition p S 0 determines L uniquely that this L has properties which enable it to be identified naturally as the centre of mass line in general relatiity Through the work of Ehlers collaborators, this too has been achieed 5

6 Lecture : Realisation The first clue towar realising the aims of Mathisson s new mechanics can be found by considering the simpler enironment of special relatiity Let us use a rectangular coordinate system so that the components g of the metric tensor are constant, but not necessarily diagonal The coordinates x can then be treated as components of a position ector Recall Mathisson s defining equation for the moments, which we now write in full as T X L ϕ D m ( s L n! L ϕ ( z(s Here L is parametrised as z (s, where for generality s is not necessarily proper time Here throughout, n is the number of indices in the set marked with dots, in this casel, we allow the possibilities n 0 n We seek to aoid the requirement for ϕ to be analytic To do so we introduce the Fourier transform ϕ defined by where k x k x F( ϕ ϕ ( ϕ ( x exp(ik x Dx For more complicated cases we shall also write the Fourier transform as If we express ϕ in terms of ϕ in the defining equation, we get n T DX d D ( i ( 4 s k L ϕ k k m s ( exp( ik x (π L n! L ϕ If we now exchange the order of the summation the k-space integration we get T ϕ DX M [ Φ ] where M [ Φ ] 4 Dk m ( s, Φ ( z( s, (π with n (, ( i L m s k k k m ( n! s L Φ ( z, ϕ ( exp( i k z Note that Φ ( z, is simply the Fourier transform of ϕ about z as origin In this form the moment defining equation no longer requires ϕ to be analytic We hae, of course, achieed this by exchanging a summation an integration that cannot in general be alidly exchanged We adopt this new form as an improed defining equation in special relatiity L To extend this definition to a cured spacetime, note that since the moments m are tensors at z(s on L, k must be considered as a ector at the same point This makes m ( s, k be a tensor field on the tangent space T z (M to the spacetime manifold M at z(s, so Φ must be likewise There is no problem in defining Fourier transforms on (M T z 6

7 since it is always a flat manifold, so we can take Φ to be the Fourier transform of a Φ that is also a tensor field on T z (M This leaes us simply to decide how Φ is to be defined in terms of ϕ When M is flat we hae a natural identification of T z (M with M, which is why we were able to take Φ ϕ in special relatiity In a cured spacetime we relate the two by means of the exponential map Exp z : T z ( M M If X Tz (M x Exp z X then the deriatie map (( Expz * X of Exp z at X maps TX ( Tz ( M isomorphically onto T z (M This mapping between tangent spaces has a unique extension to a mapping of the corresponding tensor algebras We denote this by replacing the * by in the notation By letting X ary we get a map ( Exp z from tensor fiel on T z (M to tensor fiel on M If we hae such a tensor field for each z, we can apply the corresponding map to each of them to obtain a family of tensor fiel on M parametrized by z, ie a two-point function with scalar character at z tensor character at the second point x We let Exp denote this oerall map Exp denote its inerse These can both be formalised as maps between appropriate ector bundles We see that Exp acts on two-point tensors with scalar character at one point, say z special case of this is an ordinary tensor field treated as such a two-point tensor that is independent of z We can therefore take Φ Exp ϕ For each z, this defines Φ as a tensor field on (M If M is flat we identify each of its tangent spaces with M itself then this gies Φ ϕ for all z, so recoering our starting point in special relatiity We hae now done eerything necessary to take our defining equation oer into a cured spacetime Note, howeer, that there is no longer any sense in which it is a transformation of Mathisson s original equation We hae made a real change in the definition of the moments But we can now obtain explicit expressions for them as integrals without encountering problems of analyticity For the present we continue to adopt the orthogonality condition (b of Lecture, L namely that m is orthogonal to on each of its indices L Choose a Minkowskian coordinate system on T z ( s ( M such that 0 only for 4 Then m ( s, k is independent of k 4 Recall now the result that for the Fourier transform f ( k of a function f ( of a single ariable, we hae T z f ( dk π f (0 This enables us to perform the k 4 integration in the s-integr of M [ Φ ] to show that its alue for fixed s depen on Φ ( z( s, X only through its alue on the hyperplane X 4 0, ie X 0 This hyperplane is mapped by Exp z into the hypersurface Σ(s formed by all geodesics through z (s orthogonal to It follows that for each s, the s-integr of M [ Φ ] depen on ϕ only through its alues on Σ (s 7

