Check for geodesically-followed path in a coordinate-free way

[PAllen]

Thus, sticking at that given non-metric connection, the sentence "walking in the direction you are facing" actually means "keep walking forward step-by-step in the direction at a given fixed angle to the direction shown locally by the compass needle"

which is the way you would be facing at any moment ... you choose to be always facing, e.g. east.

 

[Dale]

which is the way you would be facing at any moment ... you choose to be always facing, e.g.east.

But to do that you would have to turn the direction you are facing relative to a gyroscope or to a great circle path.

 

[PAllen]

But to do that you would have to turn the direction you are facing relative to a gyroscope or to a great circle path.

Yes, relative to those. And those would be turning relative to constant bearing.

Let me recap what I consider important points to understand:

1) There is a metric extremal definition of a geodesic. This doesn't explicitly use a connection or any definition of straightness. In the case of Riemannian metric (positive definite), you are requiring that between any two nearby points on a geodesic, there is no shorter path.

2) There is a parallel transport definition of a geodesic as a "straightest possible line". Per a connection, it says the direction of the tangent doesn't change. You need not even have a metric defined to use this definition.

3) For the unique metric compatible connection without torsion (and only for this case), the two geodesic definitions are shown to be equivalent. For any other connection, straightest path geodesics and metric extremal geodesics may be different.

4) There is at least one gravitational theory matching all current observation where there a physically significant connection is used that has torsion that defines straightest lines that are different from geodesics defined by the metric extremal definition. This is Einstein-Cartan theory. Note, the connection has no torsion in vacuum regions, thus replicating all GR vacuum predictions exactly.

5) As an analogy for this not so easy to understand situation, I proposed (not having realized @pervect had used the same example much earlier in the thread) the idea that on a sphere, constant bearing defines non-metric compatible connection and alternate notion of straightness compared to the metric compatible connection. As with Einstein-Cartan theory, each corresponds to a physical observable. Which one you sense (in either example) depends what you measure.

 

[Dale]

4) There is at least one gravitational theory matching all current observation where there a physically significant connection is used that has torsion that defines straightest lines that are different from geodesics defined by the metric extremal definition. This is Einstein-Cartan theory. Note, the connection has no torsion in vacuum regions, thus replicating all GR vacuum predictions exactly.

Oh cool, I didn't know that. (I agree with your recap on the others)

 

[pervect]

Nikodem Popławski has a number of technical and popularized writings about Einstein-Cartan theory. Einstein Cartan theory is able to handle spin 1/2 particles, where GR is either unable to handle them at all, or only handles them with great difficulty. (I'm not sure which is correct.)

Wikipedia has an article on Poplawski's papers, the implications for black holes are especially interesting. Using Einstein-Cartan theory, Poplawski theorizes that black hole collapse doesn't end in a singularity, but that due to the presence of spin 1/2 fermions in the collapsing matter and the extra torsion turns, under the extremely high density conditions during the collapse, cause the torsion terms to halt the collapse. The result, according to Poplawski's analysis, is the creation of a new universe rather than the creation of a singularity.

See for instance https://en.wikipedia.org/w/index.php?title=Nikodem_Popławski&oldid=977163437#Black_holes_as_doorways

Under normal conditions, though, the effect of torsion is unmeasurable. So the theory makes definite physical predictions that are different than GR, but it requires extreme conditions to test.

This lack of testability under normal conditions also implies that even if Einstein-Cartan theory were to be 100 % correct, we could use GR for most things. And the lack of torsion makes the math considerably simpler.

 

[cianfa72]

Thanks all for support