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rolandk
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Has anyone a derivation at hand of the Christoffel symbol by permuting of indices in a free fall system?
Roland
Roland
Yup. See - http://www.geocities.com/physics_world/ma/chris_sym.htmrolandk said:Has anyone a derivation at hand of the Christoffel symbol by permuting of indices in a free fall system?
Roland
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.dextercioby said: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 [itex] \mathbb{M}_{4} [/itex])...So how about getting it all done correctly or,don't give that link anymore and exlude it from your post.
Daniel.
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[1]P.A.M.Dirac,"General Relativity",1975.
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.dextercioby said:Alright,u didn't get it..
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.rolandk said:Has anyone a derivation at hand of the Christoffel symbol by permuting of indices in a free fall system?
Roland
The Christoffel symbol is a mathematical concept used in differential geometry and general relativity. It represents the connection between the curvature of a manifold and the coordinates used to describe it.
The Christoffel symbol is derived from the metric tensor, which describes the geometry of a manifold. It involves taking partial derivatives of the metric tensor and using them to construct a set of equations that relate the curvature of the manifold to the coordinates.
The Christoffel symbol is significant because it allows us to calculate the covariant derivative, which is necessary for understanding the curvature of a manifold. It also plays a key role in the Einstein field equations, which describe the relationship between matter and the curvature of spacetime in general relativity.
In general relativity, the Christoffel symbol is used to calculate the geodesic equation, which describes the path that a freely moving object will follow in curved spacetime. It is also used in the Einstein field equations to relate the curvature of spacetime to the energy and matter present in the universe.
Yes, the Christoffel symbol has many real-world applications, particularly in the fields of astrophysics and cosmology. It is used to model the behavior of gravitational waves, black holes, and other astronomical phenomena. It also has applications in engineering, such as in the design of spacecraft trajectories using gravitational assist maneuvers.