General Relativity in the Framework of exact gravito-electromagnetic analogies

General Relativity in the framework of exact gravito-electromagnetic analogies Luis Filipe Costa Abstract In this work we studied the several gravito-electromagnetic analogies in the literature, and presented a new exact one, based on tidal tensors. We clarified the relation between the different analogies, and further worked out some of them; namely the analogy based on inertial gravitational fields (GEM fields), which was reformulated and extended. The gravito-electromagnetic analogy based on tidal tensors stems from the tidal dynamics of the two theories: the analogy for electric type tidal tensors is manifest in the geodesic deviation, and in the analogous electromagnetic worldline deviation; the analogy for magnetic-type tidal tensors is manifest in the force exerted on spinning particles (magnetic dipoles/gyroscopes). It extends to the field equations: the gravitational field equations may be cast as a set of four algebraic equations for tidal tensors and sources, analogous to Maxwell’s equations in this formalism; plus two additional equations with no electromagnetic counterpart. This analogy is ideally suited to compare the tidal dynamics of the two interactions; key differences are made transparent in the symmetries and time projections of the tidal tensors, which are related to the phenomenon of electromagnetic induction, and the absence of analogous effects in gravity. This is extensively explored in the context of the dynamics of spinning multipole particles — a natural application of the formalism. The analogy based on inertial GEM fields stems from the space part of the geodesic equation, which can be cast exactly as consisting of a Lorentzlike part where two spatial vector fields — a gravitoelectric and a gravitomagnetic field — mimic the electromagnetic fields, plus a term involving the shear/expansion of the observer congruence which has no electromagnetic analogue. This analogy also extends to the field equations, where we have on the gravitational side six equations, four of which, again in this formalism, exhibit many similarities with Maxwell’s equations; the similarity gets particularly close for rigid frames and stationary fields. This formalism also leads to an exact gravito-electromagnetic analogy for the “precession” of the spin vector of a spinning particle (this analogy, together with the tidal tensor one, means that both equations of motion for a spinning dipole particle — the spin evolution, and the force — can be cast in exact gravito-electromagnetic analogies). At the heart of these analogies is the Mathisson-Pirani (MP) spin condition; however this condition is usually portrayed in the literature as problematic, due to its degeneracy and the famous helical motions it allows, which have been deemed unphysical. We address the problem of the spin condition and the definition of center of mass in General Relativity, and show that these claims are but misconceptions: not only the MP condition is as valid as any other, as it is the most suitable one (through the nonhelical solution) for many practical applications. As for the helical motions it equally allows, we show that they are just alternative (but equivalent) descriptions, dynamically consistent and explained through the concept of hidden momentum — analogous to the hidden momentum of electromagnetic systems. We discuss the different forms of hidden momentum, and unveil some of its counter-intuitive features. A number of other issues not well understood in the literature were clarified in the course of this work; namely the physical meaning of the magnetic part of the Riemann tensor, and the problem of the covariant equations of motion for spinning particles in electromagnetic fields, and their interpretation. 2 Resumo Neste trabalho estudamos as v´arias analogias gravito-electromagn´eticas existentes na literatura, e apresentamos uma nova, baseada em “tensores de mar´es”. Clarificamos a rela¸c˜ao entre as v´arias analogias, e evolu´ımos algumas delas, em particular a analogia baseada em campos de for¸cas inerciais (campos GEM), que foi reformulada e estendida. A analogia gravito-electromagn´etica baseada nos tensores de mar´es emana da dinˆamica dos efeitos “de mar´es” das duas teorias: a analogia entre tensores de mar´es do tipo el´ectrico manifesta-se na equa¸c˜ao de desvio geod´esico, e na equa¸c˜ao de desvio an´aloga no electromagnetismo; a analogia dos tensores do tipo magn´etico manifesta-se na for¸ca exercida em part´ıculas em rota¸c˜ao (dipolos magn´eticos/girosc´opios). A analogia estende-se `as equa¸c˜oes de campo: as equa¸c˜oes relativistas do campo gravitacional podem ser retratadas como um sistema de quatro equa¸c˜oes alg´ebricas envolvendo apenas tensores de mar´es e termos de fonte, an´alogas `as equa¸c˜oes de Maxwell quando escritas neste formalismo, mais um par de equa¸c˜oes adicionais que n˜ao tˆem an´alogo electromagn´etico. Esta analogia ´e ideal para comparar as duas interac¸c˜oes; diferen¸cas chave s˜ao transparentes nas simetrias e projec¸c˜oes temporais dos tensores de mar´es, que est˜ao relacionadas com os fen´omenos de indu¸c˜ao electromagn´eticos, e a ausˆencia de efeitos an´alogos na gravidade. Estas diferen¸cas s˜ao exploradas com grande detalhe no contexto da dinˆamica de part´ıculas com momentos multipolares — uma aplica¸c˜ao natural do formalismo. A analogia baseada nos campos de for¸cas inerciais emana da parte espacial da equa¸c˜ao das geod´esicas, que pode ser exactamente descrita como consistindo de uma parte semelhante `a for¸ca de Lorentz, onde dois vectores espaciais — os campos “gravitoel´ectrico” e “gravitomagn´etico” — mimetizam os campos electromagn´eticos, mais um termo adicional envolvendo o “shear”/expans˜ao da congruˆencia de observadores, que n˜ao tem an´alogo electromagn´etico. Esta analogia tamb´em se estende `as equa¸c˜oes de campo, onde obtemos do lado gravitacional seis equa¸c˜oes, das quais quatro, neste formalismo tamb´em, manifestam v´arias semelhan¸cas com as equa¸c˜oes de Maxwell; a semelhan¸ca ´e particularmente pr´oxima no caso de referenciais r´ıgidos em campos estacion´arios. Este formalismo leva ainda a uma analogia exacta para a “precess˜ao” do vector de spin de uma part´ıcula em rota¸c˜ao (juntando esta analogia `a dos tensores de mar´e, temos que ambas as equa¸c˜oes de movimento para part´ıculas p´olo-dipolo — a equa¸c˜ao de evolu¸c˜ao do spin, e a da for¸ca — podem ser retractadas em analogias gravito-electromagn´eticas exactas). No cora¸c˜ao destas analogias est´a a condi¸c˜ao de spin de Mathisson-Pirani (MP); todavia esta condi¸c˜ao ´e vista na literatura como problem´atica, devido `a sua degenerescˆencia e aos famosos movimentos helicoidais que ela admite, que foram considerados n˜ao f´ısicos. N´os abordamos o problema da condi¸c˜ao de spin, e da defini- ¸c˜ao de centro de massa em Relatividade Geral, e mostramos que tais afirma¸c˜oes n˜ao passam de um mal entendido: n˜ao s´o a condi¸c˜ao MP ´e t˜ao v´alida como qualquer outra, como ´e mesmo a mais adequada (atrav´es da sua solu¸c˜ao n˜ao helicoidal) para v´arias aplica¸c˜oes pr´aticas. Quanto `as solu¸c˜oes helicoidais que ela igualmente admite, mostramos que s˜ao apenas descri¸c˜oes alternativas (mas equivalentes), dinˆamicamente consistentes e explicadas pelo conceito de “hidden momentum” — an´alogo ao hidden momentum dos sistemas electromagn´eticos. Discutimos tamb´em as v´arias de formas hidden momentum, e revelamos alguns dos seus efeitos contra-intuitivos. V´arias outras quest˜oes que n˜ao eram bem compreendidas na literatura foram sendo clarificadas no decurso deste trabalho; nomeadamente o significado f´ısico da parte magn´etica do tensor de Riemann, e o problema das equa¸c˜oes de movimento covariantes para part´ıculas multipolares (em rota¸c˜ao) sob ac¸c˜ao de campos electromagn´eticos, e a sua interpreta¸c˜ao. 3 Contents 1 Description of this document 7 2 Article compilation — index 8 3 Introduction and motivation 9 3.1 The Gravito-electromagnetic analogies in the literature . . . . . . . . . . . . 10 3.1.1 Analogy based on inertial forces from linearized gravity . . . . . . . 10 3.1.2 Exact analogy based on inertial forces . . . . . . . . . . . . . . . . . 12 3.1.3 The exact analogies between the Weyl and the Maxwell tensors . . . 14 3.1.4 Some then open questions to be addressed . . . . . . . . . . . . . . . 16 3.2 Spinning multipole particles in general relativity . . . . . . . . . . . . . . . 17 3.2.1 Equations of motion for pole-dipole particles . . . . . . . . . . . . . 19 3.2.2 Equations of motion to quadrupole order . . . . . . . . . . . . . . . 23 3.2.3 The gravito-electromagnetic analogies for spinning particles in the literature ................................. 25 4 Roadmap to the papers 26 4.1 The exact GEM analogy based on tidal tensors . . . . . . . . . . . . . . . . 26 4.2 The exact GEM analogy based on fields of inertial forces . . . . . . . . . . . 27 4.3 Gravity contrasted with electromagnetism — where can they be similar . . 28 4.4 The problem of the center of mass in general relativity; Mathisson-Pirani spincondition................................... 29 4.5 Spinning test particles in general relativity . . . . . . . . . . . . . . . . . . . 30 4.6 Other issues clarified in the course of this work . . . . . . . . . . . . . . . . 31 4.7 Outcome and future directions . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 The papers summarized and discussed 33 5.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Paper #1 — “Gravitoelectromagnetic analogy based on tidal tensors” . . . 34 5.2.1 The gravitational analogue of Maxwell’s Equations . . . . . . . . . . 38 5.2.2 Gravity vs Electromagnetism . . . . . . . . . . . . . . . . . . . . . . 38 5.2.3 Matching between tidal tensors . . . . . . . . . . . . . . . . . . . . . 41 5.2.3.1 Linearized Gravity . . . . . . . . . . . . . . . . . . . . . . . 41 5.2.3.2 Ultrastationary Spacetimes . . . . . . . . . . . . . . . . . . 42 5.2.4 Conclusion. Where does it stand in the context of the literature. . . 43 5.2.5 Erratum for Paper #1 . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4 Contents 5.3 Paper #2 — Reference frames and the physical gravito-electromagnetic analogy ...................................... 45 5.3.1 Linearized theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.2 Translational vs. Rotational Mass Currents . . . . . . . . . . . . . . 49 5.3.3 Conclusion ................................ 53 5.4 Paper #3 — Mathisson’s helical motions for a spinning particle: Are they unphysical? .................................... 54 5.4.1 Equations of motion for free spinning particles in flat spacetime. Mathisson’s helical motions. . . . . . . . . . . . . . . . . . . . . . . . 54 5.4.2 Center of mass. Significance of the spin condition. . . . . . . . . . . 56 5.4.3 Kinematical interpretation of the helical motions . . . . . . . . . . . 57 5.4.4 The misconception in the literature . . . . . . . . . . . . . . . . . . . 59 5.4.5 Dynamical Interpretation of the Helical Motions . . . . . . . . . . . 59 5.4.6 Conclusion ................................ 60 5.5 Paper #4 — Spacetime dynamics of spinning particles – exact gravitoelectromagnetic analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.5.1 Equations of motion and the exact analogies . . . . . . . . . . . . . 63 5.5.2 Dynamical implications of the symmetries of the tidal tensors . . . . 66 5.5.2.1 Radial motion in Schwarzschild spacetime . . . . . . . . . . 67 5.5.2.2 Equatorial motion in Kerr and Kerr-de-Sitter spacetimes . 68 5.5.3 Time projections of the forces and work done on a test particle . . . 72 5.5.3.1 Time components in test particle’s frame . . . . . . . . . . 74 5.5.3.2 Time components as measured by static observers . . . . . 74 5.5.4 Beyond pole-dipole: the torque on the spinning particle . . . . . . . 78 5.5.4.1 Electromagnetic torque . . . . . . . . . . . . . . . . . . . . 79 5.5.4.2 Gravitational torque . . . . . . . . . . . . . . . . . . . . . . 82 5.5.4.3 Summarizing with a simple realization . . . . . . . . . . . . 83 5.5.5 Conclusion ................................ 84 5.6 Paper #5 — Gravito-electromagnetic analogies . . . . . . . . . . . . . . . . 85 5.6.1 Analogy based on tidal tensors . . . . . . . . . . . . . . . . . . . . . 86 5.6.1.1 The analogy for differential precession . . . . . . . . . . . . 90 5.6.2 The exact analogy based on GEM fields . . . . . . . . . . . . . . . . 91 5.6.2.1 Inertial forces — “gravitoelectromagnetic (GEM) fields” . . 93 5.6.2.2 Gyroscope precession . . . . . . . . . . . . . . . . . . . . . 97 5.6.2.3 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.6.2.4 Relation with tidal tensor formalism . . . . . . . . . . . . . 103 5.6.2.5 Force on a gyroscope . . . . . . . . . . . . . . . . . . . . . 105 5.6.3 Conclusion ................................107 6 Communications on the material of this thesis 117 6.1 Invited Department talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Communications in International Conferences . . . . . . . . . . . . . . . . . 117 6.3 Other oral communications . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5 Contents 7 Further publications on the material of this thesis (conference proceedings) 121 6 1 Description of this document This is a “contextualization” document as required for a PhD work in physics in the form of article compilation. It is organized as follows. In Sec. 2 the articles produced in this thesis are listed; it contains five research papers, three of them published in international journals with peer review, the other two available on-line in the form of preprint, to be submitted soon for publication in refereed journals. In Sec. 3 I briefly review what initially motivated this work, was then the state of the art. Sec. 4, “Roadmap to the papers”, is a quick guide for the papers, explaining their motivation and aim, how they fit in the context of this work, and briefly describing their main outcomes. In Sec. 5 a more detailed summary and discussion of each paper is given. It is not meant to substitute the introduction and conclusion of the papers though, to which I refer the reader for an even more comprehensive account of the results in each paper, as well as a detailed literature review. In this section the main results of each paper are outlined, but the way they are presented does not always follow rigorously the original text (this is the case with the older papers #1 and #2). Paper #1 was where we first presented the analogy based on tidal tensors; but I have been using and developing this formalism since then, my knowledge advanced accordingly, in the light of the more recent papers #3, #4 and #5; thus I describe the same results from the perspective I have today, and refer the reader to the latest papers where the ideas in Paper #1 are put on firm grounds and further developed. In Paper #2 we studied the conditions under which a similarity between gravity and electromagnetism occurs, in view of astrophysical applications, and at an approximate level; we studied the dependence on the reference frame in particular detail. This issue has been revisited, with an exact approach, in the recent Paper #5, where the results in Paper #2 were generalized and understood at a more fundamental level; and I make use of that in the discussion of this paper. 7 2 Article compilation — index Papers published in international journals with peer review 1. L. Filipe Costa, C. Herdeiro, “Gravitoelectromagnetic analogy based on tidal tensors”, Physical Review D 78, 024021 (2008) 2. L. Filipe Costa, C. Herdeiro,“Reference frames and the physical gravito-electromagnetic analogy”, Proceedings of the International Astronomical Union (IAU) vol 5(Cambridge U. Press) p 31. Preprint [arXiv:0912.2146] (2009) 3. L. Filipe Costa, C. Herdeiro, J. Nat´ario, M. Zilh˜ao, “Mathisson’s helical motions for a spinning particle: Are they unphysical?”, Physical Review D 85, 024001 (2012) Papers published in the form of preprint, to be submitted to refereed journals 4. L. Filipe Costa, J. Nat´ario, M. Zilh˜ao, “Spacetime dynamics of spinning particles — exact gravito-electromagnetic analogies”, Preprint [arXiv:1207.0470] 5. L. Filipe Costa, J. Nat´ario,“Gravito-electromagnetic analogies”, Preprint [arXiv:1207.0465] Further publications on the material of this thesis exist in the form of conference proceedings, both in international journals (in Proceedings) and in book chapters; they are listed in Sec. 7, but not included in the compilation as, except for one of them, they do not contain significant new material compared with the five research papers above (they are essentially a different way of presenting the same results; they have a few new figures and equations, but I include them in this document in Sec. 5). Note that Paper #2 above, in spite of being a Proceedings paper, and indeed being associated with a conference (the International Astronomical Union Symposium), is indeed a research paper (original material is encouraged in this publication), and has been subject to scientific refereeing. 8 3 Introduction and motivation The material in this thesis can be split in two main topics: gravito-electromagnetic analogies (with their very broad range, and related subjects), and the dynamics of spinning multipole test particles in general relativity (to which we were led in the course of this work). The pursue for analogue models to describe gravity has a long history, and many different types of models have been proposed, both classical (based on e.g. fluid dynamics, electromagnetism, light propagation in dielectric media) and quantum (based on e.g. BoseEinstein condensates); for a review and references see [6]. In this work we have studied a special class of these analogies, the ones between the gravitational and electromagnetic interactions. The parallelism is in this case drawn between two relativistic fields theories (the two classical interactions), and (in the case of the the ones dubbed “physical” below) one compares effects “alike”. In this sense one might argue that these analogies have a stronger physical component than (most) other analogue models. It is my view that they are interesting not only for providing intuition and a familiar formalism to treat otherwise more complicated gravitational problems (which is a common goal to other analogue models), but also (and especially) for the prospect of yielding a formalism allowing for a direct comparison of the two interactions, from which one might learn fundamental aspects about both of them. When we started this work, the state of the art in the field of the gravito-electromagnetic (GEM) analogies, known as “gravitoelectromagnetism”, was that there were several different analogies in the literature, but the relation between them was unclear. The best known ones were the analogy between linearized gravity and electromagnetism in Lorentz frames (based on suitably defined gravitational 3-vector fields, the “GEM fields”, that mimic the electromagnetic ones), e.g. [7, 8, 9, 10, 11, 12], and the analogies between the Maxwell and the Weyl tensor (namely their decomposition on electric and magnetic parts, the scalar invariants they form, and the Maxwell-like “higher order field equations”), e.g. [31, 32, 34, 37, 46]; there were also exact analogies based on GEM fields, e.g. [18, 19, 20, 22, 23, 25], and an exact mapping, via the Klein Gordon equation, between ultrastationary spacetimes and magnetic fields in curved spacetimes [28]. Not only the relation between the different approaches had not been established, as they seemed to lead to differing views, as for instance spacetimes which were purely magnetic from the point of view of the analogies based on GEM fields, turned out to be purely electric from the point of view of the Weyl tensor (that is the case, for instance, of the Godel and Heisenberg spacetimes). And there were issues within each of them, such as the lack of a consistent physical interpretation for the magnetic parts of the Riemann and Weyl tensors, or the regime of validity of the analogies drawn in linearized theory. I briefly review these analogies in Sec. 3.1 below. 9 3 Introduction and motivation where curlAαβ ≡µν (αAβ)ν;µ, and the index notation hµνistands for the spatially projected, symmetric and trace free part of a rank two tensor (cf. definitions in [34]): Ahµνi≡hα (µhβ ν)Aαβ −1 3hµνhαβAαβ . The analogy with Maxwell’s equations is closer if one considers the case of vacuum, and takes the linear regime; in this case Eqs. (3.33)-(3.36) become the equations in the right column of Table 3.1, originally found by Matte [31] (see also [32, 33]), which are formally similar to Maxwell’s equations in Lorentz frames, only with the electric and magnetic parts of the Riemann tensor Eαβ =RαµβνUµUν,Hαβ =?RαµβνUµUνin the place of the electromagnetic fields (note that, in vacuum, Rαβγδ =Cαβγδ). Table 3.1: Analogy between Maxwell’s equations in Lorentz frames and Matte’s equations. Electromagnetism Linearized Gravity Maxwell’s Equations Matte’s Equations Ei ,i = 0 (3.1.1a) Eij ,i = 0 (3.1.1b) Bi ,i = 0 (3.1.2a) Hij ,i = 0 (3.1.2b) iklEl,k =−∂Bi ∂t (3.1.3a) iklEj l,k =−∂Hij ∂t (3.1.3b) iklBl,k =∂Ei ∂t (3.1.4a) ilkHj l,k =∂Eij ∂t (3.1.4b) 3.1.4 Some then open questions to be addressed As mentioned above, the limit of applicability of the usual analogies based on the linearized theory was unclear; new approaches allowing for a transparent assessment of the actual physical similarities between linearized gravity and electromagnetism, and under which precise conditions they occur, were for this reason needed. The physical content of the analogy in Sec. 3.1.3 was an unanswered question in the literature, and that is mainly due to the fact that the magnetic part of the Weyl (and Riemann) tensor was not well understood (the electric part Eµν was reasonably well understood due its role in the geodesic deviation equation); in the literature concerning this approach its physical significance was either presented as an open question [42, 44, 43], or given inconsistent interpretations. It was suggested in some works to be associated with rotation [37, 38, 34, 39, 40] and gravitational radiation [43, 48, 49, 50, 51, 40]. However, immediately contradictions arise [37, 38, 34]: there are many known examples of rotating spacetimes where the magnetic part of the Weyl tensor vanishes; amongst them is the notorious example of the G¨ odel Universe. It is also clear that gravitational waves cannot be the sole source for Hµν, since the latter is generically non-vanishing in most stationary spacetimes. 