How to understand Parallel Transport

[friend]

I'm listening to Prof. Leonard Susskind's lectures on GR on youtube.com at
http://www.youtube.com/watch?v=hbmf0bB38h0&feature=relmfu

He's trying to explain how to visualize parallel transport of a vector. But I'm having a hard time of it. I think I understand it. Let me know if I got it straight:

The axes of a coordinate systems can change as you move along a line... with respect to the prior coordinate system. In other words, the basis vectors can smoothly change direction along a line. And the parallel transport of a vector ensures that the direction of a vectors follows the changes in the direction of the basis vectors as the orientation of the coordinate system changes along the line. Or, parallel transport ensures that the vector is always in the same direction with respect to the local coordinate system no matter where you are on a line. Is this right?

 

[bcrowell]

Your interpretation sounds correct to me, although I wouldn't put it quite that way. The way you're saying it, it makes it sound as though the only interest of parallel transport would be to accommodate a certain coordinate system. In fact, you can do all of differential geometry and general relativity without ever using coordinates (these are called "coordinate free methods"), but you can't do it without parallel transport. Parallel transport has a definite and nontrivial physical meaning. For example, the Gravity Probe B experiment sent a satellite into orbit carrying an absurdly precise gyroscope. Parallel transport of the gyroscope's angular momentum vector through many orbits resulted in a change of the vector's orientation relative to the stars.

 

[friend]

Your interpretation sounds correct to me, although I wouldn't put it quite that way. The way you're saying it, it makes it sound as though the only interest of parallel transport would be to accommodate a certain coordinate system. In fact, you can do all of differential geometry and general relativity without ever using coordinates (these are called "coordinate free methods"), but you can't do it without parallel transport.

So what is the intrinsic concept, would it be covariant derivative or a vector?

 

[pervect]

Maybe some specific examples will help.

If you have a metric, there is a unique notion of parallel transport that preserves dot products of vectors. This means that the length and angle of vectors are both preserved by parallel transport. It's a bit of a shortcut, but not too harmful I think, to say that this is "the" definition of parallel transport, the unique one that preserves lengths and angles.

Let (x,y) be a cartesian coordinate system on a flat plane (so the metric is diagonal), and let $\hat{x}$ be a unit vector in the x direction, and $\hat{y}$ be a unit vector in the y direction.

Then if you parallel transport $\hat{x}$ along any curve, to any position, its components will be unchanged, and hence the vector will be unchanged. You can make similar statements about $\hat{y}$. The end result of parallel transporting $\hat{x}$ in these circumstances will always be $\hat{x}$.

So on the plane, you could say that all [itex]$\hat{x}$ are "parallel" to each other.

If you've also studied Christoffel symbols, you can say that this happens because the Christoffel symbols are all zero in the cartesian coordinate system.

Now consider a polar coordinate system on a flat plane, with unit vectors $\hat{r}$ and $\hat{\theta}$

Parallel transport is independent of coordinates. If at some point we have a vector $\hat{v}$ that points in both the x and r directions, i.e. $\hat{v}$ = $\hat{x}$ = $\hat{r}$, and we parallel transport it around any curve, it will wind up by are previous result pointing in the x direction. But it may not be pointing (and probably won't be pointing) in the $\hat{r}$ direction anymore. We can also say that the Christoffel symbols in polar coordinates don't vanish.

When you try to do similar ations on the surface of a sphere, the Christoffel symbols never vanish. The case where you have a fundamentally curved geometry (like the surface of sphere) is where you really wind up needing the concept.

If a path parallel transports its tangent vector (i.e. a vector pointing along a path) to a tangent vector, it's a very special path, called a geodesic. On a sphere, these special geodesic paths are great circles.

Knowing that great circles parallel transport themselves, and that angles are preserved, you should be able to visualize what happens if you parallel transport an arbitrary vector along any great circle. And you can hopefully see how parallel transporting a vector in a closed loop around a sphere can wind up with it pointing in a different direction than when it started, and that this is not possible in a plane.

