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Uku
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Homework Statement
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)
Homework 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)
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 in 4)
[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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