16 3 Introduction and motivation The relationship between the different analogies (namely, the relationship between the electric and magnetic parts of the Weyl tensor, and the GEM fields) was another issue in need for a clarification, as they lead to seemingly different, even opposite, views of the same problems. Consider, for example, the Heisenberg spacetime (the same conclusions would be reached with e.g. the Godel Universe; I choose Heisenberg’s because it is possible to analyze it both with exact and linearized theories), whose line element is given by ds2=dt −ar2dφ2+dr2+r2dφ2+dz2. Both according to the approach based on linearized theory of Sec. 3.1.1, and with the exact approach of Sec. 3.1.2, this spacetime has zero gravitoelectric field: ~ G=~ EG= 0, and a non-zero (uniform) gravitomagnetic field; ~ BG=−a~ezin the definitions of Sec. 3.1.1, or ~ H= 2a~ezin the definitions of Sec. 3.1.2. However according to the analogy based on the Weyl tensor in Sec. 3.1.3, this is a purely electric spacetime! Indeed, for the observers ui= 0 (the same observers measuring the GEM fields above), the magnetic part of the Weyl (and Riemann) tensor vanishes:Hαβ =Hαβ = 0, and it is the electric part of the Weyl tensor that is non-zero:Eαβ 6= 0 (the non-zero components are Err = 2a2/3, Eφφ = 2r2a2/3, Ezz =−4a2/3). And when observers exist for which Eαβ 6= 0,Hαβ = 0, the spacetime is in this framework classified as purely electric, see [45, 46]. 3.2 Spinning multipole particles in general relativity The classical equations of motion for spinning charged particles (possessing only charge and intrinsic magnetic dipole moment) under the action of electromagnetic fields were first derived in the framework of Special Relativity by Frenkel [55] (see also [131]), and subsequently4by Bhabha-Corben [56, 82, 83] (for particles with both electric and magnetic moments) and Weyssenhoff-Raabe [76]. Later, more rigorous treatments were put forth by Dixon [59] and Gralla et al [64]; in [64] particles with electric and magnetic dipole moments are considered, and [59] gives equations valid to arbitrary order in the multipole expansion. In the presence of a gravitational field, the equations of motion for spinning multipole particles were first derived by Mathisson [61], for zero electromagnetic field, and accurate to quadrupole order; these equations were then re-derived by Papapetrou [79], who carried out a derivation exact (in the external field) at each step, but for pole-dipole particles only. Tulczyjew [62], Taub [117], Dixon [60] and Souriau [118, 119], carried out derivations covariant at each step, again for pole-dipole. The latter two, unlike the former, include also the electromagnetic field. Equations with both electromagnetic and gravitational fields valid first to quadrupole order [81], and then to arbitrary order [106], were given by Dixon. Some recent treatments re-derive these equations; Gralla et al [66] obtained equations to quadrupole order in Dixon’s scheme (based on the “generalized Killing vectors” of [81]); 4There were also the famous treatments by Thomas [128] and by Bargmann-Michel-Teledgi [127]; these are not covariant treatments, however, and only take into account the particle’s spin in the equation for the spin evolution, and not in the force equation. 17 3 Introduction and motivation and Nat´ario [65], by a totally independent method (based on a Lagrangian approach to the problem of the Euler top in General Relativity) derived, for arbitrary dimension, the equations for pole-dipole particles in a gravitational field. In-between these works there are treatments on free spinning pole-dipole particles in flat spacetime, most notably the work by Mathisson [74] where the famous helical motions were discovered, further worked out and re-derived by Weyssenhoff-Raabe [75, 76], and the important treatment by M¨ oller [78], where first light was shed on the issue of the spin supplementary condition5, the helical motions, and the relation with the problem of defining a center of mass for spinning bodies in relativity. The treatments are all very different (and lead also to seemingly different results, which was one of the issues we needed to clarify in our work as explained below), some more rigorous than the others, thus I cannot go through the details of each of them; so below I very briefly outline the main ideas of the multipole scheme for test bodies in general relativity, and I take the physically more consistent viewpoint of extended test bodies (not point particles), and describe them in terms of a covariant multipole expansion. Thus I follow a scheme that is closer to Dixon’s, e.g. [59, 60], yet simplified and already using our formalism and notation of papers #3 [3] and #4 [4]. A test particle is described by the moments of its charge current density 4-vector jα(the“electromagnetic skeleton”), and the moments of Tαβ (the “gravitational skeleton” [61]). In flat spacetime (and in Lorentz coordinates) they are, respectively, Jα1...αnµ(τ)≡ˆΣ(τ,U) rα1...rαnjµwσdΣσ; (3.37) tα1...αnµν(τ)≡ˆΣ(τ,U) rα1...rαnTµνwσdΣσ.(3.38) These moments are taken with respect to a reference worldline zα(τ), of proper time τand (unit) tangent vector Uα≡dzα/dτ, and a hypersurface of integration Σ(τ, u). Σ(τ, u)≡Σ(z(τ), u) is the spacelike hypersurface generated by all geodesics orthogonal to some time-like vector uαat the point zα(τ); rα≡xα−zα(τ), where {xα}is a chart on spacetime; dΣγ≡ −uγdΣ, and dΣ is the 3-volume element on Σ(τ, u). wαis a vector such that displacement of every point by wγdτ maps Σ(τ) into Σ(τ+dτ), see [115, 59, 60] for more details. In a strongly curved spacetime (or if one uses a non-rectangular coordinate system) one needs to refine the expressions above in terms of bitensors (see the formulations in [60, 81, 105, 106, 66]), as not only rα=xα−zα(τ) is not a vector6(only to first order), but also the integrals (3.37)-(3.38) above would make no sense, as they would amount to adding tensor components at different points. If only lower order moments are to be kept, and if the gravitational field is not too strong, one can still to a good approximation 5One can say that this important work by M¨ oller laid the foundations for our contribution, Paper #3 [3]. 6In the general case of a curved spacetime, the distance between two points is the length of the geodesic connecting them; rαis not even (exactly) a vector, and the closest notion to a separation vector is the bitensor −σκ(x, z), which is the vector tangent to the geodesic at zαwhose length equals that of the geodesic connecting the two points [60, 81]. Also one must distinguish a coordinate system at xαfrom the one at zα, as the basis vectors change from point to point in a curved spacetime. 18 3 Introduction and motivation [81, 140, 116, 4] set up a locally nearly Lorentz frame and compute the moments from the expressions (3.37)-(3.38). In particular, to dipole order (which is the case we are mostly interested in), the bitensors are by definition redundant, as the approximation amounts to considering Tαβ and jαnon-vanishing only in a very small region around zα(τ), so that only terms linear in rare kept; to first order, rαis a vector (see e.g. Eq. (7) of [96]), and √rαrαthe distance between two points. Also, to first order, spacetime can always be taken as flat, thus these integrals are meaningful mathematical operations and do indeed define tensors, just like in flat spacetime. 3.2.1 Equations of motion for pole-dipole particles This is the simplest case next to the monopole particle (whose sole equation of motion is the Lorentz force), and it is perhaps surprising that the problem of the equations of motion for it is still not well understood, with different methods and derivations leading to different versions of the equations, and the relation between them not being clear. And that it is the electromagnetic (not the gravitational) field that has been posing more problems. The many existing approaches are very different, so I will state the general problem in the covariant multipole scheme of the previous section. Truncating the expansion to dipole order amounts to keep only two moments of Tαβ:tαβ,tαβγ, and two moments of jα:Jα,Jαβ. In the integrals (3.37)-(3.38), to dipole order, wα≃Uα. The equations of motion will follow from the charge conservation jα ;α= 0, and from the conservation of the total energy-momentum tensor [59, 81], (Ttot)αβ ;β= 0 ⇔Tαβ ;β=Fαβjβ,(3.39) where Fαβ is the Maxwell tensor of the external (background) electromagnetic field and Tαβ the energy-momentum tensor of the particle. A straightforward solution of this apparently simple problem, in term of quantities whose physical meaning is clear at each step, is yet to be given in the literature. The following form of the equations is popularized in the literature, both concerning general relativistic treatments, e.g. [81, 106, 66] DPα dτ =qFαβUβ+1 2Fµν;αQµν −1 2Rα βµνSµν ,(3.40) DSαβ dτ = 2P[αUβ]+ 2Qθ[βFα] θ,(3.41) or, for the case Rα βµν = 0, in special relativistic treatments, e.g. [56, 83, 59]. q≡´ΣjαdΣα is the charge, Qαβ is the electromagnetic dipole moment tensor, Qαβ = 2d[αUβ]+αβγδµγUδ,(3.42) where dαand µαare the electric and magnetic dipole moments measured by the observer of 4-velocity Uα(i.e., comoving with the reference worldline). These can be written in 19 3 Introduction and motivation terms of the moments Jαβ defined in Eq. (3.37), taking uα=Uα: dα=−JαβUβ,(3.43) µα=1 2α βγδUδJβγ .(3.44) But other (less well known) version of these equations exists in the literature [60, 77]: DPα dτ =qFα βUβ+1 2Fµν;αµµν −1 2Rα βµνSµνUβ+Fα γ;βUγdβ+Fα β Ddβ dτ ,(3.45) DSαβ dτ = 2P[αUβ]+ 2µθ[βFα] θ+ 2d[αFβ] γUγ,(3.46) where µαβ =αβγδµγUδ. Pαand Sαβ are taken to be the momentum and angular momentum of the particle. But clearly one cannot be dealing with the same quantities in Eqs. (3.40)-(3.41) and (3.45)- (3.46), as the equations would not be compatible. This is the case in particular with Pα (in the case of Sαβ, to dipole order, it makes no difference, see Appendix A of Paper #4 [4]); contracting (3.41) with Uα, one obtains the expression Pα=mUα−DSαβ dτ Uβ−α θµσdθBµUσ+α θλτ µθEλUτ(3.47) whereas contracting (3.46) with Uαleads to Pα=mUα−DSαβ dτ Uβ+α θλτ µθEλUτ,(3.48) so indeed these expressions correspond to different quantities. We argue in Paper #4 [4] that it is expression (3.48) that is the physical momentum momentum of the particle, understood as the integral on a spacelike hypersurface of the particle’s energy-momentum tensor Pα=ˆΣ(τ,U) TαβdΣβ.(3.49) As for the expression (3.47), which hereafter I will denote by Pα Dix (“Dixon’s momentum”), we argue, following [110], that it is a part of the canonical momentum Pα can =Pα Dix +qAα associated to the Lagrangian of the system. In many treatments it is simply not clear what Pαor Pα Dix is. In Dixon’s treatments [81, 106, 59] it is clear from the beginning that Pα Dix is not (3.49), as electromagnetic terms are explicitly added to its definition, see e.g. Eq. (5.1) of [81]. The problem in this case is in the physical interpretation, as Pα Dix is nevertheless taken therein as the physical momentum, which leads to inconsistencies, as is explained in detail in Appendix A of Paper #4 [4]. Therein the relationship between the two formulations, and how to obtain one from the other, is clarified. 20 3 Introduction and motivation The spin supplementary condition. — Many other issues were unclear in the literature concerning these equations. Firstly, in order for Eqs. (3.45)-(3.46) to be equations of motion, the reference worldline zα(τ) must be chosen as some representative point through the body (so that its tangent Uαis the body’s 4-velocity). Even in the case Fαβ = 0, the system above is undetermined, as it has 3 more unknowns than equations. The system can be closed7by imposing a condition of the type Sαβuβ= 0, for some time-like unit vector field uα, which effectively kills off 3 components of the angular momentum. The role of the condition Sαβuβ= 0 is to specify the representative point of the body; more precisely, to choose it as being the center of mass as measured by some observer of 4-velocity uα. The choice of the vector field uαhas been subject of a long debate in the literature, sometimes put in terms of which are the “correct” and the “wrong” conditions for each type of particle (since its status as a mere gauge choice is still not generally well understood; see introduction of [87] for a comprehensive review). The three best known ones are the Corinaldesi-Papapetrou condition [89], where uα∝∂/∂t corresponds to the static observers in Schwarzschild spacetime (but can be easily generalized to other stationary spacetimes), the Frenkel-Mathisson-Pirani condition [55, 61, 80] (hereafter Pirani’s condition, as it is best known), where uα=Uα(i.e., the center of mass is computed in its own rest frame), and the Tulczyjew-Dixon condition [62, 60, 81], where uαis taken parallel to Pα. It is the point of view in most of the literature that it is preferable to have equations of motion depending not on a center of mass measured by some particular observer, but instead one that is defined only in terms of properties “intrinsic” to the particle. The latter two conditions accomplish that, and are the most widely used (especially the Dixon-Tulczyjew); the differences between the two have been discussed, and again subject of a number of misunderstandings (for a review and clarification, I refer the reader to Appendix C.2 of Paper 4 # [4]). Especially poorly understood is the Mathisson-Pirani condition, which happens to be the most important one in the context of this work, as the exact gravito-electromagnetic analogies for spinning particles require this condition to be used. This condition is degenerate, and allows for exotic helical solutions, even for a free particle in flat spacetime (in addition to the uniform straightline motion, which is also a solution). In the zero 3-momentum frame (Pi= 0), these are circular motions of radius R=vγ2S m where S=pSαβSαβ/2, m=−PαUαis the particle’s “proper mass”, and γ= 1/√1−v2; for more details see Sec. 5.40 below. The motions have been dubbed unphysical (see e.g. [76, 77]), due to the belief that R, for a given particle, can be arbitrarily large, based on the fact that γcan be arbitrarily large. That is, the representative point of a finite free body might move along circular trajectories with any radius, which is obviously contradicted by experiment and seemingly would invalidate the interpretation of this spin condition as a center of mass choice. It was our goal in Paper #3 [3] to clarify the misconception at the origin of these assertions, see Sec. 5.4.4 below, demystify the helical motions, and prove 7In the case Fαβ 6= 0, one needs also to provide evolution laws for the moments dαand µα. 21 3 Introduction and motivation that there is nothing wrong with this spin condition. The hidden momentum. — Eq. (3.48) above tells us that the particle’s momentum Pαis not parallel to the 4-velocity Uα— the particle is said to possess“hidden momentum”. This was another issue that was not well understood, and yet to be discussed in the framework of General Relativity when we started this work. The reason for the denomination hidden momentum is as follows: since Pαis not parallel to Uα, in the center of mass frame (where ~ U= 0) there will be a non-vanishing spatial momentum ~ P6= 0; since in this frame the body is, by definition, at rest, this momentum must be hidden somehow. The third term in (3.48) is what we dub the “electromagnetic” hidden momentum. It is a still not well known feature of relativistic electrodynamics, despite its discovery by Schockley & James [90] dating back from the 60’s, and having since been discussed in number of papers, e.g. [91, 90, 92, 93]. It is originated by the action of the electromagnetic field on the particle, and it is for that reason that we dub it “electromagnetic”; but it should be noted that it is purely mechanical in nature, see e.g. the simple physical model in Fig. 9 of [100] (see also Appendix D of Paper #4 [4]). The second term in (3.48) is what we dub “inertial” hidden momentum, a concept that did not exist yet when we started this work. It was introduced only recently by Gralla-Harte-Wald in [66] (where it was dubbed “kinematical” hidden momentum), which was also the first work where the problem of the non-parallelism of Uαand Pαwas addressed in a fully relativistic approach. This hidden momentum originates from the spin supplementary condition (i.e., from the field vector uαwith respect to which the center of mass is computed). We further worked out these ideas in papers #3 (where we shown that the hidden momentum explains the dynamical consistency of Mathisson’s Helical motions) and #4, and a paper [120] is now in preparation where an exact formulation in terms of the GEM inertial fields of Sec. 3 of Paper #5 [5], yielding the hidden momentum of the particle when its center of mass is computed in an arbitrary frame (that is, applying to arbitrary spin conditions), is presented. Finally, it should be noted that the hidden momentum (and now in particular the electromagnetic one) is not a feature one only needs to care about in sophisticated relativistic treatments; indeed it affects the textbook expressions for the forces exerted on a particle with electromagnetic moments. Take the case of the force exerted on a magnetic dipole; there has been a long debate in the literature concerning the correct equation for this force, and how it changes according to the two concurrent dipole models (the current loop, and the pair of monopoles; the result will differ because the former, but not the later, possesses hidden momentum); see e.g. [91, 122, 123, 124, 125]. I will not go through the details of each of these works (some of them is fair to say are not very rigorous, and contain even some mistakes). For a purely magnetic dipole (dα= 0, q= 0) in flat spacetime, and in the particle’s rest frame, the space part of Eq. (3.45) above reads ~ FEM =D~ P dτ =∇(~ B·~µ),(3.50) where ∇i(~ B·~µ)≡Bj,iµj. One is used to see this expression in textbooks, e.g. [99], and 22 3 Introduction and motivation it is correct, for a magnetic dipole taken as a small current loop. But what many authors are not aware of is that this is not m~a, because such particle possesses hidden momentum ~ Phid =~µ ×~ E; c.f. Eq. (3.48). And the force Eq. (3.50) above is the derivative of the total momentum of the particle (including the hidden one). The acceleration equation reads m~a =~ FEM −D~ Phid dτ =∇(~ B·~µ)−D dτ (~µ ×~ E)=(~µ ·∇)~ B−D~µ dτ ×~ E(3.51) which is in agreement with [91, 124, 125]. Here (~µ·∇i)~ B≡Bi,jµj, and in the last equality I used Maxwell vacuum equation ∇× ~ B=∂~ E/∂t. In some textbooks, e.g. [126], we find the expression ~ F= (~µ · ∇)~ Bfor the force on a magnetic dipole; it is thus not true in general. Such expression can only yield either D~ P/dτ for vacuum electrostatics (so that ∇× ~ B= 0 ⇒(~µ ·∇)~ B=∇(~ B·~µ)), or m~a for the case of a fixed dipole D~µ/dτ = 0. 3.2.2 Equations of motion to quadrupole order The relativistic equations of motion for spinning particles, to quadrupole order, were given first by Mathisson [61] (see also [97]) for purely gravitational fields; then Dixon derived the equations in the presence of an electromagnetic field, first in flat spacetime [59], and later for curved spacetime [81, 106]. Dixon’s treatments in [59, 106] are actually valid to arbitrary multipole order, and based on exact multipole moments, which in the case of curved spacetime require the use of bitensors. The theory of bitensors is given in [129]; see also the brief reviews in [81, 105]. As explained above, to quadrupole order, and to a good approximation, the bitensors may be dropped and one may define the moments in a way similar to the case of flat spacetime. The equations are [59, 106, 66], DPα Dix dτ =qFαβUβ+1 2Fµν;αQµν −1 2Rα βµνSµν +1 3QβγδFγδ;βα −1 6JβγδσRβγδσ;α,(3.52) DSαβ can dτ = 2(PDix)[αUβ]+ 2Qθ[βFα] θ +2m[αρµFβ]µ;ρ+4 3Jµνρ[αRβ] ρµν ,(3.53) where mαβγ ≡4Q(αβ)γ/3, Qαβγ is an electromagnetic quadrupole moment, and Jαβγδ a quadrupole moment of Tαβ, see Sec. VI of Paper #4 [4] for their definitions. The question mark is on the physical interpretation of these equations. In the literature these are portrayed as giving the “force” and the “torque” [81, 66] on the spinning particle, up to quadrupole order. We argue that they do not yield the actual force and torque on the body, because these are not equations for the physical momentum Pα, defined by Eq. (3.49), and angular momentum Sαβ, defined by Sαβ ≡2ˆΣ(τ,U) r[αTβ]γdΣγ. 23 3 Introduction and motivation For this reason, following the discussion of the previous section, I denote the “momentum” in Eqs. (3.52)-(3.53) by Pα Dix, and the angular momentum by Sαβ can, because it is argued in to be the “canonical” angular momentum [110] (for more details see Sec. 5.5.4.1). Pαis not the same as Pα Dix when an electromagnetic field is present, which is already manifest to dipole order as discussed above; and to quadrupole order, Sαβ can cannot also be taken as Sαβ, as shown by Eq. (5.93) below. Moreover, unlike the issue with the definition of momentum, which does not affect the main results in Paper #4 [4] as we deal mostly therein with magnetic dipoles (for which Pα Dix =Pα), this one affects any spinning charged body, and also the interpretation of the work of the force, and of the proper mass variation, that we obtain to dipole order (as explained in Sec. 5.5.4.1 below). On top of that, unlike the equations to dipole order, for which the alternative version (3.45)-(3.46), in terms of what we call the physical momenta, is known, in the quadrupole case, by contrast, Eqs. (3.52)-(3.53) above (given in e.g. [59, 81, 106, 66]), were the only ones available in the literature that take into account the electromagnetic field (in the framework of a covariant multipole approach). The inadequacy of Eq. (3.53), taken as a physical torque, becomes clear if one considers a spherical, uniformly charged body in flat spacetime. In that case (in vacuum) the term m[αρµFβ]µ;ρvanishes (see Sec. VIA of Paper #4 for details); all that remains is the dipole term 2µθ[βFα] θ, present in Eqs. (3.46) or (3.41), which does not change the magnitude of Sαβ can (if one also assumes µαβ =σSαβ can, as done in [81, 66]), and moreover vanishes if the magnetic field Bαis aligned with Sα can. That contradicts what we know from elementary arguments: if an electric field with a curl is present, it must torque the charged body; that is the case when the magnetic field is time-dependent, by Faraday’s law of induction ∇ × ~ E=−∂~ B/∂t. That torque has actually been computed in some non-relativistic treatments [107, 108, 109], where the following expression is presented for the torque exerted on a spinning charged ball in an electromagnetic field, e.g. Eq. (1) of [107]: ~τ =~µ ×~ B−σI d~ B dt . Here σ=q/2mis the gyromagnetic ratio, and Ithe moment of inertia of the sphere about an axis passing through its center. The first term is the usual torque on the magnetic dipole, the second is the torque due to the induced electric field, which we dub ~τind. If the sphere is uniform, σI =qα α/3, where qαβ is the charge quadrupole, given by definition (5.94) below. Thus ~τind is manifest to quadrupole order. Due to its importance in the context of this work — since a fundamental difference revealed in the tidal tensor formalism put forth in Paper #1 [1] is the absence of gravitational effects analogous to electromagnetic induction — this was a problem in need to be addressed: obtain the equations for the physical quadrupole torque, single out the torque due to electromagnetic induction and clarify how it fits in Dixon’s multipole scheme. That is done in Paper #4 [4], where, as discussed in Sec. 5.5.4.1 below, it is shown that indeed it is the part of the torque ignored in Eq. (3.53) (i.e., the space part of −DS0αβ/dτ, see notation therein) that encodes the torque due to Faraday’s induction on an arbitrary charged body (not only spherical). 