I Question about Parallel Transport

[cianfa72]

To reiterate, the tank can measure its rotation rate with a gyroscope internal to its hull and program its turret to counter-rotate at the same rate. No external references are needed.

As far as I understand this, using a gyroscope internal to the tank's hull and programming the turret to counter-rotate at the same tank's rotation rate w.r.t. it, one basically "implements" the parallel transport of the turret's attached gun along the path being taken from the tank.

Only when the tank takes the geodesic path along the Earth, it doesn't rotate at all w.r.t. the gyroscope, hence the programmed turret's counter-rotate rate vanishes.

As you pointed out, there is no external reference involved. In a sense, let me say, the gyroscope takes in account internally the metric connection (Levi-civita). Suppose to pick a different connection on the Earth as sphere (e.g. compass connection). In that case tank gyroscope's relative rotation no longer gives the relevant information about the curvature of the path being taken, hence it isn't good to use to program the turret's counter-rotation in order to parallel transport the gun according the chosen non-metric connection.

 

[A.T.]

As far as I understand this, using a gyroscope internal to the tank's hull ...

As I explained in post #25, a gyroscope doesn't work well in this 2D-space analogy.

 

[cianfa72]

As I explained in post #25, a gyroscope doesn't work well in this 2D-space analogy.

Yes, as @Orodruin said in #27, we can assume a sort of 2D gyroscope (i.e. free to rotate only around one axis).

 

[A.T.]

Yes, as @Orodruin said in #27, we can assume a sort of 2D gyroscope (i.e. free to rotate only around one axis).

As I said in post #28, constraining a standard gyroscope axis to stay tangential to the surface won't work here.

An actual 2D gyroscope would (in the idealized sense) be what the OP proposed: Make the turret rotation relative to the hull free and frictionless with turret center of mass on the axis of that rotation. This is actually the simplest idealised mechanical analogy I can think of.

 

[cianfa72]

An actual 2D gyroscope would (in the idealized sense) be what the OP proposed: Make the turret rotation relative to the hull free and frictionless with turret center of mass on the axis of that rotation. This is actually the simplest idealised mechanical analogy I can think of.

Ah ok. So suppose to start with the turret's gun pointing in one given direction. Then, along the path being taken from the tunk hull on Earth's surface, the turret's gun is parallel transported (according to the Levi-Civita connection). Suppose now the gun initially points along the direction of the path being taken from the tank hull. Then only if that path is geodesic will the gun stay aligned with it along the travel.

 

[cianfa72]

As far as I can tell, the parallel transport on the round sphere, when considering it embedded in 3D ambient euclidean space, works as follows.

Take a curve C on the sphere starting from point p and a vector v in the tangent space at p. Move v along C pointing in the same direction within the 3D ambient space (i.e. apply to it the "plain" parallel transport in euclidean space). Then take its orthogonal projection on the tangent plane at any point along the curve. Done, what you get is the parallel transport of v along C on the round sphere according to the Levi-Civita connection (i.e. metric compatible torsion-free affine connection).