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 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}##
4velocity ##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 oneshot my problem.
Is this the correct approach instead? If so, how do I prove the equation?
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 oneshot my problem.
Is this the correct approach instead? If so, how do I prove the equation?