Transverse momentum as a source of gravitoelectromagnetism

Transcription

1 Transverse momentum as a soure of gravitoeletromagnetism D. H. Delphenih Spring Valley, OH USA Abstrat: Momentum an be regarded as a mass urrent that an be used as the soure of the gravitoeletromagneti field, whih is the weak-field gravitational analogue of the lassial eletromagneti field. Typially, gravitoeletromagneti researh onsiders only mass urrents of onvetive type, whih are ollinear with the veloity vetor field, but when one looks at harged fluids that interat with a bakground eletromagneti field or harged, spinning point-like matter, suh as the Frenkel eletron, or extended spinning mass distributions, suh as the Dira eletron and the Weyssenhoff fluid, one will also enounter transverse momentum. The effets of that transverse momentum on the gravitoeletromagneti field are investigated.. Introdution. One of the most elementary things that one learns in physis is that there is a partial analogy between Coulomb s law of eletrostati attration and repulsion and Newton s law of universal gravitation. What makes the analogy partial is the fat that to date gravitational repulsion has never been observed, so one always assumes that this also implies that whereas harge an be positive or negative, gravitational mass must always be positive, in order to produe only attration. Some early attempts were made to extend Coulomb s law of eletrostatis to a more omplete law of eletrodynamis that would inlude additional magneti fores that would originate in the relative motion of the interating harges. Suh an extension was first suggested by Gauss in a note that was published only posthumously in his olleted works and then reast in a different form by Wilhelm Weber [] and Franz Neumann []. As Joseph Bertrand later showed [3], the formulas of Gauss and Weber were atually equivalent. One interesting aspet of the Gauss-Weber law was that it inluded a large parameter that played the role of the speed of propagation of the interation, whih was assumed to be that of light, and as that parameter beome infinite, the Gauss-Weber formulas would both go bak to that of Coulomb. These extensions of Coulomb s law eventually led to orresponding extensions of Newton s law, and mostly in the ontext of elestial mehanis. First, Gustav Holzmüller [4] used Weber s law of attration in a Hamilton-Jaobi approah to planetary motion, and later Tisserand [5] used a more diret approah to introduing both Gauss s and Weber s laws. Typially, the effet that researhers in elestial mehanis were looking for was a preession of the longitude of the perihelion of planetary orbits. Oliver Heaviside [6] also looked into extending Coulomb s law to a more omplete analogy between eletromagnetism and gravitation. Another signifiant differene between eletrostatis and gravitostatis besides the weak analogy between harge and mass is that many orders of magnitude separate the two types of fore. That is, the gravitational interation is so feeble in omparison to the eletrostati interation that one must typially look for gravitational effets in the ontext of astrophysis in order to be dealing with the kind of masses that would produe notieable gravitational fields and the kind of distanes at whih the effets would assert themselves (e.g., elestial mehanis, stellar interiors, and galati struture).

2 Transverse momentum as a soure of gravitoeletromagnetism. In 98, Hans Thirring published a paper in die Physikalishe Zeitshrift [7] in whih he alulated the effet of rotating distant masses in Einstein s theory of gravitation. Basially, he was addressing Mah s suggestion that the olletive rotation of the distant stars would also lead to a deformation of the surfae of the water in a buket, just as the rotation of the buket with respet to the distant stars would. At one point, he inluded a footnote that proposed that there was a lose analogy between the weak-field Einstein equations when the soure of the gravitation was a rotating mass and Maxwell s equations of eletromagnetism. In a subsequent paper with Josef Lense in Zeitshrift für Physik [8], Thirring then alulated what it now alled the Lense-Thirring effet for an elementary orbital senario. That effet, in its original form, manifested itself as a preession of the plane of an orbit around a gravitating body that would be due to the rotation of the body. In a third paper in Phys. Zeit [9], Thirring elaborated upon the formal analogy between the Maxwell s equations and Einstein s equations of gravitation in the linear approximation. There was also a orretion to the first paper that showed up in 9 [0] that was based upon some onversations between Thirring and Max von Laue and Wolfgang Pauli. In essene, the Lense-Thirring effet amounts to the possibility that the rotation of a gravitating body will produe a dragging of the frames in its viinity that will manifest itself as the preession of a gyrosope that is plaed in the gravitational field. It is not surprising that many deades passed before the Lense-Thirring effet was atually put to an experimental test. One needs only to reall that the magneti fores that are produed by an eletri urrent are typially muh weaker than the eletrostati ones to see that in the gravitational ase, where the analogue of the eletrostati fore is already quite weak, the effets of any possible gravitomagneti field that is produed by the motion of a gravitating body, suh as its rotation around an axis, would be even more impereptible. Only the advanes of experimental tehnology made suh measurements even possible. It was in 960 that Leonard Shiff [] first proposed that one ould possibly test the Lense-Thirring effet using a satellite in a low Earth orbit. That suggestion, in turn, took a few more deades to implement. The first attempt was with the LAGEOS II satellite program, whih was deployed by the Spae Shuttle in 99, and the latest test of framedragging was made during the years The result of the measurement seemed to be positive, although there was onsiderable debate regarding the atual auray of the measurement. During the years , the Gravity Probe B improved upon the auray of the LAGEOS II result ( ); the rate of preession that Gravity Probe B measured was 37. ± 7. milliseonds of ar per year. It is important to notie that even though the Lense-Thirring effet was first derived from Einstein s theory of gravitation (but in the linear approximation), sine the gravitational fields of most elestial bodies, suh as the Earth, Sun, and Moon, are atually quite weak in the eyes of general relativity, one would expet that the Maxwellian analogy would be entirely suffiient to desribe the phenomenon in most astrophysial problems. Just as one an seek to extend the sope of Maxwell s equations to something more analogous to Einstein s equations, one an take the opposite approah and look for how ( ) For detailed disussions of the experimental test of the Lense-Thirring effet, as well as more referenes on the subjet, one an onfer the survey artiles by Bahram Mashoon [] and Ruggiero and Tartaglia [3], as well as the book [4] by Ignaio Ciufolini and John Arhibald Wheeler. Note, however, that the last two referenes predated the atual satellite experiments.

