Reference Frames and the Physical Gravito-Electromagnetic Analogy
aa r X i v : . [ g r- q c ] D ec Relativity in Fundamental AstronomyProceedings IAU Symposium No. 261, 2009S. A. Klioner, P. K. Seidelman & M. H. Soffel, eds. c (cid:13) Reference Frames and the PhysicalGravito-Electromagnetic Analogy
L. Filipe O. Costa and Carlos A. R. Herdeiro , Centro de F´ısica do Porto e Departamento de F´ısica da Universidade do PortoRua do Campo Alegre 687, 4169-007 Porto, Portugal email: [email protected] , email: [email protected] Illustrations by Rui Quaresma ( [email protected])
Abstract.
The similarities between linearized gravity and electromagnetism are known since theearly days of General Relativity. Using an exact approach based on tidal tensors, we show thatsuch analogy holds only on very special conditions and depends crucially on the reference frame.This places restrictions on the validity of the “gravito-electromagnetic” equations commonlyfound in the literature.
Keywords.
Gravitomagnetism, Frame Dragging, Papapetrou equation
1. Gravito-electromagnetic analogy based on tidal tensors
The topic of the gravito-electromagnetic analogies has a long story, with differentanalogies being unveiled throughout the years. Some are purely formal analogies, like thesplitting of the Weyl tensor in electric and magnetic parts, e.g. Maartens-Basset 1998; butothers (e.g Damour et al. 1991, Costa-Herdeiro 2008, Jantzen et al. 1992, Nat´ario 2007,Ruggiero-Tartaglia 2002) stem from certain physical similarities between the gravita-tional and electromagnetic interactions. The linearized Einstein equations (see e.g.Damour et al. 1991, Ruggiero-Tartaglia 2002, Ciufolini-Wheeler 1995), in the harmonicgauge ¯ h ,βαβ = 0, take the form (cid:3) ¯ h αβ = − πT αβ /c , similar to Maxwell equations inthe Lorentz gauge: (cid:3) A β = − πj β /c . That suggests an analogy between the trace re-versed time components of the metric tensor ¯ h α and the electromagnetic 4-potential A α . Defining the 3-vectors usually dubbed gravito-electromagnetic fields, the time com-ponents of these equations may be cast in a Maxwell-like form, e.g. eqs (16)-(22) ofRuggiero-Tartaglia 2002. Furthermore (on certain special conditions, see section 2)geodesics, precession and forces on gyroscopes are described in terms of these fields in aform similar to their electromagnetic counterparts, e.g. Ruggiero-Tartaglia 2002,Ciufolini-Wheeler 1995. Such analogy may actually be cast in an exact form using the3+1 splitting of spacetime (see Jantzen et al. 1992, Nat´ario 2007).These are analogies comparing physical quantities (electromagnetic forces) from onetheory with inertial gravitational forces (i.e. fictitious forces, that can be gauged away bymoving to a freely falling frame, due to the equivalence principle); it is clear that (non-spinning) test particles in a gravitational field move with zero acceleration DU α /dτ = 0;and that the spin 4-vector of a gyroscope undergoes Fermi-Walker transport DS α /dτ = S σ U α DU σ /dτ , with no real torques applied on it. In this sense the gravito-electromagneticfields are pure coordinate artifacts, attached to the observer’s frame.However, these approaches describe also (not through the “gravito-electromagnetic”fields themselves, but through their derivatives; and, again, under very special condi-1 L. Filipe O. Costa & Carlos A. R. Herdeirotions) tidal effects, like the force applied on a gyroscope. And these are covariant effects,implying physical gravitational forces.Herein we will discuss under which precise conditions a similarity between gravity andelectromagnetism occurs (that is, under which conditions the physical analogy ¯ h µ ↔ A µ holds, and Eqs. like (16)-(22) of Ruggiero-Tartaglia 2002 have a physical content). Forthat we will make use of the tidal tensor formalism introduced in Costa-Herdeiro 2008.The advantage of this formalism is that, by contrast with the approaches mentionedabove, it is based on quantities which can be covariantly defined in both theories — tidalforces (the only physical forces present in gravity) — which allows for a more transparentcomparison between the electromagnetic (EM) and gravitational (GR) interactions. Table 1.
