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

4 Dezembro (quinta-feira) 14:30
Sala P6