3 Transverse momentum as a soure of gravitoeletromagnetism. 3 Einstein s equations might lead to field equations of Maxwell type. The main shool of thought in regard to that is based upon the Bianhi identities for the Weyl tensor. Sine we will not employ that approah, we shall only refer to some referenes on the subjet [, 5]. The purpose of this study is to investigate what would be involved with extending the Maxwellian analogy with weak-field gravitation even further by allowing the possibility that some material media are, in a sense, polarizable by gravitoeletromagneti fields in the same way that eletri and magneti dipoles an form in eletromagneti media. In partiular, we are mostly onerned with the gravitomagneti analogue of magneti dipoles, sine the apparent absene of negative masses that would omplete the eletrostati analogy would seem to prelude the existene of gravitoeletri dipoles, as well. Our basi thesis is that sine energy-momentum an be regarded as a mass urrent that would serve as the soure of the gravitoeletromagneti field, one might onsider that in addition to the onvetive or longitudinal momenta that take the relativisti forms of m 0 u or ρ 0 u for point-like and extended masses, respetively, there is suh a thing in nature as transverse momentum, whih an typially originate in the interation of a harge with a bakground eletromagneti field or take the form of a relativisti quantum sort of effet that is most definitive in the ontext of the Dira eletron, whih inluded its spin. One would then naturally wonder about the orresponding gravitomagneti field that would be produed by the transverse omponent of the mass urrent in the same way that one onsiders polarization urrents in the eletromagneti ontext, as well as the possible physial effets that might be aessible to experimental testing. The next setion of this paper summarizes the relevant notions from Maxwell s theory that will be applied in the gravitoeletromagneti analogy. The third setion then goes over that analogy and disusses the physial interpretation of the basi analogue fields. The fourth setion disusses the general onept of transverse momentum and gives various examples of how it an our naturally in physial models. The fifth setion then adds the transverse momentum to the mass urrent that generates the gravitoeletromagneti field and examines the onsequenes. Finally, there is a disussion of the limitations of the present analysis and the possible extensions in its sope.. Maxwellian eletromagnetism. Before we present the basi analogy between Maxwellian eletromagnetism and weak-field gravitation, we shall first summarize the key fats of Maxwellian eletromagnetism. We shall first present things in the vetor alulus formulation and then in terms of the more modern alulus of exterior differential forms. a. Vetor alulus formulation of the field equations. To many physiists, the most ommon way of expressing Maxwell s equations for eletromagnetism is the form that is found in Jakson [6]; in Gaussian units, they are:

4 Transverse momentum as a soure of gravitoeletromagnetism. 4 E + B D 4π = 0, B = 0, H = J, D = 4πσ, (.) t t in whih E is the eletri field strength, D is its orresponding eletri exitation (often alled the displaement ), H is the magneti field strength, B is its orresponding exitation (often alled the magneti flux density ), σ is the eletri harge density, and J is the eletri urrent. When one is dealing with stati fields, the time derivatives will drop out, and one will be left with: E = 0, D = 4πσ, H = 4 π J, B = 0, (.) in whih the order of equations has been hanged in order to show that one now has two sets of deoupled equations for the stati eletri and magneti fields. So far, the system of linear, first-order partial differential equations (.) that we have presented for the four spatial vetor fields E, D, H, B is underdetermined, sine there are eight equations for twelve omponent funtions when σ and J are given. One an redue the twelve omponents to six independent ones by introduing an eletromagneti onstitutive law for the medium in whih the fields exist, whih will take the funtional form: D = D (E, H), B = B (E, H). (.3) Hene, one has six equations that relate twelve funtions, whih will leave six independent ones. One now has an overdetermined system of eight equations for six unknown funtions. One first notes that a natural onsequene of the equations is essentially the onservation of harge: σ + J = 0. (.4) t That ondition represents an identity that redues the number of independent equations by one. In order to add one more identity that would make the equations welldetermined, one needs to introdue an eletromagneti potential -form, whih we shall define shortly. When one is dealing with the lassial vauum, the relationship between the field strengths and the exitations that they produe is the simplest possible one: D = ε 0 E, B = µ 0 H, (.5) in whih ε 0 is the vauum dieletri onstant, and µ 0 is the vauum magneti permeability. However, in polarizable eletromagneti media, the relationship between field strengths and their orresponding exitations an be muh more involved. In partiular, E an produe both eletri and magneti dipoles as a result of its presene in the

5 Transverse momentum as a soure of gravitoeletromagnetism. 5 medium, as an H. Indeed, the relationship (.5) would be typial of media in whih no suh dipoles are produed. We shall return to elaborate upon this onept at the end of this setion. b. Field equations in terms of differential forms. Sine the mid-nineteenth Century, mostly due to the work of Frobenius, Clebsh, and Darboux on the Pfaff problem, as well as Grassmann s introdution of exterior algebra (but in a different form from its modern definition), the basi ideas of what for a while ame to be alled the symboli alulus or absolute alulus were growing into a replaement for the vetor alulus that was more broad-ranging in its appliations. Its definition was formalized by Élie Cartan and Edouard Goursat at about the same period of time (viz., 9). Cartan applied the alulus of exterior differential forms to geometri sorts of things, suh as the geometry of moving frames [7], while Goursat applied it to more analytial ones, suh as the Pfaff problem [8], and eventually Cartan, along with Kähler, disussed the integrability of systems of exterior differential equations, whih was a generalization of both the Pfaff problem and the Cauhy-Kowalevski theorem for partial differential equations. Meanwhile, the appliations of differential forms to physis were also inreasing in number and general familiarity. Suh researhers as Godbillion, Souriau, and Gallisot were applying the alulus of differential forms to analytial mehanis (mostly by way of Hamiltonian mehanis), while Debever made one of the early appliations of differential forms to general relativity after Cartan. The appliation of differential forms to Maxwell s theory of eletromagnetism was impliit in Minkowski s relativisti formulation of it, but the expliit mention of its relationship to what Cartan and Goursat were doing did not ome until later. This is not the plae to introdue the alulus of differential forms to the uninitiated, so we refer them to the books by Henri Cartan [9], Walther Thirring [0], and Theodore Frenkel [] for that introdution. Hene, we shall start by simply summarizing the formulation of Maxwell s equations in terms of differential forms and then show how the gravitational analogy omes about. For the benefit of the physiists, we shall often give both the basis-free form of the geometri objets and their omponent form in terms of a natural oframe field (dx µ, µ = 0,, 3) that is defined by a oordinate system (U, x µ ) on the spae-time manifold, whih shall be simply Minkowski spae M 4 = (R 4, η). Our sign onvention for the salar produt η is that its omponents in an orthonormal oframe will be: η µν = diag[+,,, ]. (.6) Typially, x 0 = t will be the time oordinate while {x i, i =,, 3} will be the spatial ones. The line [x 0 ] in M 4 that is generated by x 0 will be alled the time line, while the three-dimensional linear spae Σ that is spanned by the x i will be alled spae. Spae is also the annihilating hyperplane of the -form dx 0 then. This oordinate system then defines a time+spae splitting of the spae-time i.e., a diret sum deomposition M 4 = [t] Σ into a one-dimensional time line [t] and a three-dimensional spae Σ. However,

