Calculation Involving Projection Tensor in Minkowski Spacetime

[crime9894]

Homework Statement

I am asked to calculate the expression in Minkowski spacetime

Relevant Equations

Projection tensor $P^{\alpha\beta}=\eta^{\alpha\beta}+U^{\alpha}U^{\beta}$
4-velocity $U^{\mu}$
Minkowski Metric $\eta^{\alpha\beta}$ signature $(-+++)$

In Minkowski spacetime, calculate $P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}$.

I had calculated previously that $P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}$
When I subsitute it back into the expression
$P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}$
$=(\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma})U^{\beta}\partial_{\beta}U^{\alpha}$
$=U^{\beta}\partial_{\beta}U^{\gamma}+U_{\alpha}U^{\gamma}U^{\beta}\partial_{\beta}U^{\alpha}$
But I think hit a dead end. Could it be further simplified?

Later, I look back into my lecture slides again and I saw this "geodesics equation $U^{\upsilon}\nabla_{\upsilon}U^{\mu}=0$" written at a corner. I haven't reach geodesics yet and I can't find relevant source on confirming this equation.

I believe it reduce to $U^{\upsilon}\partial_{\upsilon}U^{\mu}=0$ in flat spacetime and would one-shot my problem.
Is this the correct approach instead? If so, how do I prove the equation?

 

[vanhees71]

Note that $U^{\nu} \partial_{\nu} U^{\mu}=\mathrm{D}_{\tau} U^{\mu}$ is the "material time derivative". This is 0 for "dust", i.e., for non-interacting "particles" only. For an ideal or viscous fluid it's not!

Concerning implification of your expression, note that $U_{\alpha} U^{\alpha}=-1=\text{const}$. What does that imply for the 2nd term in your result?

 

[crime9894]

Concerning implification of your expression, note that $U_{\alpha} U^{\alpha}=-1=\text{const}$. What does that imply for the 2nd term in your result?

I see! Thank you.
I could prove $U_{\alpha}\partial_{\beta}U^{\alpha}=0$ and eliminate second term.
As for the first term, I don't think it could proceed further. Am I done?