Derivation of Christoffel symbol

1. rolandk

2
Has anyone a derivation at hand of the Christoffel symbol by permuting of indices in a free fall system?

Roland

2. pervect

8,155
Staff Emeritus
If you have the equations for geodesic motion in a coordinate basis, you can "read off" the Christoffel symbols from the equation using the geodesic deviation equation, i.e if $$x^a(\tau)$$ is a geodesic, and we represent the differentaition with respect to $$\tau$$ by a dot, we can write

$$\ddot{x^a} + \Gamma^a{}_{bc} \dot{x^b}\dot{x^c} = 0$$

MTW gives the example on pg 345 in "Gravitation"

If
$$\ddot{\theta} - sin(\theta) cos(\theta) (\dot{\phi})^2 = 0$$
then
$$\Gamma^{\theta}{}_{\phi \phi} = -sin(\theta}) cos(\theta)$$

and the other Christoffel symbols are zero.

I'm not sure if this is what you're looking for, though.

3. dextercioby

12,327
Section 6.4 of [1] gives exacly what u want.Isn't that

$$\Gamma^{\sigma}{}_{\mu\rho}=:e_{\mu}{}^{m}D_{\rho}e_{m}{}^{\sigma}$$

,where the covariant derivative uses the ordinary $\partial_{\rho}$and the spin-connection...?

Daniel.

---------------------------------------------
[1]Pierre Ramond "Field Theory:A Modern Primer",Addison-Wesley,2-nd ed.,1989

Last edited: May 7, 2005
4. pmb_phy

Yup. See - http://www.geocities.com/physics_world/ma/chris_sym.htm

This is strictly a mathematical derivation so there is no mention of free-fall. However the condition which is equivalent to it is in Eq. (3).

Please note that one does not "derive" the Christoffel symbols (of the second kind). They are "defined." Once they are defined then one demonstrates relationships between them and other mathematical objects such as the metric tensor coefficients etc. In the link above the Christoffel symbols are defined in the same way Kaplan defines them in his advanced calculus text. In my page that is given in Eq. (8). I know of at least 3 different ways to define them though.

Pete

Last edited: May 7, 2005
5. dextercioby

12,327
Well,Pete,either u or Dirac[1] have it all mixed up.I'd go for you,as Dirac got a Nobel prize and i've been taught GR from his book[1].

Your formula #2 is valid for contravariant vectors ([1],eq.3.3,page 6) (a.k.a.vector,which is defined on the tangent bundle to a flat/curved $\mathbb{M}_{4}$)...So how about getting it all done correctly or,don't give that link anymore and exlude it from your post.

Daniel.

--------------------------------------------------
[1]P.A.M.Dirac,"General Relativity",1975.

6. pmb_phy

What's with the atttitude dude?? I don't delete portions of my web pages due to the comment of a readed who is a bit ignorant on the subject. As for "giving it" to you, I wasn't. I was giving it to Roland.

As I said, their are several definitions of the Christoffel symbols. Dirac uses one definition, Kaplan another, Ohanian yet another, Lovelock and Rund yet another. However, I'm looking at the text you refer to and I don't see what you're talking about. That equations is not directly related to this topic. Also what you refer to as "valid for contravariant vectors" (really the components of such a vector in a particular coordinate system) is identical in meaning to what I refered to above as the Christoffel symbols of the first kind. The other ones are the Christoffel symbols of the first kind.

So how about first learning the subject completely before you make another attempt at correcting me in such a rude manner?

7. dextercioby

12,327
Alright,u didn't get it...Let's take it slowly:What's formula #2 about...?

Daniel.

8. pmb_phy

So why bother with me? Seems that you're unwilling to entertain the possibility that you made an error. In any case Eq. #2 in my page is the transformation properties of the components of a contravariant vector.

However I see your point in Dirac. It was confusing since he never named them. Dirac and Ramond, two authors you quote, use different definitions of the Christoffel symbols of the second kind.

However it appears that you're having a bad day since I find your comments to be irritating.

Later

9. pmb_phy

So why bother with me? Seems that you're unwilling to entertain the possibility that you made an error. In any case Eq. #2 in my page is the transformation properties of the components of a contravariant vector.

However I see your point in Dirac. It was confusing since he never named them. Dirac and Ramond, two authors you quote, use different definitions of the Christoffel symbols of the second kind.

However it appears that you're having a bad day since I find your comments to be irritating. A long time ago I found that discussing anything with someone posting in such a grating manner not worth posting to. Li'fe's too short and I have many other irritating things which deserve more attention.

Later

10. dextercioby

12,327
You specifically use "covariant vector",and 2-ice...And now u turn it and use "contravariant" vector...What should i understand,that I'm having a bad day...?

Daniel.

11. pmb_phy

There is a minor point I'd like to make to add to those of mine above. There are four symbols in tensor analysis which are tightly related. In certain circumstances they are identical. In most circumstances you'll see in GR they are identical. Two of the symbols are referred to as the Christoffel symbols (of the first and second kind) and the affine connection symbols (of the first and second kind). The affine connection has a capital gamma as a kernal lettter. There is also two terms referred to as "affine geometry" and "metric geometry.

See
http://www.geocities.com/physics_world/ma/affine_vs_metric_geometry.htm

I'm not 100% sure which symbols are which since I'm not sure everyone uses the same definition of each. In any case they are the same when the manifold has both a connection and metric defined on it and the metric geodesics and the affine geodesics are the same.

Pete

Last edited: May 8, 2005