6 Transverse momentum as a soure of gravitoeletromagnetism. 6 one does not need to define a oordinate system in order to define a time+spae splitting ( ). One first redefines E and B as a spatial -form and a spatial -form, resp.: E = E i dx i, B = B ij dx i ^ dx j. (.7) The omponents B ij of the -form B relate to the omponents B i of the vetor field B by way of: B ij = ε ijk B k, B i = ε ijk B jk, (.8) in whih the Levi-Civita symbols ε ijk and ε ijk equal + when ijk is an even permutation of 3, when the permutation is odd, and 0 otherwise. This relationship between the spatial vetor B and the spatial -form B amounts to the spatial version of the Poinaré isomorphisms # s : Λ k Σ Λ 3 k Σ, whih are defined when one hooses a volume element V s Λ 3 Σ for R 3. Here, we are using Λ k Σ to represent the linear spae of k-vetor fields on M 4 and Λ k Σ to represent the linear spae of k-forms on it. That is, an element b of Λ k Σ (k = 0,,, 3) will represent a ompletely-antisymmetri, totally-ontravariant tensor field of rank k on Σ: i b = b k! ik i i ( k i i x ), (.9) while an element α of Λ k Σ will represent a ompletely-antisymmetri, totally-ovariant tensor field of rank k on Σ: i ik α = i i dx dx k k! α. (.0) The symbol ^ then represents the exterior produt in all of this i.e., the ompletelyantisymmetrized tensor produt. A volume element V s on Σ is then a non-zero 3-form, suh as: V s = dx ^ dx ^ dx 3 = i j k ijk dx dx dx 3! ε. (.) For eah k, the Poinaré isomorphism # s : Λ k Σ Λ 3 k Σ then takes the k-vetor field b, as in (.9), to the (3 k)-form: # s b = i b V s, (.) in whih i b is the interior produt operator. If b is a vetor field and the k-form ω takes the form α ^ ^ α k then: ( ) For more details on the geometry of spae-times with time+spae splittings (also alled +3 splittings), one an onfer the author s artile [] and the referenes that are ited in it.

7 Transverse momentum as a soure of gravitoeletromagnetism. 7 i b ω = α (b) α ^ ^ α k α (b) α ^ α 3 ^ ^ α k + ± α k (b) α ^ ^ α k, or, more onisely: i b ω = k i+ ( ) ( ) αi b α αi αk, (.3) i= in whih the aret signifies that the -form in question has been omitted from the produt. A k-form suh as α ^ ^ α k is alled deomposable. Not all k-forms are typially deomposable, and one extends this definition of i b to all k-forms by linearity. That is, if the k-form β is a linear ombination interior produt i b β will be defined by: p m= m λ ω of p deomposable k-forms ω m then the m i b β = p m λ m ib ω. (.4) m= As it happens, sine Σ is three-dimensional, all k-forms on Σ will be deomposable, anyway, but the generalizations of these definitions to four dimensions will be useful later on. In order to extend i b from vetor fields b to k-vetor fields, one starts with deomposable k-vetor fields of the form b = b ^ ^ b k and an l-form α (l k) and defines: i b α = ib α b i i α k b b. (.5) = k In partiular, notie the inversion of the order in the sequene of vetors b,, b k. In the ase when α is the non-zero three form V s on Σ, if b represents eah of the elements in the sequene b, b i i, bij i ^ j, b 3 ^ ^ 3, in turn, then # s b will be equal to: b V s, ε ijk b k dx i ^ dx j, ε ijk b jk dx i, b 3, respetively. In partiular, from (.8), we see that the -form B relates to the vetor field B by way of: B = # s B. (.6) One an assemble the spatial -form E and the spatial -form B into a -form F: F = dt ^ E B (.7) that one alls the eletromagneti field strength -form. Note that we are now assuming that the omponents of E and B are also funtions of t, as well as the spatial variables. The first set of Maxwell equations in (.) an then be absorbed into: d^f = 0, (.8)

8 Transverse momentum as a soure of gravitoeletromagnetism. 8 in whih d^ : Λ k M 4 Λ k+ M 4 is the exterior derivative operator on Minkowski spae. In order to see how that happens, one first substitutes F from (.7) into (.8) to first get: d^f = dt ^ d^e d^b. If one temporarily reverts to the loal omponents of E and B then one an verify that: d^e = dt ^ t E + d s^e, d^b = dt ^ t B + d s^b, in whih d s^ : Λ k Σ Λ k+ Σ is the spatial exterior derivative operator. In partiular: All of this makes: d s^e = ( i E j j E i ) dx i ^ dx j, (.9) d s^b = 3 ( i B jk + j B ki + k B ij ) dx i ^ dx j ^ dx k. (.0) d^f = dt ^ ( t B + d s^e) d s^b, and sine the two terms in this are linearly-independent, the olletive vanishing of d^f would be equivalent to: t B + d s^e = 0, d s^b = 0. (.) In order to show that these equations are equivalent to the first two Maxwell equations (.), one needs only to show how E relates to d s^e and how B relates to d s^b. In fat, one has: E = # s d s^e, B = # s d s^b. (.) In order to see that first relationship, it is suffiient to look at the omponents of d s^e, as in (.9), and then note that: # s (dx i ^ dx j ) = ε ijk k. (.3) At this point, it helps to point out something that is usually overlooked in purelymathematial treatments of differential forms, namely, the fat that when an n- dimensional manifold M admits a volume element V (hene, it or rather its tangent bundle must be orientable), and therefore a set of Poinaré isomorphisms # : Λ k M Λ n k M, whih are defined as usual by: #b = i b V, (.4) one an define an adjoint operator to d^ by using #, namely: div = # d^ #. (.5) This operator does, in fat, generalize the divergene of a vetor field, sine one finds that if X = X µ µ is a vetor field on M then: div X = µ X µ. (.6)

