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According to General Relativity, the way mass curves spacetime depends on its motion. Just like the gravitational field of a mass at rest resembles, under certain conditions, the Coulomb electric field, mass currents give rise to effects which strongly resemble magnetism.
In this talk we present a new approach (Phys. Rev. D 78 024021, 2008) to explore the physical analogy between General Relativity and Electromagnetism, based on tidal tensors of both theories. Our proposal goes well beyond the previous approaches found in literature, since it leads to an exact, covariant, and fully general form for the physical gravitational analogues of 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.
The tidal tensor formalism allows for a comparison between gravity and electromagnetism in terms of quantities common to both theories, making transparent the similarities and key differences between the two interactions. Among the latter the absence of gravitational induction effects (that have been predicted in the literature) analogous to the electromagnetic ones, which 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.
Two special cases of matching between gravitational and electromagnetic tidal tensors are discussed: one matches linearized gravitational tidal tensors to exact electromagnetic tidal tensors in Minkowski spacetime; in the other we reveal an exact matching between magnetic gravitational tidal tensors from ultra-stationary spacetimes and magnetic tidal tensors from electromagnetism in curved spaces.
Analogous scalar invariants built from tidal tensors of both theories are also discussed.
Our approach clarifies some issues concerning other gravito-electromagnetic analogies commonly found in the literature: it sheds light on the debate about the limit of validity of the analogy based on linearised theory (eg. [arXiv: gr-qc/0207065]), and solves conceptual difficulties in the physical interpretation of the analogy based on the splitting of the Weyl tensor in electric and magnetic parts (eg. [arXiv: gr-qc/9704059]), clarifying at the same time the relationship between these two analogies.
It also achieves an unification within gravito-electromagnetism, by revealing that the analogy known from the 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 [arXiv: hep-th/0309199].
A gravito-electromagnetic analogy based on tidal tensors
Luís Filipe P. O. Costa - Centro de Física do Porto, Universidade do Porto