Calculation Involving Projection Tensor in Minkowski Spacetime

  • #1
crime9894
5
2
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?
 

Answers and Replies

  • #2
vanhees71
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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?
 
  • #3
crime9894
5
2
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?
 

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