The gravito-electromagnetic analogy based on tidal tensors.Electromagnetism GravityWorldline deviation: Geodesic deviation: D δx α dτ = qm E αβ δx β , E αβ ≡ F αµ ; β U µ (1a) D δx α dτ = − E αβ δx β , E αβ ≡ R αµβν U µ U ν (1b)Force on magnetic dipole: Force on gyroscope: F βEM = q m B βα S α , B αβ ≡ ⋆F αµ ; β U µ (2a) F βG = − H βα S α , H αβ ≡ ⋆R αµβν U µ U ν (2b)Maxwell Equations: Eqs. Grav. Tidal Tensors: E αα = 4 πρ c (3a) E αα = 4 π (2 ρ m + T αα ) (3b) E [ αβ ] = F αβ ; γ U γ (4a) E [ αβ ] = 0 (4b) B αα = 0 (5a) H αα = 0 (5b) B [ αβ ] = ⋆ F αβ ; γ U γ − πǫ αβσγ j σ U γ (6a) H [ αβ ] = − πǫ αβσγ J σ U γ (6b) ρ c = − j α U α and j α are, respectively, the charge density and current 4-vector; ρ m = T αβ U α U β and J α = − T αβ U β are the mass/energy density and current (quantities measured by the observer of 4-velocity U α ); T αβ ≡ energy-momentum tensor; S α ≡ spin 4-vector; ⋆ ≡ Hodge dual. We use ˜ e = − The tidal tensor formalism unveils a new gravito-electromagnetic analogy, summarizedin Table 1, based on exact and covariant equations. These equations make clear keydifferences, and under which conditions a similarity between the two interactions mayoccur.Eqs. (1) are the worldline deviation equations yielding the relative acceleration of twoneighboring particles (connected by the infinitesimal vector δx α ) with the same U α (and the same q/m ratio, in the electromagnetic case). These equations manifest thephysical analogy between electric tidal tensors: E αβ ↔ E αβ .Eq. (2a) yields the electromagnetic force exerted on a magnetic dipole moving with4-velocity U α , and is the covariant generalization of the usual 3-D expression F EM = ∇ ( S . B ) q/ m (valid only in the dipole’s proper frame); Eq. (2b) is exactly the Papapetrou-Pirani equation for the gravitational force exerted on a spinning test particle. In both (2a)and (2b), Pirani’s supplementary condition S µν U ν = 0 is assumed (c.f. Costa-Herdeiro 2009).These equations manifest the physical analogy between magnetic tidal tensors: B αβ ↔ H αβ .Taking the traces and antisymmetric parts of the EM tidal tensors, one obtains Eqs. eference Frames and the Physical Gravito-Electromagnetic Analogy F αβ ; β =4 πj α ; i.e., they are, respectively, covariant forms of ∇· E = 4 πρ c and ∇× B = ∂ E /∂t +4 π j ;Eqs. (4a) and (5a) are the space and time projections of the electromagnetic Bianchiidentity ⋆F αβ ; β = 0; i.e., they are covariant forms for ∇ × E = − ∂ B /∂t and ∇ · B = 0.These equations involve only tidal tensors and sources, which can be seen substitutingthe following decomposition (or its Hodge dual) in (4a) and (6a): F αβ ; γ = 2 U [ α E β ] γ + ǫ αβµσ B µγ U σ . (1.1)It is then straightforward to obtain the physical gravitational analogues of Maxwell equa-tions: one just has to apply the same procedure to the gravitational tidal tensors, i.e.,write the equations for their traces and antisymmetric parts (that is more easily donedecomposing the Riemann tensor in terms of the Weyl tensor and source terms, seeCosta-Herdeiro 2007 sec. 2), which leads to Eqs. (3b) - (6b). Underlining the analogywith the situation in electromagnetism, Eqs. (3b) and (6b) turn out to be the time-timeand and time-space projections of Einstein equations R µν = 8 π ( T µν − g µν T αα ), andEqs. (4b) and (5b) the time-space and time-time projections of the algebraic Bianchiidentities ⋆R γαγβ = 0. 