# Schnutz Special Relativity Tensors Question

• I
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!

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.

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?

robphy
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$$P_{\mu\nu}=\eta_{\mu\nu}- \frac{U_{\mu}U_{\nu}}{\eta_{\alpha\beta}U^{\alpha}U^{\beta}}$$

• fengqiu
$$P_{\mu\nu}=\eta_{\mu\nu}- \frac{U_{\mu}U_{\nu}}{\eta_{\alpha\beta}U^{\alpha}U^{\beta}}$$
Thanks for that, but I don't understand why you do this?

robphy
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Does this operator do what you want it to do, independent of signature convention?
Use it and see.

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.

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.

• fengqiu
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

Do you mean Schutz, A First Course in General Relativity?