- #1
Safinaz
- 255
- 8
- Homework Statement
- Any help how to simplify these terms of Ricci tensor:
- Relevant Equations
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R_{\alpha\mu} R_{\gamma \nu} g^{\alpha \gamma} + R_{\mu \beta} R_{\nu \delta} g^{\beta \delta} + g^{\alpha \gamma} g^{\beta \delta} \left( R_{\alpha\beta} \frac{ \nabla_\gamma \delta \Gamma^\rho_ {~ \delta \rho} - \nabla_\rho \delta \Gamma^\rho_ {~ \gamma \delta} }{ \delta g^{\mu\nu} }
+ R_{\gamma\delta} \frac{ \nabla_\alpha \delta \Gamma^\rho_ {~ \beta \rho} - \nabla_\rho \delta \Gamma^\rho_ {~ \alpha \beta} }{ \delta g^{\mu\nu} } \right)
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So that they become:
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g^{\sigma \rho} \nabla_\sigma \nabla_\rho R ~g_{\mu\nu} + R ~R_{\mu\nu} - \nabla_\mu \nabla_\nu R
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g^{\sigma \rho} \nabla_\sigma \nabla_\rho R ~g_{\mu\nu} + R ~R_{\mu\nu} - \nabla_\mu \nabla_\nu R
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