1.1. Gravity vs ElectromagnetismCharges — the gravitational analogue of ρ c is 2 ρ m + T αα ( ρ m + 3 p for a perfect fluid) ⇒ in gravity, pressure and all material stresses contribute as sources. Ampere law — in stationary (in the observer’s rest frame) setups, ⋆F αβ ; γ U γ vanishesand equations (6a) and (6b) match up to a factor of 2 ⇒ currents of mass/energy sourcegravitomagnetism like currents of charge source magnetism. Symmetries of Tidal Tensors — The GR and EM tidal tensors do not genericallyexhibit the same symmetries, signaling fundamental differences between the two interac-tions. 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 sta-tionary field), the electric tidal tensor E αβ possesses an antisymmetric part, which is thecovariant derivative of the Maxwell tensor along the observer’s worldline; there is also anantisymmetric contribution ⋆F αβ ; γ U γ to B αβ . These terms consist of time projectionsof EM tidal tensors (cf. decomposition 1.1), and contain the laws of electromagnetic in-duction. The gravitational tidal tensors, by contrast, are symmetric (in vacuum, in themagnetic case) and spatial, manifesting the absence of analogous effects in gravity. Gyroscope vs. magnetic dipole — According to Eqs. (2), both in the case of the mag-netic dipole and in the case of the gyroscope, it is the magnetic tidal tensor, as seen by thetest particle ( U α in Eqs. (2) is the gyroscope/dipole 4-velocity), that determines the forceexerted upon it. Hence, from Eqs. (6), we see that the forces can be similar only if thefields are stationary (besides weak) in the gyroscope/dipole frame, i.e., when it is at “rest”in a stationary field. Eqs. (2) also tell us that in gravity the angular momentum S playsthe role of the magnetic moment µ = S ( q/ m ); the relative minus sign manifests thatmasses/charges of the same sign attract/repel one another in gravity/electromagnetism,as do charge/mass currents with parallel velocity.
2. Linearized Gravity
If the fields are stationary in the observer’s rest frame, the GR and EM tidal tensorshave the same symmetries, which by itself does not mean a close similarity between thetwo interactions (note that despite the analogy in Table 1, EM tidal tensors are linear, L. Filipe O. Costa & Carlos A. R. Herdeirowhereas the GR ones are not). But in two special cases a matching between tidal tensorsoccurs: ultrastationary spacetimes (where the gravito-magnetic tidal tensor is linear, seeCosta-Herdeiro 2008 Sec. IV) and linearized gravitational perturbations, which is thecase of interest for astronomical applications.Consider an arbitrary electromagnetic field A α = ( φ, A ) and arbitrary perturbationsaround Minkowski spacetime in the form † ds = − c (cid:18) − c (cid:19) dt − c A j dtdx j + (cid:20) δ ij + 2 Θ ij c (cid:21) dx i dx j . (2.1) Tidal effects. — The GR and EM tidal tensors from these setups will be in generalvery different, as is clear from equations (3-6), and as one may check from the explicitexpressions in Costa-Herdeiro 2008.But if one considers time independent fields, and a static observer of 4-velocity U µ = cδ µ , then the linearized gravitational tidal tensors match their electromagnetic counter-parts identifying ( φ, A i ) ↔ (Φ , A i ) (in expressions below colon represents partial deriva-tives; ǫ ijk ≡ Levi Civita symbol): E ij ≃ − Φ ,ij Φ ↔ φ = E ij , H ij ≃ ǫ lki A k,lj A↔ A = B ij . (2.2)This suggests the physical analogy ( φ, A i ) ↔ (Φ , A i ), and defining the “gravito-electro-magnetic fields” E G = −∇ Φ and B G = ∇× A , in analogy with the electromagnetic fields E = −∇ φ, B = ∇ × A . In terms of these fields we have E ij ≃ ( E G ) i,j and H ij ≃ ( B G ) i,j ,in analogy with the electromagnetic tidal tensors E ij = E i,j and B ij = B i,j .The matching (2.2) means that a gyroscope at rest (relative to the static observer) willfeel a force F αG similar to the electromagnetic force F αEM on a magnetic dipole, which