# Parallel Transport

## Main Question or Discussion Point

I'm a bit confused about parallel transport. We demand that the absolute covariant derivative of our (generalised) coordinates is zero along some curve in some frame. Does this really make our vector stay parallel? What if these coordinates were angles or some other general curvilinear coordinate system?
Now that I think about it, the same can be said about the principle of covariance - why should the equations of motion be the same for angles as they are for regular position coordinates? Obviously I am missing something here...

## Answers and Replies

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atyy
Science Advisor
You define a rule of parallel transport, and if you transport a vector by that rule, it is parallelly transported by definition. In GR, parallel transport is defined so that it has nice properties wrt the metric. http://en.wikipedia.org/wiki/Levi-Civita_connection

Im confused too and still don't get that part of diffgeom. Is it just very bad terminology?

If we use the word paralell, that's pretty intuitive too many people and implies things ARE paralell. If they aren't then why not just call it perpendicular transport instead and just say. 'Oh yes, they are not really perpendicular, we just define them to be...'

Or have i COMPLETELY misunderstood everything as allways?

Parallel transport just keeps the vector at the same angle it started, irrespective of the worldline that's transporting it. Fermi-Walker transport keeps the vector at the same angle to the worldline. On a geodesic, they coincide.

Ok I haven't actually done differential geometry I'm working from Ray d'Inverno's book on GR(I'm planning on doing differential geometry next year). So you're saying parallel transport is defined with respect to the connection? It still seems like if I were to transport a vector along a curve keeping polar coordinates constant (r, theta and phi) then my vector wouldn't stay parallel.
What about the original question on covariance? Why should the laws of physics be the same for all coordinates (angles, distances etc)?

It still seems like if I were to transport a vector along a curve keeping polar coordinates constant (r, theta and phi) then my vector wouldn't stay parallel.
What about the original question on covariance?
It's the tangent vector of the worldline that stays parallel to itself. That's the definition of a geodesic. The geodesic equation is written in terms of connections which occur in the covariant derivative D. You don't keep coordinates constant, but angles and lengths.

$$\frac{D}{D\lambda}\frac{dx^a}{d\lambda}=0$$

Why should the laws of physics be the same for all coordinates (angles, distances etc)?
They wouldn't be much use otherwise.

jambaugh
Science Advisor
Gold Member
Ok I haven't actually done differential geometry I'm working from Ray d'Inverno's book on GR(I'm planning on doing differential geometry next year). So you're saying parallel transport is defined with respect to the connection? It still seems like if I were to transport a vector along a curve keeping polar coordinates constant (r, theta and phi) then my vector wouldn't stay parallel.
What about the original question on covariance? Why should the laws of physics be the same for all coordinates (angles, distances etc)?
This is the whole problem you can't simply "hold coordinates constant" as this is coordinate dependent. The first question is (on a general manifold) what do we mean by "parallel" at different positions on the manifold. The first answer is "we don't know" which means we must define a connection. Even this doesn't work in an absolute sense and we must define instead parallel relative to some path between the two points i.e. parallel transport. See the definition of parallel transport as defining the geometry of the space and not something we derive.

Hmm...

The first question is (on a general manifold) what do we mean by "parallel" at different positions on the manifold. The first answer is "we don't know" which means we must define a connection.
Ok, but then i Look up the definition of 'connection' here:

http://en.wikipedia.org/wiki/Connection_(mathematics [Broken])

The first line says 'In geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner'

So the notion of a connection is described in terms of the word 'paralell'. This short circuits my brain since you just said the opposite?

and if I look at the notion of 'paralell transport' :

http://en.wikipedia.org/wiki/Parallel_transport

first line is: 'In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection'

Again, paralell is just used here. But then i look at:

http://en.wikipedia.org/wiki/Parallel_(geometry [Broken])

Under 'construction' it says 'Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.'

That makes a bit more sense, but then again, further down it says:

'The angle the parallel lines make with the perpendicular from that point to the given line is called the angle of parallelism. The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space'

So, this uses the curvature... I think it makes more sense now somehow, but I cant find the
right way to explain it to myself consistently.

