Note on the torsion tensor

In commenting on letters responding to his Einstein article (PHYSICS TODAY, November 2005, page 31, and April 2006, page 10), Steven Weinberg states that he "never understood what is so important physically about the possibility of torsion in differential geometry." He basically argues that torsion "is just a tensor" and could be treated like any additional tensor field in the context of general relativity.

In my opinion, however, a decisive point was overlooked. Torsion is not just a tensor, but rather a very specific tensor that is intrinsically related to the translation group, as was shown by Élie Cartan¹ in 1923–24. In fact, in the Yang–Mills sense, it is the field strength of the translations. Torsion is related to translations and curvature to Lorentz rotations. As one consequence, torsion cracks an infinitesimal parallelogram in the spacetime continuum and gives rise to a closure failure described by a vector (in dislocation theory in solids in three dimensions, it is the Burgers vector).

The simplest gravitational theory with torsion, the Einstein–Cartan theory, is a viable one.² Incidentally, torsion could be measured by the precession of nuclear spins, even though the effects are expected to be minute in the present-day cosmos.³

References

1. 1. E. Cartan, Riemannian Geometry in an Orthogonal Frame, trans. from Russian, World Scientific, Hackensack, NJ (2001), section 87.
2. 2. F. Gronwald, F. W. Hehl, [LINK].
3. 3. C. Lämmerzahl, Phys. Lett. A 228, 223 (1997) [INSPEC], available at [LINK].

Weinberg replies: Sorry, I still don't get it. Is there any physical principle, such as a principle of invariance, that would require the Christoffel symbol to be accompanied by some specific additional tensor? Or that would forbid it? And if there is such a principle, does it have any other testable consequences?