(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I am given a symmetric tensor A, meaning [tex]A^{\mu\nu}=A^{\nu\mu}[/tex] and I am given an asymmetric tensor B, meaning [tex]B_{\mu\nu}=-B_{\nu\mu}[/tex]

Now I need to show that:

[tex]A^{\mu\nu}B_{\mu\nu}=0[/tex]0)

2. Relevant equations

We know that an asymmetric tensor can be written as:

[tex]A^{\mu\nu}=\frac{1}{2}(T^{\mu\nu}-T^{\nu\mu})[/tex]1)

3. The attempt at a solution

This is what I have written down from the class:

We can use 1) to write:

[tex]B_{\mu\nu}=\frac{1}{2}(B_{\mu\nu}-B_{\nu\mu})[/tex]

Now we multiply this by [tex]A^{\mu\nu}[/tex]:

[tex]A^{\mu\nu}B_{\mu\nu}=\frac{1}{2}(A^{\mu\nu}B_{\mu\nu}-A^{\mu\nu}B_{\nu\mu})[/tex]2)

Here comes the point which I am confused: the lecturer has written that we can use

[tex]\mu\leftrightarrow\nu[/tex]3)to change the indexes on 2) such as:

[tex]A^{\mu\nu}B_{\mu\nu}=\frac{1}{2}(A^{\mu\nu}B_{\mu\nu}-A^{\mu\nu}B_{\mu\nu})=\frac{1}{2}A^{\mu\nu}(B_{\mu\nu}-B_{\mu\nu})=0[/tex]4)

Right, but here is the question. I know that tensor B is antisymmetric, meaning that

[tex]B_{\mu\nu}=-B_{\nu\mu}[/tex]!! How can I then use 3) to change the indexes? That should give me a change of signs in4)

[tex]...=\frac{1}{2}(B_{\mu\nu}+B_{\mu\nu})\neq0[/tex]

So... how is 0) properly shown? Have I missed some principle at 3)?

EDIT:Is it just that in 4) [tex]-B_{\nu\mu}=B_{\mu\nu}[/tex] ? But that does not work either... because if I plug that in, then I still get B+B

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# Homework Help: Special relativity: proof of symmetry concept

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