Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

[minits]

TL;DR Summary

Covariant derivative of the Riemann tensor evaluated in Riemann normal coordinates

Hello everyone,

in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the Christoffel symbols seem to vanish. I assume this results from the note that it is evaluated in Riemann normal coordinates. I know one can choose the coordinate system so that a given Christoffel-symbol vanishes but in this case there are so many to handle so that I am not convinced this is working. Can someone please give me an input on how to make myself clear that it works? Thanks in advance!

 

[martinbn]

Can you be more specific. What doesn't seem to work? It looks perfectly fine.

 

[Gaussian97]

Summary:: Covariant derivative of the Riemann tensor evaluated in Riemann normal coordinates

Hello everyone,

in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the Christoffel symbols seem to vanish. I assume this results from the note that it is evaluated in Riemann normal coordinates. I know one can choose the coordinate system so that a given Christoffel-symbol vanishes but in this case there are so many to handle so that I am not convinced this is working. Can someone please give me an input on how to make myself clear that it works? Thanks in advance!

Riemann normal coordinates are, by definition, coordinates in which at a specific point $x_0$ the metric is minkowskian and (all) the Christoffel symbols vanish.
Therefore covariant derivatives evaluated at this point $x_0$ are just ordinary derivatives.

 

[minits]

Thanks for your answer! I thought in this case the Riemann tensor should vanish as well due to it´s definition in form of products and derivatives of Christoffel symbols but one will probably simply use the argument that tensors are independent of any basis.

 

[PeterDonis]

I thought in this case the Riemann tensor should vanish as well due to it´s definition in form of products and derivatives of Christoffel symbols

The Christoffel symbols vanish at the origin of Riemann normal coordinates, but their derivatives do not vanish at that point. So the Riemann tensor, since it includes derivatives of the Christoffel symbols, does not vanish at that point.

 

[minits]

Ah ok thanks for your answer!