Christoffel Coefficient Defined: Exploring Beyond the Metric Tensor

[CPL.Luke]

so I'm taking general relativity right now and we've just gone over the covariant derivaive and the riemann tensor, however we haven't yet defined explicitly what the christoffel coefficients are except in relation to the metric tensor, I remember that there is another way of defining them explicitly however we haven't done it in class and I'm not entirely sure that we are going to, does anybody know this method?

 

[dextercioby]

The "Christoffel coefficients", or, in a more pedantic way, the components of the Riemann-Christoffel connection in the coordinate basis, are only defined on a metric manifold, which are usually what we encounter in GR. There are other, more general connections on an arbitrary manifold, but for GR purposes, they are not useful, at least at an elementary level.

 

[pmb_phy]

so I'm taking general relativity right now and we've just gone over the covariant derivaive and the riemann tensor, however we haven't yet defined explicitly what the christoffel coefficients are except in relation to the metric tensor, I remember that there is another way of defining them explicitly however we haven't done it in class and I'm not entirely sure that we are going to, does anybody know this method?

The affine coennection and the Christoffel symbols and the and are defined diferently. Each in general, has a different set of geodesics. That is to say that in general a manifolds will in general have to kinds of geodesics, an affine geodesic and a metric geodesic (which uses the Christoffel symbols). The two classes of geodescics will coincide if the components of each are equal to each other and so most people don't know the difference. Which connection are you referring to? If it truly is the Christofell symbols then it is the symbols which appear in the geodesic equation but those symbols are represented by braces and not a capital Gamma.

The quantity which connects a displacement and a chage in a vector to a change in that vector during that displacement is called an affine connection by definition. The affine connecion. If the equation of motion of a free particle is derived by the principle of stationary action (also known, incorrectly, as Least Action a principle inspired by religious beliefs!) then the Christoffel Symbols are what are found in the geodesic equation (not the Christoffel symbol unless that manifold has certain properties).

Best regards

Pete

 

[pervect]

Wikipedia states and my recollection is that there are an infinite number of affine connections, but that a metric singles out a "natural choice", the Leva-Civita connection.

[add]IIRC, the distingushing feature of the Leva-Civita connection is that the dot product (defined by the metric) of vectors is unchanged by parallel transport (which is defined by the connection). But I may be missing some minor details.

 

[cesiumfrog]

The Christoffel symbols measure how straight the coordinate axes are, so it is inevitable that (in terms of those chart coordinates) the values of the symbols should be calculated from derivatives of the metric tensor. $\nabla_i e^k \equiv \Gamma^k_{\phantom k ij} e^j$

 

[pmb_phy]

The Christoffel symbols measure how straight the coordinate axes are, so it is inevitable that (in terms of those chart coordinates) the values of the symbols should be calculated from derivatives of the metric tensor. $\nabla_i e^k \equiv \Gamma^k_{\phantom k ij} e^j$

I'm aware of that definition but I don't understand why you say that should be calculated from derivatives of the metric tensor?? I don't see that in your equation. Can you clarify for me please? Thank you.

Pete

 

[cesiumfrog]

Can you clarify for me please?

There's more detail here, if you want to follow the derivation yourself.