Deriving the Definition of the Christoffel Symbols

[Physicist97]

In Sean Carroll's Lecture Notes on General Relativity (Chapter 3, Page 60), in the chapter on Curvature, he derives the definition of the Christoffels Symbols by assuming the connection is metric compatible and torsion free. He then takes the covariant derivative of the metric and cycles through the indices to arrive at the usual definition of the Christoffel Symbols, that is ${\Gamma}^{\sigma}_{\mu\nu}=1/2g^{\sigma\rho}({\partial}_{\mu}g_{\nu\rho}+{\partial}_{\nu}g_{\rho\mu}-{\partial}_{\rho}g_{\mu\nu})$ , but why is it not possible to derive the definition of the Christoffel Symbols this way. Assuming metric compatibility , ${\nabla}_{\mu}g_{\nu\sigma}={\partial}_{\mu}g_{\nu\sigma}-{\Gamma}^{\lambda}_{\mu\nu}g_{\lambda\sigma}-{\Gamma}^{\lambda}_{\mu\sigma}g_{\nu\lambda}=0$ . From here you can subtract the partial derivative from both sides and multiply by a negative to get you, ${\Gamma}^{\lambda}_{\mu\nu}g_{\lambda\sigma}+{\Gamma}^{\lambda}_{\mu\sigma}g_{\nu\lambda}={\partial}_{\mu}g_{\nu\sigma}$ . Now multiplying both sides by $g^{\lambda\sigma}$ leaves ${\Gamma}^{\lambda}_{\mu\nu}+{\delta}^{\sigma}_{\nu}{\Gamma}^{\lambda}_{\mu\sigma}=g^{\lambda\sigma}{\partial}_{\mu}g_{\nu\sigma}$ . The delta is the Kronecker Delta, ${\delta}^{\sigma}_{\nu}=g^{\lambda\sigma}g_{\lambda\nu}$ and it is 1 when ${\sigma}={\nu}$ and 0 otherwise. The Kronecker Delta will simply change the ${\sigma}$ of the Christoffel Symbol to a ${\mu}$ , thus getting you ${\Gamma}^{\lambda}_{\mu\nu}=1/2g^{\lambda\sigma}{\partial}_{\mu}g_{\nu\sigma}$ . How is any of what I did wrong, other than arriving at what I assume is a wrong definition of the Christoffel Symbol? All I did was assume metric compatibility, and then solved for the Christoffel Symbol. I would appreciate some clarification to whether this is wrong, and if so why :) .

 

[strangerep]

$$\Gamma^\lambda_{\mu\nu} g_{\lambda\sigma} + \Gamma^\lambda_{\mu\sigma} g_{\nu\lambda} ~=~ \partial_\mu g_{\nu\sigma}$$

Note that $\lambda$ is a dummy (summation) index on the LHS.

Now multiplying both sides by $g^{\lambda\sigma}$ leaves [...]

That step is illegal, since you're trying to use $\lambda$ when it's already a summation index in the original expression. The free indices in the first expression are $\mu, \nu, \sigma$, so you can only contract something new with them, not with the existing summation index $\lambda$.

 

[Physicist97]

Thanks for the quick reply, I didn't know about that rule for summed-over indices. I was curious, I've been looking around for awhile and can't seem to find an equation for the Christoffel Symbols in terms of the metric that does not assume torsion freeness. Is there any?

 

[ShayanJ]

Thanks for the quick reply, I didn't know about that rule for summed-over indices. I was curious, I've been looking around for awhile and can't seem to find an equation for the Christoffel Symbols in terms of the metric that does not assume torsion freeness. Is there any?

Read this!
But the fact that you didn't know about the dummy indices, means you need to learn more tensor analysis before going to more advanced stuff.

 

[strangerep]

I've been looking around for awhile and can't seem to find an equation for the Christoffel Symbols in terms of the metric that does not assume torsion freeness. Is there any?

I presume you "..for the connection coefficients in terms of the metric that does not assume torsion freeness". I vaguely recall many attempts at extended theories involving some kind of torsion, but I haven't paid much attention to them since they seem not physically relevant (imho). Usually, there's some other kind of fundamental field or structure, not just the usual Riemannian metric. E.g., in Einstein's nonsymmetric unified field theory, both the metric and connection are assumed to be nonsymmetric. After a very long calculation, one can determine the nonsymmetric connection in terms of the nonsymmetric metric. But that theory is now merely an historical curiosity, without relevance to modern physics.

[Edit: I haven't yet read the Jensen paper linked by Shyan -- I probably should.]

 

[ShayanJ]

I presume you "..for the connection coefficients in terms of the metric that does not assume torsion freeness". I vaguely recall many attempts at extended theories involving some kind of torsion, but I haven't paid much attention to them since they seem not physically relevant (imho). Usually, there's some other kind of fundamental field or structure, not just the usual Riemannian metric. E.g., in Einstein's nonsymmetric unified field theory, both the metric and connection are assumed to be nonsymmetric. After a very long calculation, one can determine the nonsymmetric connection in terms of the nonsymmetric metric. But that theory is now merely an historical curiosity, without relevance to modern physics.

What about Einstein-Cartan theory? People don't seem to find it irrelevant.
Of course GR is good enough as far as experimental verification is concerned but we should pay attention to any reasonable alternative as long as there is no single agreed-upon approach to finding a theory of quantum gravity.

 

[strangerep]

What about Einstein-Cartan theory? People don't seem to find it irrelevant.

I meant "irrelevant for real world, experimental physics" -- I know that some people retain interest and hope that something might come of it. But, for context, have you read the little book that reprints all of the letters between Einstein and Cartan during the development of the E-C theory? At the end, Einstein says that he no longer thinks the theory is relevant for physics (or words to that effect) -- which may explain why he moved on to other attempts such as the nonsymmetric UFT.

I don't have a problem with people continuing to work on such things, at least for a while, and provided not too much public money is being absorbed in such attempts. The tricky part is to know when to recognize that a line of research is not bearing edible fruit (by which I mean realistically testable predictions).

 

[ShayanJ]

I meant "irrelevant for real world, experimental physics" -- I know that some people retain interest and hope that something might come of it. But, for context, have you read the little book that reprints all of the letters between Einstein and Cartan during the development of the E-C theory? At the end, Einstein says that he no longer thinks the theory is relevant for physics (or words to that effect) -- which may explain why he moved on to other attempts such as the nonsymmetric UFT.

I don't have a problem with people continuing to work on such things, at least for a while, and provided not too much public money is being absorbed in such attempts. The tricky part is to know when to recognize that a line of research is not bearing edible fruit (by which I mean realistically testable predictions).

I understand your concern. But when I consider your comments together with other approaches to quantum gravity(string theory, LQG), it seems to me those other approaches aren't doing better(or maybe its better to say theories with torsion aren't doing worse!).

 

[strangerep]

I understand your concern. But when I consider your comments together with other approaches to quantum gravity(string theory, LQG), it seems to me those other approaches aren't doing better(or maybe its better to say theories with torsion aren't doing worse!).

I agree -- although perhaps there is some hope for experimental testing via cosmology (CMB), as Marcus suggested elsewhere.

Anyway, this subdiscussion is becoming tangential to the topic of this thread.