24 3 Introduction and motivation 3.2.3 The gravito-electromagnetic analogies for spinning particles in the literature Analogies between the equations of motion for gyroscopes in a gravitational field and magnetic dipoles in an electromagnetic field have been unveiled in different forms throughout the years. This is the case for both the force equation (center of mass motion) and the spin evolution equation of these test particles in external fields. Below I shall describe what was the state of the art prior to our paper devoted to the subject, Paper #4 [4]. The analogy for the force was first found by Wald [10] in the framework of linearized theory, who showed that the gravitational force exerted on a spinning pole-dipole test particle (hereafter a gyroscope), whose center of mass is at rest in a stationary field, takes the form (3.5), analogous to the force on a magnetic dipole. The analogy was later cast in an exact form, Eq. (3.13), by Nat´ario [19], using the exact gravitoelectromagnetic (GEM) inertial fields from the so-called 1+3 “quasi-Maxwell” formalism that I briefly review in Sec. 3.1.2. The force was seen therein to consist of an electromagnetic-like part in the form ˜ ∇(~ H·~ S)−~ S(˜ ∇· ~ H), plus a term (~ S·~ H)~ Ginterpreted as the weight of the energy of the gravitomagnetic dipole; the limit of validity of the analogy was thereby extended to arbitrarily strong stationary fields and when the gyroscope’s worldline is tangent to any time-like Killing vector field (it comprehends e.g. circular trajectories with arbitrary speed in axisymmetric spacetimes). And in Paper #1 [1] we put forth the exact, covariant and fully general analogy relating the two forces, made explicit in the tidal tensor formalism. The analogy between the so-called “precession” of a gyroscope in a gravitational field and the precession of a magnetic dipole under the action of a magnetic field was noticed long ago, in the framework of linearized theory, see Eq. (3.4), by a number of authors [140, 147, 7, 141]. The analogy was later cast in an exact form, Eq. (3.12), in the framework of the GEM inertial fields, e.g. [140, 25, 19]; it is not covariant, holding only in a specific frame comoving with the particle, but, in the formulation in [25] (see Paper #4 [4] for details), the test particle can be moving with arbitrary velocity in an arbitrary field. Finally, it had recently been found by Gralla-Harte-Wald [66] an analogy, at an approximate level, between what we call the“inertial”hidden momentum Pα hidI (dubbed“kinematical” in [66]), under Dixon-Tulczyjew spin condition SαβPβ= 0, and the electromagnetic hidden momentum Pα hidEM. The approximate expressions are ~ PhidI ≈ −(1/M)~ S×~ F(with ~ F=D~ P/dτ), and ~ PhidEM ≈~µ ×~ E(we show in Paper #4 [4] that the analogy can be cast in an exact form, using the Mathisson-Pirani condition instead). 25 4 Roadmap to the papers is a lot to be learned from a comparative study of the two interactions; and 2) the analogies are useful from a practical point of view, as they provide intuition and a familiar formalism to treat otherwise more complicated gravitational problems. Indeed, the formalisms developed, by being exact and general, provide a powerful set of tools that allows to study gravitomagnetic effects in arbitrarily strong fields, and also add new phenomena to the “gravitoelectromagnetism” category — a major addition being the motion of spinning pole-dipole particles in General Relativity, which can be exactly described in the framework of gravito-electromagnetic analogies, as we have shown in [4]. This is just the first major application of the formalism, on which we plan to build on (a first glimpse of the application of the tidal tensor formalism to the study of gravitational radiation, and the physical insight it brings, is given in [5]; and a paper on the use of the curvature scalar invariants will soon be published [30]), with many further applications being planned for the coming years. 32 5 The papers summarized and discussed 5.1 Notation and conventions 1. Signature and signs. We use the signature −+ ++; αβσγ ≡√−g[αβγδ] denotes the Levi-Civita tensor, and we follow the orientation [1230] = 1 (i.e., in flat spacetime, 1230 = 1). ijk ≡ijk0is the 3-D alternating tensor. We use the convention for the Riemann tensor: Rα βµν = Γα βν,µ −Γα βµ,ν +.... 2. Sometimes we use the abbreviation αβγ ≡αβγδUδ, where Uαis the 4-velocity of the test particle’s CM. 3. Greek letters α, β, γ, ... denote 4-D spacetime indices, Roman letters i, j, k, ... denote 3-D spatial indices. Following the usual practice, sometimes we use component notation Tαβ to refer to a tensor T. We use arrows for 3-vectors ~v except in Paper #2 where bold fonts vare used instead. In papers #3-#5 we use bold fonts to denote tensors T(including 4-vectors U) . 4. Time and space projectors. (>u)α β≡ −uαuβ, (hu)α β≡uαuβ+gα βare, respectively, the projectors parallel and orthogonal to a unit time-like vector uα; may be interpreted as the time and space projectors in the local rest frame of an observer of 4-velocity uα.hαidenotes the index of a spatially projected tensor: Ahαiβ... ≡(hu)α βAµβ.... 5. Tensors resulting from a measurement process. (Au)α1..αndenotes the tensor A as measured by an observer O(u) of 4-velocity uα. For example, (Eu)α≡Fα βuβ, (Eu)αβ ≡Fαγ;βuγand (Eu)αβ ≡Rανβνuνuµdenote, respectively, the electric field, electric tidal tensor, and gravito-electric tidal tensor as measured by O(u). Analogous forms apply to their magnetic/gravitomagnetic counterparts. For 3-vectors we use notation ~ A(u); for example, ~ E(u) denotes the electric 3-vector field as measured by O(u) (i.e., the space part of (Eu)α,written in a frame where ui= 0). When uα=Uα(i.e., the particle’s CM 4-velocity) we drop the superscript (e.g. (EU)α≡Eα), or the argument of the 3-vector: ~ E(U)≡~ E. 6. Electromagnetic field. The Maxwell tensor Fαβ and its dual ?Fαβ decompose in terms of the electric (Eu)α≡Fα βuβand magnetic (Bu)α≡?Fα βuβfields measured by an observer of 4-velocity uαas Fαβ = 2u[α(Eu)β]+αβγδuδ(Bu)γ; (5.1) ?Fαβ = 2u[α(Bu)β]−αβγσuσ(Eu)γ.(5.2) 33 5 The papers summarized and discussed 7. Static observers. In stationary, asymptotically flat spacetimes, we dub “static observers” the rigid congruence of observers whose worldlines are tangent to the temporal Killing vector field ξ=∂/∂t; may be interpreted as the set of points rigidly fixed to the “distant stars” (the asymptotic inertial rest frame of the source). For the case of Kerr spacetime, these correspond to the observers of zero 3-velocity in Boyer-Lindquist coordinates. This agrees with the convention in e.g. [73]. Note however that the denomination “static observers” is employed with a different meaning in some literature, e.g. [94], where it designates rigid, vorticity-free congruences (existing only in static spacetimes). In the case of the electromagnetic systems in flat spacetimes, by static observers we mean the globally inertial rest frame of the sources. 8. GEM and GEM fields. GEM is the acronym for “gravitoelectromagnetism”. By “inertial GEM fields”, we mean the fields of inertial forces that arise from the 1+3 splitting of spacetime: the gravitoelectric field, which plays in this framework a role analogous to the electric field of electromagnetism, and the gravitomagnetic field, analogous to the magnetic field. Different notations and conventions are used in the literature for these fields; and it is also the case for the papers in this compilation. In papers #1 and #2, the gravitoelectric and gravitomagnetic fields were denoted, respectively, by ~ EGand ~ BG, and defined (following e.g. [12]) such that the geodesic equation in linearized stationary fields reads, in its space components, d2x dt2=−~ EG−2~v ×~ BG, cf. Eq. (3.2) above. In papers #3-#5, we denoted the exact GEM fields by ~ Gand ~ H, and used a convention (following e.g. [19]) such that the exact geodesic equation, for stationary fields, has space components ˜ D~ U dτ =Uˆ 0Uˆ 0~ G+~ U×~ H; cf. Eq. (3.11). In linear regime, the correspondence between the two definitions is ~ G≈ −~ EG;~ H≈ −2~ BG. Accordingly, different conventions for the GEM “potentials” were used. Φ,~ A,Θij of papers #1 and #2 correspond to, respectively, −Φ,−~ A/2,−ξij of paper #5. In what follows I will discuss each paper using its original notation. 5.2 Paper #1 — “Gravitoelectromagnetic analogy based on tidal tensors” In this paper we first presented an exact analogy between gravity and electromagnetism that stems from the tidal dynamics of both theories, based on mathematical objects that we dubbed “tidal tensors”. 34 5 The papers summarized and discussed We were interested in comparing the two interactions in a way as transparent as possible; the rationale behind our approach was that such comparison should based on physical, covariant forces that are present in both theories. The electromagnetic Lorentz force has no physical counterpart in gravity,as monopole point test particles in a gravitational field move along geodesics, without any real force being exerted on them. Also the spin vector of an ideal gyroscope in a gravitational field undergoes Fermi-Walker transport (i.e., follows the compass of inertia) without any real torque being exerted on it. Thus the analogies between the Lorentz force and the geodesic equation, and between the precession of a magnetic dipole and the “precession” of a gyroscope, which are well known from the linearized theory approaches reviewed in Sec. 3.1.1, and exist also in exact versions (reviewed in Sec. 3.1.2), are not suited for our purpose. They draw a parallelism between the electromagnetic fields Eα, Bαand fields of inertial forces Gα,Hα(fictitious forces, that vanish in locally inertial frames), i.e., they compare physical forces from one theory, with reference frame artifacts from the other. Tidal forces, by their turn, are covariantly present in both theories, and their mathematical description in terms of “tidal tensors” is the basis of this approach. Tidal forces manifest themselves in two basic effects: the relative acceleration of two nearby monopole test particles, and in the net force exerted on dipoles. These notions of multipole moments are the ones given in Sec. 3.2, i.e., the multipole moments of the current density vector jα= (ρc,~ j) in electromagnetism, and the moments of the energy momentum tensor Tαβ in gravity. From the latter, only the moments of the projection Jα=−TαβUβ(i.e., the moments of the mass/energy 4-current density) have an electromagnetic counterpart. Monopole particles in the context of electromagnetism are those whose only non-vanishing moment is the total charge; dipole particles are particles with nonvanishing electric and magnetic dipole moments (i.e., respectively, the dipole moments of ρcand ~ j). Monopole particles in gravity are particles whose only non-vanishing moment of Tαβ is tαβ (see definition (3.38) above), of which only the momentum Pα=−tα βUβcontributes to the equations of motion; they correspond to the usual notion of point test particles, which move along geodesics. There is no gravitational analogue of the intrinsic electric dipole, as there are no negative masses; but there is an analogue of the magnetic dipole moment, which is the“intrinsic”angular momentum (i.e. the angular momentum about the particle’s center of mass), usually dubbed spin vector/tensor. A particle possessing only pole-dipole gravitational moments corresponds to the notion of an ideal gyroscope. We thus have two physically analogous effects suited to compare gravitational and electromagnetic tidal forces: worldline deviation of nearby monopole test particles, and the force exerted on magnetic dipoles/gyroscopes. An exact gravito-electromagnetic analogy, summarized in Table 5.1 , emerges from this comparison. Eqs. (5.1.1) are the worldline deviations for nearby test particles with the same1tangent 1We want to emphasize this point, which, even today, is not clear in the literature. Eqs. (5.1.1) apply only to the instant where the two particles have the same (or infinitesimally close, in the gravitational case) tangent vector. For both electromagnetism and gravity, in the more general case that where velocity of the two particles is not infinitesimally close, the deviation equations include more terms (which depend on both particles’ 4-velocity, thus in their relative velocity); the relative acceleration is not, in either case, given by a simple contraction of a tidal tensor with a separation vector, see [1, 27, 96]. A more 35 5 The papers summarized and discussed Table 5.1: The gravito-electromagnetic analogy based on tidal tensors. Electromagnetism Gravity Worldline deviation: Geodesic deviation: D2δxα dτ2=q mEα βδxβ, Eα β≡Fα µ;βUµ(5.1.1a) D2δxα dτ2=−Eα βδxβ,Eα β≡Rα µβν UµUν(5.1.1b) Force on magnetic dipole: Force on gyroscope: Fβ EM =Bβ αµα, Bα β≡?Fα µ;βUµ(5.1.2a) Fβ G=−Hβ αSα,Hα β≡?Rα µβν UµUν(5.1.2b) Maxwell Equations: Eqs. Grav. Tidal Tensors: Eα α= 4πρc(5.1.3a) Eα α= 4π(2ρm+Tα α) (5.1.3b) E[αβ]=1 2Fαβ;γUγ(5.1.4a) E[αβ]= 0 (5.1.4b) Bα α= 0 (5.1.5a) Hα α= 0 (5.1.5b) B[αβ]=1 2? Fαβ;γUγ−2παβσγjσUγ(5.1.6a) H[αβ]=−4παβσγJσUγ(5.1.6b) ρc=−jαUαand jαare, respectively, the charge density and current 4-vector; ρm=Tαβ UαUβand Jα=−Tα βUβ are the mass/energy density and current (quantities measured by the observer of 4-velocity Uα); Tαβ ≡ energy-momentum tensor; Sα, µαare the spin and magnetic moment 4-vectors; ?≡Hodge dual. We use 1230 =√−g. vector (and the same ratio charge/mass in the electromagnetic case), separated by the infinitesimal vector δxα. They tell us that the so-called electric part of the Riemann tensor Eα β≡Rα µβνUµUνplays in the geodesic deviation equation (5.1.1b) the same physical role as the tensor Eαβ ≡Fαγ;βUγin the electromagnetic worldline deviation (5.1.1a). Eαβ describes the tidal effects produced by the electric field Eα=Fα γUγas measured by the test particle of 4-velocity Uα. We can define it as a covariant derivative of the electric field as measured in the inertial frame momentarily comoving with the particle: Eαβ =Eα;β|U=const . Hence we dub it the “electric tidal tensor”, and its gravitational counterpart the “gravitoelectric tidal tensor”. There is a magnetic counterpart to this analogy. Consider a purely magnetic dipole, that is, a particle whose only non-vanishing electromagnetic moment, as measured in the particle’s CM frame, is the magnetic moment µα= (0, ~µ), defined by Eq. (3.44) above. The force exerted on it is [59, 60, 63, 66] Fα EM =DPα dτ =Fλν;αµλν =Bβαµβ,(5.3) where µαβ ≡αβγδµγUδand Bαβ ≡?Fαγ;βUγis the “magnetic tidal tensor”, describing the tidal effects produced by the magnetic field Bα=?Fα γUγas measured by the particle of 4-velocity Uα. detailed discussion of this important issue will be presented elsewhere. 36 5 The papers summarized and discussed The gravitational force exerted on a gyroscope (i.e., a spinning pole-dipole particle) is given by the Mathisson-Papapetrou equation: Fα G≡DPα dτ =−1 2Rα βµνUβSµν.(5.4) If the Mathisson-Pirani spin condition SαβUβ= 0 holds, we have Sµν =µντλSτUλ, where Sαis the spin 4-vector, defined as being the 4-vector with components (0,~ S) in the CM frame; substituting in Eq. (5.4) above, we obtain Eq. (5.1.2b) of Table 5.1, revealing the physical analogy Bαβ ↔Hαβ. That is, the magnetic part of the Riemann tensor Hα β≡?Rα µβνUµUνplays in the gravitational force (5.1.2b) the same physical role as Bαβ in the electromagnetic force (5.1.2a); for this reason we dub it “gravitomagnetic tidal tensor”. Note the relative minus sign between Eqs. (5.1.2a) and (5.1.2b); it reflects the fact that masses/charges of the same sign attract/repel, implying that parallel charge/mass currents attract/repel. In Paper #1 [1] this magnetic analogy was the final addition to the results of Table 5.1, and was presented as an application of the formalism (i.e., an analogy based “derivation” of Mathisson-Papapetrou-Pirani equation). There were however relevant issues left unaddressed, and for this reason the treatment therein is not fully satisfactory. The gravitational part of the analogy was based on the well established Eq. (5.4); however this equation needs to be supplemented by a spin condition as explained in Sec. 3.2.1; and this is an old, but important problem in this context because, in order to be exact, the analogy requires the Mathisson-Pirani spin condition to hold, as explained above. It turns out that this condition was poorly understood and portrayed as problematic in the literature, due to its degeneracy and the exotic helical solutions it allows. In the later work [3] (Paper #3 [3]) we clarify these issues, explain the meaning of the spin condition, and demystify the helical motions. The electromagnetic part of the analogy, Eq. (5.1.2b) of Table 5.1, was obtained therein from a covariantization of the textbook, non-relativistic, 3-D expression ~ FEM =∇(~µ ·~ B); and then the result checked with expression (11.26) of [63]. However in the latter result (in spite of being relativistic) only space components of the force were given; furthermore, in the derivation in [63], the physical meaning of the moments of the current defined therein is not clear (as customary, unfortunately, in most literature concerning multipole equations), and given by guessing. Especially because the physical content explored in Sec. V of [1] concerns mostly the time components of the forces, a more solid foundation for these results was in order. This is done in the recent work [4] (Paper #4), where the same exact analogy is shown to stem from the rigorous equations of motion for pole-dipole particles in gravitational and electromagnetic fields, and a number of issues and subtleties involving them (some of them not well understood in the literature, even today) are clarified. Namely the physical meaning of the forces above (which are not trivial, DPα/dτ 6=maαin general!), as well as the momentum (which is not mUα, due to the “hidden momentum”) and the mass of the test particle (which is not a constant). 37 5 The papers summarized and discussed 5.2.1 The gravitational analogue of Maxwell’s Equations If we take the traces and anti-symmetric parts of the electromagnetic tidal tensors, we obtain Eqs. (5.1.3a)-(5.1.6a) of Table 5.1, which are Maxwell’s equations Fαβ ;β= 4πjα(a); ?F αβ ;β= 0 (b),(5.5) written in tidal tensor form. That is, Eqs. (5.1.3a) and (5.1.6a) are, respectively, the time and space projections (with respect to Uα) of the Maxwell equations with sources (5.5a); and (5.1.4a) and (5.1.5a) are, respectively, the space and time projections of the source-free equations (5.5a) (i.e., the electromagnetic Bianchi identity). This is explicitly shown using the projectors in [5] (Paper #5). Eqs. (5.1.3a)-(5.1.6a) may be cast as equations involving only tidal tensors and sources, which can be seen decomposing: Fαβ;γ= 2U[αEβ]γ+αβµσBµ γUσ or by noting that the pair of Eqs. (5.1.4a), (5.1.6a), may be condensed in the equivalent pair βγ αδUδE[γβ]=−BαβUβ; (a) βγ αδUδB[γβ]=EαβUβ+ 4πjα(b) (5.6) In a Lorentz frame in flat spacetime, since Uα ;β=Uα ,β = 0, we have Eγβ =Eγ;β,Bγβ = Bγ;β; and (using Uα=δα 0) Eqs. (5.6) can be written in the familiar textbook forms ∇× ~ E=−∂~ B/∂t and ∇× ~ B=∂~ E/∂t + 4π~ j, respectively. Likewise, Eqs. (5.1.3a) and (5.1.5a) reduce in this frame to the familiar forms ∇·~ E= 4πρcand ∇· ~ B= 0, respectively. By performing, on the gravitational tidal tensors, the same operations that lead to Eqs. (5.1.3a)-(5.1.6a) (i.e., taking the traces and anti-symmetric parts) we obtain the analogous set of Eqs. (5.1.3b)-(5.1.6b). Eqs. (5.1.3b) and (5.1.6b) turn out to be exactly the time-time and and time-space projections of Einstein equations with sources (5.7a): Rγ αγβ ≡Rαβ = 8πTαβ −1 2gαβTγ γ(a); ?Rγα γβ = 0 (b).(5.7) And Eqs. (5.1.4b) and (5.1.5b) are, respectively, the time-time and space-time projections of the algebraic Bianchi identity R[αβγ]δ= 0 ⇔?Rγα γβ = 0. Again, this is explicitly shown using the projectors in [5]. 5.2.2 Gravity vs Electromagnetism Eqs. (5.13a)-(5.16a) are strikingly similar to Eqs. (5.13b)-(5.16b) when the fields do not vary along the observer’s worldline. Otherwise, they tell us that the two interactions must differ significantly, since the tidal tensors do not have the same symmetries. Sources — Eqs. (5.1.3) tell us that the source of Eαβ is ρc, and its gravitational analogue, the source of Eαβ, is 2ρ+Tα α(ρ+ 3pfor a perfect fluid), manifesting the well known fact that in gravity, by contrast with electromagnetism, pressure and stresses act as sources of the field. The magnetic/gravitomagnetic tidal tensors are analogously sourced 38 5 The papers summarized and discussed by the charge/mass-energy spatial currents jhµi/Jhµi, as shown by Eqs. (5.1.6). Note that, when the fields do not vary along the observer’s worldline, ?Fαβ;γUγvanishes and equations (5.1.6a) and (5.1.6b) match up to a factor of 2, identifying jµ↔Jµ. Symmetries and time projections of tidal tensors — The gravitational and electromagnetic tidal tensors do not generically exhibit the same symmetries; moreover, the former are spatial, whereas the latter have a time projection (with respect to the observer measuring them), signaling fundamental differences between the two interactions. In the general case of fields that are time dependent in the observer’s rest frame (that is the case of an intrinsically non-stationary field, or an observer moving in a stationary field), Eαβ possesses an antisymmetric part, which is the covariant derivative of the Maxwell tensor along the observer’s worldline; Eαβ, by contrast, is always symmetric. As discussed above, Eαβ is a covariant derivative of the electric field as measured in the the momentarily comoving reference frame (MCRF); and Eq. (5.6a) is a covariant way of writing the Maxwell-Faraday equation ∇× ~ E=−∂~ B/∂t. Therefore, the statement encoded in the equation E[αβ]= 0 is that there is no physical, gravitational analogue to Faraday’s law of induction (in the language of GEM vector fields of Paper #5 [5], we can say the the curl of the gravitoelectric field ~ Gdoes not manifest itself in the tidal forces, unlike its electromagnetic counterpart). To see a physical consequence, let δxαin Eq. (5.1.1a) — the separation vector between a pair of particles with the same q/m and the same 4-velocity Uα— be spatial with respect to Uα(δxαUα= 0); and note that the spatially projected antisymmetric part of Eµν can be written in terms of the dual spatial vector αµ:E[hµihνi]=µνγδαγUδ. Then the spatial components (5.1.1a) can be written as (using Ehµihνi=E(hµihνi)+E[hµihνi]): D2δxhµi dτ2=q mhE(hµihνi)δxν+µνγδαγUδδxνi⇔D2δ~x dτ2=q mh←→ E·δ~x +δ~x ×~αi, (5.8) the second equation holding in the frame Ui= 0, where we used the dyadic notation ←→ E of e.g. [67]. From the form of the second equation we see that q~α/m is minus an angular acceleration. Using relation (5.6), we see that αµ=−Bµ βUβ; and in an inertial frame ~α =∂~ B/∂t =−∇× ~ E. In the gravitational case, since Eµν =E(µν)=Ehµihνi, we have D2δxhµi dτ2=D2δxµ dτ2=−E(µν)δxν⇔D2δ~x dτ2=−←→ E·δ~x . (5.9) That is, given a set of neighboring charged test particles, the electromagnetic field“shears” the set via E(µν), and induces an accelerated rotation2via the laws of electromagnetic induction encoded in E[µν]. The gravitational field, by contrast, only shears3the set, since 2By rotation we mean here absolute rotation, i.e, measured with respect to a comoving Fermi-Walker transported frame. See Paper #5 [5]. 