# Homework Help: General Relativity asymmetry identity

1. Jan 9, 2017

### binbagsss

1. The problem statement, all variables and given/known data

I have $R^{u}_{o b u } F^{o}_a - R^{u}_{oau}F^{o}_{b}$

and I want to show that this equal to $2R_{o[aF^{o}_b]}$

where $[ ]$ denotes antisymmetrization , and $F_{uv}$ is a anitymstric tensor

2. Relevant equations

Since $F_{uv}$is antisymetric the antisymetrization $2R_{o[aF^{o}_b}]$ reduces to $3(R^{u}_{o b u } F^{o}_a - R^{u}_{oau}F^{o}_{b} + R_{oo}F^a_b)$

3. The attempt at a solution

Contracting to get the Ricci tensor and using that $F_{uv}$ is antisymetric

$R^{u}_{o b u } F^{o}_a - R^{u}_{oau}F^{o}_{b}=R_{o b } F^{o}_a + R_{oa}F^{o}_{b}$
$=R{o b } F^{a}_o + R_{oa}F^{o}_{b}$

I can see that there needs to be a $R_{oo}F^a_b$ term added somewhere, which I cant see where this is going to come from unless this is zero ? which I cant see that it is, and even then Id get a factor of $3$ rather than the $2$ needed.

2. Jan 9, 2017

### TSny

Go to this link and then scroll up into the previous section about 10 lines.
https://en.wikipedia.org/wiki/Ricci_calculus#Differentiation

It says,
As with symmetrization, indices are not antisymmetrized when they are not on the same level, for example;
$$A_{[\alpha }B^{\beta }{}_{\gamma ]}={\dfrac {1}{2!}}\left(A_{\alpha }B^{\beta }{}_{\gamma }-A_{\gamma }B^{\beta }{}_{\alpha }\right)$$
I don't know if you are meant to adopt this convention.

3. Jan 10, 2017

### binbagsss

ahh right thank you,
and you can surely get this from the antisymmetrization of indices on the same level by raising an index?

4. Jan 10, 2017

### TSny

I don't think so unless I'm overlooking something.

Suppose we have a tensor $C_{\alpha}{} ^{\mu}{} _{\beta}$. Then we can antisymmetrize over $\alpha$ and $\beta$ to produce another tensor $C_{[\alpha}{} ^{\mu}{} _{\beta]}$.

But it would not generally be true that $C_{[\alpha}{} ^{\mu}{} _{\beta]} = g^{\mu \tau} C_{[\alpha \tau \beta]}$.

Instead, you would have to write $C_{[\alpha}{} ^{\mu}{} _{\beta]} = g^{\mu \tau} C_{[\alpha |\tau |\beta]}$ where we use another convention that indices located between vertical bars are to be ignored in the antisymmetrization.

That's how I see it, anyway.