8 Now define a scalar function τ (x by τ ( x s if x Σ( s let field such that w τ Then we hae a corresponding decomposition T ϕ DX Σ ( s T ϕ w dσ w be any ector where d Σ is the surface element on Σ (s It follows that the two s-integran are equal, so that T ϕ w dσ 4 Dk m ( s, Φ ( z( s, Σ ( s (π It is now straightforward to express ϕ in terms of Φ so to identify m ( s, k as an integral oer Σ (s By exping the resulting integr as a series in k we may obtain L explicit expressions for the moments m as integrals of Note that no use T oer Σ (s has been made here of being tangent to L We could equally well hae taken it to be an arbitrary timelike ector field n along L we shall make this change to the orthogonality conditions from now on Without loss of generality we shall take n to be a unit ector We shall not gie further detail here but instead turn to the ariational equation This is obtained by taking ϕ ω ( for an arbitrary ector field ω We begin by inestigating to what extent ω is determined if we only know ϕ Since the difference of two solutions for the same ϕ is a Killing ector field, ω is not unique if the spacetime admits Killing ectors It is easily shown that ω satisfies ω du & & + ω x x R x x along all affinely parametrised geodesics x (u, where x& dx du curly brackets around three indices are defined by & & { { } + The sign conention for the curature tensor is such that [ ] ω R ω By integrating the equation for ω along all geodesics through a fixed point z, we can therefore find ω eerywhere if we only know ω ω at z, as these alues determine the required initial conditions for any geodesic But ( ω ϕ, so in fact we only need ϕ } ( [ ] z z ω ( z B ( z ω ( Gien a base point z, this construction will determine a ector field ω from an arbitrary tensor field ϕ, for any alues of antisymmetric B at z We shall let λ ( z, x be the ector field so obtained when we take 0 B 0 It is of course 8

9 also a functional of the field ϕ, but we shall leae this dependence implicit Similarly we let ξ ( z, x be the ector field so obtained when we take ϕ 0 but leae B arbitrary, its dependence on these two tensors at z again being left implicit If ϕ, B are constructed as aboe from a gien ω then we hae ω ( x λ ( z, x + ξ ( z, x for any z Note that the equation satisfied by ξ is the equation of geodesic deiation To continue, we need to be able to differentiate fiel such as Φ ( z, X in a coariant manner, with respect to both z M X T (M We gie the required formulae for a field Ψ ( z, X so as to illustrate the terms that arise from both contraariant coariant indices We define Ψ z Ψ where the Lei-Ciita connection Γ Γ ε Ψ X z X ε X Ψ Ψ + Γ Ψ Γ Ψ is ealuated at z The first of these differs from the usual coariant deriatie only through the addition of one term inoling a deriatie with respect to X The second nee no connection terms since the components of X form a rectangular coordinate system on the flat tangent space It can be shown that commutes with Fourier transformation that X 0 With this notation it can be shown that Exp ( λ Μ Λ ( { Ξ Exp ξ G + B X G B ( ( G X ( where G Exp g, G { G } G } Λ Exp λ, Μ Exp λ G Λ B are ealuated at z are the alues used in the construction of g ξ Indices The of tensors on T z (M are raised lowered with (z, which is the flat metric on this tangent space, is why we need different symbols for the lifts of the contraariant coariant forms of the ector field λ ( z, x The field Ξ ( z, X is defined by the aboe equation If ϕ ( ω as is used in the ariational equation then these results gie Φ + G + Ξ Λ { } ( Μ The right h side is particularly simple as its first term only inoles partial differentiation its second term is completely determined by the parameters B at z The left 9

