Schnutz Special Relativity Tensors Question

  • #1
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There's a question in Schnutz - A first course in special relativity
Consider a Velocity Four Vector U , and the tensor P whose components are given by
Pμν = ημν + UμUν .
(a) Show that P is a projection operator that projects an arbitrary vector V into one orthogonal to U . That is, show that the vector V⊥ whose components are
Vα ⊥ = Pα βVβ = (ηα β + UαUβ)Vβ is
(i) orthogonal to U

Now I've attempted the solution and it is the following

PβαVα = Vβ+UβUαVα

So now if I calculate

Vα ⊥ ⋅ U = VαUα+UαUαUαVα

which is orthogonal if c=1 ... as |U|^2= -c^2

but.. this is just in the metric -+++ , if I change metrics to +--- then it won't be orthogonal? Also it's not orthogonal if c=/=1 .. which doesn't seem right to me either
how can that be?

Thank for you help!

Adam
 

Answers and Replies

  • #2
robphy
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In general, you should include a factor of ##U\cdot U## in the denominator.
That is to say, you should do the normalization explicitly, in your signature.
 
  • #3
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In general, you should include a factor of ##U\cdot U## in the denominator.
That is to say, you should do the normalization explicitly, in your signature.
Hmmm, how do you mean?
do you mean I need to normalise ##U\cdot U## with itself?
 
  • #4
robphy
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[tex]P_{\mu\nu}=\eta_{\mu\nu}- \frac{U_{\mu}U_{\nu}}{\eta_{\alpha\beta}U^{\alpha}U^{\beta}}[/tex]
 
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  • #5
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[tex]P_{\mu\nu}=\eta_{\mu\nu}- \frac{U_{\mu}U_{\nu}}{\eta_{\alpha\beta}U^{\alpha}U^{\beta}}[/tex]
Thanks for that, but I don't understand why you do this?
 
  • #6
robphy
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Does this operator do what you want it to do, independent of signature convention?
Use it and see.
 
  • #7
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Does this operator do what you want it to do, independent of signature convention?
Use it and see.
I think it should, but I can't get it to work out.
The operator is given in the question in the text book.
 
  • #8
Orodruin
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I think it should, but I can't get it to work out.
The operator is given in the question in the text book.
Yes, and the textbook uses all of the conventions which makes the operator a projection operator. If it did not, it would have had to write out the form given in post #4.
 
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  • #9
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Yes, and the textbook uses all of the conventions which makes the operator a projection operator. If it did not, it would have had to write out the form given in post #4.
Ahhh right I see, that makes sense!

thanks for the help guys
 
  • #10
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Do you mean Schutz, A First Course in General Relativity?
 

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