9 Transverse momentum as a soure of gravitoeletromagnetism. 9 In partiular, one sees that for the spatial vetor field B, one will have: so: div s B = ds # s ds # s B = # s d s B = µ B µ, B = # s div s B = ( µ B µ ) V s. (.7) One finds that, in general, if b is a k-vetor field of the form (.9) then the omponents of div b will take the form: (div ) i ik b ii i = k i b. (.8) Like the operator d^, one an easily show that: div div = 0. (.9) However, the operator div does not have any distintive properties in regard to the exterior produt, as d^ does; i.e., it is not an anti-derivation. We are now in a position to deal with the seond set of Maxwell equations. One first defines D to be a spatial vetor field and H to be a spatial bivetor field and assembles them into the eletromagneti exitation bivetor field on Minkowski spae: H = t ^ D + H. (.30) If we define the eletri harge-urrent density vetor field by way of: j = σ t + J (j t = s, j i = J i ) (.3) then we an summarize the seond two Maxwell equations i.e., the soure equations as: div H = 4π j. (.3) This time, we have defined the spae-time volume element to be the non-zero 4-form: V = dx 0 ^ V s = dx 0 ^ dx ^ dx ^ dx 3 µ 0 µ 3 = ε µ dx dx 0 µ. (.33) 3 4! Straightforward, but tedious alulations will show that (.3) is equivalent to: # s d s^h t D = 4 π J, divs D = 4π σ. (.34) These equations an be ompared to the seond set of equations in (.).

10 Transverse momentum as a soure of gravitoeletromagnetism. 0 A onsequene of (.3) is derived from (.9), and takes the form of the onservation of eletri harge: div j = 0 (0 = µ j µ = t σ + i j i ). (.35) From the version of the Poinaré lemma that pertains to the div operator, there must exist a bivetor field b suh that: j = div b (.36) (if only loally).. Potential -forms. The first Maxwell equation, in the form (.8), in onjuntion with the Poinaré lemma, implies that there is a -form: suh that: A = φ dt A s, (.37) F = d^a. (.38) One would all suh a -form an eletromagneti potential -form. The first Maxwell equation then beomes an identity that follows from the fat that d = 0 for any possible A. However, when one substitutes d^a for F in the onstitutive law, the seond Maxwell equation (.3) will take the form: div H (d^a) = 4π j, (.39) whih represents four equations for the four unknown funtions that take the form of the omponents of A. However, sine one also has the identity (.35), the system will beome underdetermined by one variable, whih an be aounted for by making a hoie of a gauge for A; i.e., replaing A with A + dλ, means that the gauge degree of freedom amounts to the free hoie of funtion λ. When one does the atual exterior differentiation of A in the form (.37), one will get: d^a = dφ ^ dt d^a s = dt ^ (dφ + t A s ) d s^a s. Equating this to F in the form (.7) will give: E = (dφ + t A s ), B = d s^a s, (.40) and with the usual identifiation of the differential operators with their analogues in vetor alulus, one will get: E = ( φ + Aɺ ), B = A, (.4)

11 Transverse momentum as a soure of gravitoeletromagnetism. whih is the way that one would find the assoiation of field strengths and potentials presented in many onventional texts on eletromagnetism. For a topologially-general spae-time manifold, a -form suh as A would exist only loally (i.e., on a neighborhood of eah point), but for Minkowski spae, whih is ontratible, it will exist globally. Furthermore, it will not be unique, sine one an add any losed -form χ (i.e., d^χ = 0) to A and produe another -form that will give F upon exterior differentiation. That freedom to alter A by a losed -form without hanging the resulting F is referred to as gauge invariane, and a hoie of A or χ is referred to as a gauge. Typially, the losed form is represented as an exat form (i.e., χ = dλ), whih is possible loally, in general, and globally in the present ase of Minkowski spae. The hoie of gauge that will be most interesting to us is the Lorentz gauge ( ), whih imposes the ondition that the vetor field A that is metri-dual to the -form A must have vanishing divergene: div A = 0. (.4) d. Lorentz fore law. The fore that the ombined eletri and magneti fields exerts upon a point-like mass m with a harge of q that moves with a veloity of v relative to the soure of the fields is given by the Lorentz law: F = q (E + v B). (.43) If one thinks of the motion of q as an eletri urrent that takes the form of: then the Lorentz fore law an also be written: J = q v (.44) F = q E + J B. (.45) One an think of the urrent J as it is defined in (.44) as being of onvetive type; i.e., it is ollinear with the veloity, as if q were arried along by v. The Lorentz fore law is even more onisely formulated in terms of the -form F: f = i j F (f ν = jµ F µν ). (.46) In order to see the equivalene of (.46) with (.45), substitute (.7) for F and get: i j ( dt ^ E) i j B = ( i j E) dt + J t E i j B = E(J) dt + q E i J B, ( ) Interestingly, this gauge was not named after the Duth physiist Hendrik Antoon Lorentz, but a Russian one by the name of Ludwig Valentin Lorentz, who has been otherwise passed over by the history of gauge field theory.