inthis case take the very simple forms (time components are zero): F EM = q mc ∇ ( B . S ); F jG = − c H ij S i ≈ − c ( B G ) i,j S i ⇔ F G = − c ∇ ( B G . S ) . (2.3)Had we considered gyroscopes/dipoles with different 4-velocities, not only the expres-sions for the forces would be more complicated, but also the gravitational force wouldsignificantly differ from the electromagnetic one, as one may check comparing Eqs. (12)with (17)-(20) of Costa-Herdeiro 2008. This will be exemplified in section 2.1.The matching (2.2) also means, by similar arguments, that the relative accelerationbetween two neighboring masses D δx i /dτ = − E ij δx j is similar to the relative accel-eration between two charges (with the same q/m ): D δx i /dτ = E ij δx j ( q/m ), at theinstant when the test particles have 4-velocity U α = cδ α (i.e., are at rest relative to thestatic observer O ). Gyroscope precession. — The evolution of the spin vector of the gyroscope is givenby the Fermi-Walker transport law, which, for a gyroscope at rest reads DS i /dτ = 0;hence, we have, in the coordinate basis, Eq. (2.4a). The last term of Eq. (2.4a) vanishesif we express S in the local orthonormal tetrad e ˆ α : S i = S ˆ i e i ˆ i , where to linear order e i ˆ i = δ i ˆ i − Θ i ˆ i /c ; in this fashion we obtain Eq. (2.4b), which is similar to the precessionof a magnetic dipole in a magnetic field d S /dt = q S × B / mc : dS i dt = − c Γ i j S j = − c (cid:20) ( S × B G ) i + 1 c ∂ Θ ij ∂t S j (cid:21) ( a ); dS ˆ i dt = − c ( S × B G ) ˆ i ( b ) . (2.4) † In the previous sections we were putting c = 1. In this section we re-introduce the speed oflight in order to facilitate comparison with relevant literature. eference Frames and the Physical Gravito-Electromagnetic Analogy Geodesics. — The space part of the equation of geodesics U α,β U β = − Γ αβγ U β U γ isgiven, to first order in the perturbations and in test particle’s velocity, by ( a i ≡ d x i /dt ): a = ∇ Φ + 2 c ∂ A ∂t − c v × ( ∇ × A ) − c " ∂ Φ ∂t v + 2 ∂ Θ ij ∂t v j e i . (2.5)Comparing with the electromagnetic Lorentz force: a = qm (cid:20) −∇ φ − c ∂ A ∂t + v c × ( ∇ × A ) (cid:21) = qm h E + v c × B i , (2.6)these equations do not manifest, in general, a close analogy. Note that the last termof (2.5), which has no electromagnetic analogue, is, for the problem at hand (see nextsection), of the same order of magnitude as the second and third terms. But when oneconsiders stationary fields, then (2.5) takes the form a = − E G − v × B G /c analogousto (2.6).Note the difference between this analogy and the one from the tidal effects consideredabove: in the case of the latter, the similarity occurs only when the test particle seestime independent fields (fields ≡ derivatives of potentials/of metric perturbations); forgeodesics, it is when the observer (not the test particle!) sees a time independent potential ( φ ) /metric perturbations (Φ , Θ ij ).2.1. Translational vs. Rotational Mass Currents
The existence of a similarity between gravity and electromagnetism thus relies on thetime dependence of the mass currents: if the currents are (nearly) stationary, for instancefrom a spinning celestial body, the gravitational field generated is analogous to a magneticfield; an example is the gravitomagnetic field due to the rotation of the Earth, detected onLAGEOS data by Ciufolini et al. (and which is also the subject of experimental scrutinyby the Gravity Probe B and the upcoming LARES missions). But when the currentsseen by the observer vary with time — e.g. the ones resulting from translation of thecelestial body, considered in Soffel et al. 2008 — then the dynamics differ significantly.
