# Calculation Involving Projection Tensor in Minkowski Spacetime

• crime9894
In summary, the conversation discusses the calculation of ##P^{\gamma}_{\alpha}U^{\beta}\partial_{\beta}U^{\alpha}## in Minkowski spacetime, using the previously calculated equation ##P^{\gamma}_{\alpha}=\delta^{\gamma}_{\alpha}+U_{\alpha}U^{\gamma}##. The speaker reaches a dead end and considers using the geodesics equation ##U^{\upsilon}\nabla_{\upsilon}U^{\mu}=0##, but is unsure how to prove it. They also mention that for non-interacting particles, the material time derivative is 0 and therefore the second term in their result should be
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?

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?

etotheipi
vanhees71 said:
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?

vanhees71

## 1. What is the projection tensor in Minkowski spacetime?

The projection tensor in Minkowski spacetime is a mathematical tool used to project a vector onto a specific direction or subspace. It is a symmetric and traceless tensor that is used to project vectors onto the space-like or time-like subspaces in Minkowski spacetime.

## 2. How is the projection tensor calculated?

The projection tensor is calculated by taking the dot product of the vector with the metric tensor in Minkowski spacetime. This dot product is then multiplied by the inverse of the metric tensor to obtain the projection tensor.

## 3. What is the significance of the projection tensor in Minkowski spacetime?

The projection tensor is significant in Minkowski spacetime as it allows for the decomposition of vectors into their space-like and time-like components. This is important in many applications, including relativity and quantum mechanics.

## 4. Can the projection tensor be used in other spacetimes?

Yes, the projection tensor can be used in other spacetimes, but it may have different properties and calculations depending on the specific spacetime. It is most commonly used in Minkowski spacetime, but can also be applied in other spacetimes with appropriate modifications.

## 5. How does the projection tensor affect calculations in Minkowski spacetime?

The projection tensor affects calculations in Minkowski spacetime by allowing for the separation of vectors into their space-like and time-like components. This simplifies calculations and allows for a better understanding of the dynamics of objects in Minkowski spacetime.

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