10 h side is well defined for a general field ϕ (x as this completely determines λ ( z, x hence also Λ ( z, X We capitalise on this simplicity by modifying the definition of our moments to take adantage of it We change the defining equation to D [ T ϕ X M Φ + Λ { }] G The additional term anishes in a flat spacetime since G is then constant, so we are only modifying the graitational contribution to the moments The ariational equation now becomes Since Fourier transformation gies we hae where t M M Μ + Ξ ] 0 F [ ( ( ( Μ ik( Μ (π [ ( Μ ] 4 Dk t ( s, ik m ( s, ( i n n! Μ k Lk t L L ( L with t 0 t ( n + m for n 0 L For n we can decompose m through the use of symmetry operations into two L tensors, one of which is t The other can be taken as the appropriate one from the set J ζlε ( m for n ζlε [ ] ζlε [ ] m For n 0 the m s t s are equialent as either can be expressed in terms of the other Recall that the orthogonality conditions (b of lecture which were imposed to ensure uniqueness of the moments are now L (b n m 0 for n since we generalised the original choice by using an arbitrary timelike unit ector field along L in place of the tangent ector The set of these for n has a similar decomposition into the two sets ζlε (b n J 0 for n ( L (b n m 0 for n ζ We see that although set (b inoles only the J s, set (b does not inole only the t s We can achiee this separation if we replace set (b by the set L ( L (b3 n t n m 0 for n These hae the same symmetry properties as those of set (b hence they impose the same number of constraints They hae the adantage, howeer, of constraining precisely 0 0

11 those parts of the m s that occur in the ariational equation We therefore make this change, which is of course a further change to the definition of the moments We cannot extend (b3 to include n since for this case (b is symmetric but (b3 is not, so this would increase the number of constraints we hae already shown that the original set (b suffices to ensure uniqueness of the moments We shall for the present omit any constraint for n shall see in due course precisely what freedom this leaes in the definition of the moments We now show that as a consequence of the constraints (b3, the ariational equation can be soled exactly We first separate out the two terms in t ( s, whose t tensors are not constrained by (b3 We integrate these oer k then express the result in terms of the m s ω instead of the t s λ If we also use the expression for Ξ gien aboe, the ariational equation can be put in the form where K ( m ω ( z + m (π 4 F Dk Μ n 3 ω ( z + F ( i n k n! + Lk t B λ L m 4 G Dk (π, ( K 0 + L + ( i t G Dk L i G m 4 [ ] 4 [ ] Dk (π (π Let, as before, Σ (s be the hypersurface formed by all geodesics through z(s that are orthogonal at z to n (s The method used aboe in connection with the original orthogonality condition (b then shows that the k-space integral in this form of the ariational equation depen on λ ( z( s, x only through its alues on Σ (s The construction of λ shows in turn that its alues on Σ (s are determined by the alues of ω (x on Σ (s We will show that all the other terms in the s-integr of the ariational equation can also be put in a form with this restricted dependence still dependent on ω only through the alue of it its first two deriaties at z Once this is done, since the ector field ω is arbitrary it follows that the s-integr must anish separately for each alue of s But this integr can be regarded as a generalised function, ie a continuous linear functional, on M itself rather than on Σ (s Its anishing therefore implies that the infinite series in the k- space integral must be identically zero, as must be the coefficients of ω its first two deriaties in the other terms of this integr Note that these terms correspond to terms in the infinite series that are constant, linear quadratic in k it is because such terms are absent that the series the remainder of the integr must anish separately We now follow this through t each point of L we define the projection operators where χ ( n P χ n Q P is the unit tensor Then the alues at z(s of ε ω, Q ω Q Q ( ω + h can all be ealuated from knowledge of ω (x on Σ (s n ω ζ ( ε ε ζ extrinsic curature tensor (or second fundamental form of Here h is symmetric is the Σ (s, defined by