3If the two particles were connected by a “rigid” rod then the symmetric part of the electric tidal tensor would also, in general, torque the rod; hence in such system we would have a rotation even in the gravitational case, see [140] pp. 154-155. The same is true for a quasi-rigid extended body; however, even in this case the effects due to the symmetric part are very different from the ones arising from electromagnetic induction: first, the former do not require the fields to vary along the particle’s worldline, they exist even if the body is at rest in a stationary field; second, they vanish if the body is spherical, which does not happen with the torque generated by the induced electric field, see [4]. 39 5 The papers summarized and discussed E[µν]= 0. Further physical evidence for the absence of a physical gravitational analogue for Faraday’s law of induction is given in Sec. VI. of paper #4 [4]: consider a spinning spherical charged body in an electromagnetic field; and choose the MCRF (in order to keep things simple; see [4] for the general covariant treatment); if the magnetic field is not constant in this frame, by virtue of equation ∇× ~ E=−∂~ B/∂t, a torque will in general be exerted on the body by the induced electric field, changing its angular momentum and kinetic energy of rotation. By contrast, no gravitational torque is exerted in a spinning “spherical” body (i.e., a particle whose multipole moments in a local orthonormal frame match the ones of a spherical body in flat spacetime) placed in an arbitrary gravitational field; its angular momentum and kinetic energy of rotation are constant. There is also an antisymmetric contribution ?Fαβ;γUγto Bαβ; in vacuum, Eq. (5.1.6a) is a covariant form of ∇× ~ B=∂~ E/∂t; hence the fact that, in vacuum, H[αβ]= 0, means that there is no gravitational analogue to the antisymmetric part B[αβ](i.e., the curl of ~ B) induced by the time varying field ~ E. Some physical consequences of this fact are as follows. Eq. (5.1.6a) implies, via (5.1.2a), that whenever a magnetic dipole moves in a nonhomogeneous field, it measures a non vanishing B[αβ](thus also Bαβ 6= 0), and therefore (except for very special orientations of the dipole moment µα) a force will be exerted on it; in the gravitational case, by contrast, the gravitational force on a gyroscope is not constrained to be non-vanishing when it moves in a non-homogeneous field; it is found that it may actually move along geodesics, as is the case of radial motion in Schwarzschild spacetime, or circular geodesics in Kerr-dS spacetime. For more details see Paper #4 [4]. The spatial character of the gravitational tidal tensors, contrasting with their electromagnetic counterparts, is another difference in the tensorial structure related to the difference in the symmetries (and thus to the laws of electromagnetic induction), as can be seen from Eqs. (5.6) and (5.1.6a). Physically, this is manifest (for instance) in the fact that the electromagnetic force on a magnetic dipole has a non-vanishing projection along the particle’s 4-velocity Uα, which is the rate of work done on it by the induced electric field (and is reflected in a variation of the particle’s proper mass, as shown in Paper #4 [4]). This is more easily seen be seen if we imagine the dipole as a small current loop, as depicted below: where Φ is the magnetic flux through the loop and ~ Eis the induced electric field. Thus Fα EM Uαis minus the power transferred to the dipole by Faraday’s induction, due to a time varying magnetic field ~ B(as measured in the particle’s proper frame). An equivalent, but manifestly covariant derivation of this result is given in Paper #4 [4]. 40 5 The papers summarized and discussed This effect has no counterpart in gravity. Since Hαβ is a spatial tensor, we always have Fα GUα= 0 which means that no work is done on the gyroscope, as measured in its proper frame (and its proper mass is constant). Hence, the spatial character of the gravitational tidal tensors precludes induction effects analogous to the electromagnetic ones. 5.2.3 Matching between tidal tensors It is important to realize that the existence of this exact gravito-electromagnetic analogy does not mean that the interactions are similar. This is a functional analogy: despite playing analogous roles in the dynamics of both theories, gravitational and electromagnetic tidal tensors are generically very different. In their symmetries and time projections, as we have seen above, and also in the more obvious fact that the electromagnetic tidal tensors are linear in terms of the electromagnetic potential (and fields), whereas the gravitational tidal tensors are non-linear in the metric tensor (and in the “gravitoelectromagnetic” fields of Paper #5 [5]). Nevertheless, we found that a matching occurs in certain special cases: linearized gravity under certain conditions, and an exact matching in the so-called ultrastationary spacetimes. 5.2.3.1 Linearized Gravity Take arbitrary perturbations around Minkowski spacetime in the form, ds2=−(1 −2Φ) dt2−4Ajdtdxj+ [ˆgij + 2Θij]dxidxj(5.10) (ˆgij ≡euclidean metric in an arbitrary coordinate system), and an arbitrary electromagnetic field Aα= (φ, ~ A) in Minkowski spacetime ds2=−dt2+ ˆgij(xk)dxidxj. The explicit expressions for gravitational and electromagnetic tidal tensors from these setups, as measured by an arbitrary observer of 4-velocity uα= (u0, ui), are given in Eqs. (11)-(12) and (14)-(20) of Paper #1 [1]. Comparing these expressions we see that they will in general be very different, even to linear order. But if one takes time independent electromagnetic potentials/gravitational perturbations, and a “static observer”, Uµ=δµ 0(i.e., an observer with zero 3-velocity in the coordinates of (5.10)) then the linearized gravitational tidal tensors match their electromagnetic counterparts: Eij ≃ −Φ;ij Φ↔φ =Eij,Hij ≃ˆlk iAk;lj A↔A =Bij .(5.11) This can be illustrated by an elementary example of analogous physical systems — the gravitational field of a spinning mass and the electromagnetic field of a spinning charge: in the far field limit (where the non-linearities of the gravitational field are negligible), for an observer at rest with respect to the central body, the gravitational tidal tensors asymptotically match the electromagnetic ones, cf. Eqs. (8)-(9) of [1]. But if the observer moves, the electromagnetic tidal tensors it measures will be very different from the gravitational ones (for explicit expressions, see [27], sec. 2.2.1; or, for the case of non-spinning bodies, Eqs. (2.11)-(2.16) of Paper #2 [2]). 41 5 The papers summarized and discussed Gyroscope precession. — The evolution of the spin vector of the gyroscope is given by the Fermi-Walker transport law, which, for a gyroscope at rest reads DSi/dτ = 0; hence, we have, in the coordinate basis, dSi dt =−cΓi 0jSj=−1 c(S×BG)i+1 c ∂Θij ∂t Sj.(5.22) Comparing with the equation for the precession of a magnetic dipole under the action of a magnetic field dS dt =1 cµ×B(5.23) we see that there is an extra term in the gravitational equation, arising from the shear of the reference frame (that generically exists when the gravitational field is time-dependent), as discussed above. But this term can be made to vanish by choosing a suitable frame. Consider the orthonormal tetrad eˆα, and let eαˆαdenote the transformation matrix relating it to the coordinate basis eα≡∂α:eˆα=eαˆαeα. Take eˆαsuch that eˆ 0is the observers 4-velocity uα, and the spatial triads eˆ ifollow as much as possible the coordinate basis vectors ei. That is, the eˆ ico-rotate with the observer congruence (i.e., they rotate with respect to Fermi-Walker transport with an angular velocity that equals the vorticity of the congruence, see Secs. 3 and 5 of Paper #5 [5] for detailed explanation), but without suffering the shear and expansion effects of the later. To linear order, eαˆα, and its inverse eˆα α, are given by eˆ 0= (1 −φ)e0;eˆ i=ei+Θˆ j i c2ej+ 2Ai ce0; e0= (1 + φ)eˆ 0;ei=eˆ i−Θˆ j i c2eˆ j−2Ai ceˆ 0. (5.24) Thus ei ˆ i=δiˆ i−Θiˆ i/c2; expressing Sin the tetrad, Si=Sˆ iei ˆ i, we obtain an expression similar to the electromagnetic one (5.23): dSˆ i dt =−1 c(S×BG)ˆ i.(5.25) Thus, in the special case of gyroscope precession, the linear gravito-electromagnetic analogy holds even if the fields vary with time. This might be somewhat surprising, because if we think about the magnetic dipole as a spinning charged body in a time dependent magnetic field, an induced electric field would arise that should torque the body, changing it angular momentum S. Which, in the light of the conclusions of Paper #1, does not have a gravitational counterpart. As discussed in detail in Sec. VI of Paper #4 [4], the apparent paradox is an artifact of the dipole approximation, which neglects the torque exerted by the induced electric field. The latter depends on the particle’s second moment of the charge, that is of quadrupole order. To dipole order the total torque reduces to Eq. (5.23), which preserves the magnitude of S, only causing the dipole to precess (regardless of the time-dependence of the field). That is: the analogy holds because electromagnetic induction effects are neglected in this degree of approximation. 48 5 The papers summarized and discussed 5.3.2 Translational vs. Rotational Mass Currents The existence of a similarity between gravity and electromagnetism, both in terms of tidal and inertial effects, requires time independent fields, as shown above. To what pertains gravitomagnetic effects, this is a statement about the time dependence of the mass currents in the chosen reference frame: if the they are (nearly) stationary, for example from a rotating celestial body, the gravitational field generated is analogous to a magnetic field; such is the field detected on LAGEOS Satellites data [52], in the Gravity Probe B mission [53], and presently under experimental scrutiny by the LARES mission [54]. But when they vary with time — e.g. the ones resulting from translation of the celestial body, considered in [69] — then the dynamics differ significantly. Rotational Currents. — We start by the basic example of analogous systems already considered in Sec. III of Paper #1 [1]: the electromagnetic field of a spinning charge (charge Q, magnetic moment µs) and the gravitational field (in the far region r→ ∞) of a rotating celestial body (mass m, angular momentum J), see Fig. 5.1. Figure 5.1: Spinning charge vs. spinning mass The electromagnetic field of the spinning charge is described by the 4-potential Aα= (φ, A), given by (5.26a). The spacetime around the spinning mass is asymptotically described by the linearized Kerr solution, obtained by putting in (5.17) the perturbations (5.26b) : φ=Q r,A=1 c µs×r r3(a); Φ = M r,A=1 c J×r r3,Θij = Φδij (b).(5.26) Let Obe a static observer (i.e., at rest with respect to the central bodies); the gravitational tidal tensors it measures asymptotically match the electromagnetic ones, identifying the appropriate parameters: Eij ≃M r3δij −3Mrirj r5 M↔Q =Eij;Hij ≃3 c(r.J) r5δij + 2r(iJj) r5−5(r.J)rirj r7J↔µs =Bij (all the time components are zero for this observer). This means that Owill find a similarity between physical (i.e., tidal) gravitational and electromagnetic forces: the gravitational force Fi G=−HjiSj/c exerted on a gyroscope carried by Ois similar to the force Fi EM =Bjiµj/c on a magnetic dipole; and the worldline deviation D2δxi/dτ2=−Eijδxi of two masses dropped from rest is similar to the deviation between two charged particles with the same q/m. 49 5 The papers summarized and discussed Moreover, in the frame of the static observers O, test particles will be seen moving on geodesics described by equations analogous to the electromagnetic Lorentz force, see Fig. 5.1. Translational Currents. — For observers ¯ Omoving relative to the mass/charge of Fig. 5.1, however, the electromagnetic and gravitational interactions will look significantly different. Consider the frame obtained from the frame of the static observers Oby applying a boost of constant coordinate velocity w; and let us denote7the boosted frame by ¯ O. For simplicity we specialize here to the case where J=µ= 0, so that the mass/charge currents in the frame ¯ Oarise solely from translation. To obtain the electromagnetic 4-potential A¯αin ¯ O, we apply the boost A¯α= Λ¯α αAα= (¯ φ, ¯ A), where Λ¯α α≡∂¯x¯α/∂xα, using the expansion of Lorentz transformation (as done in e.g. [72]): t=¯ t1 + w2 2c2+3w4 8c4+1 + w2 2c2¯x.w c2;x=¯x+1 2c2(¯ x.w)w+1 + w2 2c2w¯ t , (5.27) yielding, to order c−2,A¯α= (¯ φ, ¯ A), with ¯ φ=Q(1 + w2/2c2)/r and ¯ A=−Qw/rc. To obtain A¯αin the coordinates (¯xi,¯ t) of ¯ O, we must also express r(which denotes the distance between the source and the point of observation, in the frame O) in terms of R≡ |¯r +w¯ t|, i.e., the distance between the source and the point of observation in the frame ¯ O. Using transformation (5.27), we obtain: r−1=R−1[1 −(w.R)2/(2R2c2)], and finally the electromagnetic potentials measured in ¯ O: ¯ φ=Q R1 + w2 2c2−(w.R)2 4R2c2;¯ A=−1 c Q Rw.(5.28) The metric of the spacetime around a point mass, in the coordinates of ¯ O, is also obtained using transformation (5.27), which is accurate to Post Newtonian order, by an analogous procedure. First we apply the boost g¯α¯ β= Λα¯αΛβ¯ βgαβ to the metric (5.26) (with A= 0); then, expressing rin terms of R, we finally obtain (note that, although we are not putting the bars therein, indices α= 0, i in the following expressions refer to the coordinates of ¯ O): g00 =−1+2 M Rc2+4Mw2 Rc4−M(w.R)2 c4R3≡ −1 + 2¯ Φ c2; g0i=4Mwi Rc3≡ −2¯ Ai c2;gij =1+2 M Rc2δij ≡1+2¯ Θ c2δij ,(5.29) where we retained terms up to c−4in g00, up to c−3in gi0, and c−2in gij, as usual in Post-Newtonian approximation. Note that in the fields (5.28)-(5.29) we kept terms to second order in w(and the transformation (5.27) originating them was non-linear). This is because, in order to consistently take into account the gravitomagnetic force on a test 7This is a slight notation abuse, as one should in general distinguish observer from frame, see Sec. 3.1 of Paper #5 [5] where this issue is discussed in detail. Herein the situation is simple because we are dealing with Post-Newtonian frames [68], differing between each other only by boosts, thus both the observer congruence and the corresponding spatial frame are always well defined. 50 5 The papers summarized and discussed particle, −2v×BG/c, even to first order in the velocity of the test particle v, one needs to keep terms up to second order8in the translational velocity of the source (−w); see Eq. (5.36) below. We neglected non-linear terms in M, as done in [69]; the metric (5.29) is equivalent to expressions (11) of [69] (where an additional gauge choice, Eq. (19) of [72], was made), for the case of a single source. It also matches Eqs. (5) of [70] to linear order in M(and again for the case of one single source). Note that the metric (5.29), like the electromagnetic potential (5.28), is now time dependent, since R(¯ t) = ¯r +w¯ t. The gravitational tidal tensors measured by the observers ¯ Oare (Eα0=E0α=Hα0= H0α= 0): Eij =−¯ Φ,ij −2 c ∂ ∂¯ t¯ A(i,j)−1 c2 ∂2 ∂¯ t2¯ Θδij =Mδij R31 + 3w2 c2−9 2 (R.w)2 c2R2−3MRiRj R51 + 2w2 c2−5(R.w)2 2c2R2 −3Mwiwj c2R3+6Mw(iRj)(R.w) c2R5; (5.30) Hij =lk i¯ Ak,lj −1 cl ij ∂¯ Θ,l ∂¯ t=M cR33k ij wk−3 R2(R.w)k ij Rk−6 R2(R×w)iRj,(5.31) which significantly differ from the electromagnetic ones (E0α=B0α= 0): Eij =−¯ φ,ij −1 c ∂ ∂¯ t¯ Ai;j=Ei,j =Qδij R31 + w2 2c2−3 4 (R.w)2 c2R2−3QRiRj R51 + w2 2c2−5(R.w)2 4c2R2 −Qwiwj 2c2R3+3Qw[iRj](R.w) c2R5; (5.32) Ei0=−1 c ∂ ∂¯ t¯ φ;i−1 c2 ∂2¯ Ai ∂¯ t2≡1 c ∂Ei ∂¯ t=Q cR3wi−3(R.w)Ri R2; (5.33) Bij =lm i¯ Am;lj ≡Bi,j =Q cR3k ij wk−3 R2(R×w)iRj; (5.34) Bi0=1 c ∂Bi ∂¯ t=−3Q c2R5(R.w)(R×w)i.(5.35) In particular, unlike their gravitational counterparts, Eαβ and Bαβ are not symmetric, and have non-zero time components. Note that the differing terms causing this are of the same order of magnitude as the others, thus cannot be neglected in any consistent approximation. The space part of the geodesic equation for a test particle of velocity vis: a=∇¯ Φ + 2 c ∂¯ A ∂¯ t−2v×(∇× ¯ A)−3 c2 ∂ ∂¯ tM Rv(5.36) =−M R31 + 2w2 c2−3(R.w)2 2c2R2R+3M(R.w) c2R3w−4M c2R3v×(R×w) + 3 c2 M R3(R.w)v, 8Otherwise, if we assumed w2≈0 together with v2≈0, then vw ≈0, and Eq. (5.36) would reduce to the first term (the Newtonian acceleration) 51 5 The papers summarized and discussed which matches equation (10) of [70], or (7) of [71], again, in the special case of only one source, and keeping therein only linear terms in the perturbations and test particle’s velocity v. Comparing with its electromagnetic counterpart m qa=E+v c×B=Q R31 + w2 2c2−3(R.w)2 4c2R2R−1 2 Q(R.w) c2R3w+Q c2R3v×(R×w) we find them similar to a certain degree (up to some factors), except for the last term of (5.36). That term signals a difference between the two interactions, because it means that there is a velocity dependent “acceleration” which is parallel to the velocity; that is in contrast with the situation in electromagnetism, where the velocity dependent accelerations arise from magnetic forces, and are thus always perpendicular to v. As expected from Eqs. (5.25) (and by contrast with the other effects), the precession of a gyroscope carried by ¯ O, Eq. (5.37b) takes a form analogous to the precession of a magnetic dipole, dS d¯ t=q 2m Q c2R3[S×(R×w)] , if we express Sin the local orthonormal triad eˆ ias defined in Sec. 5.3.1, such that Si= (1 −M/R)Sˆ i: dSˆ i d¯ t=2M c2R3[(R×w)×S]ˆ i.(5.37) The triad of axes eˆ iis in this case fixed relative to the “distant stars” (i.e., non rotating relative to inertial frames at infinity); thus Eq. (5.37) yields the precession of the gyroscope relative to the distant stars, which is the situation of interest for astrophysical applications, namely the measurements performed by the Gravity Probe B (GPB) [53]. Indeed, the precession angular velocity above, Ω=2M c2R3R×w, or equivalently, Ω=BG/c, cf. Eq. (5.25), can be regarded as the sum of two terms: the geodetic (or de Sitter) precession, measured by the GPB (in addition to the Lense-Thirring one), which amounts to 3Ω/4, plus the Thomas precession, which amounts to Ω/4. See e.g. Eqs. (40.33) of [73], or Eqs. (3.4.38) of [7]; see also Sec. IVB of Paper #4 [4] where a related issue is discussed. Finally, a problem that was not addressed in this paper, and is usually overlooked in the literature concerning both the linearized and Post-Newtonian approaches, is the following: we said above that Eq. (5.37) yields the precession of a gyroscope with respect to a system of axes eˆ ithat is non-rotating with respect to an inertial frame at infinity; but how can one ensure that, as it amounts to comparing systems of vectors at different points in a curved spacetime? This is an highly non-trivial problem, which is studied in a exact approach in Secs. 3.1 and 3.3 of Paper #5. The conclusion is that one may determine the relative rotation of two tetrads at different points, exactly, only in spacetimes admitting shearfree observer congruences. In the problem at hand, the boosted frame indeed shears, but not 52 5 The papers summarized and discussed to this degree of accuracy, as explained above; the traceless shear is neglected, only the expansion (which preserves angles) is manifest in the boosted metric (5.29). 5.3.3 Conclusion In this work, and in view of the analogies from linearized theory reviewed in Sec. 3.1.1, whose limit of applicability is not always clear, we dissected under which specific conditions gravitational dynamics (for weak fields) becomes similar to electromagnetism, with a special emphasis to setups of recent and present experimental interest. We have concluded that the actual physical similarities between gravity and electromagnetism (on which the physical content of such approaches relies) occur only on very special conditions. In the framework of linearized theory and Post-Newtonian approximations (and with the type of frames commonly used therein), this is a requirement of time-independence of the fields; the frame in which such independence is required depends on the type of effect. For tidal effects, like the forces on gyroscopes/dipoles, the similarity manifest in Eqs. (5.18) (and the analogy based on GEM fields therein) holds only when both the potentials (gravitational/electromagnetic) and their gradients are timeindependent in the test particle’s frame. In the example of analogous systems considered in Sec. 5.3.2, this means that the center of mass of the gyroscope/magnetic dipole must not move relative to the central body. In the case of the analogy between the equation for the geodesics and the Lorentz force law (see Fig. 5.1), as manifest in equation (5.19), it is in the the observers’ (not the test particle!) frame, that the time independence of the potentials, is required. In the case of the tidal effects, as mentioned above, the restriction is not only on the potentials, but also in its gradients; here I would like to remark that this makes a difference. Consider this basic example, a test particle in circular motion around the Coulomb field of a point charge. In the inertial frame momentarily comoving (MCRF) with the particle, the potential φis constant; but not the electric field E(it is constant in magnitude, but time-varying in direction), and due to that the electromagnetic tidal tensors can no longer be similar to the gravitational ones (for instance of the analogous situation, a particle in circular motion around a Schwarzschild black hole). In this framework this can be understood as follows: take the magnetic tidal tensor Bαβ; as discussed in Sec. 5.2, it is a covariant derivative of the magnetic field as measured in the MCRF: Bαβ =Bα;β|U=const = (BMCRF)α;β. Since in this frame the electric field is time-varying, by virtue of Maxwell equation ∇× ~ B=∂~ E/∂t, this means that ~ Bhas a curl, thus B[αβ]6= 0. By contrast with the gravitational analogue, where H[αβ]= 0. This can be stated in this way: the symmetries of the electromagnetic tidal tensors differ from the gravitational ones when Fαβ;γUγ6= 0, i.e., when the electromagnetic field is not covariantly constant along the observer’s (in this case the particle’s) worldline; this is what Eqs. (5.1.6a), (5.1.4a) of Table 5.1 tell us. And indeed for circular motions around a coulomb charge, Fαβ;γUγ6= 0, despite dφ/dτ =φ;αUα= 0. In Sec. 7 of Paper #5 [5] the analysis herein is refined and generalized for the exact case. Finally, as a consequence of this analysis, a distinction, from the point of view of the 53 5 The papers summarized and discussed analogy with electrodynamics, between effects related to (stationary) rotational mass currents, and those arising from translational mass currents, becomes clear: albeit in the literature both are dubbed “gravitomagnetism”, one must note that, while the former are clearly analogous to magnetism, in the case of the latter the analogy is not so close. 