- #1
JustinLevy
- 895
- 1
How can one work out what terms like:
[tex](g^{cd}R^{ab}R_{ab})_{;d}[/tex]
are in terms of the divergence of the Ricci curvature or Ricci scalar?
One student noted that since:
[tex]G^{ab} = R^{ab} - \frac12 g^{ab}R[/tex]
[tex]{G^{ab}}_{;b} = 0[/tex]
that we could maybe use the fact that
[tex]G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12 R^{ab}g_{ab}R - \frac12 R R_{ab}g^{ab} + \frac14 RRg^{ab}g_{ab} = R^{ab}R_{ab} [/tex]
to help? We weren't sure where to go next.
Can someone better with tensor manipulations show how we could work this out?
[tex](g^{cd}R^{ab}R_{ab})_{;d}[/tex]
are in terms of the divergence of the Ricci curvature or Ricci scalar?
One student noted that since:
[tex]G^{ab} = R^{ab} - \frac12 g^{ab}R[/tex]
[tex]{G^{ab}}_{;b} = 0[/tex]
that we could maybe use the fact that
[tex]G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12 R^{ab}g_{ab}R - \frac12 R R_{ab}g^{ab} + \frac14 RRg^{ab}g_{ab} = R^{ab}R_{ab} [/tex]
to help? We weren't sure where to go next.
Can someone better with tensor manipulations show how we could work this out?
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