Undergrad 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.

 

[cianfa72]

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.

Do you think the above actually makes sense ?

 

[PeterDonis]

Do you think the above actually makes sense ?

If you can Lie drag a vector field, how to Lie drag the integral curves of that vector field should be obvious.

 

[cianfa72]

Just to be clear: take a timelike congruence in spacetime. Fix an event/point on each of them and take three spacelike vectors (in the tangent space at that point) that are mutually orthogonal each other and to the (timelike) tangent vector to the congruence's worldline passing through that point.

Now the FW transport process applied to the above vectors defines a frame field. By its very definition such a frame field's vectors turn out to be FW transported along the timelike congruence.

 

[PeterDonis]

Just to be clear: take a timelike congruence in spacetime. Fix an event/point on each of them and take three spacelike vectors (in the tangent space at that point) that are mutually orthogonal each other and to the (timelike) tangent vector to the congruence's worldline passing through that point.

Now the FW transport process applied to the above vectors defines a frame field. By its very definition such a frame field's vectors turn out to be FW transported along the timelike congruence.

Yes, all this is true and well known. Is there a question you have about this topic that hasn't been addressed yet?

 

[cianfa72]

Is there a question you have about this topic that hasn't been addressed yet?

No, thanks for your time !

 

[PeterDonis]

No, thanks for your time !

You're welcome!