5.4 Paper #3 — Mathisson’s helical motions for a spinning particle: Are they unphysical? Both the analogy between the electromagnetic force on a magnetic dipole and the gravitational force on a gyroscope (a spinning pole-dipole particle), introduced in Paper #1 (and put on firm ground in Paper #4), and the analogy between the precession of a magnetic dipole in a electromagnetic field and the “precession” of a gyroscope, known from the approaches based on exact GEM inertial fields of Sec. 3.1.2 (see also papers #4 and #5), require the so-called Mathisson-Pirani spin condition to hold. The very notions of rotation and compass of inertia in relativity, and the physical meaning of the Fermi-Walker transport law, rely also on it. It turns out that the problem of the spin supplementary condition is an old one, and not well understood even today. This is even more so in the case of Mathisson-Pirani condition, due to its degeneracy and the exotic solutions it allows. In particular the famous helical motions for a free particle in flat spacetime (where the particle’s center of mass accelerates without the action of any force), which were regarded with a lot of skepticism, and deemed unphysical, due to the belief that the radius of the helices could be arbitrarily large [76, 75, 60, 77]. In this work, which initially started out as an Appendix of Paper #4 [4], we clarify these issues, explain the helical motions, and show that there is nothing wrong or unphysical with the Mathisson-Pirani condition. 5.4.1 Equations of motion for free spinning particles in flat spacetime. Mathisson’s helical motions. As explained in Sec. 3.2, in a multipole expansion, a body is represented by a set of moments of Tαβ, called “inertial” or “gravitational” moments (forming the so called [61] “gravitational skeleton”), and the moments of jα(the electromagnetic skeleton). In this work we are interested in free particles, so only the former contribute to the equations of motion. The moments are taken about a reference worldline zα(τ), which could in principle be arbitrary, but will be chosen below as a suitably defined center of mass. The case of pole-dipole particles corresponds to truncating the expansion at dipole order. In this case the equations of motion involve only two moments of Tαβ, the momentum Pα, and the angular momentum Sαβ defined as (see e.g. [97, 60, 59]): Pα≡ˆΣ(τ,u) TαβdΣβ,(5.38) Sαβ ≡2ˆΣ(τ,u) r[αTβ]γdΣγ.(5.39) 54 5 The papers summarized and discussed Here Pα(τ) is the 4-momentum of the body; Sαβ(τ) is the angular momentum about a point zα(τ) of the reference worldline; Σ(τ, u)≡Σ(z(τ), u) is the spacelike hypersurface generated by all geodesics orthogonal to some time-like vector uαat the point zα(τ); rα≡xα−zα(τ), where {xα}is a chart on spacetime; dΣγ≡ −uγdΣ, and dΣ is the 3-volume element on Σ(τ, u). In the case of free particles in flat spacetime (i.e., without any further fields), the equations of motion that follow from the conservation law Tαβ ;β= 0 are [76, 79, 60, 59, 81, 97]: DPα dτ = 0 (a),DSαβ dτ = 2P[αUβ](b).(5.40) Contracting (5.40b) with Uαwe obtain an expression for the momentum: Pα=mUα−DSαβ dτ Uβ,(5.41) where m≡ −PαUα. Eqs. (5.40) form an indeterminate system. Indeed, there are 13 unknowns (Pα, 3 independent components of Uα, and 6 independent components of Sαβ) for only 10 equations. This is where the spin condition comes into play. A supplementary spin condition of the type Sαβuβ= 0, for some unit timelike vector uα(τ), effectively kills off 3 components of the angular momentum, thereby closing the system. Such condition has the role of specifying the representative point of the body (i.e., the worldline of reference relative to which Sαβ is taken); as I shall show below, it demands it to be the center of mass as measured in the rest frame of the observer of velocity uα. In this way Uαis the center of mass 4-velocity and mdenotes the proper mass, i.e., the energy of the body as measured in the center of mass frame. Note from Eq. (5.41) that the momentum Pαis not, in general, parallel to Uα; the spinning particle is said to possess “hidden momentum” [66, 4], which plays a key role in this discussion, as explained in Sec. 5.4.5 below. Mathisson’s helical solutions [74] arise when one uses the condition SαβUα= 0. In this case Sαβ =αβµνSµUν; (5.40c) becomes Pα=mUα+Sαβaβ, where aα=DUα/dτ; and DSα/dτ = 0. The solution of (5.40) under this condition turns out to be degenerate; it describes the famous helical motions, which, in the Pi= 0 frame, correspond to clockwise (i.e. opposite to the spin direction) circular motions with radius R=vγ2S m(5.42) and speed von the xy plane. Taking their center as the spatial origin of the frame, they read: zα(τ) = γτ, −Rcos vγ Rτ, R sin vγ Rτ,0(5.43) These motions were interpreted in [74] (for the case of an electron) as the classical counterpart of the Dirac equation ‘zitterbewegung’. However, the fact that γcan be arbitrarily large has led some authors (see e.g. [76, 77]) to believe that, according to (5.43), a given free body might move along circular trajectories with any radius; for this reason these 55 5 The papers summarized and discussed A B xCM ¯xCM ~v =−v~ey ∆~x Figure 5.2: Center of mass of a spinning particle (~ S=S~ez, orthogonal to the page) as evaluated by two different observers. Observer Oof 4-velocity uα=Pα/M is at rest with respect to center of mass xi CM ≡xi CM (u) it measures (i.e., xi CM is a proper center of mass). Observer ¯ O, moving with velocity ~v =−v~eyrelative to O, sees the points on the right hemisphere (e.g. point B) moving faster than the points in the left hemisphere (e.g. point A), and, therefore, for ¯ O, the right hemisphere will be more massive than the left one. This means that the center of mass ¯xi CM ≡xi CM (¯u) as evaluated in the moving frame of ¯ Ois shifted to the right (relative to xi CM ). The shift is exactly ∆~x =~ S?×~v/M. solutions have been deemed unphysical. The same arguments were used to imply that the the frequency of these motions, given by ω=m γ2S,(5.44) only coincides, for the case of an electron, with Dirac’s zitterbewegung frequency ω= 2Me/~, in the limit γ→1. Both these assessments are misconceptions as shown below. 5.4.2 Center of mass. Significance of the spin condition. In order for (5.40) to be equations of motion for the body, zα(τ) must be taken as its representative point. The natural choice for such point would be the body’s center of mass (CM); however, in relativity, the CM of a spinning body is an observer dependent point. This is illustrated in Fig. 5.2. As mentioned above, a spin condition of the type Sαβuβ= 0 (for some unit time-like vector uα) amounts to choosing zα(τ) as the center of mass xα CM(u) measured by the observer O(u) of 4-velocity uα. This is easily seen in the rest frame of O(u) (the ui= 0 frame). In such frame Sαβuβ=Sα0u0; thus, from Eq. (5.39): Si0= 2 ˆΣ(τ,u) r[iT0]γdΣγ=ˆxiT00d3x−m(u)zi,(5.45) where m(u)≡ −Pαuαdenotes the mass as measured in the frame O. The first term of (5.45) is by definition m(u)xi CM (u), where xi CM (u) are the coordinates of the center of mass as measured by O, and so xi CM (u)−zi=Si0 m(u)⇔xα CM (u)−zα=−Sαβuβ m(u).(5.46) 56 5 The papers summarized and discussed Figure 5.3: Kinematical explanation of the helical motions allowed by SαβUβ= 0: every point within a disk of radius S?/M, centered at xα CM(P), is a centroid corresponding to some observer; and it is also a proper center of mass if it rotates with angular velocity ω=M/S?in the opposite sense of the spinning body (solid red lines). Thus the condition Sαβuβ= 0 is precisely the condition xα CM (u) = zα, i.e., that the reference worldline is the center of mass as measured in this frame. In order to see how the center of mass changes with the observer, take uα=Pα/M, i.e., the reference worldline is the center of mass as measured in the zero 3-momentum frame, that we denote by xα CM(P). And let Sαβ ?denote the angular momentum with respect to xα CM (P). Consider now another observer ¯ Omoving relative to Owith 4-velocity ¯uα= ¯u0(1,~v); for this observer the center of mass will be at a different position, as depicted in Fig. 5.2. It is displaced by a vector ∆xα=−Sαβ ?¯uβ/m(¯u) relative to the reference worldline zα, where m(¯u)≡ −Pγ¯uγdenotes the mass of the particle as measured by ¯ O. That is, ∆xi=(~ S?×~v)i M,(5.47) with M≡√−PαPα. Hence the set of all possible CM’s measured by all observers O(¯u) fills a disk of radius Rmax =S∗/M centered at xα CM(P). This is the minimum size a particle can have without violating the dominant energy condition (i.e., without possessing matter/energy flowing faster than light). The latter implies ρ≥ |~ J|, where ρ≡T00 and Ji≡T0i; let bbe the largest dimension of the body. Using the definition of Sαβ ?in [3], we may write, in the Pi= 0 frame, S?=ˆ~r ×~ Jd3x≤ˆr|~ J|d3x≤ˆρrd3x≤Mb ⇔b≥S? M=Rmax .(5.48) Thus the disk of CM’s, within which all the helical motions are contained, is always smaller than the body. 5.4.3 Kinematical interpretation of the helical motions The Mathisson-Pirani condition SαβUα= 0 amounts to choosing for zαthe center of mass xα CM(U) as measured in its own rest frame, i.e., the frame Ui= 0. Such CM is dubbed a “proper center of mass”. It turns out that, contrary to what one might expect, such 57 5 The papers summarized and discussed exact gravito-electromagnetic analogies studied in this work arise. The rigorous equations of motion, that follow from the conservation equation (5.55) and the charge conservation jα ;α= 0, have been derived in a number of independent treatments [60, 59, 81, 106, 64, 66]; the relationship between them, as well as the physical interpretation of the terms involved is an important clarification made in this work; it is discussed detail in Appendixes A (see also B) of Paper #4 [4]. Writing them in terms of the physical momentum and angular momentum, given by definitions above, they read, in tidal tensor form: DPα dτ =qEα+Bβαµβ−HβαSβ+Eαβdβ+Fα β Ddβ dτ ; (5.60) DSαβ dτ = 2P[αUβ]+ 2µθ[βFα] θ+ 2d[αFβ] γUγ,(5.61) where Fαβ is the background Maxwell tensor. The first term in (5.60) is the Lorentz force; the second and third terms terms are the forces discussed in Paper #1 [1]: the force Bβαµβ≡Fα EM due to the tidal coupling of the electromagnetic field to the magnetic dipole moment, and the third, −HβαSβ≡Fα G, is the Mathisson-Papapetrou spin-curvature force. The last two terms are the force exerted on the electric dipole, consisting of a tidal term Eαβdβgoverned by the electric tidal tensor, and of a non tidal term Fα βDdβ/dτ. Analogy based on tidal tensors.— The force equation (5.60) manifests the physical analogy Bαβ ←→ Hαβ we found in Paper #1 (now being extracted from the rigorous, fully covariant equations of motion): both the electromagnetic force on a magnetic dipole and the gravitational force on a gyroscope are determined by a contraction of the spin/magnetic dipole 4-vector with a magnetic type tidal tensor. Bαβ is a covariant derivative, keeping Uαfixed (covariantly constant), of the magnetic field Bα=?Fα βUβmeasured by the test particle: Bαβ =Bα;β|U=const.; i.e., it is a derivative of Bαas measured in the inertial frame momentarily comoving with the particle. Precession analogy based on GEM fields.— Another exact analogy arises from the spin evolution equation (5.61). Take now, for simplicity, purely magnetic dipoles (i.e., dα= 0); if the Mathisson-Pirani condition SαβUβ= 0 holds, Eq. (5.61) can be written as DFSµ dτ =µαβνUνµαBβ,(5.62) where Bαis the magnetic field as measured by the test particle, Bα=?F αβUβ, and DF/dτ denotes the Fermi-Walker covariant derivative (see e.g. [73, 7, 63] for an explanation of this derivative). This is the relativistic generalization of the the familiar textbook torque τ=~µ ×~ B. Consider now an orthonormal frame eˆαcomoving with the test particle, i.e. 64 5 The papers summarized and discussed U=eˆ 0. In such frame, Sˆ 0= 0 and Uˆα=δˆα ˆ 0, and equation (5.62) reduces to: DSˆ i dτ = (~µ ×~ B)ˆ i⇔dSˆ i dτ =~ S×~ Ω + ~µ ×~ Bˆ i(5.63) where ~ Ω is angular velocity of rotation of the spatial axes eˆ irelative to the tetrad FermiWalker transported along the particle’s CM worldline (may be interpreted as their rotation relative to a system of local comoving guiding gyroscopes, defining the so-called compass of inertia). This equation manifests the analogy ~ Ω↔~ B. If instead of a local tetrad one considers an extended frame, such that the time axis of the tetrads is tangent to a congruence of observers, and the spatial triads eˆ ico-rotate with the observers (this is set up by demanding ~ Ω = ~ω, where ~ω is the vorticity of the congruence), which is physically the most natural and relevant frame9, then ~ Ω becomes one half of the gravitomagnetic field ~ Hof the corresponding frame. The details on this are given in Sec. 3 of Paper #5 [5]. In this way we obtain a generalized version (now valid for arbitrary fields) of the analogy in Sec. (3.1.2) dSˆ i dτ =1 2~ S×~ H+~µ ×~ Bˆ i (5.64) Momentum of the particle: analogy based on GEM fields.— The momentum of the particle is not parallel to its 4-velocity, it is said to possess “hidden momentum”, Pα hid = (hU)α βPβ. Contracting (5.61) with Uα, and using the spin condition SαβUβ= 0, one obtains an expression for Pα: Pα=Pα kin +Pα hidI +Pα hidEM; Pα kin =mUα;Pα hidEM ≡α βγδµβEγUδ;Pα hidI ≡ −α βγδSβaγUδ.(5.65) Thus the hidden momentum consists of two parts: the “inertial” one Pα hidI discussed in Sec. 5.4.5 above, and which is pure gauge, and another part Pα hidEM that arises in when an electromagnetic field is present (albeit being purely mechanical in nature, see e.g. the model in Fig. 9 of [100]). Eαis the electric field as measured in the particle’s CM frame; and −aα=Gαis the gravitoelectric field in this frame, see Eq. (3.18) above. Thus we have another exact analogy based on GEM fields: the inertial hidden momentum is analogous to the “electromagnetic” hidden momentum, with ~ Splaying the role of ~µ, and the gravitoelectric field playing the role of the electric field. To make it more explicit, we write the momentum in the particle’s CM frame (where Ui= 0), and in vector notation (P0 hid = 0): ~ Phid =~ P=−~ S×~a +~µ ×~ E=~ S×~ G+~µ ×~ E . (5.66) 9It is this type of frame that is useful in the experimental detection of gravitomagnetism, such as in the Gravity Probe B mission [53], where one needs to define a frame whose axes are everywhere fixed to the “distant stars”. For shear-free congruences, this amounts to say that the triads eˆ ipoint to fixed neighboring observers, as is the case of the frames defined in Sec. 3.1.2 for the case of stationary spacetimes. 65 5 The papers summarized and discussed Table 5.2: Analogy between the electromagnetic force on a magnetic dipole and the gravitational force on a gyroscope Electromagnetic Force Gravitational Force on a Magnetic Dipole on a Spinning Particle Fβ EM =Bβ αµα; (5.2.1a) Bα β≡?Fα µ;βUµ Eqs. Magnetic TT Bα α= 0 (5.2.2a) B[αβ]=1 2? Fαβ;γUγ−2παβσγjσUγ(5.2.3a) BαβUα= 0; BαβUβ=βγ αδE[βγ]Uδ(5.2.4a) Fβ G=−Hβ αSα; (5.2.1b) Hα β≡?Rα µβνUµUν Eqs. Gravitomagnetic TT Hα α= 0 (5.2.2b) H[αβ]=−4παβσγJσUγ(5.2.3b) HαβUα=HαβUβ= 0 (5.2.4b) Note that this analogy holds for the Mathisson-Pirani condition, with other spin conditions Pα hidI has a different form, cf. Eq. (5.54). Mass of the particle. — The “proper mass” of the particle is defined as the scalar m= −PαUα, and represents the energy of the particle as measured in the frame where its center of mass is at rest. It is conserved in a purely gravitational field, if the Mathisson-Pirani condition holds; but it is not conserved in general in the presence of an electromagnetic field. Using (5.60), we obtain (see Paper #4 for details) dm dτ =−µγ DBγ dτ +Eγ Ddγ dτ .(5.67) The first term is essentially10 the rate of work done on the magnetic dipole by Faraday’s law of induction, already discussed in Sec. 5.2.2 above (see figure therein). We shall see below that if the test particle is a rigid spinning particle, this corresponds to a variation of kinetic energy of rotation. The second term corresponds to the work done on the magnetic dipole by the electric field when the dipole vector varies, e.g., when the dipole rotates. Note the important difference between the two terms: the first is non-zero only when the magnetic field varies along the particle’s worldline; the second has nothing to do with induction effects, it is non-zero only when the dipole varies (regardless of the variation of the electric field). 5.5.2 Dynamical implications of the symmetries of the tidal tensors According to Table 5.2, both in the case of the electromagnetic force on a magnetic dipole, and in the case of the gravitational force on a gyroscope, it is the magnetic tidal tensor, 10If Pα hidEM = 0, then µµDBµ/dτ =BγαUαµγ=Fα EMUα; otherwise there is an extra term (quadratic in µ) originating from the electromagnetic hidden momentum. 66 5 The papers summarized and discussed as seen by the test particle of 4-velocity Uα, that determines the force exerted upon it. The explicit analogy in Table 5.2 is thus ideally suited to compare the two forces, because in this framework it amounts to comparing Bαβ to Hαβ. The most important differences between them are: i) Bαβ is linear in the electromagnetic potentials and vector fields, whereas Hαβ is not linear in the metric tensor, nor in the GEM “vector” fields (for a detailed discussion of this aspect, see Sec. 3 of [5]); ii) in vacuum,H[αβ]= 0 (symmetric tensor), whereas Bαβ is generically not symmetric, even in vacuum; iii) time components: Hαβ is a spatial tensor (with respect to the observer measuring it), whereas Bαβ is not. These last two differences, which are clear from equations (5.2.3)-(5.2.4), are the ones in which we are mostly interested in the present work. In this section we start by the physical consequences of the symmetries, and in the next section we discuss the time projections. Eq. (5.2.3a), in vacuum (jα= 0) reduces to B[αβ]=1 2?Fαβ;γUγ.(5.68) There is an important statement encoded in this equation that can be stated as follows: since it is the tensor Bαβ measured by the particle that yields the force Fα EM, whenever the particle sees a varying field (DFαβ/dτ 6= 0), Bαβ is non-vanishing (B[αβ]6= 0 ⇒Bαβ 6= 0) and therefore a force will be exerted on it (except possibly for very special orientations of ~µ). In particular, whenever a magnetic dipole moves in a non-homogeneous field, a force will be exerted on it (again, except for very special ~µ’s). In the gravitational case, since H[αβ]= 0, analogous effects to not occur, and therefore, even in non-homogeneous fields, there are velocity fields for which Hαβ = 0, i.e., for which gyroscopes feel no force. There are even geodesic motions for spinning particles. This is exemplified below. 5.5.2.1 Radial motion in Schwarzschild spacetime Consider a magnetic dipole in the field of a static point charge Q, and with a purely radial initial velocity Uα=U0(1,~v). The particle sees a varying field (DFαβ/dτ 6= 0); thus, by virtue of Eq. (5.68) it measures a non-vanishing tensor Bαβ, and therefore (except for the special case ~v k~µ) a net force will be exerted on the dipole; explicitly: F0 EM = 0; Fi EM =B[αi]µα=γQ r3(~v ×~µ)i.(5.69) This is unlike what one might naively expect, as the radially moving dipole sees a vanishing magnetic field Bα; taking the perspective of the frame comoving with the particle, this is explained through the laws of electromagnetic induction: the moving dipole “sees” a time-varying a electric field; by virtue of Eq. (5.68) (which is a covariant form for ∇× ~ B= ∂~ E/∂t), that will induce a curl in ~ B, i.e., an antisymmetric part in the magnetic tidal tensor Bαβ. For this configuration, actually Bαβ =B[αβ], i.e., the force Fα EM comes entirely from the antisymmetric part of Bαβ. Therefore, since Hαβ is symmetric (in vacuum), H[αβ]= 0, we expect, in the spirit of the analogy, the force to vanish in the analogous gravitational setup. This is exactly the 67 5 The papers summarized and discussed Figure 5.5: An illustration of the physical consequences of the different symmetries of the tidal tensors. A gyroscope dropped from rest in Schwarzschild spacetime will move radially along a geodesic towards the source, with no force exerted on it. A magnetic dipole in (initially) radial motion in a Coulomb field, by contrast, feels a force. Due to the hidden momentum, the force is approximately opposite to the acceleration! case. If one considers a gyroscope in radial motion in Schwarzschild spacetime, the force is exactly zero: Fα G=−HβαSβ= 0 This means that a gyroscope in radial motion moves along a geodesic (for instance, a gyroscope dropped from rest will fall into the singularity moving in a straight line). Finally (this is not the topic of this section, but is nevertheless interesting), we note, in the electromagnetic system, this counterintuitive consequence of the hidden momentum Pα hidEM: if one assumes ~µ =σ~ S, the acceleration is m0aα=B[αβ]µβ+α βγδ D dτ (Sβaγ)Uδ≈B[αβ]µβ=−Fα EM , approximately opposite to the force! 5.5.2.2 Equatorial motion in Kerr and Kerr-de-Sitter spacetimes We found another manifestation of the absence of a gravitational counterpart to the antisymmetric part of the magnetic tidal tensor B[αβ](i.e., of induction effects analogous to the electromagnetic ones) comparing the forces on gyroscopes in equatorial motions in Kerr and Kerr-de-Sitter spacetimes, to the ones of magnetic dipoles in equatorial motions in the field of a spinning charge. In the equatorial plane of the Kerr, and Kerr-dS spacetimes, there are observers for which Hαβ = 0; it is so when the observer’s angular velocity is vφ=Uφ Ut=a a2+r2≡vφ (H=0) .(5.70) 68 5 The papers summarized and discussed Figure 5.6: a) Equatorial plane of a spinning charge. Black arrows: velocity field ~v(B=0), which makes the magnetic field Bαvanish; magnetic dipoles in straightline motion, momentarily with such velocities, do not precess relative to the distant stars. b) Equatorial plane of Kerr spacetime. Black arrows: velocity field ~v(Ω?=0) for which a gyroscope in straightline motion momentarily does not “precess” (with respect to the distant stars); asymptotically it matches its electromagnetic counterpart. Red arrows: velocity field ~v(H=0) which makes the gravito-magnetic tidal tensor Hαβ vanish; this means that gyroscopes moving along trajectories tangent to ~v(H=0) feel no force, Fα G= 0. If Λ >0 (KerrdS spacetime) circular geodesics for gyroscopes do even exist. ~v(H=0) has no electromagnetic analogue: for a moving dipole always Bαβ 6= 0, by virtue of B[αβ]=?DFαβ/dτ,generically implying Fα EM 6= 0 . 