# Test-particles motion in general space-time with *torsion* : geodesics or auto-parallel curves?

Consider a general curved space-time with torsion. In the standard Einstein-Cartan-Kibble-Sciama theory (ECKS or ECSK), torsion is non-dynamical and doesn't propagates in free space. But a more general theory could allow torsion to be dynamical and propagates. In general, geodesics (the shortest or extremal length curves) and auto-parallels (the straightest curves) are different curves in space-time.

In classical general relativity (without torsion), test-particles without spin should follow geodesics. This is a statement of inertia motion.

But with torsion, what curve should a test-particle follow ? A geodesic or a auto-parallel curve ?

It can be shown that if torsion is totally antisymetric (which is just a special case), geodesics and auto-parallel curves are the same. But in general they aren't.

I feel that Newton's inertia principle is really about the straightest curves, and not the shortest (or extremal) curves.

Is there any indication, clue or argument, that the spineless test-particles should follow auto-parallel curves in space-time, instead of geodesics ? Would it be more natural in some way ?

If the auto-parallel curves are more fundamental (from an inertia point of view), then would it imply that the lagrangian method for fields is falling apart as a general principle, since it's (i.e. was) motivated by the inertia principle ?

• Isn't the variational principle related to the shortest proper time, assuming the curve on the space-time manifold is parametrized by proper time? This variational principle should replace whatever principle occurring in Newtonian mechanics (Newton's inertial principle holds for motion in a 0 vector total field of forces, and this is unrelated to notions of "straightest" or "shortest" curves). – DanielC Oct 11 '17 at 13:50
• – Qmechanic Oct 11 '17 at 13:57
• I'm voting to close this question as off-topic because this identical question has been asked here: physics.stackexchange.com/q/318200 – Dvij D.C. Oct 11 '17 at 15:32
• @Dvij: that's not what off-topic means: it could be marked as a duplicate, but even that is problematic as the question it is a duplicate of has no anwers – Christoph Oct 11 '17 at 15:39
• @Christoph I agree. But there is a bug in the StackExchange network which prevents one to do so in certain cases. See: physics.meta.stackexchange.com/q/9949 – Dvij D.C. Oct 11 '17 at 16:19

According to Kleinert, Pelster (doi, arxiv), due to the closure failure of parallelograms in spaces with torsion, the action principle needs to be modified, leading to the appearance of an additional torsion term in the Euler-Lagrange equations $$\frac{\partial L}{\partial q^\lambda(\tau)} - \frac{d}{dt} \frac{\partial L}{\partial \dot q^\lambda(\tau)} = 2S_{\lambda\mu}{}^\nu(q(\tau)) \dot q^\mu(\tau) \frac{\partial L}{\partial \dot q^\nu(\tau)}$$