- #1

crime9894

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- 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.

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.

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?**