69 5 The papers summarized and discussed This means that gyroscopes carried by such observers feel no force (regardless of the orientation of the spin vector ~ S). The angular velocity vφ (H=0) does not correspond to any circular geodesic for material particles in the Kerr spacetime; circular geodesics “are too fast”, their angular velocity dies off as r−3/2, whereas vφ (H=0) goes as r−2. But in Kerr-dS, the repulsive Λ “slows down” the circular geodesics, and makes possible the existence of circular geodesics for which Hαβ = 0. The angular velocity that makes Hαβ vanish in the equatorial plane of Kerr-dS is the same Eq. (5.70) above; and it is a rotation in the same sense of the black hole; thus the geodesics obeying this condition are found equating (5.70) to the equation for prograde circular geodesics in Kerr-dS, (vφ geo)+=−Ma +Λ 3ar3±qMr3−Λ 3r6 r3−a2M+Λ 3a2r3. We show the solutions to exist numerically; that is, in Kerr-dS, it is possible for gyroscopes to move along geodesic orbits (for arbitrary ~ S). This type of motions, and the velocity field (5.70) have no electromagnetic counterpart; due to the laws of electromagnetic induction, cf. Eq. (5.68), Bαβ, can never vanish for a moving particle (nor does it vanish for a particle at rest in this field), hence a force will always be exerted on the dipole (except for very special, fixed, orientations of ~µ). The formal analogy between the scalar invariants of Fand Ras a guide. — In our study of the equatorial motions, and in particular to find the velocity field (5.70) above we made use of the formal analogy between the electromagnetic invariants (3.22) and the quadratic invariants of the Weyl tensor, expressions (3.31) above. The electromagnetic invariants have the following physical interpretation [63, 18, 30]: 1. if EαBα6= 0, then the electric Eαand magnetic Bαfields are both non-vanishing for all observers; 2. if EαEα−BαBα>0 (<0) and EαBα= 0, there are observers uαfor which the magnetic field Bα(the electric field (Eα) is zero. In the case of the Weyl tensor (or the Riemann tensor, in vacuum) the quadratic invariants (3.31) are not sufficient for an analysis like the one we did for the electromagnetic case. Whereas the invariants (3.22) are the only two independent scalar invariants of Fαβ, in the case of Rαβγδ there are 14 independent invariants in general, which in vacuum reduce to four: the invariants (3.31), plus two cubic invariants, given by A≡1 16Rαβ λµRλµρσRρσ αβ, B ≡1 16Rαβ λµRλµρσ ? Rρσ αβ and usually combined in the complex quantity J≡A−iB. Define also the complex quantity I≡(R·R+i ? R·R)/8. It turns out (cf. [45, 46, 30]) that one obtains formally equivalent statements to 1-2 above, replacing Fby R,provided that the condition M≡I3/J2−6≥0 (real or infinite) is added to 2); that is: 70 5 The papers summarized and discussed 1. ?R·R6= 0 ⇒Eαγ and Hαγ are both non-vanishing for all observers; 2. ?R·R= 0, R·R>0, with M≥0⇒there are observers for which Hαγ vanishes (“Purely Electric” spacetime)11. Further details and comments on this classification shall be given in [30]. The analysis above is for vacuum, where C=R; in order to obtain similar statements valid generically, one only has to replace Rby C. The invariant structure of the electromagnetic of a spinning charge (charge Q, magnetic moment µs) is:          ~ E2−~ B2=Q2 r4−µ2 s(5 + 3 cos 2θ) 2r6>0, ~ E·~ B=2µsQcos θ r5(= 0 in the equatorial plane) (5.71) which tells us that in the equatorial plane θ=π/2 there are observers measuring the magnetic field Bα(not the tidal tensor!) to be locally zero, since ~ E·~ B= 0 and ~ E2−~ B2>0 therein. The angular velocity of such observers is Uφ/Ut=µs/(Qr2). If we additionally assume that the charge and mass are identically distributed in the body, its gyromagnetic ratio is µs/J =Q/2M, and we obtain the angular velocity vφ=Uφ Ut=J 2Mr2≡vφ (B=0) ,(5.72) which asymptotically matches, up to a factor of 2, the velocity (5.70) above. This analogy proves illuminating for the gravitational problem. In the case of Kerr spacetime, which is of Petrov type D, the condition M≥0 (real) is satisfied, since I3= 6J2 (see e.g. [46]). Thus one only has to worry about the invariants (3.31), which have the structure:          EαγEαγ −HαγHαγ ≈6M2 r6>0 EαγHαγ ≈18JM cos θ r7(= 0 in the equatorial plane) (5.73) formally analogous to its electromagnetic counterpart (5.71). Note in particular that the result EαγHαγ = 0 for the equatorial plane (θ=π/2) is exact. It was in this way that we concluded that in the equatorial plane,there are observers for which the magnetic tidal tensor vanishes, whose angular velocity we found to be given by Eq. (5.70)above (see Paper #4 [4] for more details). At this point it is important to stress the following: in spite of the striking similarities in the invariant structures (5.71) and (5.73), and in the velocity fields (5.72) and (5.70), 11The case ?R·R= 0, ?R·R<0would mean that there would be observers for which Eαγ vanishes (but no “Purely Magnetic” vacuum solutions are known, and it has been conjectured that they do not exist, see e.g. [111, 112]). 71 5 The papers summarized and discussed this is a purely formal analogy; in one case we are talking about velocities for which the magnetic field Bαvanishes, in the other about the vanishing of the gravito-magnetic tidal tensor Hαβ. The physical effects are actually opposite: in the first case magnetic dipoles with such velocities do not undergo Larmor precession, D~ S/dτ = 0, cf. Eq. (5.62), but they feel a force Fα EM 6= 0 (Bαβ never vanishes, as discussed above). In the gravitational case, the gyroscope feels no force: Fα G= 0, but it precesses (relative to the local comoving tetrad non-rotating relative to the distant stars): ˜ D~ S/dτ =~ S×~ H, cf. Eq. (5.63). For completeness, we also investigated the physical gravitational counterpart of the velocity for which magnetic dipoles do not precess; it involves some subtleties (it comes down to finding the velocity that a gyroscope, momentarily in “straightline” motion, must have in order to not precess relative to the distant stars) but asymptotically it matches (5.72). 5.5.3 Time projections of the forces and work done on a test particle A fundamental difference between the gravitational and electromagnetic interactions concerns the time projections of the forces Fα Gand Fα EM in the different frames. These encode the work done by the force in the given frame. In order to understand it, and its relation with the particle’s energy, consider a congruence of observers O(u) with 4-velocity uα, and let Uαdenote the 4-velocity of a test particle. The following relation generically holds [25]: Uα=γ(uα+vα); γ≡ −uαUα=1 √1−vαvα ,(5.74) where vα=Uα/γ −uαis the velocity of the test particle relative to the observers O(u); in the frame ui= 0, viis the ordinary 3-velocity. The energy of the test particle relative to O(u) is E≡ −Pαuα, and its rate of change (the “power equation”) dE dτ =−Fαuα−Pαuα;βUβ,(5.75) where Fα≡DPα/dτ denotes the 4-force. Thus we see that the variation energy of the particle relative to O(u) consists of two terms: the time projection of Fαalong uα, plus a term depending on the variation of uαalong the test particle worldline. The first term is interpreted as the rate of work, as measured by O(u), done by the force on the test particle. Using uα=Uα/γ −vα, we can write it as −Fαuα=−FαUα γ+Fαvα.(5.76) In the simplest case of a point particle with no internal structure (a monopole particle) the first term is zero, since the momentum is parallel to the 4-velocity (Pα=mUα), and its mass is a constant, m=m0; hence the force is parallel to the acceleration and orthogonal to Uα. Such force is said to be spatial with respect to Uα. Thus −Fαuα=Fαvα, telling us that the time-projection of the 4-D force Fαis the familiar power (i.e., the rate of 72 5 The papers summarized and discussed work per unit of proper time τ) transferred to the particle by the 3-D force (hU)α µFµ(see e.g. [61, 114]). If the frame is inertial, so that the second term of (5.75) vanishes, then −Fαuα=dE/dτ =m0dγ/dτ, i.e., Fαvα=m0dγ/dτ is the rate of variation of kinetic energy of translation of the particle’s center of mass. This is the type of force we are more familiar with; an example is the Lorentz force on a charged particle, DPα/dτ =qFαβUβ, whose projection along uαreads −uαDPα/dτ =γqvαEα, yielding the rate of work (per unit proper time) done by the electric field on the particle moving with velocity vαrelative to O(u). However, if the particle has internal structure, as in the problem at hand (spinning multipole particles), its internal degrees of freedom may store energy, which in general will be exchanged with the energy of the external fields and the kinetic energy of the center of mass. Therefore, the proper mass of the particle m=−PαUαno longer has to be a constant, cf. Eq. (5.67). Also the momentum is not be parallel to Uα, as the particle in general possesses hidden momentum, cf. Eq. (5.65). These, together, endow Fαwith a nonvanishing time projection: FαUα6= 0. Let us turn our attention now to the second term of Eq. (5.75). Decomposing (e.g. [25, 5, 34]) uα;β=−a(u)αuβ+ωαβ +θαβ ,(5.77) where a(u)α=uα ;βuβis the acceleration of O(u) (not the particle’s!), ωαβ ≡(hu)λ α(hu)ν βu[λ;ν] the vorticity, and θαβ ≡(hu)λ α(hu)ν βu(λ;ν)the total shear of the congruence (θαβ ≡σαβ + θ(hu)αβ/3, with σαβ as usual the traceless shear and θthe expansion scalar). G(u)α= −a(u)αis thus the gravitoelectric field measured in the frame ui= 0. Decomposing Pα=mUα+Pα hid, and decomposing Uαusing (5.74), the second term of Eq. (5.75) becomes: −Pαuα;βUβ=mγ2[G(u)α−θαβvβ]vα +γPα hid hG(u)α−(ωαβ +θαβ)vβi.(5.78) This part of dE/dτ depends only on the kinematical quantities of the congruence. That is, unlike the term (5.76), which arises from the 4-force Fα, the term (5.78) does not depend on any physical quantity one can locally measure; it is locally an artifact of the reference frame, which can always be made to vanish by choosing a locally inertial one. Its importance (in a non-local sense) should not however be overlooked. To understand it, consider a simple example, a monopole particle in Kerr spacetime, from the point of view of the congruence of static observers (i.e., uαparallel to the time-like Killing vector field ξ≡∂/∂t, in Boyer Lindquist coordinates). Since the congruence is rigid, θαβ = 0; also, for a monopole particle, Pα hid = 0, and, in a gravitational field, Fα= 0 (the particle moves along a geodesic). Therefore, the energy variation reduces to dE/dτ = −Pαuα;βUβ=mγ2G(u)αvα; which is the rate of “work” (per unit proper time τ) done by the gravitoelectric “force” [19, 5, 25] mγ2G(u)α. (In the Newtonian limit, reduces to the work of the Newtonian force m~ G.) Hence we see that (5.78) is the part of (5.75) that encodes the change in translational kinetic energy of a particle (relative to static observers) 73 5 The papers summarized and discussed in the frame Ui= 0, also coincides with the canonical angular momentum obtained by differentiating the non-relativistic Lagrangian, ∂L/∂~ω (cf. Eq. (31) of [109]). Having this issue clarified, one computes the physical torque — that is, the Fermi-Walker derivative of the physical angular momentum vector Sα, τα≡DFSα dτ ⇒τσ=1 2σδ αβ Uδ DSαβ dτ ,(5.95) by subtracting the contribution of S0αβ from Eq. (5.91) above. It reads: τσ=σ αβνUνµαBβ+σ αβ m[αρµFβ]µ;ρ−1 2σ αβFα γ Dqβγ dτ +τσ ind ,(5.96) where the first term is the dipole torque, cf. Eq. (5.62), and the next terms are quadrupole contributions; the second term is the one that vanishes for a spherical body (thus not the one we are interested in), and the fourth term is the result we were looking for, τα ind =1 2σ µνE[µν]qα σ−δα σqγγ,(5.97) the torque due to the laws of electromagnetic induction, governed by the antisymmetric part of the electric tidal tensor. Indeed in the Lorentz frame momentarily comoving with the particle, we can write τi ind =−1 2(∇× ~ ECM)jqij−δijqγγ where (∇× ~ ECM)jis the curl of the electric field at the particle’s center of mass. This is precisely what one obtains computing explicitly the torque, to quadrupole order, from the integral ~τind =´ρc~r ×~ Eindd3x, see Paper #4 [4] for details. Using Eqs. (5.6), we write (5.97) in the equivalent form τα ind =1 2Bσ βUβqα σ−δα σqγγ.(5.98) Now, if the spinning body is “quasi-rigid”, we have µα=Ωβ 2δα βqγγ−qα β(5.99) where Ωαis the body’s angular velocity relative to a system of comoving Fermi-Walker transported axes. Therefore, the rate of work done on this body by the induction torque τα ind,P=τα indΩα, is: τα indΩα=−Bα βUβµα=−Fα EMUα(5.100) i.e., it equals the time projection, in the particle’s proper frame, of the dipole force Fα EM (in other words, the work done by the dipole force, as measured in the particle’s frame). This confirms that the work transferred to the particle by Faraday’s induction, that we 80 5 The papers summarized and discussed Figure 5.8: a) A spinning, positively charged spherical body being pulled by a strong magnet; ~ Eind ≡electric field induced in the body’s CM frame. b) A spinning spherical body falling into a Kerr Black hole. As the spinning charge moves in the inhomogeneous magnetic field ~ B, a torque τα ind, Eq. (5.97), is exerted on it due to ~ Eind, i.e. to the antisymmetric part E[αβ].That causes S≡√SαSα, and the body’s angular velocity Ω = S/I to vary. τα ind does work at a rate τα indΩα=Pind, which exactly matches time projection of the dipole force Fα EM. That causes the kinetic energy of rotation about the CM to decrease, manifest in a variation of proper mass dm/dτ, and cancels out the gain in translational kinetic energy, so that the total work transfer is zero (cf. Sec. 5.5.3.2). In the gravitational case no analogous induction effects occur (as expected since E[αβ]=HαβUβ= 0): no torque is exerted on the spinning particle, its angular momentum S, angular velocity Ω, and proper mass m, are constant; and there is a net work done on it by Fα Gat a rate Ptot =−Fα Guα, corresponding to an increase of translational kinetic energy. 81 5 The papers summarized and discussed discussed in Sec. 5.5.3.1, is indeed associated to a torque, which causes S2to vary as expected (since τα ind is not orthogonal to Sαin general). It is also associated with a variation of kinetic energy of rotation. In order to more easily see that, consider now a spherical body, so that the second term in (5.96) vanishes; and a configuration where the electromagnetic hidden momentum ~ PhidEM =~µ ×~ Evanishes, for example the setup in Fig. 5.8a), which causes (see Paper #4 for details) the third term of (5.96) to vanish also; in this case the total torque reduces to τσ=σ αβνUνµαBβ+τσ ind .(5.101) Assuming µα=σSα, and since Sα=IΩα(I≡moment of inertia of the sphere with respect to an axis passing through its center), it follows that the power of the total torque is τσΩσ=τσ indΩσ. Since also, from Eq. (5.95), τσ=IDFΩσ/dτ, we have that the power of τα ind equals the rate of variation of the particle’s rotational kinetic energy, IΩ2/2: τα indΩα=τσΩσ=1 2 dΩ2 dτ I . Thus we can write 1 2IdΩ2 dτ =−Fα EMUα=−DBα dτ µα, where in the last equality I used again Pα hidEM = 0. That is, for this setup, the variation of the body’s kinetic energy of rotation is the projection, along its worldline, of the dipole force Fα EM . And finally, this result confirms also that (for a purely magnetic dipole, dα= 0) the variable part of the particle’s mass that we obtained in the dipole approximation, Eq. (5.67), is kinetic energy of rotation (not potential energy, as claimed in some literature, e.g. [81, 76, 77]) — this confirms, in a relativistic covariant formulation, and in the context of Dixon’s multipole approach, the claims in [107, 108, 109, 113]. 5.5.4.2 Gravitational torque The equation for the spin evolution of an extended spinning body in a gravitational field is, up to quadrupole order [106, 66], DSκλ dτ = 2P[κUλ]+4 3Jµνρ[κRλ] ρµν (5.102) leading to τα≡DFSσ dτ =4 6Jµνρ[κRλ] ρµνσδ κλ Uδ,(5.103) where Jαβγδ is a quadrupole moment of the energy-momentum tensor Tαβ, see Paper #4 for more details. Our goal herein is to consider the gravitational analogue of the problem in Sec. 5.5.4.1. Therein we considered a spherical charged body in flat spacetime; we prescribe the analogous test body for the gravitational problem by demanding it to have an analogous multipole structure (i.e., its “gravitational skeleton” [61]) to its electromagnetic counterpart (rather than demanding its shape to be “spherical”, which in a general curved 82 5 The papers summarized and discussed spacetime is not a well defined notion. More precisely: in a local orthonormal tetrad ˆeα, such that ˆe0=Uα(i.e., the triad ˆeispans the rest space of the center of mass), the moments of Jαare the same as for a sphere in flat spacetime (thus have the same structure as the moments of jαin the electromagnetic problem above). The moments of the space part Thαihβiare negligible to a good approximation. For this type of body we have (see Sec. VIB of Paper #4 [4] for details) τσ= 0 , the equality holding for vacuum (Rµν = 0), which (as in the electromagnetic case) is the problem at hand. Thus, no gravitational torque is exerted, up to quadrupole order, in a spinning spherical body. Therefore, as expected from the discussion in the previous sections, there is no gravitational counterpart to the torque τα ind that comes from the antisymmetric part of the electric tidal tensor Eαβ (or, equivalently, from the time projection BαβUβ), and which is due to Faraday’s law of induction. The comparison with the electromagnetic analogue makes this result natural, since the gravito-electric tidal tensor Eαβ is symmetric, and the gravitomagnetic tidal tensor Hαβ is spatial, meaning that the dynamical effects which, in electromagnetism, are caused by the curl of the electric field ~ E, have no counterpart in the physical gravitational forces and torques. 5.5.4.3 Summarizing with a simple realization The main ideas in Secs. 5.5.3 and 5.5.4 can be summarized in the example of analogous systems in Fig. 5.8: a spinning spherical charge moving in the field of a strong magnet (or another spinning charged body), and a spinning “spherical” mass moving in Kerr spacetime. Starting by the electromagnetic system, a force Fα EM, Eq. (5.2.1a) of Table 5.2, will be exerted on the particle, causing it to move and gain translational kinetic energy. And as it moves in an inhomogeneous magnetic field, a torque τα ind is exerted upon it, due, from the the viewpoint of the observer comoving with the particle, to the electric field induced by the time-varying magnetic field. That torque will cause a variation of angular momentum Sα, and therefore of the angular velocity Ωα=Sα/I of the particle (measured with respect to the comoving Fermi-Walker transported tetrad). Clearly, as we see from Eq. (5.101), S2is not conserved (as would be the case in a pole-dipole approximation, see Eq. (5.62)). The variation of the magnitude Ω of the angular velocity also implies a variation of rotational kinetic energy of the particle; that variation is the projection of Fα EM along the particle’s worldline, and is reflected in a variation of proper mass dm/dτ. With respect to the “static observers” uα, the variation of rotational kinetic energy is exactly canceled out by the variation of translational kinetic energy, ensuring that a static magnetic field does not do work, so that Fα EMuα= 0, and the total energy of the particle, E=−Pαuα, is conserved. In the gravitational case, there is also a net force Fα Gon the body, cf. Eq. (5.2.1b) of Table 5.2, causing it to gain kinetic energy at a rate Ptrans =Fi Gvi. But no torque is exerted on it; up to quadrupole order we have: DFSα dτ = 0; S2= constant 83 5 The papers summarized and discussed (i.e., the spin vector of the spinning spherical mass is Fermi-Walker transported), implying also Ω = constant. This is consistent with the constancy of the proper mass (and the fact that Fα Gis spatial, meaning that no work is done by induction), because, since there is no torque, the kinetic energy of rotation is constant. Thus in this case, from the point of view of the static observers, the gain in translational kinetic energy is not canceled out by any variation of rotational kinetic energy, and therefore the stationary gravitomagnetic (tidal) field does a net work on the particle. 5.5.5 Conclusion In this work we explored the exact gravito-electromagnetic analogies in the equations of motion for spinning particles in gravitational and electromagnetic fields, that were seen to arise when the Mathisson-Pirani spin condition is employed. In special detail we explored the analogy based on tidal tensors for the force equations, that was put forth in Paper #1 [1]. We also studied the analogy for the spin precession, based on GEM fields, and, also based on it, found a new one, for the particle’s hidden momentum. A point that it is never too much to emphasize is that the existence of these exact analogies does not mean that the interactions are similar. These are functional analogies: Bαγ plays in the equation (5.2.1a) for the force exerted on a magnetic dipole the same role as Hαγ in Eq. (5.2.1b) for the gravitational force exerted on a gyroscope; also, in the appropriate frame, the gravitomagnetic field ~ Hplays in the “precession” of the gyroscope a role analogous to ~ Bin the precession of a magnetic dipole, cf. Eq. (5.64). But the analogies do not imply that these objects themselves are similar, in fact they are in general very different even in seemingly analogous setups (we give in this work many examples of that). The exact analogies are suited instead for a comparison between the interactions, as it amounts to comparing mathematical objects that play analogous dynamical roles in both theories. Such comparison unveils suggestive similarities, useful in terms of the intuition they provide. But, and especially in the case of the tidal tensor analogy, it was the differences it makes transparent that proved particularly illuminating. We had found in Paper #1 [1] that the key differences, in terms of tidal forces, between gravity and electromagnetism, are the fact that Eαβ and Hαβ are symmetric (the latter in vacuum) whereas their electromagnetic counterparts Eαβ and Bαβ are not. These differences were seen to be related with the phenomenon of electromagnetic induction, and the way it manifests itself in the electromagnetic tidal forces, which has no analogue in gravity. In this work we explored the physical consequences for the dynamics of test particles. The results in Sec. 5.5.3, concerning the time components of the force, and in Sec. 5.5.4, concerning the torque exerted on the spinning particle, are manifestations of the absence of a gravitational counterpart to the antisymmetric part of Eαβ (or, equivalently, to the projection of Bαβ along Uα); E[αβ]encodes the Maxwell-Faraday law ∇×~ E=−∂~ B/∂t; the gravitoelectric tidal tensor, by contrast, is symmetric: E[αβ]= 0, translating in an absence of analogous induction effects in the physical gravitational forces and torques. And the results in Sec. 5.5.2, showing that in a non-homogeneous gravitational field there are moving observers for which Hαβ = 0, so that gyroscopes can actually move 84 5 The papers summarized and discussed along radial or circular geodesics (in Schwarzschild and Kerr-dS spacetimes, respectively), manifest that there is no gravitational analogue the antisymmetric part B[αβ], encoding Maxwell Eq. ∇× ~ B=∂~ E/∂t. In electromagnetic systems, due to this law (more precisely, in covariant form 2B[αβ]=?Fαβ;γUγ), Bαβ is non-vanishing whenever the dipole “sees” a varying field (as is the case when the particle moves in a non-homogeneous field), and therefore (except for some special orientations of ~µ) an electromagnetic force Fα EM 6= 0 is exerted on it. We have studied in detail the work done by the fields on the particle from the point of view of different frames, which is an important physical content encoded in the time projections of the forces; and its relation with the particle’s energy (in the given frame) and proper mass. For that we needed to generalize the power law existing in the literature, extending it to the case of particles with multipole structure (which possess hidden momentum and varying mass). An interesting reciprocity was found to exist: in a frame comoving with the particle, the electromagnetic (but not the gravitational) field does work on it, causing a variation of its proper mass; conversely, for “static observers”, a stationary gravitomagnetic (but not a magnetic) field does work on the particle. We shown that there is actually a potential energy associated with this work which embodies the Hawking-Wald spin-spin interaction energy [10] (that had been found to exist in the special case of an axial fall in a Kerr black hole). In the course of this work a number of issues had to be clarified, the first of them being the equations of motion themselves, both to dipole and quadrupole order, and the physical meaning of the quantities involved. In particular the misconceptions in the literature regarding the problem of the spin supplementary condition, and the difficulties in the electromagnetic part of the equations. Some of this problems are briefly reviewed in Sec. 3.2 above (for a more comprehensive summary I refer the reader to the conclusion of Paper #4). 