- #1
binbagsss
- 1,266
- 11
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 !
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 !