My teacher of General Relativity has proposed a demonstration of the geodesic deviation equation based on normal coordinates, the problem is that for me the procedure is wrong, could you help me to find the problem?(adsbygoogle = window.adsbygoogle || []).push({});

Suppose to have a differentiable manifold M of dimension 4, and two geodesics x and y.

Define the difference y-x = E as an element of the tangent space (the geodesics are calculated at the same proper time of the geodesic x).

Now we are in normal coordinates for x, in such a way that the Christoffel simbols in x vanish.

With all these assumptions we can calculate the second covariant derivative of E along x: for me, in normal coordinates in x, the second covariant derivative of E is simply the second derivative of E respect to the proper time, because everytime that you derive E in a covariant way the term of the connection is set to zero because of the normal coordinates, but for the teacher the result is different: he obtains also an additional term that i can t find.

Where is the problem?

Guys, forgive me for my english and for the lack of rigour, i hope someone of you will help me :)

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# Geodesic deviation equation

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