I need to prove the following identity for scalar field ($\phi:M\rightarrow R$) in curved spacetime without torsion called $M$

$\nabla_{\mu}[\Box \phi \nabla^{\mu}\phi-\frac{1}{2}\nabla^{\mu}(\nabla \phi)^{2}]=(\Box \phi)^{2}- R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi- \nabla^{\mu}\nabla^{\nu}\phi \nabla_{\mu}\nabla_{\nu}\phi$.

I opened up the total derivative and the term $(\Box \phi)^{2}$ appears but the others I cannot get the right combination. I tried to use the following identity to make appears the terms $R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$, namely, $[\nabla_\mu, \nabla_\nu]X_{\alpha}=-R^{\kappa}_{\alpha\mu\nu}X_{\kappa}$, but I got stuck at some point.

Any help would be much appreciated.

I need to prove the following identity for scalar field ($\phi:M\rightarrow R$) in curved spacetime without torsion called $M$

$\nabla_{\mu}[\Box \phi \nabla^{\mu}\phi-\frac{1}{2}\nabla^{\mu}(\nabla \phi)^{2}]=(\Box \phi)^{2}- R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi- \nabla^{\mu}\nabla^{\nu}\phi \nabla_{\mu}\nabla_{\nu}\phi$.

I opened up the total derivative and the term $(\Box \phi)^{2}$ appears but the others I cannot get the right combination. I tried to use the following identity to make appears the terms $R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$, namely, $[\nabla_\mu, \nabla_\nu]X_{\alpha}=-R^{\kappa}_{\alpha\mu\nu}X_{\kappa}$, but I got stuck at some point, and the third term does not appear at all.

Any help would be much appreciated.

I need to prove the following identity for scalar field ($\phi:M\rightarrow R$) in curved spacetime without torsion called $M$

$\nabla_{\mu}[\Box \phi \nabla^{\mu}-\frac{1}{2}\nabla^{\mu}(\nabla \phi)^{2}]=(\Box \phi)^{2}- R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi- \nabla^{\mu}\nabla^{\nu}\phi \nabla_{\mu}\nabla_{\nu}\phi$.

I opened up the total derivative and the term $(\Box \phi)^{2}$ appears but the others I cannot get the right combination. I tried to use the following identity to make appears the terms $R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$, namely, $[\nabla_\mu, \nabla_\nu]X_{\alpha}=-R^{\kappa}_{\alpha\mu\nu}X_{\kappa}$, but I got stuck at some point.

Any help would be much appreciated.

I need to prove the following identity for scalar field ($\phi:M\rightarrow R$) in curved spacetime without torsion called $M$

$\nabla_{\mu}[\Box \phi \nabla^{\mu}\phi-\frac{1}{2}\nabla^{\mu}(\nabla \phi)^{2}]=(\Box \phi)^{2}- R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi- \nabla^{\mu}\nabla^{\nu}\phi \nabla_{\mu}\nabla_{\nu}\phi$.

I opened up the total derivative and the term $(\Box \phi)^{2}$ appears but the others I cannot get the right combination. I tried to use the following identity to make appears the terms $R_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$, namely, $[\nabla_\mu, \nabla_\nu]X_{\alpha}=-R^{\kappa}_{\alpha\mu\nu}X_{\kappa}$, but I got stuck at some point.

Any help would be much appreciated.