R computation from 1 independent Riemann tensor component

[binbagsss]

We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components.

The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## .

I thought it would have been ##R_{11}=R^{1}_{111} + R^{2}_{121}+R^{2}_{112}+R^{2}_{211}##

All I can think of then is that the upper indicie can only contract with the middle indice? I've never heard this before though, is that what's going on here?

Similarlly for ##R_{22}=R^{2}_{222} + R^{1}_{212}=R^{2}_{121}## is the solution.

The components ##R^{2}_{211}, R^{2}_{112}, R^{1}_{221}, R^{1}_{122}, ##are the unused components with two 2s, two 1s - if the upper indice only contracts with the middle, these would not come into the formula for any ricci vector...

If someone could let me know if I'm on the right or wrong track here,
cheers !

 

[TSny]

See http://mathworld.wolfram.com/RicciCurvatureTensor.html

 

[binbagsss]

See http://mathworld.wolfram.com/RicciCurvatureTensor.html

Ok. so it looks like only the middle indice contracts. But doesn't this differ then from when we contract vectors with the metric, here the position of the indices isn't important is it? When we contract with the metric all that matters is upper and lower? Thanks.

 

[Matterwave]

Ok. so it looks like only the middle indice contracts. But doesn't this differ then from when we contract vectors with the metric, here the position of the indices isn't important is it? When we contract with the metric all that matters is upper and lower? Thanks.

The Ricci tensor is defined as ##R_{ab}\equiv R^c_{~acb}## the contraction is in the middle index.

 

[paranormal]

I would respond by saying that both you and the solution are correct, but you are considering different things. The solution is correct in terms of the specific independent Riemann tensor component that was given, which is ##R^{1}_{212}##. This component only involves the first and second indices, so when contracting with the first index to get ##R_{11}##, only the components with two 1s and one 2 (such as ##R^{1}_{111}## and ##R^{2}_{121}##) will contribute. The other components with two 2s and one 1 (such as ##R^{2}_{211}## and ##R^{2}_{112}##) will not contribute to the calculation of ##R_{11}##.

However, you are also correct in that the components with two 2s and one 1 are not completely unused. They will contribute to other independent Riemann tensor components, such as ##R^{2}_{222}## and ##R^{1}_{221}##, which can then be used to calculate other Ricci tensor components like ##R_{22}##. So while they may not directly contribute to the calculation of ##R_{11}##, they are still important in the overall calculation of the Riemann and Ricci tensors.

In summary, both you and the solution are correct, but you are considering different aspects of the problem. It is important to understand the specific components and their relationships when working with tensors.