But anyway, is there another approach to these things that don't use the word paralell? Could we formulate the whole thing with something like 'angle preserving' etc?

Sorry if im babbling a bit, I suspect that the main problem is really that to understand this stuff better, one should study the subject much more rigorously and from within the mathematics and not try to use 'natural language' which only confuse things.

/Frederic

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jambaugh
Science Advisor
Gold Member
Hmm...

Ok, but then i Look up the definition of 'connection' here:

http://en.wikipedia.org/wiki/Connection_(mathematics [Broken])

The first line says 'In geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner'

So the notion of a connection is described in terms of the word 'paralell'. This short circuits my brain since you just said the opposite?
Parse what was said more carefully.... The connection makes precise what we mean by transporting...in a parallel and consistent manner. The active noun here is connection and it acts on the notion of "parallel" making it precise, i.e. defining it in a precise way.
and if I look at the notion of 'paralell transport' :

http://en.wikipedia.org/wiki/Parallel_transport

first line is: 'In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection'
again "parallel with respect to connection" means the connection is what decides what is "parallel".
Again, paralell is just used here. But then i look at:

http://en.wikipedia.org/wiki/Parallel_(geometry [Broken])

Under 'construction' it says 'Definition 2: Take a random line through a that intersects l in x. Move point x to infinity.'

That makes a bit more sense, but then again, further down it says:

'The angle the parallel lines make with the perpendicular from that point to the given line is called the angle of parallelism. The angle of parallelism depends on the distance of the point to the line with respect to the curvature of the space'

So, this uses the curvature... I think it makes more sense now somehow, but I cant find the
right way to explain it to myself consistently.

But anyway, is there another approach to these things that don't use the word paralell? Could we formulate the whole thing with something like 'angle preserving' etc?

Sorry if im babbling a bit, I suspect that the main problem is really that to understand this stuff better, one should study the subject much more rigorously and from within the mathematics and not try to use 'natural language' which only confuse things.

/Frederic
No need to apologize. It is this kind of quest for comprehension that you must undergo to develop understanding of the deep physics/mathematics rather than turning the crank on the calculations.

The problem with your last reference is it is a construction in Euclidean geometry...though I think they're trying to extend it to other geometries. Firstly it is talking about global parallelism rather than a local property which the connection defines. Secondly you have to know how "line" generalizes to other geometries (usually geodesic curves) and if the construction is still valid.

But it helps because we use the fact that all smooth manifolds with a geometry (metric) defined are locally flat. Understand that when we speak of vectors on a curved manifold the vectors don't reside in the manifold but in a tangent space touching the manifold at one point. Keep in mind that we define a metric on the tangent space not on the manifold itself but as we see below we can relate metric information on the tangent space to the manifold.

Directions on the tangent space at a point can be identified with curves through that point using the coordinate derivatives and coordinate differentials. The flatness of the tangent space reflects the local flatness of the manifold. We can thus take all the curves through a point and map them to lines or vectors on the tangent space at that point. We can thus speak of angles (or pseudo-angles if the metric is indefinite) between curves at that point.

OK so far? Now naturally we can have many curves passing through a point in a given direction. So without some way to connect the tangent spaces of infinitesimally close points to our original point we have trouble figuring which curves bend more or less than others. The easiest way to approach this is to imagine a random connection and ask if it is "good" or "bad" in terms of how it relates to the local metric at each point. We first must ask what the connection does.

Given two points we can "connect" their tangent spaces by defining a linear operator T(p1,p2) mapping vectors in one to vectors in the other. We presume this operator is invertible so we can map both ways. We do this for all point pairs on the manifold and we get a "Global connection" Now we assume certain "natural" conditions on this global connection. First that it is smoothly differentiable with respect to either of the points. Second that it remains invertible at least for all points in a certain open region. Thirdly we would hope that as we move continuously along a curve we can express the connection as an integral of infinitesimal transformations as we move from point to point along that curve.

This last may be a problem as we see that for an arbitrary global connection different curves may compose to yield different integrated connections between two points. If we impose the condition that this doesn't happen then this restricts our choice too much and yields the equivalent of a flat manifold (though it may have screwy topology).

