Contraction of an asymmetric tensor?

[Dixanadu]

Hey guys,

So in my notes I've got this statement written:

If tensor with no symmetry properties, $A^{\mu\nu}$, contracts to $a_{\mu\nu}$, we can write this as $A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-A^{\nu\mu})$ as $a_{\mu\nu} (A^{\mu\nu}+A^{\nu\mu}) = 0$. So I don't see how the symmetric part contracts to 0.

*Note* I do also have written that $a^{\mu\nu}=-a^{\nu\mu}$ but I am not sure if this is relevant.

I understand that you can decompose the tensor $A^{\mu\nu}$ into the sum of symmetric and anti-symmetric parts, but i don't see why the symmetric part vanishes under contraction.

If someone could explain I'd be very grateful - thank you!

 

[my2cts]

Hey guys,

So in my notes I've got this statement written:

If tensor with no symmetry properties, $A^{\mu\nu}$, contracts to $a_{\mu\nu}$, we can write this as $A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-B^{\nu\mu})$ as $a_{\mu\nu} (A^{\mu\nu}+A^{\nu\mu}) = 0$. So I don't see how the symmetric part contracts to 0.

*Note* I do also have written that $a^{\mu\nu}=-a^{\nu\mu}$ but I am not sure if this is relevant.

I understand that you can decompose the tensor $A^{\mu\nu}$ into the sum of symmetric and anti-symmetric parts, but i don't see why the symmetric part vanishes under contraction.

If someone could explain I'd be very grateful - thank you!

Your notes are inaccurate. What you tried to note down is probably:
If $a^{\mu\nu}=-a^{\nu\mu}$, then $A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-A^{\nu\mu})$,
for any $A^{\mu\nu}$.

 

[Dixanadu]

Whoops that B was meant to be an A -- error fixed! but what do you mean by inaccurate exactly? what part is wrong?

 

[my2cts]

Your notes are inaccurate. What you tried to note down is probably:
If $a^{\mu\nu}=-a^{\nu\mu}$, then $A^{\mu\nu}a_{\mu\nu}=\frac{1}{2}a_{\mu\nu}(A^{\mu\nu}-A^{\nu\mu})$,
for any $A^{\mu\nu}$.

Whoops that B was meant to be an A -- error fixed! but what do you mean by inaccurate exactly? what part is wrong?

The part where you wrote B instead of A ?

 

[Dixanadu]

Yes -- sorry about that!