- #1

JuanC97

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Hello everyone,

I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor.

I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇

You know that δg

As far as I've tought, δg

and that's really different from "

δ

Now...maybe the problem is just about notation...

The issue that really makes me think twice about it is... What does it really mean that covariant derivative of δg

Could you please rephrase ∇

Thanks btw.

I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor.

I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇

_{ρ}δg_{μν}if δ{g_{μν}} is not a tensor?"You know that δg

_{μν}can be written as (- g_{μα}g_{νβ}δg^{αβ}) since δ{δ_{μ}^{ν}}=0 so... at first glance, you could say that δg_{μν}is not a tensor since you can't lower its indices just by using two metrics BUT the truth is that those indices are not the indices of the variation so you can't just think like that, it is, in fact, a matter of notation as I will explain now:As far as I've tought, δg

_{μν}stands for "__the variation of the components__" of the metric ( i.e. δ{g_{μν}} )and that's really different from "

__the components of the variation__" ( i.e. (δ**g**)_{μν}- wich, of course, would be tensorial since that's just the difference of two metrics ), take this into account:δ

**g**= δ{g_{μν}dx^{μ}⊗dx^{ν}} = δ{g_{μν}}dx^{μ}⊗dx^{ν}+ g_{μν}δ{dx^{μ}⊗dx^{ν}} so (δ**g**)_{μν}is, in general different from δ{g_{μν}}.Now...maybe the problem is just about notation...

The issue that really makes me think twice about it is... What does it really mean that covariant derivative of δg

_{μν}that you found in the formula for the variation of the Christoffel symbols? and, even more, how can you interpret that equation in the proper form?Could you please rephrase ∇

_{ρ}δg_{μν}in a different notation or in words, or maybe interpreting as multilinear maps?.Thanks btw.

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