5.6 Paper #5 — Gravito-electromagnetic analogies This work has two main goals: 1) establish the connection between the several gravitoelectromagnetic analogies existing in the literature, and in particular between the tidal tensors and the exact GEM fields; 2) further develop these two approaches. As for the approach based on tidal tensors, we complete the tidal tensor formulation of the gravitational field equations started in Paper #1 [1]. Using the time and space projectors, we do a full splitting of the gravitational field equations (the Einstein equations with sources, plus the algebraic Bianchi identities), obtaining six equations (5 independent), four of which are the ones first derived in [1], which are analogous to Maxwell’s equations in this formalism, and two additional ones with no electromagnetic counterpart that are not given in [1]. And we add to the list of analogies in this formalism the one we found to exist in the “differential precession” of gyroscopes/magnetic dipoles. As for the analogy based on exact GEM fields, we take its most general form in the literature [25], valid for arbitrary fields, reformulate and further generalize it for arbitrary frames. We discuss in detail the inertial forces that arise in the different frames, and the 85 5 The papers summarized and discussed origin of the GEM fields; and we derive a general expression for them which generalizes the previous results in terms of an arbitrary transport law for the spatial frame. The gravitomagnetic field (i.e., the field that yields the Coriolis-like “acceleration” in the geodesic equation) is seen to consist of two contributions of independent origin: the vorticity of the observer congruence, and the rotation of the spatial triads relative to Fermi-Walker transport. This definition encompasses the many gravitomagnetic fields that have been defined in the literature. As for the field equations, we do, again, a full splitting of the gravitational and electromagnetic equations, and express them in this formalism. It turns out that from the set of six gravitational equations, four are seen to exhibit many similarities with the electromagnetic equations (that is, it is not only in the tidal tensor formalism; the similarity occurs in this formalism as well). Restricting the approach to stationary fields, we obtain on the gravitational side the “quasi-Maxwell” field equations of Sec. 3.1.2 above; and in the electromagnetic side, the equations for the electric and magnetic fields in the analogous situation: for arbitrarily accelerated and rotating frames (not in Lorentz frames, as is the usual comparison in the literature), which unveils a much closer analogy. We also build up on the work in [19] — where an analogy was found between the gravitational force on a gyroscope written in terms of GEM fields, Eq. (3.13), and the electromagnetic force on a magnetic dipole at rest in a Lorentz frame, Eq. (3.14) — by adding the corresponding electromagnetic force in the analogous conditions, i.e., in terms of the fields measured in the arbitrarily accelerating and rotating frame where the particle is at rest. Again the analogy is seen to get strikingly closer. 5.6.1 Analogy based on tidal tensors In Paper #5 [5] we revisited the analogy based on tidal tensors introduced in Paper #1 [1], and completed the tidal tensor formulation of the gravitational field equations. In [1] it was shown that, by taking the traces and antisymmetric parts of the electromagnetic tidal tensors, one obtains the Maxwell equations, and performing the same operations on the gravitational tidal tensors leads to a strikingly similar set of equations, which turn out to be some projections of the gravitational field equations. In [5], using a more robust approach, we extend this formalism to the full gravitational field equations — Einstein equations with sources plus the algebraic Bianchi identities, Rαβ = 8πTαβ −1 2gαβTγ γ(a); ?Rγα γβ = 0 (b),(5.104) which I summarize in what follows. Using the time and space projections with respect to a unit time-like vector Uα, >α β≡(>U)α β=−UαUβ;hα β≡(hU)α β=UαUβ+δα β.(5.105) we first do a full decomposition of the Riemann tensor Rαβγδ =>αρ+hα ρ... >δσ+hδσRρ..σ , 86 5 The papers summarized and discussed obtaining17 Rαβ γδ = 4E[α [γUδ]Uβ]+ 2 nµχ γδUχH[β µUα]+µαβχUχHµ[δUγ]o +αβφψUψµν γδUνFφµ.(5.106) This equation tells us that the Riemann tensor decomposes, with respect to Uα, in three spatial tensors: the gravitoelectric tidal tensor Eαβ, the gravitomagnetic tidal tensor Hαβ, plus a third tensor Fαβ ≡?R ?αγβδ UγUδ=µν αγλτ βδRµνλτ UγUδ, introduced by Bel [138], which encodes the purely spatial curvature with respect to Uα, and has no electromagnetic analogue. Eαβ and Fαβ are symmetric (and spatial), and therefore have 6 independent components each; Hαβ, is traceless (and spatial), and so has 8 independent components. Therefore these three tensors together encode the 20 independent components of the Riemann tensor. Substituting decomposition (5.106) and its Hodge dual in Eqs. (5.104a), and decomposing in time and space projections, we obtain, respectively, the time-time, time-space, and space-space projections: Eα α= 4π(2ρm+Tα α) ; (5.107) H[στ]=−4πλστγJλUγ; (5.108) Fα β+Eα β−Fσ σhα β= 8π1 2Tγ γhα β−Thαi hβi(5.109) (since Eq. (5.104a) is symmetric, these are the only non-trivial projections). Here Jα≡ −TαβUβ,ρm≡TαβUβUαare, respectively, the mass/energy current and density, as measured by an observer of 4-velocity Uα; and Thλi hθi≡hλ δhβ θTδ β. Repeating the procedure in Eqs. (5.104b), we obtain the time-time (which is the same as the space-space), time-space, and space-time projections, respectively: Hα α= 0; (a) F[αβ]= 0; (b) E[αβ]= 0 (c).(5.110) Turning now to the electromagnetic field equations (source equations, plus Bianchi identity), Fαβ ;β= 4πjα(a); ?F αβ ;β= 0 (b),(5.111) we decompose Fαβ;γand its dual in terms of the electromagnetic tidal tensors, Fαβ;γ= 2U[αEβ]γ+αβµσUσBµ γ; (5.112) ?Fαβ;γ= 2U[αBβ]γ−αβµσUσEµ γ.(5.113) 17The characterization of the Riemann tensor by these three spatial rank 2 tensors is known as the “Bel decomposition”, even though the explicit decomposition (5.106) is not presented in any of Bel’s papers (e.g. [138]). To the author’s knowledge, an equivalent expression (Eq. (4.6) therein) can only be found in [144]. 87 5 The papers summarized and discussed Table 5.3: Tidal tensor formulation of the electromagnetic and gravitational field equations. Electromagnetism Gravity Maxwell Source Equations Einstein Equations Fαβ ;β= 4πJβRµν = 8πTµν −1 2gµνTα α •Time Projection: •Time-Time Projection: Eα α= 4πρc(5.3.3a) Eα α= 4π(2ρ+Tα α) (5.3.3b) •Space Projection: •Time-Space Projection: B[αβ]=1 2?Fαβ;γUγ−2παβσγ jσUγ(5.3.6a) H[αβ]=−4παβσγJσUγ(5.3.6b) •Space-Space Projection: No electromagnetic analogue Fα β+Eα β−Fσ σhα β= 8πh1 2Tγ γhα β−Thαi hβii(5.3.7) Bianchi Identity Algebraic Bianchi Identity ?Fαβ ;β= 0 (⇔F[αβ;γ]= 0 ) ?Rγα γβ = 0 (⇔R[αβγ]δ= 0) •Time Projection: •Time-Time (or Space-Space) Proj: Bα α= 0 (5.3.5a) Hα α= 0 (5.3.5b) •Space Projection: •Space-Time Projection: E[αβ]=1 2Fαβ;γUγ(5.3.4a) E[αβ]= 0 (5.3.4b) •Time-Space Projection: No electromagnetic analogue F[αβ]= 0 Then, substituting in Eqs. (5.111), and splitting in the time and space projections, we obtain the set of four electromagnetic equations already presented in Paper #1 [1], Eqs. (5.1.3a)-(5.1.6a) of Table 5.1, and that we summarize again in Table 5.3. That is, the time and space projections of (5.111a) are, respectively Eqs. (5.3.3a) and (5.3.6a) of Table 5.3. The same procedure applied to Eq. (5.111b) yields Eqs. (5.3.4a) and (5.3.5a) as the time and space projections, respectively. We re-write them in the form Eα α= 4πρc; (5.114) E[αβ]=U[αEβ]γUγ+1 2αβµσUσBµγUγ; (5.115) Bα α= 0 ; (5.116) B[αβ]=U[αBβ]γUγ−1 2αβµσUσEµγUγ−2παβσγjσUγ,(5.117) to note that indeed Maxwell’s equations may be cast as algebraic equations involving only the two tidal tensors and the sources. 88 5 The papers summarized and discussed Thus, as illustrated in Table 5.3, the gravitational field equations consist of four equations with an electromagnetic analogue, plus two equations — the space-space projection of (5.104a), Eq. (5.109), and the time-space projection of (5.104b), Eq. (5.110b), which have no electromagnetic analogue. However this is not a set of six independent equations, as Eqs. (5.110b), (5.110c) and (5.109) are not independent; using the latter, together with (5.110b)/(5.110c), one can obtain the remaining one, (5.110c)/(5.110b). Eq. (5.109) involves, as a source, the space-space part of the energy momentum tensor, Thαihβi, which, unlike the energy current 4-vector Jα=−TαβUβ(analogous to the charge current 4-vector jα) has no electromagnetic counterpart. It has a fundamental difference18 with respect to the other gravitational field equations in Table 5.3 (and their electromagnetic analogues): the latter are algebraic equations involving only the traces and antisymmetric parts of the tidal tensors (or of Fαβ), plus the source terms; they impose no condition on the symmetric parts. But Eq. (5.109), by contrast, is an equation for the symmetric parts of the tensors Eαβ and Fαβ. It can be split in two parts. Taking the trace, and using (5.107), one obtains the source equation for Fαβ: Fσ σ= 8πρ ; (5.118) substituting back in (5.109) we get: Fα β+Eα β= 8πhα β1 2Tγ γ+ρ−Thαi hβi.(5.119) This equation tells us that the tensor Fα βis not an extra (comparing with electrodynamics) independent object; given the sources and the gravitoelectric tidal tensor Eαβ,Fαβ is completely determined by (5.119). In vacuum (Tαβ = 0, jα= 0), the Riemann tensor becomes the Weyl tensor: Rαβγδ = Cαβγδ; due to the self duality property of the latter: Cαβγδ =−? C?αβγδ, it follows that Fαβ =−Eαβ. The gravitational field equations are summarized and contrasted with their electromagnetic counterparts in Table 5.3. Eqs. (5.3.3b)-(5.3.6b) are very similar in form to Maxwell Eqs. (5.3.3b)-(5.3.6b); they are their physical gravitational analogues, since both are the traces and antisymmetric parts of tensors {Eαβ, Bαβ} ↔ {Eαβ,Hαβ}, which we know, from equations (5.1.1) and (5.1.2) of Table 5.1, to play analogous physical roles in the two theories. Note this interesting aspect of the analogy: if one replaces, in Eqs. (5.114)-(5.117), the electromagnetic tidal tensors (Eαβ and Bαβ) by the gravitational ones (Eαβ and Hαβ), and the charges by masses (i.e., charge density ρcand current jα, by mass/energy density ρand current Jα), one almost obtains Eqs. (5.3.3b)-(5.3.6b), apart from a factor of 2 in the source term in (5.3.6b) and the difference in the source of Eq. (5.3.3b), signaling that in gravity pressure and stresses contribute as sources. This happens because, since Eαβ and Hαβ are spatial tensors, all the contractions with Uα present in Eqs. (5.115) and (5.117) vanish. 18We thank Jo˜ao Penedones for drawing our attention to this point. 89 5 The papers summarized and discussed Figure 5.9: A test particle in uniform motion in flat spacetime from the point of view of three different frames: a) a frame composed of observers at rest, but carrying spatial triads that rotate with uniform angular velocity ~ Ω; b) a frame consisting of a congruence of rigidly rotating observers (vorticity ~ω), but each of them carrying a non-rotating spatial triad (i.e., that undergoes Fermi-Walker transport); c) a rigidly rotating frame (a frame adapted to a congruence of rigidly rotating observers); the spatial triads co-rotate with the congruence, ~ Ω = ~ω. Note: by observer’s rotation we mean their circular motion around the center; and by axes rotation we mean their rotation (relative to FW transport) about the frame’s origin. 96 5 The papers summarized and discussed The total inertial force is in this frame ~ FGEM =γhγ~ω ×(~r ×~ω)+2~ U×~ωi which is the relativistic generalization of the inertial force in e.g. Eq. (4-107) of [137]. Moreover, in this case the spatial connection coefficients Γˆ i ˆ jˆ kequal the ones of the 3-D spatial metric (i.e. of the spatial manifold associated to the quotient of the spacetime by the congruence), there is a well defined 3-D curve obtained by projecting the particle’s worldline on the space manifold, ~ Uis the vector tangent to it (see Fig. 5.9c) and ~ FGEM =˜ D~ U/dτ, cf. Eq. (5.128), is simply the acceleration of the curve. Finally, let me make these remarks on the usefulness of Eq. (5.127), and of our general definition of ~ H. Although the congruence adapted frame, ~ Ω = ~ω =~ H/2, might seem the most natural frame associated to a given family of observers, other frames are useful and are used in the literature, and the gravitomagnetic effects of such frames discussed therein. Eq. (5.127) yields the inertial forces of any of such frames, in particular our general definition of ~ Hencompasses all the gravitomagnetic fields defined in the different approaches. That includes the case of the “locally non-rotating frames” [133, 132], or “proper frames of the fiducial observers” [140] in Kerr spacetime discussed above, for which ~ω = 0, and ~ H=~ Ω = N−1˜ ∇×~ β; that is, all the gravitomagnetic accelerations come from Ωα(N,~ βdenote, respectively, the lapse function and the shift vector [140]). Frames corresponding to a congruence with vorticity, but where the spatial triads are chosen to be Fermi-Walker transported, ~ Ω = 0, have also been considered; in such frames ~ H=~ω (dubbed the “Fermi-Walker gravitomagnetic field” [25]). 5.6.2.2 Gyroscope precession Another main result of this GEM formalism is the exact analogy between the so-called gyroscope “precession” and the precession of a magnetic dipole, that I already presented in Sec. 5.5.1 above. Herein I will further elaborate on this subject, and show how the formalism in Paper #5 [5] helps clarifying the precise meaning of the gyroscope precession, what is it in the exact theory, and how can one setup a frame that allows us to determine the rotation of a vector relative to an inertial frame at infinity. As discussed in Sec. 5.5, if the Mathisson-Pirani condition holds, the spin vector of an ideal gyroscope (that is, a spinning pole-dipole particle) in a gravitational field is FermiWalker transported: DSα dτ =SνaνUα,(5.134) where Uαis its center of mass 4-velocity. This is the natural result: a gyroscope, which is understood as an object that opposes to changes in direction of its spin axis ~ S, has it fixed with respect to the mathematical definition of a comoving non-rotating frame. (This emphasizes the importance of acknowledging the physical validity of the Mathisson-Pirani condition, that we addressed in Paper #3 [3]). In a comoving orthonormal tetrad eˆα 97 5 The papers summarized and discussed (where Uˆ i= 0, and Sˆ 0= 0) we can write: D~ S dτ = 0 ⇔dSˆ i dτ =−Γˆ i ˆ 0ˆ kSˆ k=~ S×~ Ωˆ i. So ideal gyroscopes in a gravitational field are torque-free, and do not precess (relative to non-rotating frames). What one means in the literature by gyroscope “precession”, e.g. [7, 8, 73], and which has been measured by the Gravity Probe B mission [53], is a precession with respect to the distant stars, that is, with respect to the axes of an inertial frame at infinity. And this has a physical meaning, as it may detect the presence of frame dragging (and also the Thomas precession). That is, locally ~ Ω has no meaning; if ~ Ω6= 0, that tells us only that we are using a rotating frame to describe the motion of the gyroscope. But if the frame we are choosing has its axes fixed to an inertial frame at infinity, and still ~ Ω6= 0, then non-rotating frames at different points rotate one relative to another which indicates frame dragging (if the Thomas precession can be ruled out, e.g. if the gyroscopes are in geodesic motion). This notion of “frame of the distant stars” obviously applies only to asymptotically flat spacetimes. The question now is how can one compare systems of axes at different points in a curved spacetime, in order to determine if one rotates or not one relative to another. The answer is given by Eq. (5.126) above. As discussed above, if a rigid congruence of observers exists (as is the case in stationary spacetime), setting eˆ 0as the tangent to the congruence uα, and locking the rotation of the spatial triads to the vorticity, ~ Ω = ~ω (i.e., choosing the congruence adapted frame), the connecting vectors between neighboring observers obey ˙ Yˆ i= 0 ; that is, the tetrad vectors point to fixed neighboring observers. Thus we have a frame in which the local spatial triads carried by the observers are all locked one to another. Therefore measuring the angular velocity rotation of a vector relative to the local system of axes at point, effectively amounts to measure it with respect to any tetrad at another point. Now consider the spacetime to be asymptotically flat (besides stationary). In this case there are the so-called “static observers” (cf. point 7 of Sec. 5.1), the rigid congruence of observers whose worldlines are tangent to the time-like Killing vector field, and that at infinity coincides with the asymptotic inertial rest frame of the source — the axes of the latter define the directions fixed relative to the distant stars. Setting up the frame adapted to this congruence as explained above, yields a frame with axes everywhere locked to the distant stars, thus by measuring the precession of a gyroscope relative to any local tetrad of this frame one is in fact measuring it relative to the former. The analysis above, based on rigid congruences, applies to stationary spacetimes, such as the Kerr metric or the (approximate) gravitational field of spinning bodies, which is was the problem at hand in the Gravity Probe B mission. But what about the gravitational field generated by a system of translating bodies, which have been studied in the Post-Newtonian approximation (and whose gravitomagnetic field has also been subject of experimental test, e.g. [69, 71, 70])? These are not stationary spacetimes. However 98 5 The papers summarized and discussed a similar analysis for gyroscope precession can be done in these spacetimes, because, as can be seen from the line element (5.29) above (and discussed in detail in Paper #5 [5]), to post Newtonian order (as well as in the “gravitomagnetic limit” of linearized theory) the shear of the PN frame is negligible, only the expansion remains. In this case, for a congruence adapted frame, Eq. (5.126) reads ˙ Yˆ i≃1 3θY ˆ i, which again means that tetrad vectors point to fixed neighboring observers, and by the same construction above one can show that the so-called PN frames (the frames adapted to the ui= 0 observers in the PN metrics) are frames fixed to the distant stars. Finally, it should be noted that in most literature dealing with GEM analogies, the precession of the gyroscope is cast as being governed by the same gravitomagnetic field that yields the Coriolis acceleration ~ U×~ Hin the geodesic equation (5.127), see e.g. Eqs. (3.4), (3.2) and (3.12), (3.11), only with a relative factor of 2 between the two. It is clear in the general formulation herein that the fields involved in these effects are not the same; the field ~ Hleading to the Coriolis acceleration arises not only from the rotation ~ Ω of the frame relative to a local Fermi-Walker transported tetrad (that yields the gyroscope “precession”), but also from the vorticity ~ω of the congruence. In this sense, one can say that the Lense-Thirring effect detected in the LAGEOS satellite data [52] (and currently under scrutiny by LARES mission [54]), measuring ~ Hfrom test particle’s deflection, is of a different mathematical origin from the one which was under scrutiny by the Gravity Probe B mission [53], measuring ~ Ω from gyroscope precession, the two being made to match by measuring both effects relative to the “frame of the distant stars” (verifying ~ Ω = ~ω, and thus in this case the fields differ only by a factor of 2). It is important to bear this in mind, as in the literature GEM fields of frames which are not congruence adapted are discussed; for instance the “Fermi-Walker gravitomagnetic field” defined in [25], which is the ~ Hof a frame corresponding to a congruence with vorticity, but where the spatial triads are chosen to be Fermi-Walker transported: ~ Ω = 0. Thus there is a non-vanishing ~ H=~ω in this frame, whereas at the same time gyroscopes do not precess relative to it. 5.6.2.3 Field equations In a parallelism to what is done in Sec. 5.6.1, we split the Einstein and the Maxwell equations in their time and space projections with respect to the observer congruence, but now expressing them not in tidal tensors, but instead in terms of the EM/GEM fields as measured in such frame. I start by the electromagnetic equations. Using decomposition (5.1), we write Maxwell’s Eqs. (5.111) in terms of the electric and magnetic fields (Eu)α=Fα βuβand (Bu)α= ?Fα βuβmeasured by the congruence of observers of 4-velocity uα. All the fields below are measured with respect to this congruence, so we may drop the superscripts: (Eu)α≡Eα, (Bu)α≡Bα. For simplicity, below I choose the congruence adapted frame (~ω =~ Ω = ~ H/2); and I refer the reader to Paper #5 for the general expressions. The time and space 99 5 The papers summarized and discussed projections with respect to uαof Eq. (5.5a) read, in tetrad components, respectively, ˜ ∇· ~ E= 4πρc+~ H·~ B , (5.135) ˜ ∇× ~ B=˙ ~ E+~ G×~ B+ 4π~ j−K(ˆ iˆ j)Eˆ j~eˆ i+θ~ E . (5.136) The time and space projections of (5.5b) are, in the tetrad, ˜ ∇· ~ B=−~ H·~ E , (5.137) ˜ ∇× ~ E=−˙ ~ B+~ G×~ E+K(ˆ iˆ j)Bˆ j~eˆ i−θ~ B . (5.138) ˜ ∇is the connection defined in (5.131); since herein we are dealing with derivatives along the spatial directions, and for spatial vectors, it could be taken also as the spatial projection of the ordinary covariant derivative, since, for spatial Xα,˜ ∇αXβ= (hu)β γ∇αXγ(or, in the tetrad, ˜ ∇ˆ iXˆ j=∇ˆ iXˆ j). Eqs. (5.135)-(5.138) are equivalent to Eqs. (3.25)-(3.28), only written in a different form. In the special case of a rigid frame (K(ˆ iˆ j)=θ= 0) and time-independent fields (˙ ~ E=˙ ~ B= 0), these equations yield Eqs. (5.4.4a)-(5.4.8a) of Table 5.4. Turning now to the gravitational equations, using Tˆ 0ˆ 0=ρand Tˆ 0ˆ i=Jˆ i, and the expressions for the Riemann and Ricci tensors in terms of GEM fields given in Paper #5 [5], the time-time, time-space, and space-space components of the Einstein field equations with sources, Eq. (5.104a), read, respectively: ˜ ∇· ~ G=−4π(2ρ+Tα α) + ~ G2+1 2~ H2−˙ θ−K(ˆ iˆ j)K(ˆ iˆ j); (5.139) ˜ ∇× ~ H=−16π~ J+ 2~ G×~ H+ 2 ˜ ∇θ−2˜ ∇ˆ jK(ˆ jˆ i)~eˆ i; (5.140) 8πTˆ iˆ j−1 2δˆ iˆ jTα α=˜ Rˆ iˆ j+˜ ∇ˆ iGˆ j−Gˆ iGˆ j+˙ K(ˆ iˆ j)+K(ˆ iˆ j)θ +1 2h˙ Hˆ iˆ j+Hˆ iˆ jθ+~ H2δˆ iˆ j−Hˆ iHˆ j+K(ˆ iˆ l)Hˆ lˆ j−Hˆ l ˆ iK(ˆ lˆ j)i.