Rotational Currents. — We will start by the well known analogy between theelectromagnetic field of a spinning charge (charge Q , magnetic moment µ ) and the grav-itational field (in the far region r → ∞ ) of a rotating celestial body (mass m , angularmomentum J ), see Fig. 1 Figure 1.
Spinning charge vs. spinning mass
The electromagnetic field of the spinning charge is described by the 4-potential A α =( φ, A ), given by (2.7a). The spacetime around the spinning mass is asymptotically de-scribed by the linearized Kerr solution, obtained by putting in (2.1) the perturbations L. Filipe O. Costa & Carlos A. R. Herdeiro(2.7b) : φ = Qr , A = 1 c µ × r r ( a ); Φ = Mr , A = 1 c J × r r , Θ ij = Φ δ ij ( b ) . (2.7)For the observer at rest O the gravitational tidal tensors asymptotically match the elec-tromagnetic ones, identifying the appropriate parameters: E ij ≃ Mr δ ij − Mr i r j r M ↔ Q = E ij ; H ij ≃ c » ( r . J ) r δ ij + 2 r ( i J j ) r − r . J ) r i r j r – J ↔ µ = B ij (all the time components are zero for this observer). This me ans that O will find a sim-ilarity between physical (i.e., tidal) gravitational forces and their electromagnetic coun-terparts: the gravitational force F iG = − H ji S j /c exerted on a gyroscope carried by O issimilar to the force F iEM = qB ji S j / mc on a magnetic dipole; and the worldline devia-tion D δx i /dτ = − E ij δx i of two masses dropped from rest is similar to the deviationbetween two charged particles with the same q/m .Moreover, observer O will see test particles moving on geodesics described by equationsanalogous to the electromagnetic Lorentz force (see Fig. 1). Translational Currents. — For the observer ¯ O moving with velocity w relative tothe mass/charge of Fig. 1, however, the electromagnetic and gravitational interactionswill look significantly different. For simplicity we will specialize here to the case where J = µ = 0, so that the mass/charge currents seen by ¯ O arise solely from translation.To obtain the electromagnetic 4-potential A ¯ α in the frame ¯ O , we apply the boost A ¯ α =Λ ¯ αα A α = ( ¯ φ, ¯ A ), where Λ ¯ αα ≡ ∂ ¯ x ¯ α /∂x α , using the expansion of Lorentz transformation(as done in e.g. Will-Nordtvedt 1972): t = ¯ t (cid:18) w c + 3 w c (cid:19) + (cid:18) w c (cid:19) ¯x . w c ; x = ¯x + 12 c (¯ x . w ) w + (cid:18) w c (cid:19) w ¯ t , (2.8)yielding, to order c − , A ¯ α = ( ¯ φ, ¯A ), with ¯ φ = Q (1 + w / c ) /r and ¯A = − Q w /rc . Toobtain A ¯ α in the coordinates (¯ x i , ¯ t ) of ¯ O , we must also express r (which denotes thedistance 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 theframe ¯ O . Using transformation (2.8), we obtain: r − = R − [1 − ( w . R ) / (2 R c )], andfinally the electromagnetic potentials seen by ¯ O :¯ φ = QR (cid:18) w c − ( w . R ) R c (cid:19) ; ¯A = − c QR w . (2.9)The metric of the spacetime around a point mass, in the coordinates of ¯ O , is also obtainedusing transformation (2.8), which is accurate to Post Newtonian order, by an analogousprocedure. First we apply the Lorentz boost g ¯ α ¯ β = Λ α ¯ α Λ β ¯ β g αβ to the metric (2.7) (with A = 0); then, expressing r in terms of R , we finally obtain (note that, although we arenot putting the bars therein, indices α = 0 , i in the following expressions refer to thecoordinates of ¯ O ): g = − MRc + 4 M w Rc − M ( w . R ) c R ≡ − c ; g i = 4 M w i Rc ≡ − A i c ; g ij = (cid:20) MRc (cid:21) δ ij ≡ (cid:20) c (cid:21) δ ij , (2.10)where we retained terms up to c − in g , up to c − in g i and c − in g ij , as usualin Post-Newtonian approximation. This matches, to linear order in M , Eqs. (5) ofSoffel et al. 2008 for the case of one single source; or e.g. Eqs. (11) of Nordtvedt 1988 (in eference Frames and the Physical Gravito-Electromagnetic Analogy R (¯ t ) = ¯r + w ¯ t .The gravitational tidal tensors seen by ¯ O are ( E α = E α = H α = H α = 0 ): E ij = − ¯Φ ,ij − c ∂∂ ¯ t ¯ A ( i,j ) − c ∂ ∂ ¯ t ¯Θ δ ij = Mδ ij R » w c −
