I Physical Interpretation of Frame Field

[PeterDonis]

maybe I wasn't very clear in post #22.

Not just not very clear, but apparently confused about what you are trying to talk about. See below.

In the general case (not geodesic curve) for both parallel and FW transport we need to know the curve and the vectors involved in the transport only at the point P from which the transport is carried out.

Yes. Which means that the key point you gave in post #22, about the solution being a first-order differential equation and therefore unique, does not apply to the general case. It only applies to the particular case of parallel transport along a geodesic.

So why did you even bother mentioning the first-order differential equation solution property, if you were interested in the general case? That property doesn't even hold for parallel transport in the general case, let alone FW transport.

In other words, based on what you're saying now, your post #22, and the entire subthread based on it, was wasting both your and my time.

 

[cianfa72]

Yes. Which means that the key point you gave in post #22, about the solution being a first-order differential equation and therefore unique, does not apply to the general case. It only applies to the particular case of parallel transport along a geodesic.

No. Given any curve C, a point P on it and a vector V at P one can solve the first-order differential equation for the parallel transport of V along C from P.

 

[PeterDonis]

Given any curve C, a point P on it and a vector V at P one can solve the first-order differential equation for the parallel transport of V along C from P.

If you know the curve C, yes.

But if all you know is the vector V at point P, the "exponentiation" operation you referred to earlier in the thread gives you the geodesic through P with tangent V at P. It doesn't give you any other curve. And your "ruler" construction depends on using the unique geodesic through P with tangent V, not any other curve.

So now I'm not clear about exactly what properties of a frame field you are asking about.

 

[cianfa72]

If you know the curve C, yes.

But if all you know is the vector V at point P, the "exponentiation" operation you referred to earlier in the thread gives you the geodesic through P with tangent V at P. It doesn't give you any other curve.

Yes, of course.

So now I'm not clear about exactly what properties of a frame field you are asking about.

I'm not asking about specific properties of frame field. I'm asking about the simple case of FW transport of a set of vectors (one timelike and three spacelike mutually orthogonal) from point P along a known curve C where the timelike vector is the tangent vector of the curve C at point P.

 

[PAllen]

Yes, of course.


I'm not asking about specific properties of frame field. I'm asking about the simple case of FW transport of a set of vectors (one timelike and three spacelike mutually orthogonal) from point P along a known curve C where the timelike vector is the tangent vector of the curve C at point P.

So, FW transport along any curve will preserve all of these properties. Parallel transport will only preserve them along a geodesic. For an arbitrary curve, parallel transport will preserve mutual orthogonality, but the timelike vector will no longer be tangent to the curve.

 

[cianfa72]

So, FW transport along any curve will preserve all of these properties. Parallel transport will only preserve them along a geodesic. For an arbitrary curve, parallel transport will preserve mutual orthogonality, but the timelike vector will no longer be tangent to the curve.

Ok, indeed. My point was to stress the notion of parallel and FW transport: assign a curve C, a point P on it and a vector V at P (i.e. an element of the tangent space at P). Then the parallel or FW transport of V along C from P is well-defined.

Lie dragging is a different matter since it requires two vector fields X and Y defined on an open region (there isn't the notion of Lie dragging a vector along a curve). Btw, according to this video, even the notion of Lie dragging of a curve C (which we can think of as the integral curve of some suitable vector field X) along the flow of a vector field Y is well-defined.