What are Christoffel symbols in general relativity?

[zeromodz]

I am trying to understand everything about general relativity. I know that they have to do with how the Riemann curvature tensor uses parallel transporting a vector around a closed path. I really just don't understand the mathematics behind it. Thank you. I prefer layman's terms.

 

[arkajad]

You should start with the Euclidean plane in polar coordinates. You take a vector at some point (better not the origin) and transport it parallelly to some other point. Its polar coordinates will change (do this!). Christoffel symbols describe the infinitesimal rate of such changes for the parallel transport in different directions. In the case of the Euclidean plane parallel transport does not depend on the path. Everybody knows what "parallel" in this case. This fact is expressed in vanishing of the curvature tensor, which is expressed in terms of Christoffel symbols and their derivatives. But not always this the case. Sphere with its natural parallel transport law has a nonzero curvature tenor.

 

[Fredrik]

The best place to read about these things (not in layman's terms) is "Riemannian manifolds: an introduction to curvature", by John M. Lee.

 

[Dale]

I am trying to understand everything about general relativity. I know that they have to do with how the Riemann curvature tensor uses parallel transporting a vector around a closed path. I really just don't understand the mathematics behind it. Thank you. I prefer layman's terms.

I think of the Christoffel symbols describing how the coordinates are curved. This is opposed to the curvature tensor which describes how the manifold is curved. In a flat manifold it is still possible to use curved coordinates, but in a curved manifold it is not possible to use (globally) straight coordinates.

 

[atyy]

In addition to the above, one can think of the Christoffel symbols in a specific coordinate system as defining the rule of parallel transport. In GR, the rule of parallel transport is defined by its metric compatibility.