The Einstein-Cartan theory is a generalisation of General Relativity insofar as the condition that the metric affine connection is torsion-free is dropped. In other words, the space time is a Riemannian manifold together with the datum of a metric affine connection (which may differ from the Levi-Civita connection by a suitable contorsion tensor).

In this case, geodesics (paths that locally extremalise the length, and which are given by a variational principle) generally differ from auto-parallels. As far as I know, the trajectories of spin-less particles in Einstein-Cartan theory are usually assumed to be geodesics (rather than autoparallels) so they don't feel the difference between the given connection and the Levi-Civita connection. (By the way, is there a good reference for this statement?)

My question is, how classical particles with spin are supposed to behave? Will they also travel along geodesics with the only difference that their spin direction will evolve according to the contorsion tensor (viewed as a $so(1,3)$-valued one-form)?


1 Answer 1


No. Particles with spin will feel the torsion not only through their spin precession, since the equations of motion for them (Mathisson-Papapetrou equations) will contain the asymmetric part of the connection.

One source for the question is the review

Hehl, F. W., Von der Heyde, P., Kerlick, G. D., & Nester, J. M. (1976). General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys., 48(3), 393. (it has an online version).

From there we learn:

We have already indicated that photon and spinless test particles sense no torsion. A test particle in $U_4$ theory, one which could sense torsion, is a particle with dynamical spin like the electron. Its equation of motion can be obtained by integrating the conservation law of energy-momentum (3.12). In so doing, we obtain directly the Mathisson-Papapetrou type equation${}^{20}$ for the motion of a spinning test particle (Hehl, 1971; Trautman, 1972c)${}^{21}$ Adamowicz and Trautman (1975) have studied the precession of such a test particle in a torsion background. All these considerations seem to be of only academic interest, however, since torsion only arises inside matter. There, the very notion of a spinning test particle becomes obscure (H. Gollisch, 1974, unpublished). Only neutrinos, whose spin self interaction vanishes, seem to be possible candidates for $U_4$ test particles.

(Hehl, 1971) reference here is apparently original result on the motion of test particle with spin:

Hehl, F. W. (1971). How does one measure torsion of space-time?. Phys. Lett. A, 36(3), 225-226. (http://dx.doi.org/10.1016/0375-9601(71)90433-6)

For a more modern notation (tetrad formalism) for the mentioned Mathisson-Papapetrou equations you can use the thesis:

Laskoś-Grabowski, P. (2009). The Einstein–Cartan theory: the meaning and consequences of torsion. Master's thesis pp. 17-19

The references there should provide all further information.


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