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 ?

  • $\begingroup$ 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). $\endgroup$
    – DanielC
    Oct 11, 2017 at 13:50
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    $\begingroup$ I'm voting to close this question as off-topic because this identical question has been asked here: physics.stackexchange.com/q/318200 $\endgroup$
    – ACat
    Oct 11, 2017 at 15:32
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    $\begingroup$ @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 $\endgroup$
    – Christoph
    Oct 11, 2017 at 15:39
  • $\begingroup$ @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 $\endgroup$
    – ACat
    Oct 11, 2017 at 16:19

1 Answer 1


My vote would go to the auto-parallels as well, which seems to me the correct generalization of the idea that bodies should persist in their state of motion.

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 $$ \frac{\partial L}{\partial q^\lambda(\tau)} - \frac{d}{dt} \frac{\partial L}{\partial \dot q^\lambda(\tau)} = 2S_{\lambda\mu}{}^\nu(q(\tau)) \dot q^\mu(\tau) \frac{\partial L}{\partial \dot q^\nu(\tau)} $$

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    $\begingroup$ The arguments from the paper cited in your answer are pretty strong and convincing : inertia and locality imply that a spinless particle should follow an auto-parallel curve, instead of a geodesic. From the paper : "Because of its inertia, a particle will change its direction in a minimal way at each instant of time, which makes its trajectory as straight as possible. If it were to choose a path which minimizes the length of the orbit it would have possessed some global information of the geometry." $\endgroup$
    – Cham
    Oct 11, 2017 at 17:38

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