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lerus

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- TL;DR Summary
- How to prove that metric tensor is covariant constant

I'm reading "Problem Book In Relativity and Gravitation".

In this book there is a problem

7.5 Show that metric tensor is covariant constant.

To prove it, authors suggest to use formulae for covariant derivative:

Aαβ;γ=Aαβ,γ−AσαΓβγσ−AσβΓαγσ

after that they write this formulae for tensor g and after that it is easy to prove that

gαβ;γ=0

I think I found another way to to prove it and I like it much more :).

It is always possible to find a coordinate system such that for some given point in the manifold, not only is the coordinate basis orthonormal but additionally all first-order partial derivatives of the metric components vanish.

It means that in this coordinate system:

gαβ,γ=0

and all

Γαγσ=0

It means that it this coordinate system

gαβ;γ=0

But this is a tensor equation it means it will be correct in any coordinate system.

Is this approach correct?

In this book there is a problem

7.5 Show that metric tensor is covariant constant.

To prove it, authors suggest to use formulae for covariant derivative:

Aαβ;γ=Aαβ,γ−AσαΓβγσ−AσβΓαγσ

after that they write this formulae for tensor g and after that it is easy to prove that

gαβ;γ=0

I think I found another way to to prove it and I like it much more :).

It is always possible to find a coordinate system such that for some given point in the manifold, not only is the coordinate basis orthonormal but additionally all first-order partial derivatives of the metric components vanish.

It means that in this coordinate system:

gαβ,γ=0

and all

Γαγσ=0

It means that it this coordinate system

gαβ;γ=0

But this is a tensor equation it means it will be correct in any coordinate system.

Is this approach correct?