# I Schnutz Special Relativity Tensors Question

Tags:
1. Apr 17, 2016

### 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 ⊥ = 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!

2. Apr 17, 2016

### 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.

3. Apr 17, 2016

### fengqiu

Hmmm, how do you mean?
do you mean I need to normalise $U\cdot U$ with itself?

4. Apr 17, 2016

### robphy

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

5. Apr 17, 2016

### fengqiu

Thanks for that, but I don't understand why you do this?

6. Apr 17, 2016

### robphy

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

7. Apr 17, 2016

### fengqiu

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. Apr 18, 2016

### Orodruin

Staff Emeritus
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.

9. Apr 18, 2016

### fengqiu

Ahhh right I see, that makes sense!

thanks for the help guys

10. Apr 18, 2016

### pixel

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