92 ( R . w ) c R – − MR i R j R » w c − R . w ) c R – − Mw i w j c R + 6 Mw ( i R j ) ( R . w ) c R ; (2.11) H ij = ǫ lki ¯ A k,lj − c ǫ lij ∂ ¯Θ ,l ∂ ¯ t = McR » ǫ kij w k − R ( R . w ) ǫ kij R k − R ( R × w ) i R j – , (2.12) which significantly differ from the electromagnetic ones ( E α = B α = 0 ): E ij = − ¯ φ ,ij − c ∂∂ ¯ t ¯ A i ; j = E i,j = Qδ ij R » w c −
34 ( R . w ) c R – − QR i R j R » w c − R . w ) c R – − Qw i w j c R + 3 Qw [ i R j ] ( R . w ) c R ; (2.13) E i = − c ∂∂ ¯ t ¯ φ ; i − c ∂ ¯ A i ∂ ¯ t ≡ c ∂E i ∂ ¯ t = QcR » w i − R . w ) R i R – ; (2.14) B ij = ǫ lmi ¯ A m ; lj ≡ B i,j = QcR » ǫ kij w k − R ( R × w ) i R j – ; (2.15) B i = 1 c ∂B i ∂ ¯ t = − Qc R ( R . w )( R × w ) i . (2.16) Note in particular that, unlike their gravitational counterparts, E αβ and B αβ are notsymmetric, and have non-zero time components. The antisymmetric parts E [ ij ] = E [ i,j ] and B [ ij ] = B [ i,j ] above are (vacuum) Maxwell equations ∇ × E = − (1 /c ) ∂ B /∂t and ∇ × B = (1 /c ) ∂ E /∂t , implying that a time varying electric/magnetic field endows themagnetic/electric tidal tensor with an antisymmetric part. For instance, a time varyingelectric field will always induce a force on a magnetic dipole. The fact that E αβ and H αβ are symmetric reflects the absence of analogous gravitational effects. The time component B i means that the force on a magnetic dipole (magnetic moment µ = q/ m ) will havea time component ( F EM ) = (1 /c ) µ .∂ B /∂t , which (see Costa-Herdeiro 2009 sec. 1.2) isminus the power transferred to the dipole by Faraday’s law of induction (and is reflectedin the variation of the dipole’s proper mass m = − P α U α /c ). Again, this is an effectwhich has no gravitational counterpart: H α = H α = 0, thus ( F G ) = 0, and the propermass of the gyroscope is a constant of the motion.The space part of the geodesic equation for a test particle of velocity v is: a = ∇ ¯Φ + 2 c ∂ ¯ A ∂ ¯ t − v × ( ∇ × ¯ A ) − c ∂∂ ¯ t „ MR « v (2.17)= − MR » w c − R . w ) c R – R + 3 M ( R . w ) c R w − Mc R v × ( R × w ) + 3 c MR ( R . w ) v , which matches equation (10) of Soffel et al. 2008, or (7) of Nordtvedt 1973, again, in thespecial case of only one source, and keeping therein only linear terms in the perturbationsand test particle’s velocity v . L. Filipe O. Costa & Carlos A. R. HerdeiroComparing with its electromagnetic counterpart „ mq « a = E + v c × B = QR » w c − R . w ) c R – R − Q ( R . w ) c R w + Qc R v × ( R × w ) we find them similar to a certain degree (up to some factors), except for the last term of(2.17). That term signals a difference between the two interactions, because it means thatthere is a velocity dependent acceleration which is parallel to the velocity; that is in con-trast with the situation in electromagnetism, where the velocity dependent accelerationsarise from magnetic forces, and are thus always perpendicular to v .As expected from Eqs. (2.4) (and by contrast with the other effects), the precessionof a gyroscope carried by ¯ O , Eq. (2.18b) takes a form analogous to the precession ofa magnetic dipole, Eq. (2.18a), if we express S in the local orthonormal tetrad e ˆ i , nonrotating relative to the inertial observer at infinity, such that S i = (1 − M/R ) S ˆ i : d S d ¯ t = q m Qc R [ S × ( R × w )] ( a ); dS ˆ i d ¯ t = 2 Mc R [( R × w ) × S ] ˆ i ( b ) . (2.18)If instead of the gyroscope comoving with observer ¯ O (with constant velocity w ), wehad considered a gyroscope moving in a circular orbit, then an additional term wouldarise in analogy with Thomas precession for the magnetic dipole; for a circular geodesicthat term amounts to − /
