There is out there an approach to deal with extended bodies in General Relativity due to Dixon that seems to be quite well known and believed to be the correct way to describe such phenomena on this framework.

On the other hand, there's also the Einstein-Cartan gravity theory, which allows the affine connection which spacetime is endowed to have non-zero torsion, which AFAIK, ends up coupling to the spin of matter.

  1. Is there any work out there which tries to extend Dixon's approach to extended bodies to Einstein-Cartan gravity?

  2. Has anyone tried to extend the proposed framework and found any new observable and relevant effect when the torsion is allowed?

I'm looking for papers on the subject.

EDIT: I forgot to link the papers. The papers on which Dixon develops his theory are:

  1. Dynamics of extended bodies in general relativity. I. Momentum and angular momentum - In this paper, considering one extended body of energy-momentum tensor $T_{ab}$ on a spacetime $(M,g)$ with electromagnetic field $F_{ab}$ with Killing vector fields $K_a$ preserving $F_{ab}$ Dixon proposes, on the analysis of the conservation laws induced by $K_a$, definitions for momentum and angular momentum for the extended body.
  2. Dynamics of extended bodies in general relativity - II. Moments of the charge-current vector - In this paper, Dixon develops one general definition for multipole moments of tensor fields describing properties of extended bodies. He discusses uniqueness and existence. The main result is that there is one set of multipole moments for the charge-current vector $J_a$ called the reduced set of moments, which encodes in the simples possible manner the conservation condition $\nabla_a J^a =0$.
  3. Dynamics of extended bodies in general relativity III. Equations of motion - In this paper, Dixon reviews the result for the existence and uniqueness of the reduced multipole moments of $J_a$ and develops the analogous construction for $T_{ab}$ finding reduced moments for the energy-momentum tensor. In terms of these moments the conservation condition $\nabla_{a}T^{ab}=J_a F^{ab}$ yields the definitions from the first paper, together with the equations of motion that can be expanded to any desired multipole order.

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    $\begingroup$ As far as my understanding goes, EC gravity can be seen as the middle step between GR and SUGRA, so in this context why would extended bodies be relevant? $\endgroup$ – DanielC Sep 24 '17 at 21:53
  • $\begingroup$ @riemannium sure, I've forgotten to do so. I've linked the papers with a brief summary of what is done in each of them. $\endgroup$ – user1620696 Mar 9 '18 at 17:02

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