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Spinning test particles in gravitational and electromagnetic fields

Filipe Costa, José Natário, Miguel Zilhão

1) The problem of the spin supplementary condition.

- The equations of motion for spinning pole-dipole test particles in general relativity, which follow from the conservation of the energy-momentum tensor, are undetermined up to a spin supplementary condition. The latter, even today, is still not very well understood; it has the role of requiring the reference worldline (relative to which the moments are taken) to be the center of mass as measured by some observer (unlike in Newtonian mechanics, in Relativity the center of mass is an observer dependent point).

- The existence of exact gravito-electromagnetic analogies (on which we are especially interested) in the equations of motion for spinning particles, relies on the so-called Mathisson-Pirani spin condition.
This condition states that the reference worldline is the center of mass as measured in its own rest frame. The condition is however degenerate, allowing for exotic helical motions even for a free particle in flat spacetime. These motions have been deemed unphysical, due to the belief that the radius of the helices was arbitrarily large, and for this reason this spin condition is usually portrayed as problematic.

- In [PRD 85, 024001 (2012)] we have shown that these claims are actually a misconception, arising from a subtle (but crucial) mistake in some derivations in the literature. The radius of the helices is finite and always contained within the "`disk of centroids"' (the disk formed by all the possible positions of the centers of mass measured by the different observers), and the helical solutions are just equivalent, albeit more complicated (comparing with the non-helical one, that the same condition equally allows) descriptions of the motion of a spinning body.

- Interestingly, the dynamics of the helical motions (as well as other exotic motions allowed by the infinite possible spin conditions) are seen to be explained through the same concept of hidden momentum that was recently proposed in [PRD 81, 104012 (2010)] as an explanation for the bobbings observed numerical simulations of binary systems. This hidden momentum is formally analogous to the hidden momentum of electromagnetic systems --- another not well understood feature of relativistic electrodynamics, despite its discovery dating back from the 60's [PRL 18, 876 (1967)] , Such analogy proves insightful for the understanding of these motions.

2) The exact GEM analogies

We are presently studying the dynamics of spinning particles subject to gravitational and electromagnetic fields, from the framework of two exact gravito-electromagnetic analogies: the analogy based based on tidal tensors [PRD 78, 024021 (2008)], and the analogy based on inertial gravitational forces, e.g. [GRG 39, 1477 (2007); Ann. Phys. 215, 1 (1992)].

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