Raising and lowering Ricci Tensor

[unscientific]

Taken from Hobson's book:

How is this done? Starting from:

R_{abcd} = -R_{bacd}

Apply $g^{aa}$ followed by $g^{ab}$

g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd}
g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd}
R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd}

Applying $g_{aa}$ to both sides:
g_{aa}R^{aa}_{cd} = - g^{ab}g^{aa} g_{aa} R_{bacd}
R^a_{acd} = -g^{ab} R_{bacd}
R^a_{acd} = - R^a_{acd} = 0

Is there a quicker way?

 

[samalkhaiat]

Taken from Hobson's book:

How is this done? Starting from:

R_{abcd} = -R_{bacd}

Apply $g^{aa}$ followed by $g^{ab}$

g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd}
g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd}
R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd}

Applying $g_{aa}$ to both sides:
g_{aa}R^{aa}_{cd} = - g^{ab}g^{aa} g_{aa} R_{bacd}
R^a_{acd} = -g^{ab} R_{bacd}
R^a_{acd} = - R^a_{acd} = 0

Is there a quicker way?

g^{aa} has no meaning. The metric is symmetric tensor, g^{a b} = g^{b a}, so when you contract it with some antisymmetric tensor like R_{a b c d e f g} = - R_{b a c d e f g}, you get zero:
g^{ab} R_{a b c d} = - g^{b a} R_{b a c d}, \ \Rightarrow \ R^{b}{}_{b c d} = - R^{a}{}_{a c d} .
So, you have something equal to minus itself. It must be zero.

 

[unscientific]

g^{aa} has no meaning. The metric is symmetric tensor, g^{a b} = g^{b a}, so when you contract it with some antisymmetric tensor like R_{a b c d e f g} = - R_{b a c d e f g}, you get zero:
g^{ab} R_{a b c d} = - g^{b a} R_{b a c d}, \ \Rightarrow \ R^{b}{}_{b c d} = - R^{a}{}_{a c d} .
So, you have something equal to minus itself. It must be zero.

Thanks for replying, I think I almost got the hang of it.
Quick but unrelated question:
Suppose I start off with $R_{abcd}$, how do I get to $R_{bc}$?

I'm thinking $g^{aa}R_{abcd} = R^a _{bcd}$ then $-R^a_{bdc}$ then $-g_{de} g^{ea} R^a_{bdc} = -\delta_d^a R^a_{bdc} =- R^a_{bac} =- R_{bc}$.

 

[samalkhaiat]

Quick but unrelated question:
Suppose I start off with $R_{abcd}$, how do I get to $R_{bc}$?

This depends on the book you use. I contract the first with the last index R_{a b} = g^{c d} R_{c b a d} = R^{d}{}_{b a d} others do it between the first and the 3rd.

 

[unscientific]

This depends on the book you use. I contract the first with the last index R_{a b} = g^{c d} R_{c b a d} = R^{d}{}_{b a d} others do it between the first and the 3rd.

I don't understand how you can do that... I'm confused.

 

[unscientific]

This depends on the book you use. I contract the first with the last index R_{a b} = g^{c d} R_{c b a d} = R^{d}{}_{b a d} others do it between the first and the 3rd.

Starting with the Bianchi Identity:

\nabla_e R_{abcd} + \nabla_c R_{abde} + \nabla_d R_{abec} = 0

I raise index $a$ by multiplying $g^{aa}$:

\nabla_e R^a_{bcd} + \nabla_c R^a_{bde} + \nabla_d R^a_{bec} = 0

I'm trying to understand their steps:

 

[unscientific]

This depends on the book you use. I contract the first with the last index R_{a b} = g^{c d} R_{c b a d} = R^{d}{}_{b a d} others do it between the first and the 3rd.

It's alright, I got it - It can be done by contracting $a$ with $c$ then multiplying $g^{be}$.

 

[samalkhaiat]

Starting with the Bianchi Identity:

\nabla_e R_{abcd} + \nabla_c R_{abde} + \nabla_d R_{abec} = 0

I raise index $a$ by multiplying $g^{aa}$:

\nabla_e R^a_{bcd} + \nabla_c R^a_{bde} + \nabla_d R^a_{bec} = 0

I told you, we don't have an object like g^{aa}. If you have a tensor T_{abc} and you want to raise the first index, you multiply your tensor with g^{a d}, i.e. T^{d}{}_{b c} = g^{a d} T_{a b c}.

 

[samalkhaiat]

Multiply the BI with g^{ad}. Let us do it term by term:
The first term g^{ad} \nabla_{e} R_{abcd} = \nabla_{e} g^{ad} R_{abcd} = \nabla_{e} R^{d}{}_{bcd} = \nabla_{e} R_{bc} . The second term g^{ad} \nabla_{c} R_{abde} = \nabla_{c} R^{d}{}_{bde} = - \nabla_{c} R_{be} . And the third term is easy, just raise the index on the covariant derivative g^{ad} \nabla_{d} R_{abec} = \nabla^{a} R_{abec} = \nabla_{a} R^{a}{}_{bec} .

 

[unscientific]

Multiply the BI with g^{ad}. Let us do it term by term:
The first term g^{ad} \nabla_{e} R_{abcd} = \nabla_{e} g^{ad} R_{abcd} = \nabla_{e} R^{d}{}_{bcd} = \nabla_{e} R_{bc} . The second term g^{ad} \nabla_{c} R_{abde} = \nabla_{c} R^{d}{}_{bde} = - \nabla_{c} R_{be} . And the third term is easy, just raise the index on the covariant derivative g^{ad} \nabla_{d} R_{abec} = \nabla^{a} R_{abec} = \nabla_{a} R^{a}{}_{bec} .

The ricci tensor is defined as $R_{bc} = R^a_{bac}$ in my notes. I'm trying to work with their notation.