The vanishing of the covariant derivative of the metric tensor

[Michael Price]

I don't know the Physics so this is a naive question.

The local Lorentz frame seems to be interpreted to say that the Space-Time metric can be represented in normal coordinates as diagonal ±1 with first partial derivatives equal to zero at a central point. But doesn't this assume that the connection and the metric are compatible? Why does one do a proof?

It also seems that the Christoffel symbols are assumed to to be zero at the central point. Doesn't this require the affine connection to be torsion free? If not, I don't see how the Christoffel symbols can all vanish.

As a counter example: in spherical polars in a flat space and flat spacetime, the Christoffel symbols are not zero.

 

[Orodruin]

As a counter example: in spherical polars in a flat space and flat spacetime, the Christoffel symbols are not zero.

This has nothing to do with the central point of normal coordinates. The Christoffel symbols of normal coordinates do indeed vanish at the central point and this does require the connection to be torsion free (at least at the central point).

 

[Michael Price]

This has nothing to do with the central point of normal coordinates. The Christoffel symbols of normal coordinates do indeed vanish at the central point and this does require the connection to be torsion free (at least at the central point).

You're right, I wasn't thinking of normal coordinates which I should have been.