Diagnosing an Equation in General Relativity

[Adam35]

Hello,

since g^μν g_μν = 4 where g = diag[1,-1,-1,-1], see:
https://www.physicsforums.com/threads/questions-about-tensors-in-gr.39158/

Is the following equation correct?

x^μ x_μ = g^μνx_ν g_μνx^ν = g^μν g_μνx^ν x_ν= 4 x^μ x_μ

If not, where is the problem?

Cheers,
Adam

 

[Samy_A]

Hello,

since g^μν g_μν = 4 where g = diag[1,-1,-1,-1], see:
https://www.physicsforums.com/threads/questions-about-tensors-in-gr.39158/

Is the following equation correct?

x^μ x_μ = g^μνx_ν g_μνx^ν = g^μν g_μνx^ν x_ν= 4 x^μ x_μ

If not, where is the problem?

Cheers,
Adam

While x^μ = g^μνx_ν and x_μ=g_μνx^ν are both correct, when you put them together you can't use v twice as the dummy summation index.
This is correct, I think:
x^μ x_μ = g^μνx_ν g_μσx^σ

 

[haushofer]

Indeed, the problem is you use the same dummy index twice, which is not part of the rules.

 

[Orodruin]

This is correct, I think:
xμ xμ = gμνxν gμσxσ

Yes, that is correct. On the other hand, it is unclear why you would want to do that particular manipulation.

To OP: It is important not to get your indices mixed up. This can happen by the same error displayed here or others. You should avoid using the same summation indices when inserting different expressions into a single one or using the same name for a summation index and a free one. As a rule of thumb, your expressions should contain at most two of each index (and then one up and one down). If you have free indices in an equality, you need to have the same free indices on both sides.

 

[Ibix]

This is kind of a tensor equivalent of the simple algebraic proof that 2=1, isn't it? The trick is that you've done something illegitimate in the middle. One could interpret $g^{\mu\nu}x_\nu g_{\mu\nu}x^\nu$ in multiple ways. You initially write it to mean
$$\sum_\mu\left(\left(\sum_\nu g^{\mu\nu}x_{\nu}\right) \left(\sum_\nu g_{\mu\nu}x^{\nu}\right)\right)$$where I do not intend to imply any summation convention. But you then use it to mean (again with no implied summation)
$$\sum_\mu\left(\left(\sum_\nu g^{\mu\nu}g_{\mu\nu}\right) \left(\sum_\nu x^\nu x_\nu\right)\right)$$These are two different things. They only appear to be the same because you used ambiguous notation. As others have noted, dummy indices must appear twice and only twice in any summed term to avoid this kind of thing.

 

[pervect]

While x^μ = g^μνx_ν and x_μ=g_μνx^ν are both correct, when you put them together you can't use v twice as the dummy summation index.
This is correct, I think:
x^μ x_μ = g^μνx_ν g_μσx^σ

Looks good to me. I believe that $g^{\mu\nu} g_{\mu\sigma}$ is $\delta^\nu{}_\sigma$ where $\delta$ is the kronecker delta function which equal one if the indices are equal and zero if they are unequal.

 

[Adam35]

Hi all,

thank you very much, especially to Samy_A for a very quick answer. Now it is completely clear to me. To Orodruin: I did this manipulation exactly because I thought that I can. The result was 1=4 as Ibix said which sound weird to me and I was not able to tell what was wrong. Now I know.

No more than two same indices in one multiplication.

Maybe the equation (in the first post) is a good example for students. Shows where one can find himself (1 = 4) if not obey the discussed rule.

Very nice forum by the way. :-)