Question about christoffel symbols

In summary, the conversation discusses two different approaches for finding the components of geodesic equations, one being the tedious method and the other utilizing Lagrange's equations. Without the line-element, only the first method can be used. The conversation also provides equations for calculating the Christoffel symbols and suggests asking for clarification if needed.
  • #1
m.medhat
37
0
hello,
i have a question about christoffel symbols . if we have :-
[PLAIN]http://www.tobikat.com [Broken] [Broken]
how can I derive these equations :-
[PLAIN]http://www.tobikat.com [Broken] [Broken]
please i want the answer be clear .


with very thanks...
 
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  • #2
m.medhat said:
hello,
i have a question about christoffel symbols . if we have :-
[PLAIN]http://www.tobikat.com [Broken] [Broken]
how can I derive these equations :-
[PLAIN]http://www.tobikat.com [Broken] [Broken]
please i want the answer be clear .


with very thanks...

You can have two different approaches in finding the components of geodesic equation, as arranged in the bottom picture. One is to calculate these all through tedious and lengthy geodesic equations or to simply take the path of Lagrange and utilize his equations to obtain what you want. But this latter one needs the line-element and since I don't see any (though I can guess what it could be), so you only seem to be left with the first method. So given those Christoffel symbols, just use the equations

[PLAIN]http://upload.wikimedia.org/math/d/f/9/df9964e9250e597ed0f1f23f3d1ddb21.png.[/URL] [Broken]

Remember that here you must take the affine parameter [tex]\lambda[/tex] in place of t.

Ask where you feel stuck.

AB
 
Last edited by a moderator:
  • #3
thank you very much
 

1. What are Christoffel symbols and what do they represent?

Christoffel symbols are a set of mathematical quantities used in differential geometry and tensor calculus. They represent the connection between the coordinate system and the underlying geometry of a curved space or manifold.

2. Why are Christoffel symbols important in general relativity?

In general relativity, the curvature of spacetime is described by the metric tensor. The Christoffel symbols are used to calculate the components of this tensor, which in turn determine the curvature and thus the gravitational effects in the theory.

3. How are Christoffel symbols calculated?

Christoffel symbols are calculated using derivatives of the metric tensor and the inverse metric tensor. This involves taking partial derivatives of the metric tensor and using them to construct a set of equations known as the geodesic equation.

4. What is the difference between upper and lower indices in Christoffel symbols?

In Christoffel symbols, one index is raised and the other is lowered. This indicates the contravariant and covariant nature of the symbols, respectively. The upper index represents the derivative with respect to the coordinate of the tangent vector, while the lower index represents the derivative with respect to the coordinate of the cotangent vector.

5. How are Christoffel symbols related to the curvature of a manifold?

The Christoffel symbols are related to the curvature of a manifold through the Riemann curvature tensor. This tensor is constructed using the Christoffel symbols, and provides a measure of the intrinsic curvature of the manifold at a given point.

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