12 Transverse momentum as a soure of gravitoeletromagnetism. but sine: one will get: i J B = i J # s B = i J i B V s = # s (B ^ J) = # s (J ^ B), f = E(J) dt + q E + # s (J ^ B). (.47) The F in (.45) is essentially the spatial part of this, while the temporal omponent E(J) represents the power that is being absorbed or radiated by J. e. The polarization of eletromagneti media. Now, let us return to the relationship between the eletromagneti exitation bivetor field H and the eletromagneti field strength -form F. One will have an invertible map C : Λ M Λ M that is a diffeomorphism of eah fiber Λ,x M with eah fiber Λ x M, and in partiular it will take the eletromagneti field strength F to the eletromagneti exitation bivetor field: H = C (F). (.48) Suh a map is alled an eletromagneti onstitutive law, although in order for C to be an algebrai operator on -forms, one must assume that the medium in question is dispersionless ( ) in both time and spae. If there were dispersion, in that sense, then the map C would beome an integral operator. In the omponent formulation of C, we an introdue the zero-point field Z (x) = C( x,0), and haraterize the onstitutive law C in the omponent form: The maps C x : Λ,x M [C (F)] µν = Z µν (x) + Cµνκλ (x, F) F κλ. (.49) Λ x M will be linear isomorphisms iff: Z µν (x) = 0, C µνκλ (x, F) = C µνκλ (x), (.50) and that would make C a linear eletromagneti onstitutive law. The absene of a funtional dependeny of the omponents C µνκλ (x, F) on the point x in spae-time would make it homogeneous, but one should be autioned that suh a property is defined only in speialized frame fields. Indeed, it is only when the transition funtion h : M GL(4) is onstant that things that are onstant in one frame field will be onstant in another one. Although it is possible to disuss Maxwellian eletromagnetism without any referene to a spae-time metri, nonetheless, if one wishes to ompare the onstitutive law that is defined by C with the one that is defined by the lassial eletromagneti ( ) Here, we must point out a soure of onfusion in the theory of eletromagnetism, namely, the word dispersion is used in two largely-unrelated ways: The present usage refers to the possibility that the state of exitation at a point depends upon its state at neighboring points in spae and time. The other usage relates to the way that the frequeny of a wave gets oupled to its wave number.

13 Transverse momentum as a soure of gravitoeletromagnetism. 3 vauum then one will need to introdue a metri g. That will make the assoiation of - forms with bivetor fields take the form of raising both indies, namely, the linear isomorphism ι g : Λ M Λ M that takes the -form F = F µν dx µ ^ dx ν to the bivetor field F = Fµν µ ^ ν, suh that: F µν = g µκ g νλ F κλ = (gµκ g νλ g µλ g νκ ) F κλ. (.5) Stritly speaking, this map needs to inlude ε 0 and µ 0 in order for it to truly represent the eletromagneti onstitutive law of the lassial vauum, but we shall ignore that for the moment. However, it is important to note that sine C will define a dispersion law (in the sense of wave motion) that an redue to a Lorentzian struture in some ases, one must have some way of explaining how the tensor field g (or really the map ι g ) relates to the dispersion law of C. Presumably, ι g is an asymptoti limit of C that relates to something like the absene of eletromagneti fields. If that relationship between ι g and C is physially meaningful then one an haraterize the differene between them by: µ C i g. (.5) When this (not-neessarily-invertible) map is applied to the field strength -form F, the resulting bivetor field: µ (F) = C (F) i g F (µ µν = Z µν + Cµνκλ F κλ F µν ) (.53) an be defined to be the polarization of the medium that results from its exitation by the field strength F. One an also haraterize the differene between C (F) and i g F in terms of the exitation bivetor field H = C (F): µ (F) = H ι g F (µ µν = H µν F µν ). (.54) Before we go further, it is important to give the onept of the polarization of an eletromagneti medium a more empirial basis. In reality, an eletromagneti medium is polarizable iff the presene of an eletromagneti field F in it provokes the formation of eletri or magneti dipoles (or both). Typially, most onventional media are either dieletri or magneti of some desription (i.e., diamagneti, paramagneti, or ferromagneti), but not both, whih is why, for instane, most optial models assume that optial media are dieletris (if not insulators) that are not magnetially polarizable, and onversely, most magneti materials are ondutors, not dieletris. We shall return to this piture of polarizability when we attempt to extend gravitoeletromagnetism analogously. If one has a time+spae deomposition of spae-time (or really, its tangent bundle), so one an express H in the form (.30) and ι g F in the form:

14 Transverse momentum as a soure of gravitoeletromagnetism. 4 ι g F = t ^ E + B, (.55) then that will make the polarization bivetor field take the time+spae form: µ (F) = t ^ P + M = t ^ (D ε 0 E) + (µ 0 H B); (.56) i.e., the of eletri polarization vetor field P and the magnetization bivetor field M are defined by: P = D ε 0 E, M = µ 0 H B. (.57) When one has deomposed H into a sum i g F + µ (F), one an similarly deompose the divergene of H: div H = div i g F + div µ. (.58) When this is equated to 4π j, the seond of Maxwell s equations an be rewritten in the form: div i g F = 4π j div µ, (.59) whih suggests that one an also regard the effet of the polarization of the medium as being something that indues a polarization urrent: 4π j p div µ ( 4π j ν p = µ µ µν ). (.60) If we take the divergene of both sides of (.59) then we will see that we still have: so sine: div j = 0, (.6) div j p = 0, (.6) we must always have that not only is the total urrent: j tot j + j p (.63) onserved olletively, but its omponents must be also onserved individually. With µ in the form (.56), the polarization urrent will deompose into eletri and magneti ontributions: 4π j p = div ( t ^ P) div M. Sine P and M are both spatial tensor fields, one an say that:

15 Transverse momentum as a soure of gravitoeletromagnetism. 5 and div ( t ^ P) = # d^ # ( t ^ P) = # d^ i = # (dt ^ t # s P + d^s # s P) = # s t # s t P V = # d^ i i V = # d^ # s P P t P + # d^s # s P = [ Pɺ (div s P) t ], div M = # d^ #M = # d^ (dt ^ # s M) = # (dt ^ d^ # s M) = # (dt ^ d^s # s M) = # s d^s # s M = div s M. That will make: 4π j = [ (div s P) t + P ɺ ] + div s M. (.64) The temporal omponent of this is the polarization harge density: σ p = div s P, (.65) while the spatial part is the polarization urrent (properly speaking): j s = Pɺ + div s M. (.66) 3. Gravitoeletromagnetism. Theoretially, gravitoeletromagnetism (whih we shall abbreviate by the aronym GEM) amounts to a formal analogy between Maxwell s equations for eletromagnetism and the weak-field equations for gravitation. In order to justify the assoiation of orresponding fields, we shall first disuss weak-field gravitation in terms of vetor alulus, whih is often how the stati ase is disussed in physis, and then reast it in terms of differential forms. a. Vetor alulus formulation. The analogy between Coulomb s law of eletrostati interation and Newton s law of gravitation is essentially based upon the first pair of stati equations in (.). One ould then make them the equations of a stati Newtonian gravitational field by assoiating both E and D with the gravitational aeleration field g and assoiating the eletri harge density σ with the mass density ρ. Furthermore, the hange in oupling onstant amounts to replaing 4π with Newton s gravitational onstant, whih we shall denote by G 0. That makes: g = 0, g = G 0 ρ. (3.) The GEM field amounts to the analogue of H and B that results from the presene of the mass urrent j s in the same way that B, and therefore H, will result from the presene of the eletri urrent J. Hene, we define the GEM field to be ω, whih replaes both H and B, while j s replaes J and G 0 / replaes 4π. Altogether, we have:

16 Transverse momentum as a soure of gravitoeletromagnetism. 6 g E = D, ω H = B, ρ σ, j s J, G 0 / 4π. (3.) That gives: G ω = 0 s j, ω = 0 (3.3) in the stati ase, and if we return to the dynami ase then we will have: ω g + = 0, ω = 0, ω t g G = 0 s t j, g = G 0 ρ. (3.4) The fat that the oupling onstant between the soure urrent j s and its field ω is G 0 / shows how feeble the strength of that field will be and why the existene of ω was observed experimentally only reently; in MKS units, G 0 / = m / kg. b. GEM in terms of differential forms. In order to put (3.4) into the language of differential forms, we simply alter the basi analogy (3.) to take the form: F G = (dt ^ g ω), H G = t ^ g ω, j = ρ t + j s. (3.5) In effet, the onstitutive law that ouples G to G is defined by the Lorentzian metri isomorphism ι η, although one should note that with the sign onvention that we hose for η in (.6), the spatial -form g will go to what we are alling g, while the spatial - form ω will go to ω. The field equations will then be: or in omponent form: d^g = 0, div G = G 0 j, G = i η G, (3.6) λ G µν + µ G νλ + ν G λµ = 0, µ G µν = G 0 j ν, G µν = (ηµκ η νλ η µλ η νκ ) G κλ. (3.7) Substituting the time+spae forms of G and G will give: t ω + d s^g = 0, d s^ω = 0, div s g = G 0 ρ, t g + div s ω = G 0 j s, (3.8) or in omponent form: t ω ij + ( i g j j g i ) = 0, i ω jk + j ω ki + k ω ij = 0, (3.9) i g i = G 0 ρ, t g ij + i ω ij = G 0 j j, (3.0)

17 Transverse momentum as a soure of gravitoeletromagnetism. 7 whih should be ompared with (3.4). In the stati ase, (3.8) and (3.9), (3.0) will take the form: and G d s^g 0 = 0, div s g = G 0 ρ, d s^ω = 0, div s ω = j s (3.) i g j j g i = 0, i g i = G 0 ρ, (3.) i ω jk + j ω ki + k ω ij = 0, i ω ij = G 0 j j, (3.3) resp., whih an be ompared to (3.) and (3.3). The first equation in (3.6) implies the existene of a (loal) potential -form: u = ψ dt u s. (3.4) Like its eletromagneti analogue, it will not be unique, but will be subjet to the same gauge freedom. Taking the exterior derivative of u will give: d^u = dψ ^ dt d^u s = dt ^ ( t u s + d s ψ) d s^u s. Equating orresponding terms in the right-hand side of G in (3.5) will yield: g = t u s d s ψ, ω = d s^u s, (3.5) and in terms of omponents that will take the form: g i = t u i i ψ, ω ij = ( i u j j u i ). (3.6) Hene, sine g is a linear aeleration and ω is an angular veloity, we would be dimensionally orret in identifying ψ with an aeleration potential, while u s is the spatial part of a proper-time oveloity -form. Sine the fields g and ω are also defined at the points where there is no mass to be aelerating or rotating, one must larify what the values of those fields would represent. Basially, in the absene of gravitomagnetism, g would be the free-fall aeleration of any non-zero mass that is plaed at the point. Presumably, the angular veloity ω would then represent a sort of free-rotation speed for a gravitomagneti dipole that is plaed at the point. (We shall larify that statement shortly.). Fore on a moving mass. The gravitational analogue of the Lorentz fore law omes about by making the replaements (3.) in (.43), but with m instead of ρ and m v instead of j s, whih gives: F = m (g + v ω), (3.7)

18 Transverse momentum as a soure of gravitoeletromagnetism. 8 whih an be expressed in the relativisti form: f = i j G (f ν = j µ G µν ). (3.8) One then sees that the ontribution to the fore that omes from the gravitomagneti field ω is essentially that of a Coriolis fore (exept for the missing fator of ). Note that even though the fores that are imparted by the eletrostati field E of an individual harge often overshadow those of its magnetostati field B, nonetheless, one must notie that industrial eletromagnets are more ommonly used for heavy lifting and magneti levitation than industrial eletrets. In other words, the olletive effet of many elementary magneti dipoles seems to be more useful in pratie than that of many individual eletri harges. Hene, reasoning by analogy, one should not dismiss the possible empirial signifiane of the gravitomagneti field out of hand. We shall return to a disussion of the meaning of gravitomagneti dipoles later, but first we need to examine the GEM analogue of polarization urrent, namely, transverse momentum, and give some examples of how it shows up in theoretial physis. 4. Transverse momentum. The most ommon way of defining momentum in lassial mehanis is the onvetive form of that dynamial quantity. In the ase of a non-relativisti point-mass m that moves along a urve in spae with a veloity vetor v, one defines the non-relativisti momentum vetor by: P = m v. (4.) One then sees that the momentum vetor will be ollinear with the veloity vetor, by definition. This an also be referred to as longitudinal momentum. If the mass is extended over a finite spatial volume and desribed by a mass density ρ, while the motion is defined by a ongruene of spatial urves with a veloity vetor field v then one an define a orresponding momentum density vetor field: p = ρ v. (4.) One again, this would be a onvetive or longitudinal definition of momentum density. Similarly, when one goes on to relativisti mehanis, the main alterations that are neessary are that the urve or ongruene of urves in spae that are parameterized by the time oordinate t must beome a time-like world-line or ongruene of time-like world-lines in spae-time, resp., that are parameterized by proper time τ, so the spatial veloity vetor field u must beome a time-like vetor field u on spae-time, in whih typially the derivative is with respet to proper time. When a world-line has been parameterized by proper-time, the effet of that is to make: u = η (u, u) =. (4.3)