(5.141) where Hij =ijkHkis the dual of ~ H. Eqs. (5.139)-(5.140) are the gravitational analogues of the electromagnetic equations (5.135) and (5.136), respectively; Eq. (5.141) has no electromagnetic counterpart. As for the the algebraic Bianchi identities (5.7b), the time-time (equal to space-space, as discussed in Sec. 5.7), space-time and time-space components become, respectively: ˜ ∇· ~ H=−~ G·~ H; (5.142) ˜ ∇× ~ G=−˙ ~ H−~ Hθ +Hˆ jK(ˆ iˆ j)~eˆ i; (5.143) K(ij)Hj=−?˜ Rj ji .(5.144) Eqs. (5.142)-(5.143) are the gravitational analogues of the time and space projections of 100 5 The papers summarized and discussed the electromagnetic Bianchi identities, Eqs. (5.136)-(5.138), respectively20; Eq. (5.144) has no electromagnetic analogue. The 3-D curvature tensor ˜ Rˆ iˆ jˆ kˆ lin the equations above is the restriction to the spatial directions of the curvature of the connection ˜ ∇, given in the tetrad by ˜ Rˆ l ˆ iˆ jˆ k≡Γˆ l ˆ jˆ k,ˆ i−Γˆ l ˆ iˆ k,ˆ j+ Γˆ l ˆ iˆmΓˆm ˆ jˆ k−Γˆ l ˆ jˆmΓˆm ˆ iˆ k−Cˆm ˆ iˆ jΓˆ l ˆmˆ k,(5.145) and ˜ Rˆ iˆ j≡˜ Rˆ lˆ iˆ lˆ jis the Ricci tensor associated to it; this tensor is not symmetric in the general case of a congruence possessing both vorticity and shear. Eq. (5.144) states that if the observer congruence has both vorticity and shear/expansion, then ˜ Rijkl does not obey the algebraic Bianchi identities for a 3D curvature tensor. In some special regimes the interpretation of ˜ Rˆ iˆ jˆ kˆ lis simple. In the quasi-Maxwell limit of Sec. 3.1.2 — that is, rigid (K(αβ)= 0), congruence adapted (~ Ω = ~ω) frames — it is the curvature tensor of the spatial metric γij (which yields the constant infinitesimal distances between neighboring observers of the congruence). In the case that the vorticity is zero (~ω = 0), the congruence is hypersurface orthogonal, and ˜ Rˆ iˆ jˆ kˆ lgives the curvature of these hypersurfaces. This remarkable aspect should be noted: all the terms in the Maxwell equations (5.135), (5.136) and (5.138) have a gravitational counterpart in (5.139), (5.142) and (5.143), respectively, substituting {~ E, ~ B}→{~ G, ~ H}(up to some numerical factors). As for (5.136), there are clear gravitational analogues in (5.140) to the terms ~ G×~ Band the current 4π~ j, but not to the remaining terms. It should nevertheless be noted that, as shown in Paper #5, in the Post-Newtonian regime (or in the “GEM limit” of linearized theory), the term 2˜ ∇θof (5.140) embodies a contribution analogous to the displacement current term ˙ ~ Eof (5.136). The gravitational equations contain, as one might expect, terms with no parallel in electromagnetism, most of them involving the shear/expansion tensor K(αβ). Special cases. “Quasi-Maxwell” regime (1+3 formalism) Since most of the differing terms involve K(αβ), the similarity gets closer if we take the “quasi-Maxwell” regime, i.e., stationary fields, and a frame adapted to a rigid congruence of stationary observers K(αβ)=θ= 0. The field equations in this regime are given in Table 5.4 . Therein we drop the hats in the indices, for the following reason: as discussed in Sec. 5.6.2.1, in this regime there is a natural 3-D Riemannian manifold on the quotient space (measuring the fixed distance between neighboring observers). This manifold has metric γij, in the notation of Sec. 3.1.2. We thus interpret the spatial fields ~ Gand ~ H as vector fields on this 3-D Riemannian manifold. The operator ˜ ∇becomes the covariant derivative of γij (as Γi jk =(3) Γi jk, i.e., the 4-D spatial connection coefficients equal the connection coefficients for γij), and ˜ Rij its Ricci tensor, which is symmetric (contrary to the general case). The equations in this “quasi-Maxwell” regime exhibit a striking 20Eqs. (5.142)-(5.143) are equivalent to Eqs. (7.3) of [25]; therein they are obtained through a different procedure, not by projecting the identity ?Rγα γβ = 0 ⇔R[αβγ]δ= 0, but instead from the splitting of the identity d2u= 0 ⇔u[α;βγ]= 0. Noting that u[α;βγ]=−R[αβγ]λuλ, we see that the latter is indeed encoded in the time-time and space-time parts (with respect to uα) of the former. 101 5 The papers summarized and discussed Table 5.4: GEM formulation of the electromagnetic and gravitational field equations, for stationary fields. Stationary fields, rigid, congruence adapted frame: ~ Ω = ~ω =~ H/2 (quasi-Maxwell formalism) Electromagnetism Gravity Maxwell Source Equations Einstein Equations Fαβ ;β= 4πJβRµν = 8πTµν −1 2gµνTα α •Time Component: •Time-Time Component: ˜ ∇· ~ E= 4πρc+~ H·~ B(5.4.4a) ˜ ∇· ~ G=−4π(2ρ+Tα α) + ~ G2+1 2~ H2(5.4.4b) •Space Components: •Time-Space Components: ˜ ∇× ~ B=~ G×~ B+ 4π~ j(5.4.5a) ˜ ∇× ~ H= 2~ G×~ H−16π~ J(5.4.5b) •Space-Space Component: No electromagnetic analogue ˜ ∇iGj−GiGj+1 2~ H2γij +˜ Rij = 8π1 2γijTα α+Tij (5.4.6) Bianchi Identity Algebraic Bianchi Identity ?Fαβ ;β= 0 (⇔F[αβ;γ]= 0 ) ?Rγα γβ = 0 (⇔R[αβγ]δ= 0) •Time Component: •Time-Time (or Space-Space) Component: ˜ ∇· ~ B=−~ H·~ E(5.4.7a) ˜ ∇· ~ H=−~ H·~ G(5.4.7b) •Space Components: •Space-Time Components: ˜ ∇× ~ E=~ G×~ E(5.4.8a) ˜ ∇× ~ G= 0 (5.4.8b) similarity with their electromagnetic counterparts, Eqs. (5.4.4a)-(5.4.8a) of Table 5.4, in spite of some natural differences that remain — numerical factors, the source and terms in (5.4.4b) with no electromagnetic counterpart. We note in particular that, by simply replacing {~ E, ~ B}→{~ G, ~ H}in (5.4.5a)-(5.4.8a), one obtains, up to some numerical factors, Eqs. (5.4.5b), (5.4.7b)-(5.4.8b). Of course, the electromagnetic terms involving products of GEM fields with EM fields, are mimicked in gravity by second order terms in the gravitational field. This is intrinsic to the non-linear nature of the gravitational field, and may be thought of as manifesting the fact that the gravitational field sources itself. The results in Table 5.4 complete the usual approach in the literature dealing with this regime, e.g. [19, 22, 23], where the gravitational Eqs. (5.4.4b)-(5.4.8b) are presented, but not the electromagnetic equations (5.4.4a)-(5.4.8a); the former are usually compared with the Maxwell equations in Lorentz frames. In Table 5.4, by contrast, analogous situations are compared: gravitational and Maxwell’s equations in terms of fields both measured in accelerating and rotating frames. Finally, it should be mentioned that there is another notable limit of Eqs. (5.139)-(5.144), 102 5 The papers summarized and discussed which the case that the frame is adapted to an hypersurface orthogonal (i.e., vorticity free) congruence, leading to the well known ADM “3+1 formalism” (see e.g. [143, 25]), obtained by setting ~ H= 0 in the equations above. Namely Eq. (5.139) becomes the so-called “Hamiltonian constrain”, Eq. (5.140) the “momentum constrain”, and Eq. (5.141) the equation for the evolution of the extrinsic curvature K(αβ)of the hypersurfaces orthogonal to the congruence; see Paper #5 [5] for details. 5.6.2.4 Relation with tidal tensor formalism One of the motivations of this work was to establish the connection between the inertial GEM fields herein and the tidal tensors of Secs. 5.2 and 5.6.1. The two analogies are intrinsically different; the latter stems from tensor equations, whereas the former from fields of inertial forces, i.e., artifacts of the reference frame. A relationship between the two formalisms exists nevertheless, and we are finding it of great interest, due the importance of using the two formalisms together in some applications, to be presented elsewhere (e.g. [30]). In an arbitrary frame one can express the gravitational tidal tensors in terms of the GEM fields, using the expressions for the tetrad components of Riemann tensor given in Sec. 3.4.2 of Paper #5 [5]. The expressions obtained are to be compared with the analogous electromagnetic situation, i.e., the electromagnetic tidal tensors computed from the fields as measured in an arbitrarily accelerating, rotating, and shearing frame (in flat or curved spacetime). General expressions for an arbitrary choice of the spatial frame are given in [5], herein I will assume the congruence adapted frame (~ω =~ Ω = ~ H/2). I will start by the electromagnetic tidal tensors; since in this section all the fields and tensors will be measured with respect to the congruence of observers uα, I use the abbreviated notation Eαβ ≡(Eu)αβ =Fαµ;βuµ, and Bαβ ≡(Eu)αβ =?Fαµ;βuµ. It follows that Eαγ =Eα;γ−Fαβuβ;γ;Bαγ =Bα;γ−?Fαβuβ;γ. Using decompositions (5.1), we obtain the tetrad components (Eˆ 0ˆ i=Bˆ 0ˆ i= 0): Eˆ iˆ j=˜ ∇ˆ jEˆ i−1 2h~ B·~ Hδˆ iˆ j−Bˆ jHˆ ii−ˆ lˆm ˆ iBˆmK(ˆ lˆ j); (5.146) Bˆ iˆ j=˜ ∇ˆ jBˆ i+1 2h~ E·~ Hδˆ iˆ j−Eˆ jHˆ ii+ˆ lˆm ˆ iEˆmK(ˆ lˆ j); (5.147) Eˆ iˆ 0=dEˆ i dτ +1 2(~ H×~ E)ˆ i+ (~ G×~ B)ˆ i; (5.148) Bˆ iˆ 0=dBˆ i dτ +1 2(~ H×~ B)ˆ i−(~ G×~ E)ˆ i.(5.149) Turning now to gravitational tidal tensors, again we use the abbreviated notation Eαβ ≡ (Eu)αβ =Rαµβνuµuν,Hαβ ≡(Hu)αβ =?Rαµβνuµuν. Using the tetrad components of the 103 5 The papers summarized and discussed Riemann tensor given in Paper #5 [5], we obtain (Eˆ 0ˆα=Eˆαˆ 0=Hˆ 0ˆα=Hˆαˆ 0= 0): Eˆ iˆ j=−˜ ∇ˆ jGˆ i+Gˆ iGˆ j+1 4~ H2γij −HjHi+1 2ˆ iˆ jˆ k dHˆ k dτ +ˆ lˆ jˆmHˆmK(ˆ iˆ l) −d dτ K(ˆ iˆ j)−δˆ lˆmK(ˆ iˆ l)K( ˆmˆ j); (5.150) Hˆ iˆ j=−1 2h˜ ∇jHi+ (~ G·~ H)γij −2GjHii+ˆ lˆm ˆ i˜ ∇ˆ lK(ˆ jˆm).(5.151) Note the formal similarities with the electromagnetic analogues (5.146)-(5.147). All the terms present in Eij and Bij, except for the last term of the latter, have a correspondence in their gravitational counterparts Eij,Hij, substituting {~ E, ~ B} → −{~ G, ~ H}and correcting some factors of 2. However, the gravitational tidal tensors contain additional terms, which (together with the differing numerical factors) encode the crucial differences in the tidal dynamics of the two interactions. The fourth and fifth terms in (5.150) have the role of canceling out the antisymmetric part of ˜ ∇ˆ jGˆ i, that is, canceling out the contribution of the curl of ~ Gto the gravitoelectric tidal tensor, as can be seen from Eq. (5.143). Note in particular the term −˙ Hi, which has no counterpart in the electric tidal tensor (5.146); in Eq. (5.143), that term shows up “inducing” the curl of ~ G, in a role analogous to ˙ B in the equation (5.138) for ˜ ∇ × ~ E, which might lead one to think about gravitational induction effects in analogy with Faraday’s law of electromagnetism. The fact that it is being subtracted in (5.150), means, however, that the curl of ~ Gdoes not translate into physical, covariant forces. For instance, it does not induce rotation in a set of free neighboring particles (see Eq. (5.9) above and discussion therein), nor does it torque an extended rigid body, as discussed in Sec. 5.5.4 (see sec. VI of Paper #4 [4] for more details). There are some interesting special regimes where the relation between the tidal tensors and the inertial fields becomes simpler. One is the “quasi-Maxwell” regime of of Secs. 3.1.2 and 5.6.2.3; i.e., stationary spacetimes, and a frame adapted to a rigid congruence of stationary observers. The gravitational tidal tensors as measured in such frame can be expressed entirely in terms of the gravito-electric ~ Gand gravitomagnetic ~ Hfields; the non-vanishing components are: Eij =−˜ ∇jGi+GiGj+1 4~ H2γij −HjHi; (5.152) Hij =−1 2h˜ ∇jHi+ (~ G·~ H)γij −2GjHii(5.153) (in accordance with the discussion in Sec. 5.6.2.3, the hats in the indices are dropped since these tensors may be expressed in an arbitrary, coordinate or not, basis on the spatial manifold γij). The non-vanishing components of the electromagnetic tidal tensors are, under the same conditions, Eij =˜ ∇jEi−1 2h~ B·~ Hγij −BjHii(a) Ei0=1 2(~ H×~ E)i+ (~ G×~ B)i(b) (5.154) 104 5 The papers summarized and discussed Bij =˜ ∇jBi+1 2h~ E·~ Hγij −EjHii(a) Bi0=1 2(~ H×~ B)i−(~ G×~ E)i(b) (5.155) Thus again, even in the stationary regime, the electromagnetic tidal tensors have nonvanishing time components, unlike their gravitational counterparts. The spatial parts, however, are very similar in form; note that replacing {~ E, ~ B} → −{~ G, ~ H/2}in (5.155), the time components vanish, and one almost obtains the space part (5.153), apart from the factor of 2 in the third term; and that a similar substitution in (5.154) almost leads to (5.152), apart from the term GiGj, which has no electromagnetic counterpart. The gravitational and electromagnetic tidal tensors are nevertheless very different, even in this regime; namely in their symmetries. Eij is not symmetric, whereas Eij is (the second and third terms in (5.152) are obviously symmetric; and that the first one also is can be seen from Eq. (5.4.8b) of Table 5.4). As for the magnetic tidal tensors, note that, by virtue of Eq. (5.4.5b), the last term of (5.153) ensures that, in vacuum, the antisymmetric part H[i;j](i.e., the curl of ~ H) is subtracted from Hi;jin (5.31), thus keeping Hij symmetric, by contrast with Bij. This can be seen explicitly by noting that in vacuum (5.153) can be put in the equivalent form: Hij =−1 2hHi;j−H[i;j]+ (~ G·~ H)γij −2G(jHi)i, where we used H[i;j]= 2G[jHi], as follows from Eq. (5.4.5b). Another interesting regime to consider is the weak field limit, where the non-linearities of the gravitational field are negligible, and compare with electromagnetism in inertial frames. From Eqs. (5.146)-(5.149), the non-vanishing components of the electromagnetic tidal tensors measured by observers at rest in an inertial frame are: Eij =Ei,j ;Ei0=dEi dτ ;Bij =Bi,j ;Bi0=dBi dτ , i.e., they reduce to ordinary derivatives of the electric and magnetic fields. The linearized gravitational tidal tensors are, from Eqs. (5.150)-(5.151): Eij ≈ −Gi,j +1 2ijk dHk dτ −d dτ K(ij); (a) Hij ≈ −1 2Hi,j +lm iK(jm),l .(b) (5.156) Thus, even in the linear regime, the gravitational tidal tensors cannot, in general, be regarded as derivatives of the gravitoelectromagnetic fields ~ Gand ~ H. 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URL: http://centra.ist.utl.pt/index.php?option=com jcalpro&Itemid= 72&extmode=view&extid=28 •“A gravito-electromagnetic analogy based on tidal tensors —the physical meaning of the second order scalar invariants of the Riemann tensor” L. Filipe O. Costa Invited Seminar at Dipartimento di Ingegneria Aerospaziale e Astronautica, Sapienza Universita’ di Roma, Roma – Italy, 1 July 2009. •“Spacetime dynamics of spinning particles — exact gravito-electromagnetic analogies” L. Filipe Costa (Speaker), J. Nat´ario, M. Zilh˜ao Invited Seminar at Department of Mathematical Analysis, Ghent University, Ghent-Belgium, 21 February 2012 6.2 Communications in International Conferences •“A gravito-electromagnetic analogy based on tidal tensors” L. Filipe P. O. Costa (Speaker), Carlos A. R. Herdeiro –Oral Communication at Seventh Alexander Friedmann International Seminar Gravitation and Cosmology, Jo˜ao Pessoa – Brazil, 30 June to 04 July 2008. URL: http://www.fisica.ufpb.br/eventos/friedmann2008/friedmann2008new2.htm 117 6 Communications on the material of this thesis –Poster presented at Spanish Relativity Meeting 2008 (ERE2009), Salamanca – Spain, 14-20 September 2008. URL: www.usal.es/ere2008 –Lecture presented at VIII SIGRAV Graduate School in Contemporary Relativity and Gravitational Physics, Villa Olmo (Como) – Italy, 11-15 May 2009 URL: http://www.centrovolta.it/sigrav2009/ –Oral Communication at 12th Marcel Grossman Meeting (Parallel Session MAGT3 B — Theoretical Issues in General Relativity), Paris – France, 12-18 July 2009 URL: http://www.icra.it/MG/mg12/en/ •“Reference frames and the physical gravito-electromagnetic analogy” L. Filipe P. O. Costa, Carlos A. R. Herdeiro Poster presented at IAU (International Astronomical Union) Symposium 261 —“Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis”, Virginia Beach – USA, 27 April - 1 May 2009 Link: http://www.aas.org/divisions/meetings/iau/programme.php •“Tidal tensor approach to gravitomagnetism” L. Filipe P. O. Costa (Speaker), Carlos A. R. Herdeiro Lecture presented at 1st LARES (Laser Relativity Satellite) Workshop, Roma – Italy, 3-4 July 2009 Talk available online at: http://www.lares-mission.com/ILSW2009.html •“The gravitational force on a gyroscope and the electromagnetic force on a magnetic dipole as analogous tidal effects” L. Filipe P. O. Costa (Speaker), Carlos A. R. Herdeiro Oral Communication at Spanish Relativity Meeting 2009 (ERE2009), Bilbao –Spain, 7-11 September 2009. URL: http://www.ehu.es/ere2009/website/modules/pageworks/index.php?page=8&ve=95 •“Aspects of the motion of gyroscopes around Schwarzschild and Kerr black holes — exact gravito-electromagnetic analogies” L. Filipe P. O. Costa (Speaker), Carlos A. R. Herdeiro Oral communication at II Workshop on Black Holes, Instituto Superior T´ecnico, Lisboa – Portugal, 21-22 December 2009. Talk available on-line at http://centra.ist.utl.pt/˜bhw/ •“Spinning test particles in general relativity — exact gravito-electromagnetic analogies” L. Filipe Costa (Speaker) and C. A. R. Herdeiro Oral communication at 19th International Conference on General Relativity and Gravitation (GR19), Mexico City –Mexico, July 2010 118 6 Communications on the material of this thesis •“Gravitomagnetism and the significance of the curvature scalar invariants” Luis Filipe Costa (Speaker), C. A. R. Herdeiro, and Lode Wylleman (2010) Oral Communication at Spanish Relativity Meeting 2010 (ERE2010),Granada – Spain, 6-10 September 2010. URL: http://www.iaa.es/ere2010/website/modules/pageworks/index.php?page=8&ve=269 •L. Filipe Costa (Speaker), C. Herdeiro, J. Nat´ario, M. Zilh˜ao “Mathisson’s helical motions — are they unphysical?” Oral Communication at Spanish Relativity Meeting 2011 (ERE2011),Madrid – Spain, 29th August - 2nd September 2011 Talk available online at http://teorica.fis.ucm.es/ERE2011/Program.html •L. Filipe Costa “Hidden momentum in general relativity” Oral Communication at Spanish Relativity Meeting 2012 (ERE2012),Guimar˜aes – Portugal, 3-7 September 2012 URL: http://w3.math.uminho.pt/˜ERE2012/website/modules/tinyd0/ 6.3 Other oral communications •“Spacetime dynamics of spinning particles — exact gravito-electromagnetic analogies” L. Filipe O. Costa (Speaker), J. Nat´ario, M. Zilh˜ao CFP Journal Club - Centro de F´ısica do Porto, Porto - Portugal, 9 May 2012 Link: http://faraday.fc.up.pt/cfp/events/journal-clubs/spacetime-dynamics -of-spinning-particles-exact-gravito-electromagnetic-analogies/ •“Aspects of the motion of gyroscopes around Schwarzschild and Kerr black holes — exact gravito-electromagnetic analogies” L. Filipe O. Costa (Speaker), Carlos A. R. Herdeiro Talk presented at MAP-FIS PhD Research Conference 2009/2010,University of Aveiro,Aveiro –Portugal, 15 January 2010 Link: http://www.map.edu.pt/fis/Workshop •“A gravito-electromagnetic analogy based on tidal tensors” L. Filipe O. Costa (Speaker), Carlos A. R. Herdeiro Talk presented at MAP-FIS PhD Research Conference 2008,University of Minho,Braga –Portugal, 16-17 January 2009 Link: http://www.map.edu.pt/fis/Workshop •“Uma analogia gravito-electromagn´ etica baseada nas for¸cas de mar´ e” L. Filipe O. Costa (Speaker), Carlos A. R. Herdeiro Series of seminars presented at Journal Club - Centro de F´ısica do Porto, Porto –Portugal 119 6 Communications on the material of this thesis –Part I: 22 February 2007 Link: http://faraday.fc.up.pt/cfp/events/journal-clubs/2007/umaanalogia-gravito-electromagnetica-baseada-nas-forcas-de-mare-i/ –Part II: 03 March 2007 Link: http://faraday.fc.up.pt/cfp/events/journal-clubs/2007/uma-analogia -gravito-electromagnetica-baseada-nas-forcas-de-mare-ii/ –Part III: “O movimento n˜ ao geod´ esico do girosc´ opio”, 24 July 2008 Link: http://faraday.fc.up.pt/cfp/events/journal-clubs/2008/umaanalogia-gravito-electromagnetica-baseada-nas-forcas-de-mares 120 7 Further publications on the material of this thesis (conference proceedings) •L. Filipe O. Costa, Carlos A. R. Herdeiro, “Tidal tensor approach to gravitoelectromagnetism”, International Journal of Modern Physics A 24 1695 (2009) DOI: 10.1142/S0217751X0904525X •L. Filipe O. Costa, Carlos A. R. Herdeiro “The gravitational force on a gyroscope and the electromagnetic force on a magnetic dipole as analogous tidal effects” Proceedings of the Spanish Relativity Meeting (ERE2009), 7–11 September 2009, Bilbao, Spain Journal of Physics Conf.. Ser. 229, 012031 (2010) DOI: 10.1088/1742-6596/229/1/012031 •L. Filipe O. Costa, Carlos A. R. Herdeiro “Analogy between general relativity and electromagnetism based on tidal tensors” Proceedings of the 12th Marcel Grossmann Meeting, Paris-France, 12-18 July 2009, Edited by T. Damour, R.T. Jantzen, R. Ruffini, World Scientific (2012) •L. Filipe O. Costa, Carlos A. R. Herdeiro, Lode Wylleman “Electromagnetic and Gravitational Invariants” Proceedings of the Spanish Relativity Meeting (ERE2010), 6–10 September 2010, Granada, Spain Journal of Physics: Conf. Ser. 314, 012072 (2011) DOI: 10.1088/1742-6596/314/1/012072 •L. Filipe Costa, J. Nat´ario, M. Zilh˜ao ”Mathisson’s helical motions demystified” Proceedings of the Spanish Relativity Meeting 2011 (ERE2011), Madrid – Spain, 29th August - 2nd September 2011; AIP Conf. Proc. 1458 (2011) 367-370. DOI: 10.1063/1.4734436. Preprint [arXiv:1206.7093] 121