But in considering this idea we find a more general place to start. We just define what happens during infinitesimal transformations and don't worrying about a global connection. We then allow for this path dependence. What we need is a generator $\Gamma_\mu(p)$ of a linear transformation corresponding to each basis vector $\mathbf{e}_\mu(p)$ in the tangent space at the point p. Then by moving in a direction a certain infinitesimal distance $\mathbf{dx}$ we transform tangent vectors by the amount $dx^\mu\Gamma_\mu$. Integrating along a curve then gives us a (path dependent) connection between the tangent spaces of the curves endpoints. This is the parallel transport defined by our local connection $\Gamma_\mu(p)$.

Now there is one more condition we need on this local connection and that is that it be compatible with the local metric. In particular we would hope that it preserve the lengths of vectors and (pseudo)angles between them as we move from tangent space to tangent space. This comes down to the covariant derivative of the metric being zero. We also get the condition that curves for which the local connection preserves tangent vectors are also curves with minimal arclengths as defined by integrating with the metric. "Geodesic" in terms of connection = "Geodesic" in terms of metric.

So in summary. We define (arbitrary) parallel transport (how we map tangent space vectors to tangent space vectors when we move along a curve) by choosing an (arbitrary) local connection. We then impose conditions on this connection so that it is compatible with the metric, i.e. preserving lengths and angles. This (I think) defines the parallel transport uniquely. We won't be able to define a global connection because in general transport may be path dependent which equates to the fact that transport around a loop may yield net linear transformations on the tangent space at a point on that loop. This is reflected in the Riemann curvature. The only spaces with global connections are those with zero curvature, i.e. flat Euclidean or pseud-Euclidean manifolds.

Does this make it any clearer? I glossed over some issues such as assuming we define point dependent tangent bases $\{\mathbf{e}_\mu(p)\}$ which are smoothly differentiable. This defines another implicit (arbitrary) connection and really the $\Gamma_\mu$ are the difference between this one and the connection we actually want. Hence the covariant derivative $D_\mu=\partial_\mu + \Gamma_\mu$ is what really defines the connection. This is why neither the $\Gamma$s nor the partials are "tensorial" while the covariant derivative is.

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Yes, thx! This helps a lot. Still need to chew on the lasts parts though. :) And i still think the guys who figured this out should have called it something else than 'paralell'...

jambaugh
Science Advisor
Gold Member
Yes, thx! This helps a lot. Still need to chew on the lasts parts though. :) And i still think the guys who figured this out should have called it something else than 'paralell'...
Yea, maybe "covariant transport" or "tangential transport".

I have the same problem with "state vector" in QM. Should be called "mode vector".

As for the last part of my post. In the exposition I tried to be arbitrary with the bases of the tangent spaces. The usual practice is to use a basis defined by a coordinate system. But bringing that up first confuses the issue of the arbitrariness of the connection because the default "zero" connection gives the geometry of the flat coordinate space not the geometry of the manifold on which the coordinates are being used. This is the way to see the Gamma's they express the difference in the geometry of the flat coordinate plane where the coordinates are rectilinear with the geometry of the manifold.

Even when the manifold is flat you need a variable metric and non-trivial connection terms (Gammas) for arbitrary coordinate systems as with polar coordinates.

Perhaps think of walking along a path on your manifold, and every step draw a dash-mark parallel to the last one, eg: you walk in a triangle and mark
Code:
| |  |  |
|     |
|   |
| |
|
If the manifold is curved, then you could get back where you started, and the dash marks would actually be coming in at a different angle, such as you see on the diagram of earth linked above.

You say that you can't simply "hold coordinates constant". In my notes this is exactly what it does by first changing to a locally inertial frame where the connection vanishes and then saying we can move the vector around keeping the coordinates constant. We are certainly demanding that the absolute derivative is zero - this is how we find the equation of parallel transport.

atyy
Science Advisor
You say that you can't simply "hold coordinates constant". In my notes this is exactly what it does by first changing to a locally inertial frame where the connection vanishes and then saying we can move the vector around keeping the coordinates constant. We are certainly demanding that the absolute derivative is zero - this is how we find the equation of parallel transport.
Yes, that's why many people prefer a coordinate independent way of defining it. But the coordinate independent definition of "parallel transport" that is chosen in GR can be described as keeping coordinates constant, if one goes to the local inertial frame, which is a special choice of coordinates that makes use of the metric for its definition. So either way you can see that you have picked only one of many possible sensible definitions of "parallel", and in particular, you have picked a definition that has nice properties with respect to the metric.