3. Conclusion
We conclude our paper by discussing some of the implications of our conclusions inthe approaches usually found in literature. In the framework of linearized theory, e.g.Ruggiero-Tartaglia 2002, Ciufolini-Wheeler 1995, Einstein equations are often written ina Maxwell-like form; likewise, geodesics, precession and gravitational force on a spinningtest particle are cast (in terms of 3-vectors defined in analogy with the electromagneticfields E and B ) in a form similar to, respectively, the Lorentz force on a charged particle,the precession and the force on a magnetic dipole.We have concluded that the actual physical similarities between gravity and electro-magnetism (on which the physical content of such approaches relies) occur only on veryspecial conditions. For tidal effects, like the forces on a gyroscopes/dipoles, the analogymanifest in Eqs. (2.3) holds only when the test particle sees time independent fields . Inthe example of analogous systems considered in section 2.1, this means that the centerof mass of the gyroscope/dipole must not move relative to the central body. In the caseof the analogy between the equation of geodesics and the Lorentz force law (see Fig. 1),as manifest in equation (2.5), it is in the potentials/metric perturbations , as seen by theobserver (not the test particle!), that the time independence is required. The latter con-dition is not as restrictive as the one of the tidal effects: consider for instance observersmoving in circular orbits around a static mass/charge; such observers see an unchang-ing spacetime, and unchanging electromagnetic potentials, so, for them, the equationof geodesics and Lorentz force take similar forms (such analogy may actually be castin an exact form, see Nat´ario 2007, Jantzen et al. 1992). However, those observers see atime-varying electric field E (constant in magnitude, but varying in direction), which,by means of equations (4) and (6), implies that the tidal tensors are not similar to thegravitational ones † . † The electromagnetic field F αβ is not constant along the worldline of an observer moving in eference Frames and the Physical Gravito-Electromagnetic Analogy Acknowledgments
We thank the anonymous referee for very useful comments and suggestions.
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27, 2347a circular orbit (radius R , angular velocity Ω , velocity w = Ω × R ) around a point charge. Itsvariation endows the magnetic tidal tensor with an antisymmetric part, and the electric tidaltensor with a time component: dF i /dτ = Qw i /cR = − E [ i = − ǫ ijk B [ jk ] . This means thatthey significantly differ from the GR tidal tensors seen by an observer in circular motion arounda point mass. Note that both the GR and the EM tidal tensors for these analogous problems canbe obtained from, respectively, Eqs. (2.11)-(2.12) and (2.13)-(2.16), making therein R . w = 0(corresponding to circular motion), despite the fact that these expressions were originally derivedfor an observer with constant velocity. This is because, as can be seen from their definitionsin Table 1, it is the 4-velocity U αα