Reference Frames and the Physical Gravito-Electromagnetic Analogy

arXiv:0912.2146v1 [gr-qc] 11 Dec 2009 Relativity in Fundamental Astronomy Proceedings IAU Symposium No. 261, 2009 S. A. Klioner, P. K. Seidelman & M. H. Soffel, eds. c⃝2009 International Astronomical Union DOI: 00.0000/X000000000000000X Reference Frames and the Physical Gravito-Electromagnetic Analogy L. Filipe O. Costa1 and Carlos A. R. Herdeiro2 1,2Centro de F´ısica do Porto e Departamento de F´ısica da Universidade do Porto Rua do Campo Alegre 687, 4169-007 Porto, Portugal 1email: filipezola@fc.up.pt, 2email: crherdei@fc.up.pt Illustrations by Rui Quaresma (quaresma.rui@gmail.com) Abstract. The similarities between linearized gravity and electromagnetism are known since the early days of General Relativity. Using an exact approach based on tidal tensors, we show that such 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 commonly found 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 different analogies being unveiled throughout the years. Some are purely formal analogies, like the splitting of the Weyl tensor in electric and magnetic parts, e.g. Maartens-Basset 1998; but others (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 harmonic gauge ¯h ,β αβ = 0, take the form □¯hαβ = −16πT αβ/c4, similar to Maxwell equations in the Lorentz gauge: □Aβ = −4πjβ/c. That suggests an analogy between the trace re- versed time components of the metric tensor ¯h0α and the electromagnetic 4-potential Aα. 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) of Ruggiero-Tartaglia 2002. Furthermore (on certain special conditions, see section 2) geodesics, precession and forces on gyroscopes are described in terms of these fields in a form similar to their electromagnetic counterparts, e.g. Ruggiero-Tartaglia 2002, Ciufolini-Wheeler 1995. Such analogy may actually be cast in an exact form using the 3+1 splitting of spacetime (see Jantzen et al. 1992, Nat´ario 2007). These are analogies comparing physical quantities (electromagnetic forces) from one theory with inertial gravitational forces (i.e. fictitious forces, that can be gauged away by moving 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 DSα/dτ = SσU αDU σ/dτ, with no real torques applied on it. In this sense the gravito-electromagnetic fields 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 2 L. Filipe O. Costa & Carlos A. R. Herdeiro tions) 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 and electromagnetism occurs (that is, under which conditions the physical analogy ¯h0µ ↔Aµ holds, and Eqs. like (16)-(22) of Ruggiero-Tartaglia 2002 have a physical content). For that 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 mentioned above, it is based on quantities which can be covariantly defined in both theories — tidal forces (the only physical forces present in gravity) — which allows for a more transparent comparison between the electromagnetic (EM) and gravitational (GR) interactions. Table 1. The gravito-electromagnetic analogy based on tidal tensors. Electromagnetism Gravity Worldline deviation: Geodesic deviation: D2δxα dτ 2 = q mEα βδxβ, Eα β ≡F α µ;βU µ (1a) D2δxα dτ 2 = −Eα βδxβ, Eα β ≡Rα µβνU µU ν (1b) Force on magnetic dipole: Force on gyroscope: F β EM = q 2mB β α Sα, Bα β ≡⋆F α µ;βU µ (2a) F β G = −H β α Sα, Hα β ≡⋆Rα µβνU µU ν (2b) Maxwell Equations: Eqs. Grav. Tidal Tensors: Eα α = 4πρc (3a) Eα α = 4π (2ρm + T α α) (3b) E[αβ] = 1 2Fαβ;γU γ (4a) E[αβ] = 0 (4b) Bα α = 0 (5a) Hα α = 0 (5b) B[αβ] = 1 2 ⋆Fαβ;γU γ −2πǫαβσγjσU γ (6a) H[αβ] = −4πǫαβσγJσU γ (6b) ρc = −jαUα and jα are, respectively, the charge density and current 4-vector; ρm = TαβUαUβ and Jα = −T α βUβ are the mass/energy density and current (quantities measured by the observer of 4-velo

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