 

[friend]

Maybe some specific examples will help.

If you have a metric, there is a unique notion of parallel transport that preserves dot products of vectors. This means that the length and angle of vectors are both preserved by parallel transport. It's a bit of a shortcut, but not too harmful I think, to say that this is "the" definition of parallel transport, the unique one that preserves lengths and angles.

OK, let me see if I got it. The unit tangent vector to the curve always parallel transports to the tangent vector along the curve at any point, right? I suppose then that the vector perpendicular to the tangent vector and also tangent to the tangent space is also parallel transported at any point on the curve to be perpendicular to the tangent vector but still on the tangent space, right? Then since any vector has an angel with respect to the tangent and it's perpendicular normal, then that vector maintains this angle at every point on the curve, right? Would this be an intrinsic description, since the tangent vector and its perpendicular are intrinsic, right?

 

[friend]

OK, let me see if I got it. The unit tangent vector to the curve always parallel transports to the tangent vector along the curve at any point, right? I suppose then that the vector perpendicular to the tangent vector and also tangent to the tangent space is also parallel transported at any point on the curve to be perpendicular to the tangent vector but still on the tangent space, right? Then since any vector has an angel with respect to the tangent and it's perpendicular normal, then that vector maintains this angle at every point on the curve, right? Would this be an intrinsic description, since the tangent vector and its perpendicular are intrinsic, right?

Now I think this is wrong because the curve does not in general follow the changes in the coordinate system. The revised coordinate system may twist and bend independently of any arbitrary curve someone may draw through it, right? But then if vectors are parallel transported so that they maintain the same orientation wrt the coordintate system as it bends and twists, then how is it that the tangent vector of a curve always parallel transports to be still another tangent vector somewhere else on that curve where the coordintates system may have deformed differently than the curve?

 

[A.T.]

But then if vectors are parallel transported so that they maintain the same orientation wrt the coordintate system as it bends and twists, then how is it that the tangent vector of a curve always parallel transports to be still another tangent vector somewhere else on that curve where the coordintates system may have deformed differently than the curve?

Let me try an intuitive explanation:

Every path can be approximated as a poly-geodesic, a connected chain of geodesic segments. This is just the generalization of a poly-line for curved manifolds. The geodesic segments meet at certain angles. When you are parallel transporting a vector along a geodesic segment, the angle between the transported vector, and the path tangent vector is constant. When you arrive at the end of the segment, and the next segment goes off at an angle alpha, relative to the old direction, then the angle between the transported vector, and the path tangent vector changes by -alpha. So locally, when passing the corner the direction of the transported vector doesn't change.

 

[andrien]

Weyl described the idea Parallel Transport by an affine connection.
'we shall call a point P of a manifold affinely related to it's neighbourhood if we are given the vector P' into which every at P is transformed by a parallel displacement from P to P'.'
P' is here an arbitrary point infinitely near P.

 

[friend]

Now I wonder what happens when the coordinate system changes its orthogonality at different points along a curve. If the basis vectors no longer remain orthogonal what does it mean that vectors maintain their orientation wrt the basis vectors?

 

[pervect]

OK, let me see if I got it. The unit tangent vector to the curve always parallel transports to the tangent vector along the curve at any point, right?

That's right - if and only if the curve is a geodesic. On the Earth's surface, a great circle will parallel transport it's tangent vector along the curve, for instance. However, a circle of constant lattitude won't.

I suppose then that the vector perpendicular to the tangent vector and also tangent to the tangent space is also parallel transported at any point on the curve to be perpendicular to the tangent vector but still on the tangent space, right?

Right.

Then since any vector has an angel with respect to the tangent and it's perpendicular normal, then that vector maintains this angle at every point on the curve, right? Would this be an intrinsic description, since the tangent vector and its perpendicular are intrinsic, right?

It sounds like you're on the right track. There's also a geoemetric construction that defines parallel transport from a metric that you might be interested in, called Schild's ladder.