12 h n n where n is the field of unit normals to Σ (s, which of course agrees with n (s on L In a flat spacetime Σ (s is a hyperplane so h 0 in a Minkowskian coordinate system the n quantities on which the projection operators act reduce to the first second partial deriaties of ω Since we hae P ω χ n ω any term in the s-integr inoling this can be reduced by integration by parts to one depending only on ω on L We see similarly that if either or both of the projection operators Q in the last of the aboe list is replaced by the corresponding P then an integration by parts of any term inoling this expression can be used to reduce it to one depending only on ω ω on L These steps applied in the reerse order, ie treating the second deriaties first, will achiee the reduction of the entire s-integr to one that has the required dependence only on the alues of ω (x on Σ (s We perform this reduction in stages, drawing conclusions as we go along We can immediately conclude that the infinite series in the k-space integral must be identically zero, so that t L 0 for n 3 This is sufficient to gie K 0 as the terms that remain also ealuate to zero The coefficient of from which for some tensor ( ω in the reduced expression is easily seen its anishing gies ( m must hae the form Q Q m ( ε ε m S 0 ( ( S that at this stage is not necessarily antisymmetric This gies m [ ] [ ] m ( S + S + S ω ( S ω + n ω S [ ] The first term on the right requires an integration by parts, following which the coefficient of ω can be read off its anishing gies Q + m S L 0 d s The bracketed expression therefore has the form for some ector m S + L p R p The symmetric antisymmetric parts of this gie

13 It also shows that m ( m p + d S s [ ] p [ ] ( S + L S + L ω p ω This requires another integration by parts before the coefficient of ω can be read off its anishing seen to gie d p s [ ] S R This completes the solution of the ariational equation but it has left us with some freedom in the definitions of m m, as we expected when we left one orthogonality ( condition unspecified The alue of S is arbitrary, since if we substitute the aboe expressions for m m into the moment defining equation then the two contributions ( from S combine into a total s-deriatie that integrates to zero We saw in lecture that Mathisson s orthogonality conditions led to contributions of this form in his pole-dipole ( approximation We shall make the simplest choice, S 0 This replaces the case n of condition (b aboe which was left outsting by our adoption in its place of (b (b3, being symmetric, it imposes the same number of constraints + F We now hae the monopole dipole moments of the body gien by ( m p m S ( in terms of the ector p antisymmetric tensor S, which are now precisely defined quantities that arose naturally in the deelopment rather than being constructs within some approximation The quadrupole higher moments are determined by the J s, which hae the symmetry orthogonality properties J Lεζ J ( L [ ][ εζ ] L [ εζ ] J J 0 for n 0, L[ ] εζ Lεζ 0 n J 0 for n which determine the m s with four or more indices by m Lε 4 ( n ( L ε J for n n + Mathisson took the set of m s as his graitational skeleton of the body, despite the existence of an infinite number of relations between them It seems preferable now to take the graitational skeleton as being p, S the set of J s, as the only relations between them implied by the conseration equation for the energy-momentum tensor are the two equations of motion d p s S R + F 3

14 S p [ ] + L These are the familiar pole-dipole equations with the addition of a force F torque L produced by the quadrupole higher moments Exact expressions were gien aboe for these in terms of m, which can be considered as a moment generating function The contributions to F L from p S anish identically, so the force torque are expressible in terms solely of the J s By exchanging the order of the summation in m the k-space integration we can obtain a formal multipole expansion of the force torque which can be truncated to gie a multipole approximation to any desired order The full infinite series is only a formal one, as the exchange of summation integration is not actually alid The multipole expansion is most easily expressed in terms of the extensions g, L of the metric tensor g as defined by Veblen Thomas (93 These are the tensors that reduce at the pole of a normal coordinate system to the partial deriaties of the metric tensor This is a natural construction in our context as the image under exponential map of a rectangular coordinate system in the tangent space T z (M is a normal coordinate system on M with z as its pole We obtain L F n m n! εl g, εl [ ] εlζη g m g n! { η, ζ } εl n The extensions are easily ealuated in terms of the curature tensor the lowest two are g R, g R 4, 3 (, ε 3 ( ε To complete the proof of the aboe results we still need to show that moments do exist that satisfy our new moment defining equation with the symmetry orthogonality conditions we hae required This can be done it lea to explicit expressions for p, S the J s as integrals of the energy-momentum tensor oer Σ (s It is interesting to note that the surface of integration is still uniquely determined een though the orthogonality conditions now first arise with the octopole moment This completes the realisation of the first two aims laid down in lecture We turn now to the third aim The explicit expressions for p S are simple they show that their alues at z( s L depend on L the ector field n along it only through the point z the alue of n at this point The expressions for the J s are significantly more complicated They depend not only on z n but also on their first deriaties along L This dependence arises through the ector field w introduced aboe in the splitting of an integral oer M into one oer Σ (s followed by one oer L It turns out, perhaps unexpectedly, that the expressions for dependence point p The expressions for S do not inole p z M timelike unit ector independent of n so at any point z we can choose w so are free of this deriatie S can therefore be considered as functions of a general n at z In a flat spacetime the function p ( z, n n to be parallel to p (z is This suggests 4

