# Geodesic deviation equation

1. May 24, 2015

### Andre' Quanta

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?
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 :)

2. May 29, 2015