How do I compute the connection coefficients for a given metric?

[jostpuur]

[SOLVED] computing affine connection

I've got a task of computing the components of the Riemann tensor, starting from the given Robertson-Walker metric

$$g=-dt\otimes dt + a^2(t)\Big(\frac{dr\otimes dr}{1-kr^2} + r^2\big(d\theta\otimes d\theta + \sin^2\theta d\phi\otimes d\phi\big)\Big)$$

When I searched the lecture notes, I found out, that I know how to proceed once the connection coefficients $\Gamma^{\lambda}{}_{\mu\nu}$ are known. So I want to solve what these connection coefficients are. They are defined with the equation

$$\nabla_{e_{\mu}} e_{\nu} = \Gamma^{\lambda}{}_{\mu\nu} e_{\lambda}$$

where $e_{\mu}$ are some basis of the tangent space in some point on the manifold. The mapping $(X,Y)\mapsto \nabla_X Y$ that maps pair of vectors into one vector is an affine connection.

So I searched the lecture notes for the definition of the affine connection. The mapping is defined by postulating that it is linear in both variables, and that if $f$ is some smooth real valued function on the manifold, then

$$\nabla_{fX} Y = f\nabla_X Y$$

and

$$\nabla_X (fY) = (Xf) Y + f\nabla_X Y.$$

Having found the definition, I tried to calculate

$$\nabla_{\partial_t} \partial_r \quad\quad\quad\quad\quad\quad\quad (1)$$

and I was forced to notice that I have no clue of how to do this. I know how the nabla is bilinear, and I know what happens if the tangent vectors get scaled by real valued function, but I don't know what the vector in equation (1) is supposed to be. I was unable to find any clear formula out of the lecture notes that would have told how to actually compute these vectors.

So I want to know, that what am I supposed to start doing, if I want to find out what vector the vector in (1) is. Is there a formula for this?

Or that is what I believe I want to know. If it looks like I started doing the exercise too difficult way, any hints are welcome.

 

[George Jones]

So I want to solve what these connection coefficients are. They are defined with the equation

$$\nabla_{e_{\mu}} e_{\nu} = \Gamma^{\lambda}{}_{\mu\nu} e_{\lambda}$$

It may just be that I use a different notation, but I write

$$\nabla_{e_{\mu}} e_{\nu} = \Gamma^{\lambda}{}_{\nu\mu} e_{\lambda}.$$

So I searched the lecture notes for the definition of the affine connection. The mapping is defined by postulating that it is linear in both variables, and that if $f$ is some smooth real valued function on the manifold, then

$$\nabla_{fX} Y = f\nabla_X Y$$

and

$$\nabla_X (fY) = (Xf) Y + f\nabla_X Y.$$

This defines what a connection is, but it does not pin down a unique connection. Usually, the connection is required to be torsion-free and metric-compatible as well. There is a unique torsion-free, metric compatible connection.

For example, the torsion-free property implies that, in a coordinate basis (but not necessarily in a non-coordinate basis), $\Gamma^{\lambda}{}_{\nu\mu} = \Gamma^{\lambda}{}_{\mu\nu}$. This, combined with the fact that covariant derivative of the metric is zero and some index gymnastics, gives a component equation for the connection.

This equation is probably in your notes.

 

[jostpuur]

This defines what a connection is, but it does not pin down a unique connection. Usually, the connection is required to be torsion-free and metric-compatible as well. There is a unique torsion-free, metric compatible connection.

There was a result, that said that if a torsion tensor $T^{\kappa}{}_{\alpha\beta}$ vanishes, then the connection coefficients would be related to the Christoffel symbols (as in our lecture notes) by

$$\Gamma^{\kappa}{}_{\alpha\beta} = \left\{\begin{array}{c} \kappa \\ \alpha\beta \end{array}\right\}.$$

However, I don't know how to actually calculate the torsion tensor, because when I look how they are defined, and search earlier definitions recursively, I again end up with my original problem.

I did not know if it is common for the torsion tensor to vanish, so I didn't dare to guess that it would in this case. But would it be a good guess? I know how to calculate the Christoffel symbols (as in our lecture notes), so if I assume the previous equation to be correct, I should be able to calculate something then.

Actually, what you said sounds like that we must assume the torsion tensor to vanish, because otherwise the resulting Riemann tensor would not be unique?

Now when I checked the Wikipedia's article about Christoffel symbols, I notice that they already give the $\Gamma$ notation for them right away, and actually define the Christoffel symbols to be the connection coefficients. Our lecture notes gave that {...} notation instead for the Christoffel symbols, and a different, more explicit, definition

$$\left\{\begin{array}{c} \kappa\\ \alpha\beta \end{array}\right\} := \frac{1}{2} g^{\kappa\mu}\big(\partial_{\alpha} g_{\beta\mu} + \partial_{\beta} g_{\mu\alpha} - \partial_{\mu} g_{\alpha\beta}\big).$$