Proving Orthogonality of Vector w/ Schnutz Special Relativity Tensors

[fengqiu]

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 ⊥ = P^α _βV^β = (η^α _β + U^αU_β)V^β is
(i) orthogonal to U

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

PβαVα = V^β+U^βU_αV^α

So now if I calculate

Vα ⊥ ⋅ U = V_αU^α+U_αU^αU_αV^α

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

 

[robphy]

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.

 

[fengqiu]

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]

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

 

[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]

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

 

[fengqiu]

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 textbook.

 

[Orodruin]

I think it should, but I can't get it to work out.
The operator is given in the question in the textbook.

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

 

[pixel]

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