15 that in a cured spacetime we can find an n that is parallel to p ( z, n now an implicit equation, ie we can choose p ( z, n M ( z n een though this is for some scalar M(z With this choice of only of z The Tulczyjew condition n we can consider p S 0 p S to be functions is then well defined can be expected to determine L uniquely It has been proed by Schattner (979 that subject to mild restrictions on the strength of the graitational field, these conditions do determine the field n the line L uniquely With these conditions we hae completely specified the line L the fiel M, S along it We take M, p p S as defining the mass, momentum spin of the body L as being the centre-of-mass line We now seek a relationship between p the four-elocity dz of L First recall that we hae not required s to be proper time along L change of parametrisation changes the J s by an oerall scale factor but it leaes p S unchanged We use this final freedom to choose s so that n as well as n n s distinct from our other choices in this deelopment, there is no compelling reason for this It is purely for conenience as it simplifies subsequent deelopments If we define a ector h by it can then be shown that Mh p + ( 4 ε ε ε M + R S S ( h S ( F + R h S This is the momentum-elocity relation of the theory, due to Ehlers Rudolph (977 The elocity it determines satisfies the aboe normalisation condition since n 0 The simplest special case is a flat spacetime, where the curature the graitational force torque are all zero it shows that the momentum elocity are parallel Howeer, the less triial case of a spacetime of constant curature can also be treated exactly In this case the auxiliary field ξ is a Killing ector for all alues of the parameters B, from which we get the anishing of Ξ hence also of the force F torque L On using the expression R k L n ( g g g g for the curature tensor in this case, we find from the momentum-elocity relation that n The equation of motion for p then shows that L is a geodesic This supports our identification of L as the centre-of-mass line in general relatiity, as it is what we would expect of this line from the symmetry of the spacetime This completes the realisation of our third final aim of Mathisson s new mechanics ε S 5

16 References Bielecki,, Mathisson, M Weyssenhoff, JW 939 Sur un théoréme concernant une transformation d intégrales curilignes dans l espace de Riemann Bull Int cad Polonaise des Sci et des Lett, Cl Sci Math et Nat Sér : Sci Math (939, -8 Corinaldesi, E Papapetrou, 95 Spinning test-particles in General Relatiity II Proc Roy Soc 09, Dixon, WG 964 coariant multipole formalism for extended test bodies in general relatiity Nuoo Cimento 34, Dixon, WG 970 Dynamics of extended bodies in general relatiity I Momentum angular momentum Proc Roy Soc 34, Dixon, WG 970 Dynamics of extended bodies in general relatiity II Moments of the charge-current ector Proc Roy Soc 39, Dixon, WG 974 Dynamics of extended bodies in general relatiity III Equations of motion Phil Trans Roy Soc 77, 59 9 Dixon, WG 979 Extended bodies in general relatiity: their description motion pp 56 9 of Isolated Graitating Systems in General Relatiity, ed J Ehlers North- Holl Publ Co Ehlers, J Rudolph, E 977 Dynamics of extended bodies in general relatiity: centreof-mass description quasirigidity Gen Rel Gra 8, 97-7 Mathisson, M 937 Neue Mechanik Materieller Systeme cta Phys Polonica 6, Mathisson, M 940 The ariational equation of relatiistic dynamics Proc Cambridge Phil Soc 36, Papapetrou, 95 Spinning test-particles in General Relatiity I Proc Roy Soc 09, Schattner, R 979 The centre of mass in general relatiity Gen Rel Gra 0, Schattner, R 979 The uniqueness of the centre of mass in general relatiity Gen Rel Gra 0, Tulczyjew, B Tulczyjew, W 96 On multipole formalism in general relatiity rticle in Recent Deelopments in General Relatiity, Pergamon Press, pp Tulczyjew, W 959 Motion of multipole particles in general relatiity theory cta Phys Polonica 8, Veblen, O Thomas, TY 93 The geometry of paths Trans mer Math Soc 5,