Is the Geodesic Deviation Equation Valid in Normal Coordinates?

[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 :)

 

[jvicens]

Thank you for bringing this issue to our attention. I would be happy to assist you in finding the problem with your teacher's proposed demonstration.

Firstly, it is important to note that normal coordinates are only valid locally and cannot be extended to cover the entire manifold. This means that the Christoffel symbols will not vanish for all points on the manifold, but only at a specific point where the coordinates are defined. Therefore, it is not correct to say that the Christoffel symbols vanish for all points in the normal coordinates of x.

Secondly, the second covariant derivative of E along x should not simply be the second derivative of E with respect to the proper time. This is because the covariant derivative takes into account the curvature of the manifold and the connection between points, which cannot be ignored even in normal coordinates. The additional term that your teacher obtains is most likely due to the non-zero Christoffel symbols in the second derivative of E.

Lastly, it is important to double-check the calculations and ensure that the definitions and assumptions used are correct. It is possible that there may be a mistake in the calculations, leading to the discrepancy between your result and your teacher's.

I hope this helps to clarify the issue and guide you in finding the problem. If you require further assistance, please do not hesitate to reach out to us.