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Carlos A.R. Herdeiro
Luís Filipe Costa

We present a new approach [arXiv: gr-qc/0612140] to a physical analogy between General Relativity and Electromagnetism, based on tidal tensors of both theories.

 Our proposal goes well beyond the previous approaches (to physical gravito-electromagnetism) found in literature, since it leads to an exact, covariant, and fully general form for the gravitational analogues of the Maxwell's equations. It also leads to an exact and physically enlightening derivation of Papapetrou's equation for the gravitational force exerted on a gyroscope. 

These equations reveal, in an extremely transparent fashion, both the similarities and key differences between the electromagnetic and gravitational interactions.

 An important implication of our approach is that in gravity, induction effects analogous to the electromagnetic ones, which have been predicted in the literature (e.g. [Braginsky et al, Phys. Rev. D 15, 2047 (1977)]), and whose detection was recently experimentally attempted (e.g. [arXiv: gr-qc/0610015]), cannot take place. The absence of such effects in gravity is shown to explain Hawking's [Phys Rev. Lett. 26, 1344 (1971)] spin-dependent upper bound for the energy released when two black holes collide.

 An analogy stemming from scalar invariants built from tidal tensors of both theories is also unveiled in this formalism.

 The approach based on tidal tensors clarifies several issues concerning the gravito-electromagnetic analogies commonly found in the literature:

 * It sheds light on the debate about the limit of validity of the analogy between the electromagnetic potentials and certain components of the metric tensor (studied in the popular “linear approach to gravitoelectromagnetism”, eg. [gr-qc/0207065]);

 * It solves conceptual difficulties concerning the analogy based on the splitting of the Weyl tensor in electric and magnetic parts (eg. [gr-qc/9704059]), namely the longstanding inconsistencies in the physical interpretation of the magnetic part of the Weyl tensor;

 * It clarifies the relationship between these two analogies, by placing both in a single formalism;

 * It also achieves an unification within gravito-electromagnetism, by showing that the analogy based on linearized theory originates from the same fundamental principle as the exact mapping (via the Klein-Gordon equation) between ultra-stationary spacetimes and magnetic fields in curved manifolds: the similarity between tidal tensors.

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