Dale said:
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.
