Christoffel symbol+covariant derivative+parallel transport

[jbusc]

So, I understand in order to evaluate the proper "derivative" of a vector valued function on a curved spacetime manifold, it is necessary to address the fact that the tangent space of the manifold changes as the function moves infinitesimally from one point to another. Therefore, you cannot just subtract the two vectors as you ordinarily would because they "live" in different tangent spaces, you need a "covariant" derivative. Correct so far?

Now, when Carroll addresses this in his notes he introduces the Christoffel symbols as a choice for the coefficient for the "correction" factor (i.e., the covariant derivative is the "standard" partial derivative plus the Christoffel symbol times the original tensor)

In his book Hartle introduces the Christoffel symbols much earlier, as the coefficients used when calculating geodesics. Later he covers the covariant derivative, but talks more about the concept of "parallel transport" which seems to have some connection to Christoffel symbols that I am unsure about.

So (in summary) how do these things work togther? It seems to me, the Christoffel symbols are the components of the original tensor that allow the "translation" to preserve parallelism, which allows definition of the covariant derivative, which then in turn finds it's way into geodesics. Am I on the right/wrong track?

Also, what is the signficance of the upper/lower indices on a Christoffel symbol? I see the Christoffel symbols are not tensors so obviously it is not a summation convention...or is it?

Thanks :)

 

[Daverz]

I think you're on the right path. Geodesics are those paths for which the tangent vector is parallel transported.

If you think of a manifold as embedded in a "bigger" space, the covariant derivitive can be defined as the projection of the usual derivitive parallel to the manifold. I find this illuminating, but I don't think I've ever seen it in a GR text (though I think Dirac may have done this in his little book; I no longer have a copy to check.) See this online text aimed at math students:

http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec8.html

(Here E_s is some embedding space).

To answer your second question, yes, the upper/lower distinction is kept to maintain the summation rule: sum on repeated indices only if they differ in lower/upper position.

 

[Kuwait]

on the last question, the thing that defines a tensor is the transformation property of the elements and not the summation convention.

 

[Mentz114]

I think you've got it, in the GR context.

More generally, the Christoffel symbol (CS) is an example of a connection, which enable us to compare vectors in different spaces. It's possible to have a manifold with no metric, but a connection, in which case parallelism can still be defined.

The three index structure of the CS comes naturally from the Cartan structure equations - the upper index is the principle direction, the two lower ones will mix in components from other directions. In fact, the 3-index object ( Ricci rotation coefficient ) appears as the connection that defines the curvature one-forms in terms of the basis one-forms. The CS are the transformed Ricci rotation coefficients, using the transformation tetrad that takes the orthonormal basis to the coordinate basis.

I thought you might like to know that.

 

[Kuwait]

Thanks for the information, it is indeed very interesting to know.

The natural appearance of the CS can also be seen in three dimensional curvilinear coordinates in which the contribution of the change of the basis to the total change of a vector will naturally intorduce the CS symbols. Thats the reason they disappear in the cartesian case.

 

[bcrowell]

You asked about the relationship between Carroll's description of the Christoffel symbol (a tool for parallel transport) and Hartle's (a tool for constructing geodesics).

If you start by taking them as tools for parallel transport, then you can also use them to find out everything you want to know about geodesics. This is because a geodesic is a curve that parallel-transports its own tangent vector. Here's a description of how that works: http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.7

Going the opposite way, I think describing the Christoffel symbol as encoding information about geodesics understates the amount of information they contain. It's possible to have geometrical systems where the concept of "line" (i.e., geodesic) is well defined, but parallelism isn't (e.g., http://en.wikipedia.org/wiki/Ordered_geometry ). The Christoffel symbols contain information about parallelism, not just geodesics.

 

[bcrowell]

Also, what is the signficance of the upper/lower indices on a Christoffel symbol? I see the Christoffel symbols are not tensors so obviously it is not a summation convention...or is it?

Let's see if I can say this systematically.

One way to see that the Christoffel symbol isn't a tensor is that the Christoffel symbol is essentially the thing the plays the role of a gravitational field in general relativity. Now a change of coordinates can never make a zero tensor into a nonzero tensor or vice versa. However, the equivalence principle says that a nonzero gravitational field in one set of coordinates can be a zero gravitational field in another set of coordinates. Therefore the Christoffel symbol can't be a tensor.

Intuitively, the Christoffel symbol is like the plumbing, and the tensors are like water. The Christoffel symbol defines how to transport tensors from one place to another. You can't use the plumbing to transport the plumbing.

One of the things we expect to be able to do with tensors is to add them, and since the Christoffel symbol is sort of like the gravitational field, you might think you could add Christoffel symbols the same way you can add gravitational fields. You can't do that, because GR is a nonlinear theory.

The reason for writing the Christoffel symbols with upper and lower indices is exactly so that we can apply the same summation conventions to them that we apply to everything else.

I believe it doesn't make sense to use the metric to raise or lower indices on the Christoffel symbol, but I could be wrong about that.

I believe it's also possible to have a connection that isn't compatible with a metric. For instance, I think if gravitational torsion exists, it wouldn't be describable by a metric, but it would be describable by a Christoffel symbol (which wouldn't have the usual symmetry on the lower indices). So in this sense the Christoffel symbol is more general than the metric, even though one often computes the Christoffel symbol from the metric.

 

[haushofer]

Sometimes you see people lowering ithe upper index on Christoffel symbols. Carroll on the other hand says it doesn't make sense, but that's not completely true; the upper index of the connection comes from the contravariant metric in that connection, and so it's a "tensorial index" and as far as I see there shouldn't be a problem if you want to lower that one.