19 Transverse momentum as a soure of gravitoeletromagnetism. 9 Finally, the mass m or mass density ρ must beome a proper mass m 0 or proper mass density ρ 0, resp., and one defines the energy-momentum four-vetor for a point-like mass and the energy-momentum density four-vetor for an extended mass by: P = m 0 u and p = ρ 0 u, (4.4) respetively. Atually, sine one an think of dynamis as being dual to kinematis by way of the virtual work funtional, it is often more appropriate to think of P or p as a spatial or timelike spae-time -form P or p, resp., that relates to the vetor fields by way of the spatial or spae-time metri, resp.; i.e., one lowers the index on its omponents. That puts the last two equations into the forms: P = m 0 u and p = ρ 0 u, (4.5) resp., in whih the -form u is the metri dual of u (i.e., u µ = η µν u ν ) and is referred to as oveloity. However, this onvetive or longitudinal senario is not the only one possible. Typially, in situations in whih the matter in question is harged and interating with a bakground eletromagneti field or just spinning, there will be ontributions to the momentum (or energy-momentum) that are not ollinear with veloity, and those ontributions are then referred to as transverse momentum. By definition, if the energymomentum density vetor field ( ) takes the form: then p t will represent transverse momentum iff: p = ρ 0 u + p t (4.6) η (u, p t ) = 0. (4.7) If one wishes to express this in terms of the orresponding energy-momentum density -form: p = ρ 0 u + p t (4.8) then one will have: i u p t = p t (u) = 0. (4.9) Note that as a result: p (u) = ρ 0 u (u) = ρ 0, (4.0) so the transverse momentum makes no ontribution to the rest energy density if is defined in this way. However: p = η (p, p) = (ρ 0 ) + (p t ) = ( meff ), (4.) with: ( ) Although we are making this definition for the relativisti ase, the non-relativisti definition is entirely analogous.

20 Transverse momentum as a soure of gravitoeletromagnetism. 0 m eff = [(ρ 0 ) + (p t / ) ] / = ρ 0 p t ρ0 /. (4.) (The minus sign appears due to the fat that p t is a spae-like vetor.) One might distinguish the notion of transverse momentum from an earlier one that Lorentz exhibited in his theory of the eletron [3] that takes the form of longitudinal and transverse mass. Basially, the Fitzgerald-Lorentz ontration of a mass distribution that is spherial in its rest spae into an (apparent) oblate spheroid due to its relative motion will imply that, in effet, the mass will be different in the longitudinal diretion from what it is in the diretions that are in the plane perpendiular to veloity. In partiular, if the rest mass is m 0 and the relative speed is β = v / then the longitudinal and transverse masses will be: m0 m0 m l =, m 3/ t =, (4.3) / ( β ) ( β ) respetively. One might imagine a mass matrix ( ) M ij = diag [m l, m t, m t ] that takes spatial veloities to spatial momenta, but one has to realize that suh a matrix is defined in a frame that is adapted to the veloity vetor, so the transverse omponent of veloity would be zero, by definition, as would the orresponding transverse momentum. The examples of transverse momentum that we shall examine are all relativisti ones that typially relate to relativisti quantum wave mehanis, namely, the minimal eletromagneti oupling of an external eletromagneti field to energy-momentum, the Frenkel eletron, the Dira eletron, and the Weyssenhoff fluid. a. Minimal eletromagneti oupling. When a point-like harged mass (rest mass = m 0, harge = q) moves in the presene of an external eletromagneti field F = d^a, one an also absorb the external eletromagneti field into the definition of the harge-field system by way of minimal eletromagneti oupling: P = m 0 u q A. (4.4) One will then have: P (u) = m 0 q A(u), (4.5) and as for P = (m eff ), one will have: m eff = q q m A A 0 ( u ) 3 +. (4.6) m0 m0 ( ) Mass matries are used in the ontext of the eletroweak model for partile interations, but in a different sense than the present one.

21 Transverse momentum as a soure of gravitoeletromagnetism. If one expresses A in time+spae form as in (.37) and expresses u as γ [(/) t + v], with γ = ( v / ) /, then: A (u) = γ [φ A s (v)]. (4.7) Note that in a rest frame (v = 0, γ = ), the only ontribution to A (u) will ome from the eletrostati potential. Sine A (u) does not have to vanish unless φ = A s (v), A does not have to be purely transverse, although typially it will ontain both longitudinal and transverse omponents. One an see that the Lorentz fore law beomes equivalent to the onservation of total energy-momentum: 0 = dp dτ = q da m0 uɺ. (4.8) dτ That is beause: da dτ = L ua = i u d^a + d^(i u A) = i u F + d [A(u)] = i u F. (4.9) The term d [A(u)] vanishes beause u(τ) is defined only as a funtion of τ, so the same thing will be true of the salar A(u), and therefore its differential must vanish. Hene, (4.8) will take the form: m0 uɺ = q i u F. (4.0) Things are quite different for a harged, extended mass, suh as a harged fluid. If its proper mass density is ρ 0, its harge density σ, and its proper-time-parameterized flow veloity is u then u(x) will be funtion of τ only impliitly by way of its funtional dependeny upon the spae-time point x when one selets a trajetory x(τ). The energymomentum density -form p will then take the form: p = ρ 0 u σ A. (4.) Now, when one takes the proper-time derivative of p, the result will ontain terms that depend upon the differentials of ρ 0 and σ, as well: Now: and dp dτ = d ɺ σ σ da ( ρ0 u) A. (4.) dτ dτ σɺ = L u σ = u σ = u µ µ σ, (4.3) da dτ = L ua = i u F + d (A(u)), (4.4) so the vanishing of dp / dτ would imply that:

22 Transverse momentum as a soure of gravitoeletromagnetism. d dτ (ρ σ 0 u) = iu F + [(u σ) A + σ d (A(u))]. (4.5) Note that even for proper mass and harge densities that are onstant in time and spae, there will still be an extra ontribution to the analogue of (4.0) that omes from d (A(u)); i.e.: ρ 0 uɺ = σ iu F + σ d (A(u)). (4.6) b. Frenkel eletron. In 99, one year after Dira presented his relativisti theory of the spinning eletron, Joseph Frenkel made a first attempt [4] at a relativisti theory of a harged, spinning eletron that interated with an external eletromagneti field F. His model still assumed a point-like distribution of mass, harge, and spin for the eletron, but allowed the point to rotate. Admittedly, this sounds more like a simplifying approximation, sine rotation is more naturally defined in terms of extended matter, but eventually this way of thinking ame to be regarded as the pole-dipole approximation to an extended distribution, whih an be thought of as a Lorentzian (i.e., orthonormal) frame moving along a time-like world-line. Without going into all of the details, we shall simply summarize the relevant onsequenes of Frenkel s model. The time-like veloity vetor field of the world-line is u, while its orresponding oveloity -form is u, the proper mass is m 0, and the spin is desribed by a -form S that satisfies the Frenkel onstraint : i u S = 0 (u µ S µν = 0). (4.7) This has the effet of making the motion of the Lorentzian frame purely rotational with respet to a rest frame and is based in the fat that experiments suggested that although the eletron has an intrinsi magneti dipole moment in its rest spae, nonetheless, it seems to have no eletri dipole moment. The eletromagneti dipole moment -form µ for the spinning eletron is oupled to that spin -form by way of the Uhlenbek-Goudsmit hypothesis : µ = µ B S, (4.8) in whih µ B = eħ / m e (in CGS units) is the Bohr magneton. The energy-momentum -form p that one derives for this motion is: into whih we have introdued an effetive mass: p = m eff u i as, (4.9) m eff = m 0 + F(µ), (4.30)

23 Transverse momentum as a soure of gravitoeletromagnetism. 3 whih inludes a ontribution from the potential energy of the oupling of the eletromagneti dipole moment to the external eletromagneti field and a vetor field a along the world-line whose orresponding metri-dual -form is: a = g e m i uf uɺ. (4.3) 0 The g in this is a onstant that haraterizes the state of rotation. In partiular, for nonspinning harges, g =. Although a learly has the dimensions of an aeleration, it represents a measure of the differene between the fore law that applies to a spinning harge and the usual Lorentz fore law. In partiular, for non-spinning harges the Lorentz fore law would apply, whih would make a = 0. Otherwise, one would have a variation on the Lorentz fore law that would take the form: pɺ = e i uf + df(µ). (4.3) Hene, there is an additional fore that ats upon the spinning harge by way of the oupling of its eletromagneti dipole moment µ to the inhomogeneity df in the external eletromagneti field. (Reall that in the Stern-Gerlah experiment in order to exhibit the spin of the eletron, it was neessary to employ an inhomogeneous magneti field.) If one equates (e / ) i u F in this with the orresponding expression for it that one infers from (4.3) and solves for a then one will get: g a = [ pɺ df( µ )] uɺ. (4.33) m 0 Note that even in the absene of an external field [F = 0, so m eff = m 0, a = ( g / m0 ) pɺ uɺ ], one will still have a ontribution from a when g is not equal to ; i.e., when the harged mass is spinning. In any event, the transverse momentum will take the form: p t = i as. (4.34). Dira eletron. When Paul Dira published his quantum theory of the eletron in 98 [5], one of the big obstales that it faed was the esoteri and largely unfamiliar nature of its introdution of the Clifford algebra of Minkowski spae C (4, η), at least as far as the rest of the physiists of the era were onerned. As a result of that introdution, the way that lassial observables were enrypted into the Dira wave funtion was not entirely unique or agreed upon. Typially, what the various attempts to derive lassial (but relativisti) observables from the Dira wave funtion Ψ had in ommon was that they generally started with the definition of the sixteen bilinear ovariants.

24 Transverse momentum as a soure of gravitoeletromagnetism. 4 One defines the bilinear ovariants of Ψ by first defining a representation of the sixteen-real-dimensional Clifford algebra C (4, η) in the sixteen-omplex-dimensional algebra of 4 4 omplex matries M (4, C) (although the hoie of that representation is not at all uniquely agreed-upon by physis). Four generators of C (4, η), namely, a hoie of Lorentzian frame {e µ, µ = 0,, 3}, go to four generators of its image in M (4, C), whih will be four matries {γ µ, µ = 0, 3}. A basis {E A, A =,, 6) for C (4, η) an be defined by all linearly-independent produts of the e µ, and they will then orrespond to a basis {γ A, A =,, 6} for the image of C (4, η) in M (4, C), whih will be a proper subspae of M (4, C) as a real vetor spae. Eah basis vetor E A then assoiates Ψ with a real number Ψ EAΨ that is the A th bilinear ovariant of Ψ. In this expression, we are defining the Dira onjugate wave funtion Ψ by way of: Ψ = Ψ γ 0, (4.35) in whih Ψ is the Hermitian onjugate of Ψ (i.e., the omplex onjugate of its transpose as a olumn matrix). Sine Ψ takes its values in C 4, whih has a real dimension of eight, and there are sixteen bilinear ovariants, one sees that the latter annot all be algebraially independent. In fat, the algebra of C (4, η) imposes nine identities upon the bilinear ovariants, whih were first observed by Louis de Broglie and expanded upon more rigorously by Wolfgang Pauli and his student Koffink. Atually, that only leaves seven independent ovariants, and the way that one finds the eighth one is to go on to the differential ovariants. We shall not go into the details of that here, but refer the urious to the paper of Takahiko Takabayasi [6] in whih he disusses the onversion of the Dira equation into a set of relativisti equations of motion for a relativisti spinning fluid that orresponds to the Dira wave funtion. The author s own thoughts on the topi are disussed at length in his book on ontinuum-mehanial models for wave mehanis [7]. The essential fat that we shall ite here is that Takabayasi s expression for the energy-momentum four-vetor amounts to: p = ρ 0 os θ u + div σ + i grad θ *σ. (4.36) The bivetor field σ is the spin tensor that one obtains from Ψ, while the somewhat mysterious angle θ amounts to a sort of phase angle in a plane that is spanned by u and the vetor field s = # (u ^ σ), whih is essentially the Pauli-Lubanski vetor field. The first term in the right-hand side of (4.36) is learly the onvetive part of the momentum, while the last term is reminisent of the transverse momentum in the Frenkel eletron. It is the seond term namely, div σ that will be most interesting to us in the next setion, beause it relates to a sort of polarization urrent that gets assoiated with