Ok so you mean if the covariant derivative of something along a curve is zero then that quantity can still change along the curve? It is the regular derivative that counts. This would explain a lot. And back to my original question about angles etc, I think I can see where some of my confusion was coming from there - I was having trouble with the angle of a vector when it is not located at the origin. Are these coordinates to be measured with respect to the point the vector is situated at?

atyy
Science Advisor
Ok so you mean if the covariant derivative of something along a curve is zero then that quantity can still change along the curve? It is the regular derivative that counts. This would explain a lot. And back to my original question about angles etc, I think I can see where some of my confusion was coming from there - I was having trouble with the angle of a vector when it is not located at the origin. Are these coordinates to be measured with respect to the point the vector is situated at?
There is no angle (metric) between vectors at different points. As FredericGos and jambaugh have talked about, there is no canonical notion of "parallel transport", it is something you define arbitrarily, and it might be best to call it "covariant" or "tangential" transport to be less misleading, except that everyone already calls it "parallel". Similarly, there is no canonical notion of a "vector derivative". Once you define "parallel transport" of vectors, you can use that to define the "covariant" derivative of vectors. Conversely, if you define a "covariant" derivative, that implies a definition of "parallel" transport. Try reading jambaugh's post #10, it has all this in more detail.

Ok thanks I think I have a better understanding of it now. I think my misunderstadings were actually about vectors in general curvilinear coordinate systems, which I didn't learn very well in the first place.

jambaugh
Science Advisor
Gold Member
Right...
The covariant derivative again defines the full connection as the effect of holding coordinates constant (coordinate derivatives) and then adding the correction terms (Gammas) because this alone won't do. The fact that you can at a point choose coordinates so that the Gamma's are zero is the fact that you can choose special (geodesic) coordinates for which the parallel transport becomes the "coordinate transport".

The physical interpretation of this is "choosing a locally inertial frame".

The difficulty is that we need coordinates to express motion in a traditional format but we need to get at the geometry which is independent of the choice of coordinates. If you have the high level mathematics then you simply speak of [/i]General Covariance[/i] under the diffeomorphism group (group of arbitrary continuous changes of coordinates). But this gets farther from the intuitions about what "parallel transport" means (unless you have have a highly developed intuition from working in the higher level mathematics.)

I find it helps to start with say the gauge derivation of electromagnetism first see how these issues play out with the much simple connection between the little complex number space at each point where the real vs. imaginary directions are arbitrary and you need a connection between each complex space at each point. Generalize this to a bigger group representation i.e. (classical) Yang-Mills field theory, then attack GR first by thinking of the tangent space as abstract and independent of the actual manifold then understanding that there is an additional identification namely that the vectors in the tangent space are identified with motion in the manifold via parametric derivatives. Finally one chooses systems of parameters namely coordinates and speaks of a coordinate basis of the tangent space. (This last may make more sense when you've gone through the first steps.)

The hardest part I found with getting my head around GR (an on-going process) is unlearning the many implicit facts which no longer hold when we move away from flat euclidean space. It is a long process of retraining the intuition.

Examples: Null vectors are perpendicular to themselves! The visual size in space-time drawings are wrong since we are embedding non-euclidean space onto euclidean paper. etc.

There are no shortcuts. You have to dig through the sequence of topics and do the math to build up your understanding of what is going on in the examples...
...And as always ask lots of questions.

Can anyone explain geometrically the differences between Covariant derivative and Lie derivative ?

atyy
Science Advisor
Can anyone explain geometrically the differences between Covariant derivative and Lie derivative ?
The Lie derivative is defined using a congruence of curves, so its value along one curve depends on the vector field on neighbouring curves. The covariant derivative only depends on the vector field on the particular curve. http://en.wikipedia.org/wiki/Covariant_derivative#Relation_to_Lie_derivative